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Study area and samplingSmall mammals (murid rodents and shrews) were captured using mouse-type snap traps in Tatarstan, Russian Federation (Fig. 1, Table S1). Area type (urban or rural), vegetation (forest or field) and distance from trapping points to the nearest human settlement were recorded. The distinction between forest and field was made based on the UN Food and Agriculture Organization’s criteria23,24. Each administrative division in the Tatarstan was defined to be urban or rural by the Federal Service of State Statistics of Russian Federation25. Based on these criteria, Kazan city and Naberezhnye Chelny city were classified as urban districts and Vysokogorsky district, Yelabuzhsky district, Laishevsky district, Mamadyshsky district, Nizhnekamsky district, Pestrechinsky district and Tukayevsky district were classified as rural districts. Small mammals were captured during the spring and fall periods of 2016 and 2017. Fifty traps were placed in a line every 5 m in one place. Traps were baited and left for one night. Animal suffering was minimized as snap traps cause rapid death in murid rodents and shrews. Each captured small mammal’s species, age, and sex were morphologically identified using a reference guide26, and the animals were then stored at − 20 °C until their brains were isolated.EthicsAll experiments were performed in compliance with relevant Russian and Japanese and institutional laws and guidelines and were approved by the Ministry of Health of the Russian Federation and the Animal Research Committee of Gifu University (Permit Nos. MU 3.1.1029-01, and 17060, respectively). Study was carried out in compliance with the ARRIVE guidelines (https://arriveguidelines.org).DNA extraction and PCRBrain tissue samples were prepared as described previously12. Brain samples stored at − 20 °C were transferred to a − 86 °C deep freezer. Each deep-frozen whole brain sample was homogenized in 1 ml of a 0.9% saline solution. Total DNA was extracted from the brain tissues of each small mammal using a Genomic DNA Purification Kit (Promega, Madison, WI, USA), following the manufacturer’s instructions. Nested PCR was performed with the Takara PCR Amplification Kit (Takara Bio Inc., Foster City, California, USA) according to the manufacturer’s instructions. The primer sets and PCR conditions used to detect the B1 gene from T. gondii were those described previously12.MappingSpatial referencing of the sampling sites was conducted using global positioning system navigation with a Garmin eTrex 10 device. Visualization of cartographic data and measurements of the distances from the trapping points to the nearest human settlements were performed using QGIS 3.12 software27. Geodetic coordinates were projected into planar rectangular coordinates in the Universal Transverse Mercator projection on the WGS-84 ellipsoid (Universal Transverse Mercator, zone 39N). The overview map of the European part of Russia was made in the Lambert Conformal Conic Projection. Map coordinates are represented as geodetic coordinates (WGS-84, degrees and minutes north latitude and east longitude). To visualize thematic objects (administrative boundaries, forests, agricultural lands, and water bodies), a set of vector data layers, NextGIS (Russia), was purchased from OpenStreetMap and contributors, 2021 (https://data.nextgis.com). Data license: ODbL.Dataset and statistical analysesMultivariate logistic regression was performed using the R statistical software package (version 3.6.3)28 to assess the trapping point area (urban or rural), vegetation (forest or field), small mammal species type (alien or non-alien species), age (0–2 months-old juveniles, 3–6 months-old adults or ≧ 6 months old), sex (male or female) and distance from trapping points to the nearest human settlements as risk factors for PCR positivity. According to previous reports2,13,16,17,18, four species, Mi. arvalis, A. flavicollis, A. agrarius, A. uralensis, and three species, My. glareolus, S. araneus and D. nitedula are considered alien and non-alien species, respectively. Quantitative data were replaced with 0 or 1 dummy variables, and age data were replaced by 0, 1 and 2 for juveniles, adults and elders, respectively. Multicollinearity of the explanatory variables was evaluated using Spearman’s coefficient29 calculated using dplyr, FSA and psych packages30,31,32. None of the Spearman’s coefficients were  > 0.6. To find the best fit model, a forward selection procedure was used. Predictive performance and model fitting were assessed using the area under the receiver operating characteristic (ROC) curve, area under the curve (AUC) and corrected Akaike’s information criterion (AICc) with Akaike weight (Wi). AICc and Wi were calculated using the MuMIN package33, and the AUC was calculated using the R pROC package34. P-values of  More

