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Ethical statementThe present study was discussed and approved by the ethics and animals’ welfare committee of the University of Veterinary Medicine, Vienna, Austria, in accordance with good scientific practice and national legislation (GZ: BMWFW-68.205/0151-WF/V/3b/2016 and GZ: BMWFW-68.205/0224-WF/V/3b/2016). All methods were carried out in accordance with relevant guidelines and regulations. We confirm that the study was carried out in compliance with the ARRIVE guidelines. No plants or plant parts were used in this study.Animals and study areaThe study animals were kept in an outdoor enclosure (~ 55 ha, for details see “Supplementary Material”). The study enclosure was covered with a deciduous forest, mainly Turkey oak (Quercus cerris) and pubescent oak (Quercus pubescens) and included only few meadow patches. For the present study ten adult females, were used. We concentrated on females only because the live capture and handling of males are hampered by the large size and ferocity of boars. Also, due to competition and high levels of aggression between males during rut, the stocking of the enclosure was strongly female biased. During the study period (12/2016–01/2019), the animal density was ~ 1 adult female/ha plus up to 20 males (total) of different ages. Due to this relatively high density, animals were supplemented with 1–1.5 kg corn/individual once a day (at 2:00–14:00 h) at two feeding areas, each ~ 40 × 20 m. The enclosure was part of a game reserve, which was enclosed by 2.5 m high, solid, non-transparent fencing and was closed for the public. Thus, the study site provided an environment without disturbances due to hikers, bikers or straying dogs. There were no battue hunts or other disturbances due to hunting or forest management activities during the study period in the enclosure.Animals were trapped once a year in autumn within the feeding sites to collect data on reproductive success and body condition of females and to separate some of them for implantation/explantation of loggers. While feeding, we closed the access gates and released the boars one by one trough a wooden corridor back into the enclosure. While in the wooden corridor we recorded the body mass of each individual (Gallagher SmartScale® 500, Groningen, Netherlands). Due to management reasons the juveniles (born in spring) were removed from the enclosure during this procedure.Implantation of temperature and heart rate loggersWe implanted a heart rate logger (DST centi-HRT, Star-Oddi, Gardabaer, Iceland) and two custom-built temperature loggers in each of ten female wild boars in October/November 2016 and 2017 (age 5 and 6 years). All details about surgery techniques and anaesthesia protocols are provided in the “Supplementary Material”. Explantations were carried out approximately one year after implantations. The last explanation was carried out in January 2019. One female was implanted in two consecutive years. Mean body mass at date of implantation for all females was 71.8 ± 15.5 kg.The heart rate logger was adjusted to record data at a time interval of 12 min to cover one year of data recording. To remove outliers, all initial data from these recorders were subjected to a running median over five consecutive values. The HR recorder was positioned subcutaneously, in proximity to the heart on the lateral rib cage, behind the moving area of the elbow, to avoid rubbing, or inserted and tethered into the ventral subperitoneal space caudal of the xiphoid process of the sternum.The self-built temperature loggers were covered with inert surgical wax and had a weight of ~ 8 g. Time interval of recording was 4 min, the accuracy 0.01 °C. One of the two temperature loggers had an especially flat shape (3.4 × 1.9 × 0.5 cm) to fit smoothly into the subcutaneous neck region. The second temperature logger was placed into the intraperitoneal cavity, tethered at the Linea alba (diameter = 2.1 cm, height = 1.2 cm). For details on surgery, see “Supplement”.We collected and evaluated a mean of 227.45 ± 160.69 days of heart rate recording per individual (SD, n = 11: 33 days, 58 days, 79 days, 89 days, 143 days, 189 days, 272 days, 345 days, 412 days, 421 days, 461 days), and a mean of 382.00 ± 100.17 days (SD), of subcutaneous logger recording per individual (n = 8: 143 days, 363 days, 411 days, 414 days, 419 days, 421 days, 424 days, 461 days). From the loggers implanted in the abdominal cavity we collected 338.71 ± 117.01 days (SD) per individual (n = 10: 140 days, 143 days, 363 days, 364 days, 411 days, 419 days, 421 days, 421 days, 424 days, 461 days). The hourly means