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    An ecological element for selecting enhancement stock based on the stability of nekton community structure

    Abstract

    The abundance-biomass comparison curves (ABC curves) method was adopted to analyze the temporal stability of the fish and nekton community structure in Laoshan Bay, China, during the spring and summer seasons from 2013 to 2015. This study aimed to explore the feasibility of using stable species in the nekton structure as stock enhancement candidates and to enrich the guidelines for responsible stock enhancement. Results showed that the W-statistic values of ABC curves for nekton were generally higher than those for fish across the three years’ spring and summer seasons. For fish, the biomass curves were often located below their abundance curves; in contrast, the biomass curves of nekton were generally positioned above their abundance curves or intersected with them. These findings indicated that the nekton community structure was more complex and stable than that of the fish community. Portunus trituberculatus played an important role in maintaining the stability of nekton community structure during spring and summer. Therefore, stock enhancement of P. trituberculatus in Laoshan Bay every spring could improve the stability of nekton community structure in the subsequent summer through trophic relationships. Given that ABC curves have a solid ecological theoretical basis, the stability of nekton structure could be one of joint ecological elements for selecting stock enhancement species.

    Data availability

    The datasets used and analysed during the current study available from the corresponding author on reasonable request.
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    Download referencesAcknowledgementsThe authors are grateful to the staff in the Key Laboratory of Sustainable Development of Marine Fisheries at the Yellow Sea Fisheries Research Institute for their cooperation during the sampling operation. FundingThis work was supported by the National Key Research and Development Program of China (Grant No.: 2023YFD2401101).Author informationAuthors and AffiliationsYellow Sea Fisheries Research Institute, Key Laboratory of Sustainable Development of Marine Fisheries, Ministry of Agriculture and Rural Affairs, Chinese Academy of Fishery Science, 106 Nanjing Road, 266071, Qingdao, ChinaZhong Yi Li & Qun LinShandong Provincial Key Laboratory for Fishery Resources and Eco-environment, Yellow Sea Fisheries Research Institute, Chinese Academy of Fishery Sciences, Qingdao, 266071, ChinaZhong Yi Li & Qun LinAuthorsZhong Yi LiView author publicationsSearch author on:PubMed Google ScholarQun LinView author publicationsSearch author on:PubMed Google ScholarContributionsZhong Yi Li wrote the main manuscript text. Zhong Yi Li prepared Figs. 1, 2 and 3. Qun Lin performed data analysis.Both authors reviewed the manuscript.Corresponding authorCorrespondence to
    Zhong Yi Li.Ethics declarations

    Competing interests
    The authors declare no competing interests.

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    Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
    Reprints and permissionsAbout this articleCite this articleLi, Z., Lin, Q. An ecological element for selecting enhancement stock based on the stability of nekton community structure.
    Sci Rep (2025). https://doi.org/10.1038/s41598-025-33783-0Download citationReceived: 10 April 2025Accepted: 22 December 2025Published: 30 December 2025DOI: https://doi.org/10.1038/s41598-025-33783-0Share this articleAnyone you share the following link with will be able to read this content:Get shareable linkSorry, a shareable link is not currently available for this article.Copy shareable link to clipboard
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    KeywordsLaoshan bayCommunity structureABC curvesStock enhancementSpecies selection More

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    Extreme temperature events reshuffle the ecological landscape of the Southern Ocean

    AbstractExtreme temperature events are becoming widespread with global warming, impacting phytoplankton, the foundation of the marine ecosystem. In the Southern Ocean, these impacts are not well understood, despite the key role of phytoplankton in global carbon cycling and climate. Here, we use 26 years of satellite observations and confirm previously identified impacts of marine heatwaves (MHWs) on phytoplankton in the Southern Ocean, while systematically comparing the opposite impacts of marine cold spells (MCSs). MHWs decrease phytoplankton chlorophyll-a (Chl-a) in subtropical regions (−21.11%) but less so in polar regions, with Chl-a even increasing in the Sub-Antarctic Zone ( + 22.26%). MCSs exhibit opposite patterns, enhancing Chl-a in subtropical regions ( + 32.37%) while inhibiting it in southern regions (−21.19%). These regional differences in Chl-a anomalies are mediated by distinct responses in phytoplankton size composition to MHWs and MCSs. As extreme events intensify with global warming, Southern Ocean’s phytoplankton will be disrupted, with implications for global biogeochemical cycles. These findings highlight the importance of simultaneously considering both MHWs and MCSs when assessing the ecological impacts of climate extremes.

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    Data availability

    The daily SST-CCI (version 3.0) and SIC data are provided by ESA from their website: https://doi.org/10.5285/4a9654136a7148e39b7feb56f8bb02d2. The daily Rrs-CCI and kd (version 6.0) data are provided by ESA from their website: https://rsg.pml.ac.uk/thredds/catalog-cci.html. The PAR data are provided by GlobColour from their website: https://hermes.acri.fr/index.php?class=archive. The nitrate data provided by CMEMS from their website: https://doi.org/10.48670/moi-00019. The MLD (GLORYS12V1) data provided by CMEMS from their website: https://doi.org/10.48670/moi-00021. The HPLC data provided by PANGAEA from their website: https://doi.org/10.1594/PANGAEA.938703. The HPLC data provided by ADON from their website: https://portal.aodn.org.au/search. The data used to generate the figures presented in this study are available via figshare at https://doi.org/10.6084/m9.figshare.3021702187.
    Code availability

    The analyses were performed using MATLAB and Python; the main code used in this study is available at https://doi.org/10.5281/zenodo.1722329288, with BGC-Argo data processing based on code from https://github.com/NOAA-PMEL/OneArgo-Mat and MHWs/MCSs detection using code from https://github.com/ZijieZhaoMMHW/m_m hw1.0.
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    (摆智敏), Lin Deng 
    (邓霖), Wenbo He 
    (何文博), Qilin Chunpi 
    (七林春批) & Jun Zhao 
    (赵俊)Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, Guangdong, ChinaLin Deng 
    (邓霖) & Jun Zhao 
    (赵俊)Pearl River Estuary Marine Ecosystem Research Station, Ministry of Education, Zhuhai, Guangdong, ChinaLin Deng 
    (邓霖) & Jun Zhao 
    (赵俊)Guangdong Provincial Key Laboratory of Marine Resources and Coastal Engineering, Guangzhou, Guangdong, ChinaLin Deng 
    (邓霖) & Jun Zhao 
    (赵俊)Centre for Geography and Environmental Science, Department of Earth and Environmental Sciences, Faculty of Environment, Science and Economy, University of Exeter, Penryn, Cornwall, UKRobert J. W. BrewinAuthorsZhimin Bai 
    (摆智敏)View author publicationsSearch author on:PubMed Google ScholarLin Deng 
    (邓霖)View author publicationsSearch author on:PubMed Google ScholarRobert J. W. BrewinView author publicationsSearch author on:PubMed Google ScholarWenbo He 
    (何文博)View author publicationsSearch author on:PubMed Google ScholarQilin Chunpi 
    (七林春批)View author publicationsSearch author on:PubMed Google ScholarJun Zhao 
    (赵俊)View author publicationsSearch author on:PubMed Google ScholarContributionsB.Z.M. and D.L. conducted data processing and analysis under Z.J.’s instruction. H.W.B. provided the conceptual framework. H.W.B. and Q. C. contributed data and algorithms. B.Z.M. drafted the initial manuscript, and Z.J. and R.J.W.B. reviewed the paper.Corresponding authorCorrespondence to
    Jun Zhao 
    (赵俊).Ethics declarations

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    The authors declare no competing interests.

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    Climate-driven habitat shifts And niche overlap of overexploited trees Cordia africana Lam. and Terminalia brownii Fresen in Ethiopia

    Abstract

    Cordia africana and Terminalia brownii, indigenous Ethiopian multipurpose trees of high ecological and socioeconomic value, are increasingly threatened by overexploitation and climate change. Understanding how climatic and anthropogenic factors shape their distribution is critical for sustainable management and conservation. This study identified key environmental drivers, predicted suitable habitats under current and future climates, and assessed shifts in distribution, geographic range, and niche overlap. Future climate projections were assessed under two scenarios: the medium-emission scenario (SSP2-4.5) and the high-emission scenario (SSP5-8.5), for two time horizons: the 2050s and 2070s. An ensemble modeling framework was applied using seven algorithms: boosted regression trees, random forest, generalized linear model, generalized additive model, maximum entropy, support vector machine, and multivariate adaptive regression splines. Each model was run with ten sub-sampled replicates. We evaluated model performance using the area under the curve (AUC) for C. africana (0.84) and T. brownii (0.82), and the true skill statistic (TSS), which was 0.60 for both species. Predicted suitable habitat for C. africana (20.1%) was concentrated in the western, central, and southwestern regions, while T. brownii (19.9%) was mainly distributed in the northern, eastern, and southeastern lowlands. Under future climate scenarios, suitable areas are projected to decline sharply, shrinking to 8.68% and 2.07% of Ethiopia’s land area, respectively, by the 2070s. Suitable habitats for both species are expected to contract, with potential refugia for C. africana shifting toward highland areas. Geographic and environmental overlap between the two species was minimal. Given their multipurpose use, increasing vulnerability, and limited niche and geographic overlap, our results show that the projected habitat contraction, the marked decline of T. brownii within its natural range, and the upslope shift of C. africana refugia require species-specific conservation actions. C. africana needs refugia protection and climate-adapted management, while T. brownii requires targeted measures to support its long-term viability. Conservation efforts should focus on safeguarding existing habitats, regulating harvesting, strengthening community-based forest management, and maintaining ecological connectivity. Both species will also require stricter control of human use under future climate change.

    Data availability

    The data supporting the findings of this study are available in the supplementary material.
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    Download referencesAcknowledgementsThe authors gratefully acknowledge the Department of Plant Biology and Biodiversity Management for providing research facilities and the herbarium expert for identifying the specimen and facilitating access to the herbarium specimens used in this study.FundingThere are no funding resources for this study.Author informationAuthors and AffiliationsDepartment of Plant Biology and Biodiversity Management, College of Natural and Computational Sciences, Addis Ababa University, Addis Ababa, EthiopiaDaniel MeleseDepartment of Biology, Mizan Tepi University, Tepi, EthiopiaDaniel Melese, Muluye Asnakew & Ashebir AwokeDessie Tissue Culture Center, Dessie, EthiopiaAtnafu TesfawMekdela Amba University, Tuluawulia, EthiopiaYibeltal MeslieDepartment of Biology, University of Gondar, Gondar, EthiopiaYibelu Yitayih HailieAuthorsDaniel MeleseView author publicationsSearch author on:PubMed Google ScholarMuluye AsnakewView author publicationsSearch author on:PubMed Google ScholarAshebir AwokeView author publicationsSearch author on:PubMed Google ScholarAtnafu TesfawView author publicationsSearch author on:PubMed Google ScholarYibeltal MeslieView author publicationsSearch author on:PubMed Google ScholarYibelu Yitayih HailieView author publicationsSearch author on:PubMed Google ScholarContributionsD.M. was involved in data collection, compilation, methodology, modeling, analyzing the output, and drafting the manuscript. M.A., A.A., A.T., Y.M., and Y.Y. contributed to data collection and manuscript writing. We would like to clarify that the work presented here is original research that has not previously been published and is not under consideration for publication elsewhere, in whole or in part. All authors participated in revising and approved the final manuscript.Corresponding authorCorrespondence to
    Daniel Melese.Ethics declarations

    Competing interests
    The authors declare no competing interests.

    Additional informationPublisher’s noteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Supplementary InformationBelow is the link to the electronic supplementary material.Supplementary Material 1Rights and permissions
    Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
    Reprints and permissionsAbout this articleCite this articleMelese, D., Asnakew, M., Awoke, A. et al. Climate-driven habitat shifts And niche overlap of overexploited trees Cordia africana Lam. and Terminalia brownii Fresen in Ethiopia.
    Sci Rep (2025). https://doi.org/10.1038/s41598-025-33743-8Download citationReceived: 19 September 2025Accepted: 22 December 2025Published: 30 December 2025DOI: https://doi.org/10.1038/s41598-025-33743-8Share this articleAnyone you share the following link with will be able to read this content:Get shareable linkSorry, a shareable link is not currently available for this article.Copy shareable link to clipboard
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    KeywordsAnthropogenicConservationEnsemble modelMultipurpose trees More

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    Enhancing crayfish sex identification with Kolmogorov-Arnold networks and stacked autoencoders

    AbstractCrayfish play an important role in freshwater ecosystems, and sex classification is crucial for analyzing their demographic structures. This study performed binary classification using traditional machine learning and deep learning models on tabular and image datasets with an imbalanced class distribution. For tabular classification, features related to crayfish weight and size were used. Missing values were handled using different methods to create various datasets. Kolmogorov-Arnold networks demonstrated the best performance across all metrics, achieving accuracy rates between 95 and 100%. Image data were generated by combining at least five images of each crayfish. Autoencoders were employed to extract meaningful features. In experiments conducted on these extracted features, support vector machines achieved 84% accuracy, and multilayer perceptrons achieved 82% accuracy, outperforming other models. To enhance performance, a novel architecture based on stacked autoencoders was proposed. While some models experienced performance declines, Kolmogorov-Arnold networks showed an average improvement of 3.5% across all metrics, maintaining the highest accuracy. To statistically evaluate performance differences, McNemar’s and Wilcoxon tests were applied. The results confirmed significant differences between Kolmogorov-Arnold networks, support vector machines, multilayer perceptrons, and naive Bayes. In conclusion, this study highlights the effectiveness of deep learning and machine learning models in crayfish sex classification and provides a significant example of hybrid artificial intelligence models incorporating autoencoders.

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    Data availability

    The datasets generated and/or analysed during the current study are available in the Zenodo repository: https://doi.org/10.5281/zenodo.17516963. The source codes developed for the experiments are stored in a GitHub repository at https://github.com/yasinatilkan60/Crayfish-Sex-Identification.
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    Tunc Asuroglu.Ethics declarations

    Competing interests
    The authors declare no competing interests.