of monitored heart rates of each animal over the course of the year are shown in Supplementary Fig. S1.Activity dataTo record the activity of animals, a telemetry system (Smartbow System, Zoetis, New Jersey, USA) was installed around the two neighbouring feeding areas and two close water ponds in the enclosure. The system consisted of a central solar power and computing station and ten receivers located at the height of 2–3 m. Part of the system were ear-tags (34 g; 52 mm × 36 mm × 17 mm, for details see “Supplementary Material”). The accelerometer (located inside ear-tags) measured triaxial acceleration (x, y, z). As an estimate of locomotor activity (ACT), we computed the total acceleration vector from sqrt (x2 + y2 + z2).Climate and mastThe study site in Eastern Austria (altitude 130 m) is generally characterised by a Pannonian climate. According to long-term climate records, the mean annual temperature is 10 °C in combination with a mean precipitation of 600–700 mm and 1898 h of sunshine per year (ZAMG, 1971–2000).We recorded ambient temperature (Ta) and black bulb temperature (Tab) at 2 m height directly at the study site (Vantage Pro 2 with black bulb extension, Davis Instruments, Hayward, USA).To assess the extent of the acorn mast, each autumn seven nets, 4 × 4 m, were set up to collect acorns at random locations. The nets were regularly emptied between Sept. and Nov. each year, and the collected acorns were dried and weighed. In the autumns prior to the study (2016) and during both full study years (2017/2018) there was seeding of at least part of the oaks. Over ~ 90 days in each autumn we collected 52.4 g/m2, 134.8 g/m2, and 37.5 g/m2 acorn in 2016, 2017, and 2018, respectively. Thus, 2017 was a full mast year but there were acorns available in autumn throughout the study period.Data analysisTo facilitate handling of data and to reduce autocorrelation we compiled and evaluated hourly means for all data, i.e., heart rates (HR; see Suppl. Fig. S1), intraperitoneal and subcutaneous body temperature (Tbip and Tbsc, respectively) and activity (ACT), as well as ambient air temperature (Ta) and black-bulb temperature (Tab). We further tested for effects of day of year (DOY) and hour of day (HOUR). We did not assess the influence of environmental conditions in different years, because due to logger-failures and thus scarcity of heart rates, all data were pooled for different years (with similarly warm conditions and food available year-round). Also, we did not further evaluate daily rhythms, because animals were always fed in the early afternoon, which may have influenced their timing.We investigated the effects of season (DOY), hour of day (HOUR), and Ta on the response variables HR, Tbip, Tbsc, and ACT. We additionally used Tbip, Tbsc, and ACT as predictors for HR. As many of the relationships between these were non-linear, we used general additive mixed models (GAMMs), as implemented in package mgcv60 in R61. This function fits non-linear splines to the data, which are penalized for their “wiggliness”, i.e., the number of turning points in the fit. Because the data were repeated measurements, we calculated for all response variables mixed models with an intercept for each animal ID as a random factor (using s (ID, bs = ”re”)). Hence, these mixed models allowed for differences in the mean level of heart rates, temperatures and activities, between individuals. All residuals of models were approximately normally distributed, as inspected by normal quantile–quantile plots. Hourly means of the response variables contained various degrees of autocorrelation. This was corrected by including autoregressive order 1 (AR1) error models in GAMM-functions, which successfully reduced the autocorrelation at lag 1 to nonsignificant levels. This was confirmed by comparing the autocorrelation function of model residuals (ACF) before and after their correction. To illustrate the effects of independent variables, we show population-level predictions from GAMMs. These graphs contain rug plots to illustrate the distribution of independent variables. Because these plots were too dense for all original data (resulting in black bars), we show uniform random samples (n = 1000) from each independent predictor variable.Because hourly mean data consisted of ~ 117,000 observations we used the mgcv function “bam”, which uses numerical methods designed for large datasets. To fit non-linear functions to predictors, we used the default thin plate splines. Only the cyclic variables DOY and HOUR were modelled using cubic cyclic splines, which are guaranteed to have identical start- and endpoints (e.g., at Jan 1 and Dec 31). GAMMs were always fitted using method REML. As Tbip and Tbsc were only moderately correlated (r = 0.30), both were entered simultaneously as independent variables in the model on heart rate.We did not use partial regression plots from multiple regressions that included activity. This is because activity could only be recorded partly, in the vicinity of telemetry receivers. Thus, models that include ACT as well as all other predictors simultaneously, were restricted to ~ 7% of the data. However, we still used a full multiple regression model HR for the purpose of assessing relative variable importance (of DOY, HOUR, Ta, Tbip, Tbsc, and ACT). F-values from this model provide an indication of the importance of different predictors.To model a possible role of solar radiation and basking we computed the difference between Tab and Ta, called Tdiff, which represents an index of radiation. We used again GAMMs to test if Tdiff would affect Tbip, Tbsc and HR after adjusting for effects of Ta, hour of day, and the random factor animal ID.For a comparison of species we also computed monthly means and SEMs of HR in wild boars, and created a graph of seasonal time courses in other ungulates as published in Arnold2 that were kindly provided by the author. If not stated otherwise we provide means ± SEM. More

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NEWS AND VIEWS
15 September 2021

A spotlight on seafood for global human nutrition

What role might seafood have in boosting human health in diets of the future? A modelling study that assesses how a rise in seafood intake by 2030 might affect human populations worldwide offers a way to begin to answer this.

Lotte Lauritzen

 ORCID: http://orcid.org/0000-0001-7184-5949

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Lotte Lauritzen

Lotte Lauritzen is in the Department of Nutrition, Exercise and Sports, University of Copenhagen, 1958 Frederiksberg C, Denmark.

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An adequate and sustainable supply and intake of nutritious food is essential to tackle major global health issues such as dietary deficiencies. Seafood, which in this context includes fish, shellfish and marine mammals, is rich in micronutrients (such as vitamin A, iron, vitamin B12 and calcium) needed to combat the most common such deficiencies. Seafood is also the dominant source of marine omega-3 fatty acids, which have many health-promoting effects. Writing in Nature, Golden et al.1 present ambitious research that puts seafood centre stage.

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doi: https://doi.org/10.1038/d41586-021-02436-3

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We first identify the location of organic crop fields in Kern County and then estimate whether status as organic versus conventional fields determines pesticide use (Fig. 5).Fig. 5: Methodology overview.Figure outlines the main method steps from identifying organic fields to creating the analysis data to performing the statistical analyses. All images shown are simplified, visual representations of the datasets. CDFA refers to the California Department of Food and Agriculture, while APN is the Assessor’s Parcel Number and TRS is the Township-Range-Section. Identifying organic fields combines the created CDFA organic APN, CDFA organic TRS, and organic pesticides data layers together to create the final organic versus conventional fields layer used in the analysis data section. All analysis data layers are then inputted into the various statistical analyses.Full size imageIdentifying organic fieldsWe identified organic fields using a combination of California Department of Food and Agriculture (CDFA) records and Kern County Agricultural Commissioner’s Office spatial data (“fields shapefiles”) and pesticide use records. No single source was complete, and as such, we evaluated several different approaches to identifying organic fields.California Department of Food and Agriculture (CDFA) recordsData on the location of organic fields, per the California State Organic Program, for 2013–2019 was obtained by request from the California Department of Food and Agriculture (CDFA). The CDFA, through the State Organic Program, requires annual registration of certified organic producers who have an expected gross sale of over $5000. We were specifically interested in the pesticide aspects of organic production, which is governed in our study region by the USDA’s National List of Allowed and Prohibited Substances. The National List of Allowed and Prohibited Substances delineates which synthetic substances can be used and which natural substances