    Additional informationPublisher’s noteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Supplementary InformationBelow is the link to the electronic supplementary material.Supplementary Material 1Rights and permissions
    Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
    Reprints and permissionsAbout this articleCite this articleAtilkan, Y., Kirik, B., Acikbas, E.T. et al. Enhancing crayfish sex identification with Kolmogorov-Arnold networks and stacked autoencoders.
    Sci Rep (2025). https://doi.org/10.1038/s41598-025-34095-zDownload citationReceived: 03 April 2025Accepted: 24 December 2025Published: 30 December 2025DOI: https://doi.org/10.1038/s41598-025-34095-zShare this articleAnyone you share the following link with will be able to read this content:Get shareable linkSorry, a shareable link is not currently available for this article.Copy shareable link to clipboard
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    KeywordsCrayfishSex identificationDeep learningMachine learningKolmogorov-Arnold networksStacked autoencoders More

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    A north-south hemispheric migratory divide in the butterfly Vanessa cardui

    AbstractReversed seasonality and distinct navigation cues in the Earth’s two hemispheres may shape the evolution of migratory behaviour in animals. Migratory divides—contact zones where populations have evolved alternative migratory strategies—are well-documented in birds and typically occur longitudinally. We hypothesise that insect migratory divides are less likely to emerge longitudinally, but may exist latitudinally, driven by hemisphere-specific sensory adaptations that lead to spatial and temporal isolation. Here, we examine this hypothesis in the cosmopolitan painted lady butterfly (Vanessa cardui), whose Southern Hemisphere dynamics remain unexplored. Investigating the genomes of 300 individuals across Africa and Europe, we identify a 9 Mb chromosomal inversion on chromosome 8, which exhibits strong haplotype structure aligned with hemispheric origin, with a few potential heterozygotes near the equator. The inversion harbours 336 genes, including several directly relevant to migration. Notably, one inversion breakpoint intersects the gene encoding the GABA-B receptor, which responds to the neuropeptide γ-aminobutyric acid (GABA), crucial for insect navigation. Our findings provide genomic evidence of a migratory divide in insects and highlight the role of inverted seasonality in the two hemispheres and genomic rearrangements as isolating barriers for highly mobile species.