cannot be used for pest control in US organic production. Besides substances specifically (dis)allowed on the National List, allowed substances include non-synthetic biological, botanical, and mineral inputs. Field location data were in the form of either Assessor’s Parcel Number (APN) or PLS System Township-Range-Section (TRS) values, though data were reported without systematic formatting. We harmonized the CDFA APN values to merge with the Kern County Assessor’s parcel shapefile (2017), which we then spatially joined with the Kern fields shapefiles. We followed a similar process with PLSS TRS values, which were then merged with the Kern County PLS Section shapefile, and joined to Kern field shapefiles. We refer to our final organic designation as “CDFA Organic”. Details of the data cleaning process are described in the Ancillary Data Processing Methods section below.Using pesticide use reports to refine organic field identificationAfter spot-checking pesticide use on CDFA Organic fields, it became clear we had not entirely eliminated conventional fields. This was likely due to variation in polygon geometries between PLSS Sections, Kern County Assessor parcels, and Kern agricultural fields data. To further refine our classification, we used field-level pesticide use, again from the Kern County Agricultural Commissioner’s Office. As thousands of pesticide products (active ingredients + adjuvants) are in use in Kern County, we took an iterative approach to eliminate fields using conventional pesticides. We first limited the universe of pesticides to those applied to fields that were CDFA Organic. We then identified the 50 most commonly used pesticide products by a number of applications, and manually classified each as organic or conventional. Having identified these products as described below, we matched them back in, eliminating fields that used conventional products and identifying as “PUR Organic” those that used only organic products. We repeated this process, hand identifying the most commonly used products and eliminating fields using conventional products until we had isolated fields using only organic products.To classify a product as organic or conventional, we first searched for each product’s U.S. EPA-registered product label, using the exact product name and EPA product registration number. If there was any indication on the label that the product was certified as organic by the Organic Materials Review Institute (OMRI), or said “for use in organic production” or “organic”, then the pesticide was identified as organic (n = 132). If there was no organic indication on the product label, we searched the OMRI certification database for products with identical names and manufacturers, and identified products as organic if such certifications existed (n = 39). If all ingredients were defined (i.e., no inert or undefined ingredients) and were known organic active ingredients, then the pesticide was identified as organic (n = 1) (Supplementary Data 1). We failed to find EPA-registered labels for three products and confirmed on the California Department of Pesticide Regulation website that they are either inactive or out of production (EPA registration numbers: 52467-50008-AA-5905, 36208-50020-AA, 2935-48-AA-120). Each of the three was rarely used (n  0) to be the same as the mechanisms determining the amount sprayed when some pesticides are used (pesticides when pesticides  > 0). Double-hurdle models64 are an alternative to the Tobit model that allows for the separation of these two decisions.The mechanisms underlying the two decisions (to spray, how much to spray if spraying) can differ such that different covariates can describe each process, and the same covariates are allowed to influence the two processes in different ways (i.e., sign and magnitude can differ). The first, binary decision is usually modeled with a probit model.$${{{{{rm{P}}}}}}left(y=0|{{{{{bf{x}}}}}}right)=1-Phi left({{{{{bf{x}}}}}}gammaright)$$
(1)
Then, the second decision is modeled as a linear model with pesticide use following a lognormal distribution, conditional on positive pesticide use64$$log (y)|{{{{{bf{x}}}}}},y , > , 0 sim {{{{{rm{Normal}}}}}}({{{{{bf{x}}}}}}{{{{{mathbf{upbeta }}}}}},{sigma }^{2})$$
(2)