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    IntroductionMovement plays a fundamental role in the evolution of life. Among various forms of movement, migration stands out as a particularly specialised and multifaceted behaviour – a choreography of finely tuned responses to fluctuating environmental conditions1. The evolutionary and molecular underpinnings of this complex behaviour remain largely unknown, as do the impacts of migration on species diversification2,3.Seasonality is crucial in triggering migratory behaviour, as animals rely on environmental signals such as changes in photoperiod, plant phenology, and temperature to initiate, maintain, and terminate migration. These cyclical signals are thought to modulate transcriptional and epigenetic pathways that regulate migratory responses4,5. Once migration begins, navigational adaptations become essential for maintaining directional movement towards favourable destinations. Celestial cues, both diurnal and nocturnal, along with Earth’s magnetic field, influence navigation. These signals are integrated into neural networks that form an internal compass, guiding movement across vast distances6,7.Seasonality patterns are generally consistent across longitude (east-west) but vary latitudinally (north-south), driving animal migratory movements predominantly in a north-south direction8. In vertebrates, migratory divides – zones where partially sympatric populations adopt distinct migratory strategies – have been widely described, particularly on philopatric species migrating north-south, where divides occur longitudinally9,10. These divides are often characterised by a clear separation between breeding and non-breeding seasons and areas, as well as by behavioural mechanisms such as learning and cultural transmission11.The emergence of migratory divides can have important implications for early speciation processes in birds and other vertebrates12,13,14,15,16 by exposing sympatric populations to different selective pressures, such as different parasites, predator and climate regimes9,12,17. In addition to selection, assortative mating can also contribute to reproductive isolation12,15,18. Efforts to unravel the genetic basis of population differentiation in contrasting migratory phenotypes have primarily identified single-nucleotide polymorphisms (SNPs)19,20,21,22. However, with the acquisition of contiguous genomes, large genomic alterations such as structural variants have been found to underlie the genetic architecture of migratory divides. Recent studies in fish and birds highlight the importance of chromosomal inversions23,24,25,26,27, deletions28, and long repetitive-rich regions of transposable elements (TEs)29. These major structural variants can suppress recombination over large genomic regions, facilitating the formation of islands of differentiation even in the presence of ongoing gene flow30,31. However, the conditions under which these genomic islands might expand to generalised genome-wide differentiation are not fully understood32.In insects, long-range cyclical migrations typically span multiple overlapping generations, with each generation completing only one leg of the migratory cycle, and where individuals rarely learn their routes from experience33. Each moving generation involves vast population numbers shaping complex reticular movements, formed by individuals travelling varying distances34,35,36,37. Moreover, because individuals tend to migrate and then reproduce within relatively short lifespans, breeding can take place in any suitable habitat along their migratory routes. Only a few well-known insect species, such as the monarch butterfly, seem to migrate to non-breeding destinations as part of their annual cycle. These dynamics disfavour the evolution of strict philopatry, instead resulting in broad north-south seasonal migratory movements that span extensive geographical ranges34,35,36,37. As a result, the formation of longitudinal migratory divides in insects seems unlikely, except for scenarios where major geographic barriers hinder gene flow37. Indeed, no migratory divide supported by genetic evidence has been identified in insects to date. Nevertheless, the possibility of latitudinal migratory divides with a low temporal overlap in the equator exists for species whose range extends to both hemispheres.In this study, we investigate how living in the Northern or Southern Hemisphere may influence the evolution of insect migratory strategies – a question that remains largely unexplored. Most insights into migratory patterns in animals come from species from the Northern Hemisphere, while the Southern Hemisphere has been relatively neglected and rarely compared mechanistically. Although independent migratory circuits in each hemisphere are observed in some migratory vertebrates, such as certain birds38,39,40,41 and whales42,43,44, the potential role of hemispheric barriers in driving these patterns has not yet been proposed as a unifying hypothesis. Given that butterflies and moths possess magnetoreceptor organs that may aid in navigation45,46,47, we hypothesise that reversed seasonality, a necessarily opposite response to the magnetic field, and distinct celestial cues in the two hemispheres of Earth may shape the evolution of contrasting migratory strategies. Since environmental rhythmicity displays inverted pulses between hemispheres, populations of migratory insects may adjust their biological clocks, either through genetic changes or phenotypic plasticity, and respond differently to hemisphere-specific seasonal cues, potentially leading to spatial and temporal isolation between populations. We use genomics to address the hypothesis of a migratory divide aligned with hemispheric effects in a highly migratory insect species. The painted lady butterfly, Vanessa cardui, provides an ideal system to examine these dynamics, since its distribution in Africa encompasses the Northern and Southern hemispheres. In both hemispheres, painted ladies are well-known seasonal migrants, often among the most common butterflies in a wide diversity of habitats8. Recent studies have characterised ecological and genetic connectivity between Europe and equatorial Africa35,36,37,48,49,50,51, yet migratory patterns of populations south of the equator remain unknown.Results And DiscussionObservational data suggest independent movement dynamics in the two hemispheresInsects migrate in multigenerational waves, resulting in a latitudinal progression of the species distribution through time. Earlier studies of V. cardui revealed two annual peaks of abundance in equatorial latitudes corresponding to the wet seasons, while only one major demographic peak is observed in the European part of the distribution48,50,52. Our analysis of population latitudinal progression throughout the year, based on 760,973 compiled observations of V. cardui in Europe and Africa (Supplementary Data S1), reveals similar patterns, with successive generations tracking locations with optimal conditions (Fig. 1A). As temperatures drop in the boreal autumn, V. cardui populations migrate from the Palaearctic to Africa, reaching as far south as the equatorial latitudes before migrating northwards again in early spring48,50,51. However, another peak of abundance is observed at equatorial latitudes from May to July (Fig. 1A). These individuals are unlikely to belong to the Northern population that has migrated to Europe during this period. We argue that they are part of a Southern African population expanding northward to avoid the austral winter and the progressive desiccation of subtropical zones (Fig. 1B). While these two populations might rarely overlap in time, they inhabit equatorial latitudes during the two different wet seasons. Overall, observational data suggest that individuals in the Southern Hemisphere constitute an independent migratory circuit.Fig. 1: Seasonality and latitudinal phenology of V. cardui.A The central heatmap plot shows monthly frequencies of V. cardui occurrences by latitudinal divisions. Maps illustrate differing photoperiod and temperature amplitudes across the distributional range, factors likely influencing the migratory syndrome and denoting substantial differences between hemispheres. Occurrence data comprise 768,332 observations of painted ladies obtained from public repositories and authors’ field expeditions. Peaks of presence linking consecutive months can be observed. Monthly frequencies through the year are relative to each latitudinal division. B Summary of hypothesised migratory movements in both hemispheres. Synchrony in northwards and southwards movements in the two populations may lead to allochrony in the equatorial zone, with independent populations spatially overlapping but temporally isolated. The Sahara Desert (yellow) and the tropical forest (green) are regions where V. cardui seldom breeds. Data used for monthly observations are available in Supplementary Data 1. Data underlying the temperature amplitude map are deposited into the OSF repository and are available at the following URL: https://osf.io/6u32k/. Data underlying the photoperiod amplitude map can be obtained online using the provided code (https://doi.org/10.5281/zenodo.17113173). Maps were made using free vector and raster map data from naturalearthdata.com. Butterfly illustrations by Blanca Martí.Full size imageA chromosomal inversion segregates populations in the two hemispheresTo investigate whether V. cardui individuals in the Southern Hemisphere form an independent migratory population from that found in the Palaearctic-African region, we examined genome-wide variation in 290 specimens collected along the entire latitudinal extent of Africa and Europe (Supplementary Data S2, Fig. 2A) using a reduced representation approach (see Methods). Genetic differentiation (fixation index, FST) was minimal across most of the genome, consistent with patterns observed in other migratory insects53,54,55,56. However, a single large and contiguous 9 Mb region in chromosome 8 stood out with a sharp increase in genetic differentiation between northern and southern individuals (Table S1, Fig. 3A). This region, spanning about 60% of chromosome 8, also showed pronounced divergence (dXY) and low diversity (π) (Table S1), suggesting the presence of a chromosomal inversion. A PCA based on this region showed two clusters, each corresponding to a different hemisphere (Fig. 2C). This signal persisted after linkage disequilibrium (LD) pruning (Fig. S3).Fig. 2: Genomics of a migratory divide in the butterfly Vanessa cardui.A Collection of sites for 300 V. cardui specimens. B Results from fineRADstructure, a Bayesian co-ancestry based analysis, using 2855 SNPs belonging to the region with the largest FST differentiation in chromosome 8 (excluding repetitive regions). Two clusters of high shared ancestry were recovered, corresponding to the northern and southern hemispheres. Some individuals (dark blue) were shown to be heterozygous for the inversion. C Principal-component analysis (PCA) in the differentiated region in chromosome 8 for 290 individuals using 976 SNPs. The two main clusters obtained correspond to the North and South hemisphere populations. Samples in grey correspond to potential heterozygotes. Collection sites information is available in the Supplementary Data 2. Data used for B and C are deposited into the OSF repository and are available at the following URL: https://osf.io/6u32k/.Full size imageFig. 3: Genomic characterization of the 9 Mb inverted region in chromosome 8 in Vanessa cardui in comparison to the rest of the genome.A Population statistics calculated across the genome (1,136,244 variants and 2,788,714 invariant sites) measuring differentiation (FST), absolute divergence (dXY) and diversity (π) between 60 V. cardui individuals from Northern and Southern populations. The plot illustrates the highly differentiated region in chromosome 8. Within this region, absolute divergence between these two populations shows the highest values at the beginning and end, whereas lower values of intra-population genetic diversity, measured as π, are observed for both populations. Window size was set to 100 bp for FST and 1 Mb for dXY and π. B Phylogenetic trees built with whole genome sequencing data for 5 V. cardui individuals representing major geographic regions of its distribution, and the sister species Vanessa kershawi. The tree on top represents the genome-wide tree with the inverted region excluded. The tree below is based on data exclusively from the inverted region. The representatives of the Northern hemisphere (in blue) form a monophyletic cluster, sister to the Southern Hemisphere representative (Namibia, in purple). C Heterozygosity values for the inverted region and for the rest of the genome, calculated as the ratio of variant sites to total genotyped sites. Heterozygosity varies significantly only within the inverted region, with lower values observed in the Southern cluster compared to the Northern cluster. Individuals located between these two main clusters exhibit higher heterozygosity levels, suggesting a heterozygous state (Fig. 3). Box plots depict the median (central line), the 25th and 75th percentiles (bounds of the box), and the whiskers represent the data points within 1.5 x the interquartile range from the box. Data points beyond the whiskers indicate outliers (large dots). Data underlying A–C are deposited into the OSF repository and are available at the following URL: https://osf.io/6u32k/.Full size imageFurther evidence of population differentiation was provided by a co-ancestry Bayesian clustering analysis (fineRADstructure57), which showed higher levels of co-ancestry within individuals in each hemisphere (Fig. 2B), a pattern not observed when analysing the rest of the genome (Fig. S4). Phylogenetic analyses also support this hemispheric differentiation: while a tree inferred from the inverted region grouped individuals into well-supported clades corresponding to their hemisphere of origin, no phylogenetic structure following latitude arose from a tree inferred from the remainder of the genome (Fig. S5). Remarkably, phylogenetic divergence across the Northern clade, spanning a vast longitudinal range across the Holarctic, is lower than the divergence observed between hemispheres (Figs. 3B, S5), underscoring the pronounced role of the inversion and hemispheric barriers in shaping population structure.To investigate in greater detail whether this region is an inversion, we analysed read mapping information (BreakDancer58) from a northern individual (reference genome assembly59) and a southern individual using previously published whole-genome data34 (sample 15D327, Namibia). This analysis identified inversion breakpoint coordinates at positions 4,402,570 and 13,379,476, which coincide with the region highlighted by the population genetic summary statistics. Additionally, several smaller inversions (<50 kb) were detected within the larger inverted region (Table S2), potentially reflecting artifacts introduced by mapping uncertainties60,61. However, two nested larger inversions of 3.2 Mb and 1.2 Mb (Table S2) suggest a complex history of structural rearrangements in this genomic region.Interestingly, the PCA of this region on chromosome 8 revealed that some individuals do not cluster within the northern or the southern genotype groups (Fig. 2C), suggesting that these individuals are heterozygous for the inversion. While northern homozygous individuals formed a cohesive group at one extreme of the PCA, some individuals collected in the south and near the equator were located between this cluster and the southern homozygous type. Heterozygosity in the inverted region of these individuals was significantly higher than for the northern and southern homozygous groups (H = 0.056, Kruskal-Wallis χ2(2) = 64.15, P-value < 0.001, df = 2), but not significantly different in the rest of the genome (Kruskal-Wallis χ2(2) = 3.66, p-value = 0.160, df = 2). Most of these individuals were sampled from regions around equatorial latitudes in East Africa (e.g., Ethiopia, Kenya, Zambia), and outside peak abundance seasons. These findings suggest this area is a contact zone where gene flow can occur between alternative inversion haplotypes.Chromosomal inversions lead to reduced recombination rates within the inverted region30,62. However, if genetic exchange between inversion variants occurs within an inversion via double crossovers or gene conversion, a U-shaped divergence profile is expected, with stronger divergence near the inversion breakpoints and less pronounced divergence towards the centre25,63,64. Consistent with this prediction, we observed the highest sequence divergence (dXY) between northern and southern populations near the breakpoints (Fig. 3A), supporting marginal gene exchange between migratory ranges.Altogether, the existence of a single highly divergent genomic region between Northern and Southern Hemispheres, coupled with ecological observations, strongly indicates the presence of a migratory divide between populations of V. cardui butterflies. The lack of population differentiation outside the inversion and the U-shaped divergence within the inversion suggests that gene flow occurs extensively between populations on either side of the divide. This finding highlights the role of chromosomal inversions in maintaining sets of locally adapted alleles through recombination suppression65, acting as a key mechanism leading to local adaptation in highly mobile species66,67.Recent divergence between hemispheresWe estimated a divergence time for the inversion of 0.55–0.69 mya, although this value is likely underestimated due to gene flow between the inverted and non-inverted haplotypes. In line with a recent split or ongoing gene flow between populations, none of the 32,719 SNPs were fixed between the haplotypes. These findings, combined with the fact that the divergence between V. cardui and its sister species, V. kershawi, is estimated to be much older – around 7 mya68– suggest that the inversion arose in V. cardui after the split between these two species of Vanessa.Our analysis based on paired-end read mapping information revealed no chromosomal inversions between V. kershawi and the Northern V. cardui reference genome that match the coordinates and length of the inversion detected with population data between V. cardui of northern and southern origins. Therefore, V. kershawi shares collinearity with the Northern haplotype, supporting the Southern haplotype as the derived form. Under this scenario, the emergence of a rearranged haplotype, likely in southern Africa, may have facilitated the expansion of the ancestral haplotype across the Palaearctic and subsequent diversification in the Nearctic37, an interpretation that aligns with our phylogenomic analysis (Fig. 3B). A southern origin of the inversion is also supported by the asymmetrical distribution of haplotypes: both haplotypes as well as individuals presumably heterozygous for the inversion occur in the South, whereas only the Northern haplotype is found in the North (Fig. 2A). This pattern indicates that the inversion is still segregating in the Southern population. Furthermore, the lower heterozygosity observed in Southern individuals homozygous for the inversion, compared to their Northern counterparts (Fig. 3C), may result from a reduction in genetic diversity during the emergence of the Southern haplotype. Positive selection acting on the derived haplotype could also reinforce this pattern, and the higher heterozygosity in the Northern haplotype (apparently fixed) may be maintained through intra-population recombination.Functional significance of the inversion polymorphism: role in navigation and mating behaviourInversion polymorphisms can affect gene expression and subsequently phenotypes through various mechanisms. Primarily, breakpoints can directly affect fitness by disrupting coding and regulatory sequences or altering their positions in the genome69. Additionally, reduced recombination rates associated with chromosomal inversions facilitate faster accumulation of advantageous allele combinations and thus promote local adaptation, often leading to divergent behavioural phenotypes (e.g., refs. 70,71,72,73,74,75). Inversions have been linked to differences in migratory phenotypes in fish such as the rainbow trout76 and the Atlantic cod23,25, and birds such as the common quail26 and the willow warbler27,29,77, where an inversion and an expansion of a transposable element underlies a migratory divide in Europe.Notably, here we found that both breakpoints overlapped with protein-coding genes. The distal breakpoint intersected the last intron of the subunit 2 of the type B receptor of the γ-aminobutyric acid (GABA) gene (GABAB2). GABA is a neurotransmitter involved in learning and memory that plays a key role in the neural network processing skylight information during navigation78. This network, extensively studied in migratory species like the Monarch butterfly6,7 and the desert locust79, integrates signals in the brain’s central complex to form an internal compass necessary for determining heading-direction during flight80,81,82. The neurons forming this network are GABAergic83,84,85, suggesting that alterations in GABA-B receptor patterns of expression might impact the interpretation of celestial cues. Interestingly, these neurons also receive circadian signals in fruit flies86. The proximal breakpoint was located in an uncharacterised protein from a leucine-rich repeat (LRR) family (Table S3), pointing at another intriguing candidate for future investigation. The fact that the estimated breakpoints of the inversion intersect with genes that are fully annotated and presumably intact in the Northern reference genome implies that these genes are likely disrupted in the Southern haplotype, providing additional support for the Southern haplotype being the derived form.While the precise environmental cues triggering butterfly migration are not fully understood, synergistic factors like photoperiod, temperature shifts, and host plant availability likely play crucial roles87,88,89,90. To investigate whether the inverted region is rich in genes potentially involved in migratory responses, we first conducted a Gene Ontology (GO) term analysis. The analysis, encompassing the 336 genes within the inverted region, revealed enriched functions related to morphogenesis of locomotion-related tissues, fatty acid metabolism, sensory perception and mating behaviour (Figs. S6, S7).Secondly, we investigated the association of highly differentiated SNPs within the inverted region with potentially altered gene functions. The inverted region contains 3342 highly differentiated SNPs (FST > 0.25) between southern and northern migratory ranges. Among these, 857 SNPs are located within 48 annotated protein-coding genes of varying degrees of putative impact in protein sequence and function (synonymous and non-synonymous substitutions) (Table S3). Below, we focus on functional roles of the candidate genes within the inverted region. However, we also report 10 genes containing isolated FST outlier SNPs outside the inversion in Table S4 to provide a complete overview and potential follow-up research, as some are associated with neural activity and metabolism. Among the genes containing FST outliers, several are involved in neuroendocrine regulation and behavioural control. This set comprises receptors regulating the main excitatory neuropeptides, such as the Vesicular glutamate transporter 1 and Sodium-dependent dopamine transporter. We also identified hormonal modulators of interest, like the Pituitary homeobox homologue (Ptx1), a transcription factor that activates the promoters of most pituitary hormone-coding genes91, the Juvenile hormone acid O-methyltransferase (JHAMT), a critical enzyme in the biosynthesis of juvenile hormone and repeatedly pinpointed by insect migration studies7,92,93,94 including V. cardui89,90, and the Pyrokinin-1 receptor (PK1R) of the diapause hormone, involved in pheromone biosynthesis95. The acyl-CoA desaturase Delta-9 gene, which produces the main component of the Spodoptera littoralis moth’s sex pheromone96, was also found in this gene set. In fact, several highly differentiated variants were identified in genes related to mating. Of particular interest are the dsf and siwi genes, which are encoding the dissatisfaction protein (DSF) affecting courtship behaviour, successful copulation and fertile egg-laying in Drosophila97, and the Siwi protein, which plays an essential role during spermatogenesis/oogenesis by repressing the mobility of transposable elements98. These genes are promising candidates for investigating premating mechanisms that may contribute to reproductive isolation67,99,100 and assortative mating between hemispheres.We also found highly differentiated variants in genes related to energy metabolism, such as the main triglyceride-lipase in insect fat bodies (Phospholipase A1101), and the facilitated trehalose transporter (Tret1). Trehalose is the primary sugar in insect haemolymph, and its differential expression has been linked to migratory behaviour90,102,103. Additionally, we detected highly differentiated SNPs in some ABC transporter genes and Argonaute2 (AGO2) in the 3’ and 5’ UTR, respectively, which are associated with xenobiotic resistance. The protein AGO2 is crucial for antiviral defence by targeting and degrading viral RNAs104,105, and variation on its expression could contribute to differing innate immunity signatures across hemispheres, potentially due to variation in microbial community compositions.Regulatory regions play a critical role in the evolution of novel traits, with most trait-associated loci falling in non-coding regions106. We found 1203 highly differentiated SNPs (FST > 0.25) in the vicinity of 120 genes (5 kb flanking regions), accounting for 36% of the total variants identified. However, our ability to detect genetic differentiation from RAD sequenced variants is limited, as the genes covered in our RAD sequences represent only 21% of the genes encoded within the 9 Mb inverted region. Comprehensive population analyses using whole-genome sequencing (WGS) data could elucidate the full extent of the diverged gene sequences.Hemispheric migratory divides: a speciation driver in migratory species?Migratory divides have been proposed as a driving force in speciation by exposing sympatric or parapatric populations to divergent ecological conditions or by limiting genetic exchange through assortative mating or selection against hybridization12,13,14,15,16,17,18,21,107. These divides represent boundaries between adjoined populations with locally adapted migratory traits, such as migration direction or length, often leading to geographic segregation and genetic differentiation. While this phenomenon is well-documented in many vertebrate species (despite its genetic basis being known only in a handful of species), evidence for similar patterns in migratory insects has remained elusive. Indeed, no genetic differentiation has been found between differentially migrating insect populations such as those of monarch butterflies108,109 (reviewed in110), the Bogong moth111 or the fall armyworm54.However, our study provides evidence from Vanessa cardui suggesting that migratory divides may also play a significant role in insect speciation, despite the substantial differences between vertebrate and insect migration dynamics34. A key distinction in our findings is the identification of a latitudinal migratory divide spanning two hemispheres – a major departure from previously described longitudinal migratory divides that often arose from secondary contact following post-glacial colonisation from separate glacial refugia9. The biogeographic distributions of many migratory butterflies appear restricted to either hemisphere, with some species pairs showing evidence of speciation across hemispheric divides. An example is found in Danaus butterflies: D. plexippus (the monarch) is native to Central and North America, while its sister species, D. erippus, is confined to South America, engaging in extensive parallel migrations. Similarly, in Vanessa butterflies, the sister species V. carye and V. annabella complete their migratory cycles exclusively in South and North America, respectively, while V. kershawi, the painted lady sister species, migrates within Australia and New Zealand, where V. cardui is absent.While these cases may represent instances involving complete speciation, other systems may shed light on earlier stages of the speciation continuum. Several migratory butterfly species have distributional ranges spanning both hemispheres, such as Ascia monuste, Aphrissa statira, Phoebis sennae in the Americas, and Belenois aurota and Catopsilia florella in the Afrotropical region. Although the migratory cycles and phylogeographic structures of most migratory insect species are still to be fully characterised, we hypothesise that gene flow across such extensive migratory ranges might be constrained by reversed seasonality and differential navigational cues in each hemisphere, eventually leading to adaptation to hemisphere-specific biological rhythms and cues.This pattern likely extends beyond insects. For example, long-distance bird migrants (excluding shorebirds) breeding in the Northern Hemisphere and wintering in the Southern Hemisphere are proportionally a minority, and few Southern Hemisphere breeders are found to migrate north of the Equator38,39,40,41. Similarly, populations of humpback, fin whales, and several marine migratory fish species circulate almost exclusively within the Northern or Southern hemisphere, and rarely interact42,43,44.These eco-evolutionary patterns suggest that hemispheric divides could significantly influence the speciation of widespread migratory insect species. The existence of hemispheric divides and their segregating genetic diversity should be investigated across animal migratory lineages as a potentially overlooked pattern for speciation.MethodsObservational and field dataObservational occurrences of V. cardui used to infer phenological patterns of presence and absence by latitude were retrieved from (1) the Global Biodiversity Information Facility (GBIF112, https://www.gbif.org/es/), (2) the African butterfly and moth mapping project (LepiMAP113, http://vmus.adu.org.za/), both accessed the 23rd of May 2023, and (3) the authors’ own field expeditions from 2006 to 2023 (e.g. ref. 50), including 802 observations from Africa. We excluded the downloaded observations recorded on the 1st of January, as this date appears to be disproportionately used and is likely a placeholder for unknown collection dates within a given year. We retrieved 69,134 observations in total throughout Europe and Africa, dating from 1800 to 2023 (Supplementary Data S1). Monthly data were compiled, and relative abundances for established latitudinal divisions were plotted using ggplot2114 R package (Fig. 1).We calculated the maximum and minimum yearly photoperiod relative to each latitude using the R packages geosphere115 and terra116. Photoperiod amplitude was estimated as the subtraction of the maximum and minimum yearly day length. The temperature range was estimated as the subtraction of the lowest monthly average temperature from the highest monthly average temperature within each year, where monthly temperature averages were computed for a series of 20 years (2001–2021) using data from the Climate Research Unit (CRU v4.06) database117. Analyses were conducted using R v4.1.2, and R code is provided at the GitHub repository https://github.com/GTlabIBB/MigratoryDivide118. Both photoperiod and temperature amplitudes were projected to a map using QGIS v3.34.4119 to generate Fig. 1A.SamplingPainted ladies (V. cardui) are among the world’s most widespread butterfly species. We gathered 300 individuals across 38 countries in Europe and Africa (Fig. 1; Supplementary Data S2), as well as five individuals of V. kershawi, its sister species, which inhabits Australia and New Zealand (Supplementary Data S2). These specimens are deposited in the collections of the Botanical Institute of Barcelona (IBB, CSIC, Spain), the Institute of Evolutionary Biology (IBE, CSIC-Universitat Pompeu Fabra, Spain), and the Museum of Comparative Zoology (Harvard University, USA).Sequencing and preprocessing of ddRAD dataFor population genetic analysis, we prepared ddRAD genomic libraries for the 304 collected samples (Supplementary Data S2) following the protocol of Peterson et al.120. We used EcoRI and BfaI restriction enzymes and size selection of 300 bp using a PippinPrep instrument and 2% agarose cassettes (Sage Science, Beverly, MA, USA). Libraries were sequenced in 150 bp paired-end reads on Illumina HiSeq 2500 at the Harvard University Bauer Core Facility. Sequencing reads are deposited in the ENA (European Nucleotide Archive) database under project accession no. PRJEB80315. A total of 287,916,481 reads were generated, which were demultiplexed and assembled using ipyrad v0.9.81121 by mapping on the Vanessa cardui chromosome-level reference genome59 (GCA_905220365.1). Adapters were filtered using the cutadapt software integrated into the filter_adapters parameter of ipyrad. The clustering threshold was set to 0.85, resulting in a total of 188,408 loci across all 300 individuals. We used ipyrad default filtering parameters except for the minimum number of samples per locus, which was set to 100, i.e., one third of the samples (min_samples_locus parameter). After applying the filters, the number of assembled loci was 10,672. For downstream analysis, we subsequently filtered by variant missing data in VCFtools v0.1.16122, keeping SNPs that were present in at least 80% of the individuals (–max-missing parameter) and by minimum allele count, keeping alleles present in at least 2 individuals (–mac parameter). Last, in order to keep high-quality samples, we filtered out 3 individuals for which less than 40% of the total loci were recovered using the prop_typed_ind function in adegenet v2.1.5R package123, and removed 7 individuals that shared particularly high levels of coancestry according to fineRADstructure57, and thus were potentially related. These samples were indeed collected as larvae in the same locality, further reinforcing this inference. These filters resulted in a dataset, named dataset 1, comprising 290 samples and 32,719 SNPs. For local analysis of the inverted region on chromosome 8, we selected 976 SNPs located within this region using BCFtools v1.20124.We constructed a second dataset for fineRADstructure analysis, dataset 2, using a reduced set of high quality samples. We selected the first 30 samples from each hemisphere with the lowest levels of missing data. We ran the ipyrad reference-aligned pipeline with default filtering parameters, but the minimum number of samples per locus was set to 30 (i.e., half of the dataset). Additionally, we filtered for missing data with VCFtools, retaining SNPs that were present in at least 80% of the individuals (–max-missing parameter). Initially, we recovered 109,614 loci and 546,458 SNPs across 60 individuals; after filtering, the dataset was reduced to 13,571 loci and 169,590 SNPs. For local analysis on chromosome 8, we selected 2855 SNPs.A third dataset, named dataset 3, was generated for phylogenetic inference using the 60 samples from dataset 2, along with five additional samples from the sister species Vanessa kershawi as outgroup. The ipyrad assembly parameters and filtering steps were identical to those used for dataset 2. We recovered 117,546 loci across 65 samples, which were reduced to 12,973 loci and 569,090 SNPs after filtering steps. Summary statistics resulting from all ipyrad assembly steps for dataset 1, 2 and 3 are detailed in Supplementary Data S3.Genome-wide genetic differentiation analysisTo assess population structure, we first performed a PCA analysis using the adegenet v2.1.5 R package123 on all samples (dataset 1, 290 samples, Supplementary Data S3) across three genomic partitions: (i) the whole genome (Fig. S1), (ii) the inverted region on chromosome 8 (Fig. 2B), and (iii) the whole genome excluding the inverted region (Fig. S2). Additional PCAs were performed in LD-pruned versions of these datasets to assess for potential linkage effects, using SNP pairwise comparisons in PLINK with r2 < 0.2 and r2 < 0.05 (Fig. S3).We then used fineRADstructure to infer the number of genetic clusters and recent shared ancestry of individuals on each side of the migratory divide in dataset 2, both in the entire genome (60 samples, 169,590 SNPs, Fig. S1) and in the inverted region on chromosome 8 (Fig. 2A). In both analyses, sex chromosomes were excluded. Additionally, repetitive regions were filtered out using the annotation from Shipilina et al.125 with BEDTools v2.29.2126. No LD filtering was applied for fineRADstructure analyses, as the method specifically uses haplotype linkage information from RAD data to infer recent ancestry57.We next explored patterns of genetic differentiation by calculating window-based estimates of FST, π and dXY along the genome for dataset 2. To account for the fragmented coverage of RAD loci we used Pixy v1.2.5.beta1127, which handles missing data by excluding regions lacking mapped reads, thus excluding regions without RAD loci from genome-wide calculations. In order to accommodate this strategy, we generated an AllSites VCF file from the dataset 2 alignments obtained from ipyrad. This variant calling step was implemented in SAMtools mpileup followed by BCFTools call. Additionally, we filtered the resulting file for variant missing data in VCFtools, retaining SNPs present in at least 80% of the individuals (–max-missing parameter). The resulting AllSites VCF contained 1,136,244 variant sites and 2,788,714 invariant sites. Window size was set to 100 bp for FST to enable dot plot visualisation, while 1 Mb was used for π and dXY for curve visualisation, for which no smoothing algorithm was used. Results were plotted using ggplot2.Sequencing and preprocessing of WGS dataShort read data were used for phylogenetic inference and detection of inversion breakpoints. We used five V. cardui samples from a previous study34, collected in Ethiopia (14U392; 9°23′54.096′′ N, 38°49′25.323′′ E;), Namibia (15D327; 17°52′16.086′′ N, 19°24′22.611′′ E), Nepal (07F575; 28°25′0.635′′ N, 83°49′2.114′′ E), California, USA (15A205; 38°34′30.414′′ N, 121°34′38.117′′ E), and Spain (15A710; 40°15′29.16′′ N, 1°36′20.879′′ E). A Vanessa kershawi specimen, from Australia (AAM97U335; 34°19′30.63′′ S, 117°52′36.735′′ E), was sequenced for this study. A 350 bp insert paired-end library was prepared following the Illumina TruSeq DNA PCR-Free protocol. The library was sequenced at the Swedish National Genomics Infrastructure (NGI) facilities at 150 bp from both ends using a NovaSeq 6000 S4 sequencing platform, producing 140,660,532 reads (Table S5). Sequences are available in the ENA (European Nucleotide Archive) database under project accession no. PRJEB80315.Reads were mapped against the reference V. cardui genome59 (GCA_905220365.1) with BWA v0.7.12128, except for the V. kershawi specimen, where reads were mapped using Stampy v1.0.32129 and using an estimated substitution rate of 0.0406 to account for ~7my of divergence with V. cardui68. We obtained an average of 194 million mapped reads per sample (Table S5), which were sorted using SAMtools. The consensus sequences were obtained using SAMtools mpileup followed by BCFTools Call. The final fasta files were obtained with vcfutils vcf2fq122 command using a minimum coverage of 6x. The arthropod BUSCO v3.0.2130 analysis of completeness performed in the resulting assemblies using gvolante131 yielded an average of 98.63% completeness.Identification of inversion breakpointsTo identify the inversion breakpoints, we used paired-end read mapping information from the southern V. cardui individual (Namibia) reads mapped to the northern V. cardui reference genome59 (GCA_905220365.1). The resulting alignment had a mean mapping coverage of 32.34x and mapping quality of 71.3 (Table S5), as calculated with Qualimap v2.2.1132. Structural variant detection was performed using BreakDancerMax, a pipeline to identify breakpoints implemented in the software BreakDancer58. BreakDancer identifies anomalous read pairs in order to predict structural variants, and estimates a confidence score for each variant based on a Poisson model that takes into consideration the number of supporting anomalous read-pairs, the size of the anchoring regions, and the coverage of the genome58. With the highest probability score, BreakDancer identified 14 paired-end reads with abnormally large insert sizes and inverted orientation, pinpointing the inversion breakpoints at coordinates 4,402,570 and 13,379,476 (Table S2). We used the established coordinates to extract the inverted region for all analyses.Additionally, to investigate the synteny between V. cardui and its sister species, Vanessa kershawi on chromosome 8, we repeated the same procedure using WGS data from the sequenced V. kershawi specimen. The alignment had a mean coverage of 40.24x and a mean mapping quality of 43.61 (Table S5). No chromosomal inversions corresponding to the identified breakpoints with population data in V. cardui (positions 4,402,570–13,379,476) were detected in V. kershawi using read mapping information.Divergence time estimation of the inverted regionWe calculated the divergence time between the inverted and non-inverted haplotypes using the estimated absolute divergence (dXY) from Pixy. Absolute divergence was not consistent across the inversion length. Despite reduced crossover frequency, regions within large inversions can still pair in heterozygotes, allowing for gene conversion63,64. Stronger divergence near the inversion breakpoints and less pronounced divergence towards their centres are expected, showing a U-shaped profile25,26. Therefore, we focused on the first and last Mb of the inversion where recombination is lower, and which therefore provide the most reliable estimate of accumulated divergence since the origin of the inversion. We averaged the values obtained from the first and last Mb. Then, following the approach of Sánchez-Doñoso et al.26, we converted this divergence estimate to an estimate of divergence time using the equation133:$$T=frac{{d}_{{XY}}}{2cdot mu cdot g}$$
    (1)
    Where T is the divergence time in years, dXY is the absolute divergence, μ is the mutation rate in mutations per site per generation, and g is the number of generations per year. We used the estimated mutation rate for Heliconius melpomene134 (2.9 × 10−9), and considered a range of 7 to 10 generations per year50. Note that the divergence equation assumes a molecular clock, which implies neutral sequence divergence and a constant mutation rate. However, these regions are likely not neutral and might have complex demographic histories, so the time of divergence could be underestimated. Thus, it should be interpreted as an approximate indicator of the inversion’s origin and the onset of independent evolutionary trajectories between the arrangements.Heterozygosity estimatesHeterozygosity was estimated both for the inverted region on chromosome 8 and along the genome (excluding the inverted region) in dataset 1. We calculated heterozygosity as the number of variant sites over the total number of genotyped sites for each sample. We determined the North group as the specimens clustered in the negative values of PC1, the South group as the specimens clustered at the other side of the PC1 with values > 6, and the specimens scattered between these two thresholds were considered as potential heterozygotes. The obtained heterozygosity estimates did not follow a normal Gaussian distribution (Shapiro-Wilk normality test; inverted region: W = 0.98641, p-value = 0.008075 and excluding inverted region: W = 0.98409, p-value = 0.002754). Therefore, we statistically searched for differences in heterozygosity between North, South, and potential heterozygous groups within each region using Kruskal-Wallis one-way analysis of variance.Phylogenetic analysesWe investigated the phylogenetic history of the inverted region using whole-genome data from 5 specimens of Vanessa cardui across its worldwide distribution and one Vanessa kershawi specimen as an outgroup. The V. cardui and V. kershawi reference-based assemblies were aligned using MUSCLE135, implemented in Geneious Prime 2023.0.4136 (https://www.geneious.com). Gblocks v0.91b137 was then applied on the multiple sequence alignment to remove potentially ambiguously aligned regions using default parameters, generating concatenated alignments of 4,319,737 bp for the inverted region and 330,406,868 bp for the rest of the genome. Phylogenetic relationships were then inferred through a maximum likelihood (ML) analysis using RAxML-NG v1.1.0138, with a GTR + I model of nucleotide evolution. Nodal support was evaluated by 200 bootstrap iterations using the transfer bootstrap expectation (TBE).Phylogenetic relationships were also inferred for RAD data using dataset 3. A total of 420 RAD loci alignments were obtained for the inverted region and 12,550 for the rest of the genome using the ipyrad assembly pipeline. We used Gblocks with default parameters in each alignment to discard highly ambiguously aligned regions. Sequences with more than 90% of unmapped or uncalled bases (“Ns”) were removed from the alignments using cutadapt v5.0139 –max-n. These two steps resulted in a total of 397 and 11,935 filtered RAD loci alignments for the inverted region and for the rest of the genome, which were concatenated into 88,704 bp and 2,677,821 bp alignments, respectively. We inferred phylogenetic trees with IQ-TREE v2.0140, using ModelFinder to select the best-fit evolutionary model for each alignment based on the Bayesian Information Criterion (BIC). The resulting phylogenetic trees were plotted using the ggtree141 R package.Functional analysisEnrichment of functional categories in the genes located within the inverted region was analysed using the Bioconductor package topGO v2.44.0142. The enrichment test was performed with Fisher’s exact test using the default algorithm (“weight01”), which accounts for the hierarchical structure of the GO-terms. The annotated gene set from Shipilina et al.125 was used as the functional database. We adjusted the p-values of the resulting tests using the Benjamini-Hochberg method143 to correct for multiple testing.We explored whether SNPs showing the highest differentiation between southern and northern migratory ranges were located within protein-coding genes. We selected SNPs with FST > 0.25 (93.8th percentile) in dataset 2 and predicted effect strength and position in relation to known genes using SnpEff v5.2c144. We created the custom gene database in SnpEff based on the annotation version from Lohse et al.59. Information regarding the known biological functions of these proteins and functional effects is presented in Table S3 (inside the inversion) and Table S4 (outside the inversion).Reporting summaryFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

    Data availability

    Demultiplexed and adapter-trimmed RAD-sequencing reads, and whole genome sequencing reads generated in this study have been deposited in the European Nucleotide Archive under project accession number PRJEB80315. All accession numbers for samples used in this study are listed in Supplementary Data 2, including newly generated sequences and those from García-Berro et al.34 and Suchan et al.37 Data underlying Figs. 1–3 are available in Supplementary Data 1 and in the OSF repository at the following URL: https://osf.io/6u32k/ (https://doi.org/10.17605/OSF.IO/6U32K).
    Code availability

    Scripts used to process the data are available at: https://github.com/GTlabIBB/MigratoryDivide (https://doi.org/10.5281/zenodo.17113173).
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    Reprints and permissionsAbout this articleCite this articleGarcía-Berro, A., Shipilina, D., Backström, N. et al. A north-south hemispheric migratory divide in the butterfly Vanessa cardui.
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    Growth, immunity, and antioxidant activity responses of phytobiotics and probiotics incorporated into Oreochromis niloticus diets under stressors of unchanged water

    AbstractThe purpose of this study was to examine the impacts of incorporating commercial mixture phytobiotic/probiotics as feed additives in terms of performance, survival rate proximate composition, hematological parameters, immunological response and antioxidant enzymes activity of Oreochromis niloticus reared under un-exchanged water system. Nile tilapia with an average beginning weight ranged from 48.49 to 52.50 g and were distributed in concrete ponds (1 m × 1 m × 1 m; L × W × H) at 20 fish / pond. Four treatments were performed as follows: control group (CG) fish fed a basal feed without water exchange and three groups were reared under zero water exchange with adding three different doses of commercial phytobiotic /probiotics on the basal feed (Garlex®, Superimmune®, and Gallipro 200®): Group 1: (200 mg, 500 mg, and 200 mg)/kg diet, Group 2: (400 mg, 1000 mg, and 400 mg)/ kg diet and Group 3: (600 mg, 1500 mg, and 600 mg)/ kg diet. This trial lasted for ninety days. Groups 1 and 2 had the best growth indices and survival rates, and there was not a significant (P ≤ 0.05) distinction between them. Fish fed Groups 1 and 2 showed the greatest improvement in proximate body composition, immunological responses, and antioxidant enzyme activity, including lysozyme, superoxide dismutase (SOD), catalase (CAT), and glutathione (GSH). This study has detected that feed additives of commercial mixture of 200 mg/kg Garlex®, 400 mg/kg Super Immune®, and 200 mg/kg GallPro-200® led to improved growth and physiological status of O. niloticus under un-exchanged water ponds.