where Φ is the standard normal cdf, x is a vector of explanatory variables including organic status, y is pesticide use, and ({{{{{mathbf{upbeta }}}}}}) is a vector of coefficients. We use a lognormal hurdle model rather than a truncated normal hurdle model since pesticide use is highly non-normal, and Q-Q plots suggested substantial model improvement using a lognormal rather than normal distribution. In contrast to the panel data models described in the Ancillary Statistical Methods below, our estimation equation used natural log-transformed variables for pesticides (and field and farm size) rather than inverse hyperbolic sine (IHS) transformation since only positive observations are included in the second hurdle model. Following insights derived from our panel data models (Supplementary Notes), we build on the basic hurdle model concept using the farm-by-crop family interaction as a random intercept in both the first and second hurdle. We chose the farm-by-crop family interaction rather than a crossed random effect due to computational feasibility with thousands of permits and hundreds of crops, due to similarity of results to the within estimator model (i.e., fixed effects in causal inference terminology; Supplementary Notes, Supplementary Table 2), and due to AIC/BIC (Supplementary Table 3). Further, we find evidence of heteroskedasticity from both visual inspection and Levine’s test, which adds additional complications to computing crossed random effects. Thus, we proceed with the farm-by-crop family interaction in a random intercept model with cluster robust standard errors clustered at the same grouping. In doing so, observations, where the taxonomic family of the crop was unclear, were dropped. Of the 7367 fields that were dropped due to missing crop families, 6684 were uncultivated agriculture.Our data are effectively repeated cross-sections rather than a true panel since fields are defined by the farm-site-year combination and thus generally change year-to-year or when crops rotate. We model it as such. This implies we do not require observations to have no spray in all time periods, as would be the case in a double hurdle panel model. Linking field IDs over time through spatial processing is complicated by crop rotations of different size areas. Since farmers may farm multiple fields under different management systems, as we illustrate here, and have different contractual obligations at a sub-farm level, requiring farms to never use pesticides on all fields is not reflective of on-the-ground decisions.We repeated the lognormal hurdle models individually for carrots, grapes, oranges, potatoes, and onions, which were the only widely-grown crops with more than 100 organic fields. This allowed for a different slope and intercept by crop type.We conduct several robustness checks. First, we do not have data on crop yields. However, to assess the potential implications of a yield gap on our results, we modify our per hectare rates following Ponisio et al.15 as a robustness check. We group commodities into cereals, roots and tubers, oilseeds, legumes/pulses, fruits, and vegetables and assign them the Ponisio et al.15 yield gap estimates for that group. Crops that did not fall into any of the above groups (e.g., cannabis) were provided the all-crop average from Ponisio et al.15. Second, we analyze how conventional and organic differ with respect to soil quality using a within estimator approach to account for crop-specific differences in soil quality. Third, binary toxicity metrics, while valuable given the number of chemicals and endpoints of interest here, nevertheless fail to distinguish gradations of toxicity for chemicals above (or below) the regulatory threshold. As mentioned above, the data needed to calculate many aggregate indices (e.g., Pesticide Load57 and Environmental Impact Quotient58) are not readily available for all of the chemicals in our study. For completeness, we attempted to calculate the Pesticide Toxicity Index for one well-studied endpoint, fish. We supplemented data provided in Nowell et al.41 with data from Standartox42. However, only about 70% of the chemicals used in our study matched, and pesticide products used on organic fields were more likely to lack toxicity information for one or more chemicals. We briefly discuss the highly preliminary investigation, given the non-random missing toxicity data.All spatial analyses were performed in R Statistical Software v 3.5.3 and all statistical analyses were performed Stata 16 MP. For all tests, statistical significance was based on two-tailed tests with (alpha =0.05.)Ancillary data processing methodsCleaning parcel dataTo spatially locate organic fields, we needed to match the Assessor’s parcel numbers (APNs) provided in the CDFA tabular data to APNs in the Kern County Parcel shapefile (from 2017). Over 90% of the APN entries in the CDFA data were in the format [xxx-xxx-xx], though multiple APNs were often provided in the same cell separated by line breaks, semi-colons, commas, and/or spaces. We made