    Data availability

    The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.
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    Download referencesAcknowledgementsAuthors would like to thank all staff members of Aquaculture Department, Faculty of Aquaculture and Marine fisheries, Arish University, Egypt for supporting this research. Also, authors would like to acknowledge all members and staff of Water Resources Research Station, National institute of Oceanography and Fisheries NIOF in Fayoum, Egypt for their help and support in completing this work.FundingOpen access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).Author informationAuthors and AffiliationsDepartment of Aquaculture and Biotechnology, Faculty of Aquaculture and Marine Fisheries, Arish University, Arish, EgyptMohamed F. Abdel-AzizZoology Department, Faculty of Science, Arish University, Arish, EgyptRabab M. AlkaradaweNational Institute of Oceanography and Fisheries (NIOF), PO Box 11516, Cairo, EgyptHamed H. E. Saleh, Mohamed A. Elokaby & Abdel-Moneim M. YonesAnimal Nutrition and Clinical Nutrition Department, Faculty of Veterinary Medicine, Benha University, Toukh, 13736, EgyptFatma Ragab Abouel AzmBiology Department, College of Science, Jouf University, Sakaka, 72341, Saudi ArabiaDalia S. HamzaDepartment of Zoology, Faculty of Science, Benha University, Benha, 13518, EgyptDalia S. HamzaDepartment of Zoology, Faculty of Science, Qena University, Qena, 83523, EgyptNaglaa R. A. KasemAuthorsMohamed F. Abdel-AzizView author publicationsSearch author on:PubMed Google ScholarRabab M. AlkaradaweView author publicationsSearch author on:PubMed Google ScholarHamed H. E. SalehView author publicationsSearch author on:PubMed Google ScholarFatma Ragab Abouel AzmView author publicationsSearch author on:PubMed Google ScholarMohamed A. ElokabyView author publicationsSearch author on:PubMed Google ScholarAbdel-Moneim M. YonesView author publicationsSearch author on:PubMed Google ScholarDalia S. HamzaView author publicationsSearch author on:PubMed Google ScholarNaglaa R. A. KasemView author publicationsSearch author on:PubMed Google ScholarContributionsMohamed F. Abdel-Aziz, Rabab M. Alkaradawe, Hamed H. E. Saleh, Fatma Ragab Abouel Azm, Mohamed A. Elokaby, Abdel-Moneim M. Yones, Dalia S. Hamza and Naglaa R.A. Kasem : Conceptualization, Methodology, Formal analysis, Supervision, Investigation, Resources, Writing – review, editing. Mohamed F. Abdel-Aziz: Writing – original draft.Corresponding authorCorrespondence to
    Mohamed F. Abdel-Aziz.Ethics declarations

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    All research protocols of this work were ethically approved by the National Institute of Oceanography and Fisheries’ (NIOF, Egypt) Committee for Ethical Care and Use of Animals/Aquatic Animals (NIOF-IACUC) with code number (NIOF-AQ4-F-24-R-022). The authors confirm that all methods were performed in accordance with the relevant guidelines and regulations and that the study is reported in accordance with the ARRIVE guidelines.

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    Reprints and permissionsAbout this articleCite this articleAbdel-Aziz, M.F., Alkaradawe, R.M., Saleh, H.H.E. et al. Growth, immunity, and antioxidant activity responses of phytobiotics and probiotics incorporated into Oreochromis niloticus diets under stressors of unchanged water.
    Sci Rep (2025). https://doi.org/10.1038/s41598-025-30248-2Download citationReceived: 13 August 2025Accepted: 24 November 2025Published: 29 December 2025DOI: https://doi.org/10.1038/s41598-025-30248-2Share this articleAnyone you share the following link with will be able to read this content:Get shareable linkSorry, a shareable link is not currently available for this article.Copy shareable link to clipboard
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    Assessing cetacean encounter risk in offshore racing

    AbstractLarge cetaceans face several anthropogenic threats. Among these, collisions are a major cause of anthropogenic mortality. Assessing and limiting their impact on populations is essential, as these species play an essential ecological role. All types of vessels, including offshore racing vessels, can collide with cetaceans. When a collision occurs between an offshore racing vessel and a large cetacean, the consequences are severe for both the whale, which is often injured or even killed and the vessel, which can suffer severe damage and be forced to withdraw from the race. Our study aimed to develop an encounter model that takes the characteristics of both cetaceans and racing vessels into account to estimate the number of encounters along vessel routes. The model was applied to three different routes commonly used in offshore racing: the first between Newport, USA and Skagen, Denmark; the second between Dover, England and the Gibraltar Strait; and the third between the Gibraltar Strait and Genoa, Italy. The number of encounters was estimated to be 1.7 for Route 1, 4.1 for Route 2 and 2.6 for Route 3. The model was also used to estimate the impact of routing vessels away from any exclusion zones that may be established in areas of high cetacean abundance. This routing could significantly reduce the number of encounters and offer potential solutions to reduce collisions between cetaceans and all types of vessels. The issue of collisions is becoming increasingly important and requires the development of methods to reduce the number of collisions worldwide.

    Data availability

    Vessel tracks were simulated using the qtVlm navigation software (©Meltemus 2017 – 2024, https://www.meltemus.com/), which is an open-access software, but the polars used were provided by Bañulsdesign and are confidential. Cetacean densities in the western Atlantic Ocean are publicly available at https://seamap.env.duke.edu/models/Duke/EC/. Cetacean densities near Iceland are not publicly available and should be requested from the North Atlantic Marine Mammal Commission (NAMMCO). Cetacean densities in the Northeast Atlantic Ocean are not publicly available and should be requested from the Direction Générale de l’Armement Techniques Navales. The corresponding author can provide contact details.
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    Download referencesAcknowledgementsWe are grateful to the many observers who participated in the surveys and collected all the data used to model cetacean densities, as well as to ship captains, crews, and pilots. We thank the Direction Générale de l’Armement Techniques Navales for providing cetacean densities in the Northeast Atlantic Ocean and the Mediterranean Sea. We thank the Duke Marine Geospatial Ecology Laboratory for providing cetacean densities in the Northwest Atlantic Ocean through their web portal (https://seamap.env.duke.edu/models/Duke/EC/). We thank the North Atlantic Marine Mammal Commission (NAMMCO) for providing cetacean densities estimated from the NASS survey.FundingThe study and open access were funded by Share The Ocean.Author informationAuthors and AffiliationsShare The Ocean, Larmor-Baden, 56870, FranceAuriane Virgili & Renaud BañulsBañulsdesign, Larmor-Baden, 56870, FranceSébastien Fournier, Malo Pocheau & Renaud BañulsCentre de Mathématiques Appliquées de l’Ecole Polytechnique, UMR 7641 CNRS, Inria, Institut Polytechnique de Paris, Palaiseau, 91128, FranceOlivier Le MaîtreObservatoire Pelagis, UAR 3462 CNRS, La Rochelle Université, La Rochelle, 17000, FranceVincent RidouxCentre d’Etudes Biologiques de Chizé – La Rochelle, UMR 7372 CNRS – La Rochelle Université, Villiers-en-Bois, 79350, FranceVincent RidouxNantes Université, Ecole nationale supérieure d’architecture de Nantes, Nantes, 44000, FranceRenaud BañulsAuthorsAuriane VirgiliView author publicationsSearch author on:PubMed Google ScholarSébastien FournierView author publicationsSearch author on:PubMed Google ScholarOlivier Le MaîtreView author publicationsSearch author on:PubMed Google ScholarMalo PocheauView author publicationsSearch author on:PubMed Google ScholarVincent RidouxView author publicationsSearch author on:PubMed Google ScholarRenaud BañulsView author publicationsSearch author on:PubMed Google ScholarContributionsA.V. wrote the main manuscript. A.V., S.F. and M.P. carried out the analyses. V.R., O.L.M. and R.B. contributed their expertise to the study, and R.B. supervised it. All authors reviewed the manuscript.Corresponding authorCorrespondence to
    Auriane Virgili.Ethics declarations

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    Reprints and permissionsAbout this articleCite this articleVirgili, A., Fournier, S., Le Maître, O. et al. Assessing cetacean encounter risk in offshore racing.
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    Predicting sugar beet leaf area index: evaluating performance of double sigmoid functions under different irrigation and plant density scenarios

    AbstractThe leaf area index (LAI) dynamics in sugar beet follow a double sigmoidal curve, modeled as the subtraction of two sigmoid functions. In this study, we examined the accuracy of 15 different sigmoid functions in describing the sugar beet LAI variation based on growing degree days (GDD) and days after planting (DAP) in different irrigation treatments and crop densities under direct and transplant cultivation. The results showed that the Logistic-Richards (LR) and Hill-Hill functions (HH) effectively modeled the measured LAI data over the GDD and DAP-based growing period for both direct sowing and transplant cultivation. The LR (NRMSE = 0.04, d = 0.99, MRE = -0.006) and HH (NRMSE = 0.05, d = 0.99, MRE = -0.01) achieved the best performance for direct sowing calibration based on GDD. In contrast, the Von Bertalanffy, Weibull, and Hill functions were not suitable for describing sugar beet LAI dynamics. Adjusting function coefficients to account for environmental factors such as seasonal applied water, rainfall, and planting density generally led to decreased predictive accuracy, under direct and transplant cultivation. Therefore, LR functions can be valuable for modelling sugar beet LAI variation under various irrigation treatments and crop densities throughout the growing season.