initial edits separating values into individual cells in Microsoft Excel since formatting was highly inconsistent. Observations whose APNs were not in the [xxx-xxx-xx] were modified so that their format matched. In the R environment, dashes were inserted after the third, sixth, and eighth characters (1234567895 became 123-456-78-95) for APNs that did not already contain them. Occasionally, APN numbers were provided with dashes, but with segments of incorrect length (e.g., 12-34-567). In these instances, APN segments were either trimmed from the right or padded with a zero on the left so they matched the [xxx-xxx-xx] format. This approach yielded the greatest number of matches and was checked for accuracy as described below. Additional segments (from APNs with more than two dashes and eight numeric characters) were dropped. A handful of APNs with fewer than eight numeric characters and no dashes were dropped entirely.The edited CDFA APNs were then joined with the Kern County Assessor’s parcel shapefile, creating the “CDFA organic shapefile”. In total, 1637 of 1829 individual CDFA records joined successfully. To evaluate the accuracy of joins between CDFA tabular data, Kern County parcel, and Kern County agricultural spatial data, we spot-checked ownership information using “Company” (CDFA) and “PERMITTEE” (Kern County agricultural data) values.To then identify the crop fields within the organic parcels, we performed a spatial join between the CDFA organic shapefile and the Kern County fields shapefiles. Prior to performing the join, the CDFA parcels’ dimensions were reduced with a 50-m buffer to eliminate spatial joins between CDFA parcels and crop fields that were only touching the parcel margins. Of five different buffer widths evaluated, 50 m reduced the number of false positives and negatives, as determined by comparing the “Company” and “PERMITTEE” values. We refer to the fields that match as “APN Organic”.Cleaning PLSS Township-Range-Section valuesEach year several producers reported Township, Section, and Range (TRS) values, consistent with the PLS System (PLSS), rather than APN values. We used these TRS values to identify PLSS Sections that contained organic fields.We separated any cell containing multiple TRS values and removed any prefixes such as “S”, “Section”, “Sec.”, “T”, and “R” that would prevent joining to Kern County PLSS spatial data in Excel. In the R environment, we padded the left side of the “S” value with a 0 if it was a single digit, then concatenated the three columns into a “TRS” column. We joined TRS from the CDFA tabular data to PLSS spatial data, which identified 563 Sections as containing organic fields, from 2013 to 2019, out of a total of 664 unique TRS codes in the CDFA dataset. We then performed a spatial join between PLSS Sections that contain organic fields and Kern County fields shapefiles, to identify all agriculture fields that overlap with those Sections. Additional processing using the Pesticide Use Reports is described above.Ancillary statistical methodsWe began with a pooled ordinary least squares (OLS) model that, as the name suggests, pools observations over farms, years, and crop types. However, there may be attributes of crops or farms that may be systematically different between organic and conventional, and this systematic difference could bias our pooled OLS results. To address this, we first considered propensity score approaches but were unable to find a sufficient balance of our covariate distribution between organic and conventional fields. As an alternative, we limited our sample to fields with overlapping farmers and crop types. In other words, we focused on the subset of fields that are grown by farmers producing both organic and conventional fields and to crops that are produced both conventionally and organically. However, this shrunk our dataset by two-thirds.To leverage more of our data, we next considered panel data models as a means to address unobserved variables. We consider both within-estimator models (also known as “fixed effects” in causal inference terminology, but different from the biostatistical use of the term) and random effects models (with random intercepts), seeking to capture characteristics of the crop, grower, and year. The advantage of a within-estimator approach is that the omitted variables are removed (through differencing) and thus, they can be correlated with covariates without biasing the estimation. In other words, pesticide use and all covariates are differenced from their crop-specific mean (or crop family, farmer, etc. specific mean, depending on the model). In doing so, the propensity for certain