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    IntroductionPlant growth is very important in order to evaluate and manage irrigation and agricultural operations, with the aim of increasing production. Hence, plant indices, such as the leaf area index (LAI), are crucial to determining the photosynthetic surface area of a plant, its water consumption, and its yield1. LAI measures the total one-sided leaf area per unit ground area, reflecting canopy density2. Due to the difficulty of measuring LAI directly on a large scale3, different sigmoid functions can be used to estimate LAI.A double sigmoid function generally has two sigmoidal-shaped curves, providing a better fit for modeling certain biological and ecological processes. Nonlinear regression models best describe growth curves, with parameters estimated by minimizing the residual sum of squares4. Therefore, LAI can be modeled as a double sigmoid curve, formed by the subtraction (or interaction) of two underlying sigmoid functions. Several singular sigmoid functions have one inflection point and have been used to model growth dynamics, including the Logistic5, Gompertz6, Beta7, Richards8, Von Bertalanffy9, Weibull10, and Hill11. Double sigmoid functions exhibit two inflection points where the growth rate transitions from an increasing to a decreasing trend12. By fitting growth data to sigmoid functions, researchers can better understand and predict plant performance under varying environmental and management conditions.Despite their widespread application in modeling crop growth, sigmoid functions have several limitations that need to be acknowledged. Firstly, these models assume smooth and predictable growth patterns with clear inflection points. However, actual crop development often shows irregularities due to environmental stress, nutrient variability, or pest incidence, leading to inaccurate predictions. Secondly, parameter estimation in nonlinear regression is highly sensitive to initial values, making the calibration process computationally challenging and prone to convergence issues. Thirdly, double sigmoid models, while more flexible, involve additional parameters that increase the risk of overfitting to calibration data, reducing model robustness when applied to independent datasets or under stress conditions. Finally, these models primarily represent time-dependent growth without directly accounting for dynamic environmental drivers such as humidity, wind speed, and soil water status13,14,15,16. Addressing these limitations may require hybrid modeling strategies or integrating additional meteorological and soil parameters into future predictive.A study was conducted on five maize cultivars to evaluate the performance of different nonlinear functions (Richards, Logistic, Weibull, and Gompertz) in fitting leaf growth data16. The results indicate that the Richards, Logistic, and Gompertz functions demonstrate superior performance compared to Weibull in predicting leaf growth in maize. In another study17, reported that the growth behavior of a lettuce canopy was described using three nonlinear functions: Gompertz, Logistic, and grey Verhulst18. The functions were applied to top projected canopy area (TPCA), top projected canopy perimeter (TPCP), and plant height (PH). The grey Verhulst model showed better fitting for TPCA and TPCP growth, while the Logistic model fit well the PH changes. Also19, demonstrated that the Logistic model was more suitable for describing height growth of maize than the Gompertz function, as it achieved a coefficient of determination exceeding 99%. In a recent study by20, it was demonstrated that the XGBoost function and Light GBM function outperformed the Gompertz function and Logistic function in predicting maize plant growth. The accuracy of 49 function compositions of the double sigmoid functions in describing rapeseed dry matter based on days after planting (DAP) and growing degree days (GDD) was examined12. The results indicated that the Beta-Richards and Richards-Gompertz functions accurately represented the measured dry matter during the growing season based on DAP and GDD, respectively.Sugar beet is primarily grown in arid and semi-arid regions as an irrigated crop. Adequate water supply and management are crucial for the successful production of sugar beet and white sugar due to the crop’s long growing season21. The water requirement for sugar beet cultivation is influenced by various factors such as climatic conditions, irrigation practices, growing season, crop density, genotype, and the use of nitrate fertilizer22. A water-saving technique commonly used in arid regions is deficit irrigation23, which involves applying less water than the full crop requirement for evapotranspiration24. Optimizing crop density is a key factor in enhancing both the quantity and quality of sugar beets. Crop density directly impacts root size, sugar content, yield, and mineral composition in the roots25.Based on our knowledge, few studies were conducted on determining the LAI of sugar beet. Given the significance of sugar beet in sugar production, this study was aimed to (1) estimate the LAI of sugar beet using different double sigmoid functions (Logistic-Logistic, Gompertz-Gompertz, Logistic-Hill, Weibull-Logistic, and …) under varying irrigation treatments and crop densities under direct sowing and transplant cultivations, (2) assess the LAI of sugar beet based on days after planting (DAP) and growing degree days (GDD), and (3) select the best double sigmoid functions. The findings of this study can help in estimating the LAI of sugar beet throughout its growth cycle.Materials and methodsField experimentThe impact of plant density, planting method and irrigation regime on sugar beet yield over two years (2017 and 2018) at the Research Station, School of Agricultural, Shiraz University, Iran, situated 16 km north of Shiraz was studied by26,27. In this study LAI data, depth of irrigation water, and rainfall amount were obtained from26,27. The total amount of rainfall during growing period was 22.5 mm in 2017 occurred in the first week and 55 mm in 2018 occurred in first and second week of planting period27.The experiment used a split-split plot experimental design arranged in a complete randomized block framework with three replications. The main plots were assigned to three irrigation levels: full irrigation (100%, I100), 75% of full irrigation (I75), and 50% of full irrigation (I50). Two planting methods — direct sowing (D) and transplant cultivation (T)— were used in the subplots. The sub-subplots consisted of four plant density treatments of 180,000 plants ha− 1 (P180), 135,000 plants ha− 1 (P135), 90,000 plants ha− 1 (P90), and 45,000 plants ha− 1 (P45). Data from 2017 were employed to calibrate the double sigmoid function, and its predictive performance was validated using data from 2018. For more detailed information, please refer to26,27.Mathematical functionsAmong the various mathematical models used to describe plant growth curves, sigmoid functions (such as the Logistic, Gompertz and Richards functions and …) are particularly effective. They accurately capture the three distinct phases of biological growth: the lag phase (characterized by slow initial development), the log phase (marked by rapid growth), and the stationary phase (where growth is slow and reaches plateau). The various sigmoid equations used in this research are as follows:$$:text{y}={text{y}}_{text{m}text{i}text{n}}+frac{{text{y}}_{text{m}text{a}text{x}}-{text{y}}_{text{m}text{i}text{n}}}{1+text{e}text{x}text{p}(-text{a}text{t}+text{b})}:{rm Logistic}$$
    (1)
    $$:text{y}={text{y}}_{text{m}text{i}text{n}}+left({text{y}}_{text{m}text{a}text{x}}-{text{y}}_{text{m}text{i}text{n}}right)text{e}text{x}text{p}(-text{e}text{x}text{p}left(-text{a}text{t}+text{b}right)):::{rm Gompertz}$$
    (2)
    $$:y={y}_{min}+frac{{y}_{max}-{y}_{min}}{[1+vexp{(-at+b)]}^{raisebox{1ex}{$1$}!left/:!raisebox{-1ex}{$v$}right.}}:::{rm Richards}$$
    (3)
    $$:text{y}={text{y}}_{text{m}text{i}text{n}}+({text{y}}_{text{m}text{a}text{x}}-{text{y}}_{text{m}text{i}text{n}})(1-text{e}text{x}text{p}(-text{a}{text{t}}^{text{b}}):::{rm Weibull}$$
    (4)
    $$:text{y}={text{y}}_{text{m}text{i}text{n}}+left({text{y}}_{text{m}text{a}text{x}}-{text{y}}_{text{m}text{i}text{n}}right)left(left(1+frac{text{a}-text{t}}{text{a}-text{b}}right){left(frac{text{t}}{text{a}}right)}^{frac{text{a}}{text{a}-text{b}}}right):::{rm Beta}$$
    (5)
    $$:text{y}={text{y}}_{text{m}text{i}text{n}}+left({text{y}}_{text{m}text{a}text{x}}-{text{y}}_{text{m}text{i}text{n}}right)left(frac{{text{a}}^{text{b}}}{{text{a}}^{text{b}}+{text{t}}^{text{b}}}right):::{rm Hill}$$
    (6)
    $$:y={y}_{min}+left({y}_{max}-{y}_{min}right){left{1-left[1-{left(frac{{y}_{min}}{{y}_{max}}right)}^{raisebox{1ex}{$1$}!left/:!raisebox{-1ex}{$4$}right.}right]{exp}left[frac{-at}{4{left({y}_{min}right)}^{raisebox{1ex}{$1$}!left/:!raisebox{-1ex}{$4$}right.}}right]right}}^{4}:::{rm Von: Bertalanffy}$$
    (7)
    where y represents the leaf area index (dependent growth parameter), bounded between a minimum value ymin and maximum value ymax. The independent variable t corresponds to the growth period, measured either in days after planting (DAP) or growing degree days (GDD). The curve’s shape and transition rates are governed by three constant coefficients: “a” controls the inflection point position, “b” determines the growth rate, and “v” adjusts the curve’s asymmetry. By combining singular sigmoid functions, various double sigmoid functions can be created to predict LAI as follows:$$:text{L}text{A}text{I}={text{L}text{A}text{I}}_{text{m}text{i}text{n}}+left({text{L}text{A}text{I}}_{text{m}text{a}text{x}}-{text{L}text{A}text{I}}_{text{m}text{i}text{n}}right){odot}{text{f}}_{1}left(text{x}right)-left({text{L}text{A}text{I}}_{text{m}text{a}text{x}}-{text{L}text{A}text{I}}_{text{e}text{n}text{d}}right){odot}{text{f}}_{2}left(text{x}right)$$
    (8)
    where ʘ is the operator × or / and (:{text{f}}_{1}left(text{x}right)) and (:{text{f}}_{2})(x) are one of singular sigmoid functions (Eqs. 1–7), respectively. The double sigmoid functions for predicting LAI do not exhibit a purely increasing trend; the LAI rises up to a certain point (LAImax), then stabilizes, and after a while begins to decrease, LAImin is the LAI at initial growth stage, and LAIend is the LAI at end of growing season. The double sigmoid functions are presented in Table 1. The reason for 15 selected widely used double sigmoid functions [similar to 12] was to avoid unrealistic shapes or excessive parameterization. The order of combination was also crucial as the first function primarily influences the early growth trajectory, while the second governs the senescence phase.Table 1 Double sigmoid functions.Full size tableDetermining coefficients of equationsThe coefficients (a, b, c, d and v) of the functions in Table 1 were calculated using Excel software, based on the 2017 dataset (calibration year). Solver, a powerful tool in Excel, was utilized to solve the nonlinear equations. By minimizing the error value and employing the GRG (Generalized Reduced Gradient) method as the problem-solving technique, Solver determined the optimal coefficients for the functions. The objective function was minimized by sum of square error (SSE) as follows:$$:text{S}text{S}text{E}=sum:_{text{i}=1}^{text{n}}{({text{P}}_{text{i}}-{text{M}}_{text{i}})}^{2}$$
    (24)
    where (:{text{P}}_{text{i}}) and (:{text{M}}_{text{i}}) are the predicted and measured values by the double sigmoid function, respectively and n is the number of observations.All the calculations were conducted based on days after planting (DAP), however, cumulative heat units was used in the sigmoid growth function to make the results more applicable for various environmental conditions. The cumulative heat units, determined as growing degree days (GDD), are calculated as follows:$$:text{G}text{D}text{D}=sum:({text{T}}_{text{a}text{v}text{e}}-{text{T}}_{text{b}text{a}text{s}text{e}})$$
    (25)
    where (:{text{T}}_{text{a}text{v}text{e}}) is the mean daily air temperature (°C) and (:{text{T}}_{text{b}text{a}text{s}text{e}}) is the base temperature (°C) or the air temperature threshold below which plant growth does not occur. In this study, base temperature was 2.6 °C28. As temperature is used to calculated GDD, the results will be more applicable for different regions with different climates.Linking double sigmoid function coefficients to agronomic variablesAccording to29, the values of the coefficients of double sigmoid functions (a, b, c, d and v) are influenced by factors related to agronomic management practices. The values of constants in Eqs. (1-7) were fitted to quadratic functions of plant density and the total amount of seasonal applied water and rainfall, using SPSS software.$$:text{C}text{O}text{N}={text{A}}_{1}left(text{I}+text{R}right)+{text{B}}_{1}left(text{P}right)+{text{C}}_{1}left(text{I}+text{R}right)left(text{P}right)+{text{D}}_{1}{(text{I}+text{R})}^{2}+{text{E}}_{1}{left(text{P}right)}^{2}+{text{F}}_{1}{left(text{I}+text{R}right)}^{2}left(text{P}right)+{text{G}}_{1}left(text{I}+text{R}right){left(text{D}text{P}right)}^{2}+{text{H}}_{1}{(text{I}+text{R})}^{2}{left(text{P}right)}^{2}+{text{I}}_{1}$$
    (26)
    where CON is the coefficients of a, b, c, d, and v in Eqs. (1–7), I and R are the seasonal applied water and rainfall (m), respectively, P is the plant density (plant (:{text{m}}^{-2})), and A1, B1, C1, D1, E1, F1, G1, H1, and I1 are the coefficients of multiple regression.Statistical parametersTo evaluate the model, normalized root mean square error (NRMSE), the index of agreement (d), and mean residual error (MRE) were used. These parameters are calculated as follows30:$$:text{N}text{R}text{M}text{S}text{E}={left[frac{sum:_{text{i}=1}^{text{n}}{left({text{L}text{A}text{I}}_{{text{P}}_{text{i}}}-{text{L}text{A}text{I}}_{{text{m}}_{text{i}}}right)}^{2}}{text{n}{text{L}text{A}text{I}}_{{text{m}}_{text{a}text{v}text{g}}}^{2}}right]}^{0.5}$$
    (27)
    $$:text{d}=1-frac{sum:_{text{i}=1}^{text{n}}{left({text{L}text{A}text{I}}_{{text{P}}_{text{i}}}-{text{L}text{A}text{I}}_{{text{m}}_{text{i}}}right)}^{2}}{sum:_{text{i}=1}^{text{n}}{left(left|{text{L}text{A}text{I}}_{{text{P}}_{text{i}}}-{text{L}text{A}text{I}}_{{text{m}}_{text{a}text{v}text{g}}}right|+left|{text{L}text{A}text{I}}_{{text{m}}_{text{i}}}-{text{L}text{A}text{I}}_{{text{m}}_{text{a}text{v}text{g}}}right|right)}^{2}}$$
    (28)
    $$:text{M}text{R}text{E}=frac{sum:_{text{i}=1}^{text{n}}({text{L}text{A}text{I}}_{{text{P}}_{text{i}}}-{text{L}text{A}text{I}}_{{text{m}}_{text{i}}})}{text{n}{text{L}text{A}text{I}}_{{text{m}}_{text{a}text{v}text{g}}}}$$
    (29)
    where (:{text{L}text{A}text{I}}_{{text{P}}_{text{i}}}), (:{text{L}text{A}text{I}}_{{text{m}}_{text{i}}}) and (:{text{L}text{A}text{I}}_{{text{m}}_{text{a}text{v}text{g}}}) are the predicted, measured and mean of measured leaf area index, respectively, and n is the number of observations. MRE and NRMSE with smaller values and d with higher value indicates higher precision and accuracy of the model. Values of NRMSE less than 0.1 indicate excellent estimation accuracy, values between 0.1 and 0.2 show good estimation accuracy, values between 0.2 and 0.3 indicate fair estimation accuracy, and values of NRMSE greater than 0.3 show poor estimation accuracy31.ResultsFitting functions to leaf area index based on GDDThe coefficients of the sigmoid functions (using solver tool) were determined for all 15 functions under various irrigation levels and planting densities in both direct sowing and transplant cultivation. Among 15 functions, the Logistic-Logistic, Gompertz-Gomperz, Hill-Hill, Logistic-Richards, and Logistic-Hill functions performed more accurately considering the statistical indicators (Table S1). Based on the statistical parameters in Tables S1, the Logistic-Logistic, Gompertz-Gompertz, Hill-Hill, Logistic-Richards and Logistic-Hill double sigmoid functions yielded the most favorable outcomes for both direct sowing and transplant cultivation. The estimated coefficient values and statistical results of the top five performing double sigmoid functions are provided in Tables 2 and 3 for I100, four plant densities, and two cultivation methods. Also, the estimated coefficient values of the top five performing double sigmoid functions for all irrigation treatments, plant densities and cultivation methods are provided in Tables S2-S6.Table 2 Estimated coefficient values and the statistical indicators of various double sigmoid functions based on growing degree-days for I100, four planting densities, and direct sowing, and the statistical indicators for all irrigation treatments and crop densities.Full size tableIn direct cultivation (Table 2), the Logistic-Richards function for direct sowing showed the best overall performance, with the lowest NRMSE (0.04) and a high index of agreement (0.99), indicating minimal deviation between the predicted and measured values. Similarly, the Logistic-Hill function also performed strongly under direct sowing conditions (NRMSE = 0.04, d = 0.99, and MRE= −0.004) (Table 2).Under optimal irrigation water conditions (Table 2), all models showed strong fits with NRMSE values mostly below 0.05. The Logistic-Richards, Logistic-Logistic, and Logistic-Hill functions performed slightly better under high planting densities, with minimal errors (NRMSE = 0.008 at 180,000 plants ha− 1). The Gompertz-Gompertz and Hill-Hill functions had higher NRMSE values in comparison with latter models, especially at 135,000 and 45,000 plants ha− 1 (0.04 and 0.04, respectively). Overall, all functions provided adequate fits under well-watered conditions, with Logistic-Richards and Logistic-Hill showing more consistent accuracy across planting densities.At 75% irrigation level (data not shown), model accuracy decreased slightly, but NRMSE values remained below 0.1 for all functions. Logistic-Richards, Logistic-Logistic, and Logistic-Hill maintained high accuracy (NRMSE range: 0.03–0.07). Hill-Hill had minor increases in error, while Gompertz-Gompertz showed higher sensitivity to planting density under water deficit condition, with NRMSE values reaching 0.09 at 45,000 plants ha− 1.Under severe irrigation stress, 50% level (data not shown), model performance varied significantly. Logistic-Richards consistently outperformed all others with low error values across all densities (NRMSE range: 0.04–0.09). Logistic-Hill showed comparable performance, especially at intermediate densities. Logistic-Logistic and Gompertz-Gompertz had substantial performance degradation, with Logistic-Logistic reaching a NRMSE of 0.150 at 90,000 plants.ha− 1 and Gompertz-Gompertz peaking at 0.12 at 45,000 plants ha− 1. Hill-Hill showed moderate robustness, but Logistic-based models outperformed it under stress. The findings highlight the Logistic-Richards model’s adaptability in modeling asymmetric growth patterns typical under drought stress conditions. To maintain clarity and conciseness, the model comparison was conducted specifically under the direct sowing condition.In transplant cultivation (Table 3), the Logistic-Hill function again performed competitively, achieving the lowest NRMSE (0.06) and a high index of agreement (0.99) among the functions considered, with a near-zero MRE (−0.004). Although the Logistic-Logistic function achieved the same index of agreement (0.99) under transplant conditions, it presented a higher NRMSE (0.06) and a positive MRE (0.04), indicating a slight overestimation trend. The performance of models under different irrigation water levels, planting densities, and transplant cultivation were similar to that in direct cultivation (Table 3).Table 3 Estimated coefficient values and statistical indicators of various double sigmoid functions based on growing degree-days for I100, four planting densities, and transplant cultivation, and the statistical indicators for all irrigation treatments and crop densities.Full size tableThe goodness-of-fit parameters presented in Table 2 strongly support this observation. The Logistic–Richards function demonstrated the best performance with the lowest NRMSE (0.04) and the highest index of agreement (d = 0.99), indicating its superior accuracy and reliability under direct cultivation, while the Logistic-Hill performed the best (NRMSE = 0.064, d = 0.99, MRE=−0.004) under transplant condition (supplementary figure, Fig. S1). The Hill–Hill function also performed well, with NRMSE = 0.05 and d = 0.99, showing excellent predictive agreement across various planting densities under direct cultivation. The Logistic–Hill and Logistic–Logistic functions followed closely, with NRMSE values of 0.04 and 0.06 under direct cultivation and 0.064 and 0.063 under transplant condition, respectively (Fig. S1). All functions exhibited minimal mean relative error (MRE), typically ranging from − 0.01 to − 0.004, indicating no systematic bias in their predictions. The consistency of performance across planting densities (180,000 to 45,000 plants ha− 1) under full irrigation (I100) further underscores the robustness of these models.The Beta–Beta, Logistic–Gompertz, and Beta–Logistic functions showed moderate fitting performance for both cultivation methods (Fig. 1). The Beta–Beta function had NRMSE values ranging from 0.09 to 0.11, with index of agreement (d) values of 0.99 to 0.99 and close-to-zero MRE values (−0.004 to 0.0006). The Logistic–Gompertz function had NRMSE values of 0.08 and 0.09, with d values between 0.99 and 0.99, but positive MRE values (0.02 and 0.01) indicating a tendency to overestimate LAI. The Beta–Logistic function performed the weakest, with NRMSE values of 0.15 and 0.12, d values of 0.98 and 0.99, and the largest MRE of 0.04, showing overestimation of LAI. The Beta–Beta, Logistic–Gompertz, and Beta–Logistic functions exhibited greater deviation and less reliability in capturing LAI dynamics, compared to top-performing functions like Logistic–Richards and Hill–Hill.Moreover, variation of measured and predicted leaf area index by double sigmoid functions corresponding to 100% irrigation (I100) and a planting density of 180,000 plants ha− 1 (P180) for both cultivation methods during 2017 growing season (calibration year) are presented in Fig. 1. According to Fig. 1, the Logistic–Gompertz function modeled LAI with high accuracy under transplant conditions, especially when GDD was used as the time scale. Under direct sowing, the function also performed well, though it slightly overestimated early vegetative growth. The Beta–Logistic function slightly underfitted the initial LAI rise in direct sowing, while overestimating peak values in transplant cultivation. These observations suggest that such combinations are sensitive to cultivation-specific growth dynamics and may not generalize well without calibration. Among 15 selected double sigmoid functions, certain functions consistently showed poor performance, especially when specific functions were used as the first stage in the double sigmoid sequence. These included Weibull–Weibull, Von Bertalanffy–Von Bertalanffy, Logistic–Beta, Logistic–Weibull, Weibull–Logistic, and Von Bertalanffy–Logistic. The Weibull–Weibull function had high errors (NRMSE = 0.25 for direct sowing, 0.36 for transplanting) and slight under and overestimation (MRE= −0.005 for direct sowing, 0.11 for transplanting). The Von Bertalanffy–Von Bertalanffy function showed fair and poor results (NRMSE = 0.28 for direct sowing, 0.38 for transplanting) and slight overestimation (MRE = 0.04 for direct sowing, 0.005 for transplanting). The Logistic–Beta function performed fair and poor (NRMSE = 0.26 for direct sowing, 0.36 for transplanting) with slight overestimation (MRE = 0.01 for direct sowing, and 0.01 for transplanting). The Logistic–Weibull function was more accurate for direct sowing (NRMSE = 0.21, MRE= −0.01) but less accurate for transplanting (NRMSE = 0.38, MRE = 0.13). The Weibull–Logistic function was more accurate for direct sowing (NRMSE = 0.12, MRE = 0.005) in comparison with that for transplanting (NRMSE = 0.45, MRE = 0.11). The Von Bertalanffy–Logistic function provided good fits for both planting methods (NRMSE = 0.17 for direct sowing, and 0.25 for transplanting). The sensitivity of model performance to sigmoid component order and form was evident, with some combinations producing biologically implausible growth curves. The most significant issues arose when Weibull, Von Bertalanffy, or Beta functions were used as the initial component. These combinations often led to underestimation of early vegetative growth or produced unrealistic shapes with sudden transitions or overly flattened peaks. For instance, Weibull–Logistic and Von Bertalanffy–Logistic underestimated LAI during early GDD accumulation in direct sowing, while Logistic–Beta did not align well with the decline phase in transplant and also for direct sowing. The over- or underestimation of dual functions emphasizes the importance of function order, as improper placement can distort growth phase representation, especially when the mathematical structure lacks flexibility or biological relevance.Fig. 1Variation of measured and predicted leaf area index by double sigmoid functions during 2017 growing season (calibration year). The circular and rhombus points represent the measured leaf area index for direct sowing and transplant cultivation, respectively. The solid and dashed lines represent the predicted leaf area index for direct sowing and transplant cultivation, respectively. These graphs represent the results under 100% irrigation treatment conditions and a density of 180,000 plants (:{text{h}text{a}}^{-1}).Full size imageFigure 2 (a-j) present the relationship between the measured and predicted LAI of the five best-performing function pairs previously described in detail during the calibration phase using independent data (2018 growing season as validation). Under direct sowing conditions, the Logistic-Richards function (Fig. 2d) outperformed the other models, achieving the highest coefficient of determination ((:{text{R}}^{2})= 0.98), lowest normalized root mean square error (NRMSE = 0.1474), and a high index of agreement (d = 0.98). Under transplant cultivation, the Hill–Hill model exhibited the best performance among all evaluated functions, achieving high (:{text{R}}^{2}) = 0.99, d = 0.99, NRMSE = 0.127, and MRE = − 0.05, indicating accurate and consistent prediction of LAI. According to Fig. 2a-j, the slope of linear regression (0.95 − 0.94 for direct and 0.94 − 0.92 for transplant cultivations) indicate that the all functions accurately predicted LAI values (close to 1.0) and small values of MRE showed a slight underprediction by the functions.Similar to Fig. 2, the statistical parameters between the measured and predicted LAI based on days after planting (DAP) was calculated for all functions (Data are not shown) for calibration and validation data. During the calibration phase, the Logistic–Logistic, Gompertz–Gompertz, Hill–Hill, Logistic–Richards, and Logistic–Hill functions demonstrated strong predictive accuracy. For direct sowing, NRMSE values ranged from 0.04 to 0.07, d values from 0.995 to 0.998, and MRE values from − 0.002 to − 0.08. For transplant cultivation, the corresponding values ranged from 0.06 to 0.09 for NRMSE, 0.996 to 0.998 for d values, and − 0.004 to − 0.01 for MRE. In the validation phase, performance remained strong though with slightly increased error margins. For direct sowing, NRMSE and d, ranged from 0.12 to 0.14, from 0.983 to 0.988, respectively and MRE from − 0.008 to − 0.04, respectively. For transplant cultivation, NRMSE, d, and MRE values were varied between 0.21 and 0.22, 0.973 to 0.977, and 0.01 and 0.02, respectively.Fig. 2Relationship between the measured and predicted leaf area index based on growing degree days using data of 2018 growing season (as validation year), for top five double sigmoid functions. Dash lines represent the 1:1 and solid lines show the regression line.Full size imageAdjusted coefficients of sigmoid double functionsThe coefficients of the selected functions were determined based on key influencing factors such as seasonal applied irrigation water, rainfall, and plant density. Using the 2017 dataset from the calibration phase, the coefficients of the functions were adjusted for the five best-performing models (Logistic–Logistic, Gompertz–Gompertz, Hill–Hill, Logistic–Richards, and Logistic–Hill) to reflect these environmental and management variables (Fig. 3a-h). These updated coefficients were then used to predict LAI values for the 2018 dataset in the validation phase. However, after incorporating the adjusted coefficients into the Logistic–Hill model, it was found that this function failed to accurately predict the LAI under transplant cultivation. According to Fig. 3d, the Logistic-Richards function exhibited the best performance under direct sowing conditions, achieving the highest coefficient of determination ((:{text{R}}^{2}) = 0.99) and index of agreement (d = 0.99), along with the lowest normalized root mean square error (NRMSE = 0.08) among all evaluated functions. Furthermore, the slope of the linear regression line was 0.99 (close to 1.0) and MRE was − 0.003, indicating excellent agreement between the predicted and measured LAI values under direct sowing. Under transplant cultivation, the Hill–Hill function exhibited the best performance compared to the other functions, with (:{text{R}}^{2}) = 0.98, d = 0.98, NRMSE = 0.16, and MRE = − 0.03. Additionally, as shown in Fig. 3g, the slope of the regression line for this function was 0.9683, which is close to the 1:1 line, indicating a more accurate prediction of LAI relative to the other evaluated functions.Fig. 3Relationship between the measured and predicted leaf area index based on growing degree days in calibration year (2017 growing season), after correcting the constant coefficients of double sigmoid functions. Dash lines represent the 1:1 and solid lines show the regression line.Full size imageValidation of functionsFigure 4 (a-h) shows the relationship between the measured and predicted leaf area index based on growing degree days in validation year (2018 growing season), after adjusting the constant coefficients of double sigmoid functions. Among the four evaluated functions under direct sowing conditions (Fig. 4d), the Logistic–Richards function demonstrated superior performance, with (:{text{R}}^{2}) = 0.98, NRMSE = 0.15, MRE = − 0.05, and d = 0.98. Additionally, as shown in Fig. 4d, the slope of the regression line for this function was 0.94, indicating close agreement with the 1:1 line. Under transplant cultivation, the Hill–Hill model yielded the best performance among the four functions, with statistical parameters of (:{text{R}}^{2}) = 0.98, NRMSE = 0.17, MRE = − 0.05, and d = 0.98. However, as illustrated in Fig. 4f, the slope of the Hill–Hill function was 0.9272, while the slope of the Gompertz–Gompertz function was 0.9852, suggesting that the Gompertz–Gompertz function provided a closer fit to the 1:1 line, therefore, it is more accurate in terms of slope alignment.To evaluate the impact of coefficient correction on function performance, statistical indicators were compared between the calibration and validation phases for both direct sowing and transplant cultivation. In the calibration phase for direct sowing, the Logistic–Richards function without coefficient correction showed a very low NRMSE (0.04). However, after applying coefficient correction, the NRMSE increased to 0.08, indicating a slight reduction in calibration accuracy. Similarly, under transplant cultivation, the Logistic–Logistic function (without correction) performed better during calibration (NRMSE = 0.06, d = 0.99, MRE = 0.045) than the Logistic–Logistic function (with correction) with higher error (NRMSE = 0.19, d = 0.97, MRE = 0.11). In the validation phase, a similar trend was also observed. Under direct sowing, the Logistic–Richards function showed nearly identical performance with and without correction: NRMSE = 0.15, d = 0.98 (without correction) vs. NRMSE = 0.17, d = 0.98 (with correction), indicating that coefficient correction had a negligible effect on function accuracy. Under transplant cultivation, the Hill–Hill function had slightly lower error without correction (NRMSE = 0.13, d = 0.99, and MRE= −0.05) compared to the corrected version (NRMSE = 0.17, d = 0.98, and MRE=−0.05). Overall, the results suggest that correcting the coefficients did not significantly improve—and in some cases slightly reduced—the predictive accuracy of the functions, especially during calibration. The functions calibrated without coefficient adjustment generally performed as well as or better than those with correction across both cultivation methods and evaluation phases.Fig. 4Relationship between the measured and predicted leaf area index based on growing degree days in validation year (2018 growing season), after adjusting the constant coefficients of double sigmoid functions. Dash lines represent the 1:1 and solid lines show the regression line.Full size imageDiscussionModeling the leaf area index (LAI) of sugar beet is essential for understanding crop growth dynamics, optimizing management practices, and improving yield predictions. Sigmoid functions, particularly single and double sigmoid models, are widely used to capture the characteristic S-shaped growth patterns of LAI over the growing season32. Single sigmoid functions, such as the Logistic or Gompertz models, are commonly employed to describe the cumulative LAI development during the growing season11. These models are characterized by an initial slow growth phase, a rapid exponential increase, and a plateau as the crop canopy closes and reaches its maximum LAI7. However, single sigmoid models may have limitations in accurately capturing the asymmetric nature of LAI dynamics33. Double sigmoid functions address these limitations by allowing for two distinct inflection points: one for the rising phase and another for the senescence phase5,12 This approach provides greater flexibility in fitting the observed LAI pattern, especially in crops like sugar beet with pronounced differences between the rates of leaf development and decline stags. Studies have shown that double sigmoid functions improved the fit to observed LAI data and enhanced the accuracy of phenological phase detection34,35. Phenological phase detection for sugar beet involves identifying stages such as emergence, leaf development, canopy closure, root bulking, and sugar accumulation36,37. Since LAI reflects canopy expansion and closure, modeling its growth using double sigmoid functions allows precise identification of transitions between these phases, which is essential for optimizing irrigation, nutrient management, and yield forecasting. This study assessed the effectiveness of double sigmoid functions in modeling LAI based DAP and GDD and for direct sowing and transplant cultivation with varying irrigation levels and planting densities to make the result more applicable for different regions. The top five best functions were Logistic-Logistic, Gompertz-Gompertz, Hill-Hill, Logistic-Richards, and Logistic-Hill functions, which performed the best in predicting sugar beet LAI in comparison with other functions using both DAP and GDD. However, caution is advised when using Weibull, Von Bertalanffy, and Beta functions in sigmoid double modeling, particularly in the initial stage, as they may compromise model performance and accuracy in capturing early and late LAI dynamics. Logistic-Richards achieved the best fit for direct sowing calibration data (NRMSE = 0.04 and d = 0.99), while Hill-Hill performed best under transplant cultivation using independent 2018 validation data (R² = 0.99, NRMSE = 0.13, d = 0.99). The superior performance of LR and HH models reflects the strength of double sigmoid functions in modeling nonlinear and asymmetric growth patterns. The Logistic-Richards function effectively handled variations in early growth rate and plateau behavior in direct sowing, while the Hill-Hill function was more adaptable under transplanting, where early establishment delays slightly modified the LAI curve.The Logistic–Richards function, combining Richards and Logistic functions, is effective for direct sowing due to its adaptability to varying growth rates. Although, lack of improvement from coefficient correction suggested that the original models were well-optimized for the datasets, it also highlights their limited generalizability without recalibration. The correction method may not have captured complex interactions between environmental factors and LAI, as seen in the Logistic–Hill function’s failure in transplant cultivation after correction. This sensitivity underscores the need for tailored adjustment strategies for different models in response to environmental recalibration.Growing Degree Days (GDD) is widely used because thermal time strongly correlates with crop development; however, it remains a temperature-centric metric and does not explicitly capture other environmental drivers that influence LAI dynamics. While GDD may indirectly reflect seasonal differences in rainfall or irrigation (as soil moisture affects canopy temperature and energy balance), critical meteorological variables such as relative humidity, vapor pressure deficit (VPD), wind speed, solar radiation, and leaf temperature directly affect stomatal conductance, transpiration, and leaf expansion. Future work should consider integrating additional meteorological and soil parameters into the model structure or applying advanced approaches such as nonlinear corrections, machine learning-based parameter estimation, or hybrid modeling frameworks that link empirical growth curves with mechanistic crop processes to improve adaptability across diverse environments.The results of predicting the trend of changes in leaf area index under different irrigation treatments and crop densities using various double sigmoid functions are consistent. For instance, the Logistic–Richards model exhibited superior accuracy in predicting LAI under direct sowing across irrigation and planting density treatments when calibrated with GDD. The model’s normalized root mean square error (NRMSE) ranged from 0.008 to 0.096, while the index of agreement (d) consistently exceeded 0.991, indicating a strong agreement between observed and predicted values. Mean relative error (MRE) remained low (generally below 0.09) across treatments, further confirming the robustness of the model. The highest accuracy was observed under full irrigation at the highest planting density (I1D1D), where NRMSE was 0.008 and d reached 0.99, reflecting the model’s ability to capture the smooth and symmetrical LAI trajectory under optimal growing conditions. Agronomically, this performance is attributed to more complete physiological development under full irrigation and dense planting, which supports rapid canopy closure, minimal soil evaporation, and extended photosynthetic activity. These conditions align closely with the assumptions of sigmoid models, where leaf expansion follows a biphasic pattern—rapid early growth followed by gradual senescence. Conversely, performance slightly declined under severe water deficit (I3) and low-density treatments (D3 and D4), with NRMSE values approaching 0.09. These conditions introduce irregularities such as delayed early growth, premature senescence, and reduced LAI peaks due to limited water availability and increased bare soil exposure. Lower planting densities further amplified microclimatic variability by increasing soil heat flux and canopy temperature, accelerating leaf loss under stress. Despite these challenges, the model maintained strong predictive ability (d > 0.982) even in stressed scenarios, underscoring its robustness. The results emphasize that the Logistic–Richards model, calibrated with GDD, effectively represents LAI dynamics for sugar beet under direct sowing across diverse irrigation and density treatments. However, its underperformance in extreme deficit conditions highlights a structural limitation as fixed-parameter sigmoid models cannot fully capture abrupt physiological changes caused by severe stress. Future refinements could involve integrating water stress indices, dynamic senescence parameters, or additional meteorological variables such as vapor pressure deficit, wind speed, and leaf temperature to improve adaptability under climate variability. Such improvements would enhance the model’s applicability for precision irrigation scheduling and crop management in arid and semi-arid regions.The similar performance of double sigmoid functions under direct sowing and transplant cultivation can be attributed to several factors. First, both planting methods were subjected to comparable environmental conditions during the main growth stages and received the same irrigation schedules, which are primary determinants of leaf expansion and canopy development. Adequate water availability during early vegetative growth likely minimized physiological differences between the two systems. Second, maximum LAI were achieved within a similar timeframe in both cultivation methods, reducing variation in LAI dynamics. Sugar beet’s high morphological plasticity enables transplanted plants to compensate for early growth delays by adjusting leaf size and number, ultimately matching the photosynthetic capacity of direct-sown plants. Additionally, high planting densities (e.g., 180,000 plants ha⁻¹) accelerated canopy closure for both methods, resulting in similar light interception and LAI patterns across treatments. These factors collectively explain the comparable statistical performance of double sigmoid models for direct and transplanted crops despite differences in initial establishment.It was reported by38 process-based models such as AquaCrop, DSSAT, APSIM, and SSM simulate crop development and yield by incorporating detailed representations of soil, crop, and atmospheric processes. While these models are valuable for scenario analysis, they require extensive data inputs and calibration, reducing transparency and practicality in data-scarce environments. For instance, DSSAT and APSIM require over 200 input parameters, whereas simpler models like SSM use about 55. Despite their complexity, increased parameterization does not always ensure superior robustness; simpler models such as SSM and CropSyst showed lower variability in yield prediction (CV: 8.2% and 14.3%) compared to APSIM (15.0%) and DSSAT (18.5%). Sugar beet LAI was predicted by sugar beet simulation model (SSM) developed by39 model and results showed that the SSM model was slightly underestimated the higher values of LAI resulting in decreasing model accuracy (NRMSE = 23.2% and d = 0.0.95). In contrast to latter models, the double sigmoid approach offers a transparent and computationally efficient alternative for LAI estimation. Our best models achieved high accuracy (R² > 0.98; NRMSE as low as 0.04) using only DAP or GDD, without requiring detailed soil or weather datasets. This simplicity makes double sigmoid models practical for operational use and precision agriculture in data-limited regions. However, their empirical nature limits the ability to capture dynamic soil-water-plant interactions.It was reported by16 that the Richards ((:{text{R}}^{2}) = 0.95–0.99), Logistic ((:{text{R}}^{2}) = 0.95–0.98), and Gompertz ((:{text{R}}^{2}) = 0.95–0.98) functions were found to be the most suitable functions to fit maize leaf growth across various cultivars. In contrast, the Weibull (R² = 0.88–0.93) and Morgan-Mercer-Flodin (MMF), (R² = 0.95–0.98) functions exhibited acceptable fits but were limited by highly correlated parameter estimates, reducing their suitability for precise maize leaf growth modeling. Also12, reported that the Beta-Richards and Richards-Gompertz functions provided the best fit for the measured dry matter during the growing season based on DAP and GDD, respectively. The combined use of two Von Bertalanffy or Weibull functions did not effectively capture the growth dynamics of rapeseed plants, exhibiting limited accuracy in modeling their development. Scientists such as14,29,40 found that the Logistic function was effective in predicting the height of lettuce plants, the dry matter yield of maize, and yield and dry matter of rapeseed, respectively. A study on growth modeling of beef tomatoes (‘9930) in Taiwan was conducted by41 over two seasons using Gompertz and Logistic functions based on growing degree days (GDD) and days after transplanting (DAT). The GDD-based models showed slightly better performance in estimating plant height, Leaf Area Index (LAI), and dry matter components compared to DAT-based models. The Logistic function better matched the expected biological stages than the Gompertz function. In Taiwan, tomato scions grafted onto eggplant rootstocks were studied over six seasons to enhance summer cultivation, with seedling growth modeled using Gompertz, Richards, and Logistic curves. The results showed that, the performance of the three nonlinear models did not vary greatly in the same growing season. However, a significant difference across growing seasons was observed (R² = 0.74–0.85 for calibration, 0.72–0.80 for validation) suggesting to apply models for each season, separately32. The obtained results in the current study did not show any priority by application of DAP and GDD, while using GDD makes the results more applicable for different regions.ConclusionsThis study analyzed 15 different double sigmoid functions to model sugar beet leaf area index (LAI) based on growing degree days (GDD) and days after planting (DAP) under varying irrigation treatments (I100, I75, I50) and plant densities (P180, P135, P90, P45) across direct sowing and transplant cultivation. The Logistic-Richards and Hill-Hill functions showed the best fit during the growing season, based on both GDD and DAP, as indicated by superior statistical indicators in both calibration and validation phases. The Von Bertalanffy, Weibull and Beta functions were not effective in describing the leaf area index of sugar beet variation. In terms of irrigation water levels in calibration stage, functions under full irrigation (I100) under direct sowing demonstrated optimal performance, with the Logistic-Richards function achieving a low NRMSE of 0.008 at P180. Under water stress conditions (I75, I50), the Logistic-Richards function maintained robust fits (NRMSE = 0.03–0.09). Other functions, such as Gompertz-Gompertz in calibration phase under direct sowing, exhibited higher errors under water stress (NRMSE up to 0.12), highlighting the resilience of the Logistic-Richards function. Plant density had a significant impact on the function accuracy, with higher density (P180) resulting in the best fits (e.g., NRMSE = 0.008 for Logistic-Richards in calibration phase and under direct sowing), while lower density (P90, P45) led to increased errors (NRMSE = 0.04–0.05) due to sparser canopies. Adjusting function coefficients to account for environmental factors such as seasonal applied water, rainfall, and planting density generally led to decreased predictive accuracy under direct and transplant cultivation. The findings of this study provide a reliable tool for estimating sugar beet LAI throughout the growing season, supporting precision agriculture applications such as irrigation optimization and planting schedule adjustments. However, caution is advised when applying these functions under severe water stress or low plant density, where deviations may occur. Finally, the Logistic–Richards function, with its superior accuracy and robustness, is recommended for practical use in sugar beet management based on GDD and DAP. Future research should validate these functions under diverse field conditions and across multiple growing seasons to confirm their robustness. Incorporating additional environmental factors, such as light intensity, soil water content, or nutrient availability, may improve model performance, particularly for coefficient-corrected versions. Advanced modeling technique, such as machine learning, could offer greater flexibility in capturing complex LAI dynamics.