crops (crop family, farmer) to be grown organic or to be fast or slow adopters of new technologies is removed. The disadvantage is that characteristics shared by all fields of a crop (e.g., value) are lost in the differencing, and more importantly, that the differencing is not easily translated to nonlinear models that we employ later in the analysis. Random effects are more easily translated to nonlinear models. The disadvantage of random effects is the strong assumption that the unobserved variables are uncorrelated with the covariates18,65, which is required for random effects coefficient estimates to be unbiased. Here, we see the difference in coefficient estimates between the within-estimator and random effects models are quite small (Supplementary Table 2).Random effects particularly crossed random effects with thousands of permits and hundreds of crops, introduce computational challenges due to large, sparse matrices. Further, we find evidence of heteroskedasticity from both visual inspection and Levine’s test, which adds additional complications to computing crossed random effects. We proceed using the farm-by-crop family interaction in a random intercept model with cluster robust standard errors clustered at the same grouping based on AIC/BIC (Supplementary Table 3), computational feasibility, and similarity to the within-estimator results (Supplementary Table 2). Observations, where the taxonomic family of the crop was unclear, were dropped in any models including family in either the random effects or the cluster robust standard errors. Of the 7367 fields that were dropped due to missing crop families, 6684 were uncultivated agriculture.In the panel data models, we used IHS transformations to accommodate highly non-normal pesticide (and field and farm size) data. IHS is very similar to natural log transformation66 but is defined at zero, which is important given a sizable fraction of our observations have zero pesticide use. As with log–log transformations, IHS–IHS transformation can be interpreted as elasticities. We pre-multiply pesticide use by 100 to improve estimation66, though this does not affect interpretation. As described above, we leverage insights on model specification from the panel data models, but rely on the double hurdle models to parse apart the decision to spray from the decision of how much to spray.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

The absence of the first principles for biological systems in general, and in particular for vegetation populations where phenomena are interconnected makes their mathematical modelling complex. The theory of vegetation pattern formation rests on the self-organisation hypothesis and symmetry-breaking instability that provoke the fragmentation of the uniform cover. The symmetry-breaking instability takes place even if the environment is isotropic31,33,35. This instability may be an advection-induced transition that requires the pre-existence of the environment anisotropy due to the topography of the landscape34,39,40. Generally speaking, this transition requires at least two feedback mechanisms having a short-range activation and a long-range inhibition. In this respect, we consider three different vegetation models that are experimentally relevant systems: (i) the generic interaction redistribution model describing vegetation pattern formation which incorporates explicitly the facilitation, competition and seed dispersion nonlocal interactions (ii) the local nonvariational partial differential model described by a nonvariational Swift–Hohenberg type of model equation, and (iii) the reaction–diffusion system that incorporate explicetely water transport.The interaction-redistribution approachThe integrodifferential modelThis approach consists of considering a well-known logistic equation with nonlocal plant-to-plant interactions. Three types of interactions are considered: the facilitative (M_{f}(mathbf {r},t)), the competitive (M_{c}(mathbf {r},t)), and the seed dispersion (M_{d}(mathbf {r},t)) nonlocal interactions. To simplify further the mathematical modelling, we consider that the seed dispersion obeys a diffusive process (M_{d}(mathbf {r},t)approx nabla ^{2}b(mathbf {r},t)), with D the diffusion coefficient, b the biomass density, and (nabla ^{2}=partial ^2/partial x^2+partial ^2/partial y^2) is the Laplace operator acting in the (x,y) plane. The interaction-redistribution reads$$begin{aligned} M_{i}=expleft{ frac{xi _{i}}{N_{i}}int b(mathbf {r}+mathbf {r}’,t)phi _i(r,t)dmathbf {r}’right} , { text{ with } } phi _i(r,t)= exp(-r/L_{i}) end{aligned}$$
(1)