    Data availability

    Data sets generated during the current study are available from the corresponding authors upon request.
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    Download referencesAcknowledgementsAuthors also appreciate the support from Shiraz University Research Council, Drought Research Center, the Center of Excellent for On-Farm Water Management, and Iran National Science Foundation (INSF).FundingThis work was funded by Shiraz University, Grant #2GCB1M222407. Authors also appreciate the support from Shiraz University Research Council, Drought Research Center, the Center of Excellent for On-Farm Water Management, and Iran National Science Foundation (INSF). Second author has received research support.Author informationAuthors and AffiliationsWater Engineering Department and Drought Research Center, School of Agriculture, Shiraz University, Shiraz, IranSaba Hashempour Motlagh Shirazi, Fatemeh Razzaghi, Ali Reza Sepaskhah & Maryam KhozaeiIrrigation Department, Fasa University, Fasa, IranAli ShabaniAuthorsSaba Hashempour Motlagh ShiraziView author publicationsSearch author on:PubMed Google ScholarFatemeh RazzaghiView author publicationsSearch author on:PubMed Google ScholarAli ShabaniView author publicationsSearch author on:PubMed Google ScholarAli Reza SepaskhahView author publicationsSearch author on:PubMed Google ScholarMaryam KhozaeiView author publicationsSearch author on:PubMed Google ScholarContributionsS. H. : The conception and design of the study, acquisition of data, analysis, interpretation of data, and writing the first draft of manuscript; F.R. : The conception and design of the study, acquisition of data, analysis and interpretation of data, supervising, and revising the article critically; A. S.: The conception and design of the study, acquisition of data, analysis, and revising the article critically; A. R. S. : The conception and design of the study, and revising the article critically; M. K. : The conception and design of the study. All authors read the final draft and approved to be submitted.Corresponding authorCorrespondence to
    Fatemeh Razzaghi.Ethics declarations

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    Reprints and permissionsAbout this articleCite this articleShirazi, S.H.M., Razzaghi, F., Shabani, A. et al. Predicting sugar beet leaf area index: evaluating performance of double sigmoid functions under different irrigation and plant density scenarios.
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    KeywordsLogistic-Richards functionsHill-Hill functionGompertz-Gompertz functionsWater stressGrowing degree daysPlanting methods More