where (i=f,c). (xi _i) represents the strength of the interaction, (N_i) is a normalisation constant. We assume that their Kernels (phi _i(r,t)) are exponential functions with (L_i) the range of their interactions. The facilitative interaction (M_{f}(mathbf {r},t)) favouring vegetation development. They involve the accumulation of nutrients in the neighbourhood of plants, the reciprocal sheltering of neighbouring plants against climatic harshness which improves the water budget in the soil. The range of the facilitative interaction (L_f) operates on the crown size. The competitive interaction operates over a length (L_c) and involves the below-ground structures, i.e., the rhizosphere. In nutrient-poor or/and in water-limited territories, lateral spreading may extend beyond the radius of the crown. This extension of roots relative to their crown size is necessary for the survival and the development of the plant in order to extract enough nutrients and/or water from the soil. When incorporating these nonlocal interactions in the paradigmatic logistic equation, the spatiotemporal evolution of the normalised biomass density (b(mathbf {r}, t)) in isotropic environmental conditions reads14$$begin{aligned} partial _{t} b(mathbf {r},t)=b(mathbf {r},t)[1-b(mathbf {r},t)]M_{f}(mathbf {r},t)- mu b(mathbf {r},t)M_{c}(mathbf {r},t)+Dnabla ^{2}b(mathbf {r},t). end{aligned}$$
(2)
The normalisation is performed with respect to the total amount of biomass supported by the system. The first two terms in the logistic equation with nonlocal interaction Eq. (2) describe the biomass gains and losses, respectively. The third term models seed dispersion. The aridity parameter (mu) accounts for the biomass loss and gain ratio, which depends on water availability and nutrients soil distribution, topography, etc. The homogeneous cover solutions of Eq. (2) are: (b_{o}=0) which corresponds to the state totally devoid of vegetation, and the homogeneous cover solutions satisfy the equation$$begin{aligned} mu =(1-b)exp (Delta b), end{aligned}$$
(3)
with (Delta =xi _{f}-xi _{c}) measures the community cooperativity if (Delta >0) or anti-cooperativity when (Delta 0). The solution (u_{-}) is always unstable even in the presence of small spatial fluctuations. The linear stability analysis of vegetated cover ((u_{+})) with respect to small spatial fluctuations, yields the dispersion relation$$begin{aligned} sigma (k)=u_{+}(kappa -2u_{+})-(nu -gamma u_{+})k^{2}-alpha u_{+}k^{4}. end{aligned}$$
(8)
Imposing (partial sigma /partial k|_{k_{c}}=0) and (sigma (k_{c})=0), the critical mode can be determined$$begin{aligned} k_{c}=sqrt{frac{gamma -nu /u_{c}}{2alpha }}, end{aligned}$$
(9)
where (u_{c}) satisfies (4alpha u_{c}^2(2u_{c}-kappa )=(2gamma u_{c}-nu )^2). The corresponding aridity parameter (eta _{c}) can be calculated from Eq. (7).The reaction–diffusion approachThe second approach explicitly adds the water transport by below ground diffusion. The coupling between the water dynamics and the plant biomass involves positive feedbacks that tend to enhance water availability. Negative feedbacks allow for an increase in water consumption caused by vegetation growth, which inhibits further biomass growth.The modelling considers the coupled evolution of biomass density (b(mathbf {r},t)) and groundwater density (w(mathbf {r},t)). In its dimensionless form, this model reads33$$begin{aligned} frac{partial b}{partial t}= & {} frac{gamma w}{1+omega w}b-b^{2}-theta b+nabla ^{2}b, end{aligned}$$
(10)
$$begin{aligned} frac{partial w}{partial t}= & {} p-(1-rho b)w-w^{2}b+delta nabla ^{2}(w-beta b). end{aligned}$$
(11)
The first term in the first equation describes plant growth at a constant rate ((gamma /omega)) that grows linearly with w for dry soil. The quadratic nonlinearity (-b^{2}) accounts for saturation imposed by poor nutrients soil. The term proportional to (theta) accounts for mortality, grazing or herbivores. The mechanisms of dispersion are modelled by a simple diffusion process. The groundwater evolves due to a precipitation input p. The term ((1-rho b)w) in the second equation accounts for the evaporation and drainage, that decreases with the presence of vegetation. The term (w^{2}b) models the water uptake by the plants due to the transpiration process. The groundwater movement follows the Darcy’s law in unsaturated conditions; that is, the water flux is proportional to the gradient of the water matric potential41. The matric potential is equal to w, under the assumption that the hydraulic diffusivity is constant41. To model the suction of water by the roots, a correction to the matric potential is included; (-beta b), where (beta) is the strength of the suction. More