The micro-CT results from Arthrorhynchus agree perfectly with the previously known light microscope and transmission electron microscope images2. This emphasizes that microtomography is a good technique to visualize the type of fungal attachment to the host and especially the penetration of the cuticle, apart from the study of thallus in amber fossils17. As Jensen et al. (2019) demonstrated the presence of a haustorium in Arthrorhynchus using scanning electron microscopy, we are confident that the lack of penetration and haustorium in Rickia found by micro-CT is real. This is also in agreement with results from the scanning electron microscopical investigation of the attachment sites of R. gigas, which exhibits no indication of penetration and are very similar to those of R. wasmannii previously shown18.Despite the absence of a haustorium, and hence without any obvious means of obtaining nutrition, Rickia gigas is quite a successful fungus, being often abundant on several species of Afrotropical millipedes of the family Spirostreptidae10. It was originally described from Archispirostreptus gigas, and Tropostreptus (= ‘Spirostreptus’) hamatus20, and was subsequently reported from several other Tropostreptus species19.A further challenge for Laboulbeniales growing on millipedes is that infected millipedes, in some species even adults, may moult, shedding the exuviae with the fungus, as has been observed by us on an undescribed Rickia species on a millipede of the genus Spirobolus (family Spirobolidae).The question of how non-haustoriate Laboulbeniales obtain nutrients has been discussed by several authors18, including staining experiments using fungi of the non-haustoriate genus Laboulbenia on various beetles21. Whereas the surface of the main thallus was almost impenetrable to the dye applied (Nile Blue), the smaller appendages could sometimes be penetrated21. The dye injection into the beetle elytra upon which the fungi were sitting, actually spread from the elytron into the fungus, thus indicating that in spite of the lack of a haustorium, the fungus is able to extract nutrients from the interior of its host21.Such experiments have not been performed on Rickia species, but the possibility that nutrients may pass from the host into the basis of the fungus cannot be excluded. For this genus, or at least R. gigas, there may, however, be an alternative way to obtain nutrients: the small opening in the circular wall by which the thallus is attached to the host may allow nutrients from the surface of the millipede or from the environment to seep into the foot of the fungus. However, further experiments are needed in order to evaluate this hypothesis. Moreover, we should not exclude a potential role of primary and secondary appendages in Laboulbeniales nutrition, as we still do not understand exactly their functional role on the fungus life cycle11.The predominant position of the Laboulbeniales on the host might be related to the absence or presence of a haustorium. Thus, the haustoriate species of the genus Arthrorhynchus are most frequently encountered in large numbers on the arthrodial membranes of the host’s abdomen, although some thalli are found on legs2,22. At the arthrodial membranes the cuticle is more flexible and therefore might be easier to penetrate by a parasite. Furthermore, most tissues providing/storing nutrition (e.g., fat body) are located within the abdomen. In contrast, non-haustoriate fungi as are often located on more stiff and sclerotized body-parts like the genus Rickia on the legs or body-rings of millipedes7,20,23 or the genus Laboulbenia on the elytra of beetles21,24. A reason for this might be that the non-haustoriate forms, which are only superficially attached to the host need a more or less smooth surface for adherence and can easily become detached from a flexible surface, which is movable in itself, like the arthrodial membrane, while the haustoriate forms are firmly anchored within the hosts’ cuticle.Whereas the vast majority of the more than 2000 described species of Laboulbeniales show no sign of host penetration, haustoria have been reported from some other genera18, including Trenomyces parasitizing bird lice25,26, Hesperomyces growing on coccinellid beetles and Herpomyces on cockroaches (formerly a Laboulbeniales and now in the order Herpomycetales10), with pernicious consequences on the hosts’ fitness18,27. Micro-CT studies on these genera could help to understand the host penetration. In order to fully understand how Laboulbeniales obtain nourishment, although other approaches are, also needed—for the time being it remains a mystery how the non-haustoriate Laboulbeniales sustain themselves. More

Collection and domestication of the wild populationsThe academic permission for collections and research on medicinal plants was obtained from the Head of Biotechnology Department, Research Institute of Modern Biological Techniques, University of Zanjan, Zanjan, Iran. The study complies with all relevant guidelines. Some populations of wild spinaches were harvested during spring season 2013 from the mountain habitat of this wild plant in the Tarom region of Zanjan province from an altitude of 2500–3000 m and were transferred to the greenhouses conditions. The domestication and cultivation experiments were conducted at Research Institute of Modern Biological Techniques, University of Zanjan, 1579° m above sea level, with 48° 28′ longitude and 36° 40′ latitude, from April 2013 to August 2020. The resulted seeds were cultured on pots to produce adequate seeds. The seedlings were transferred to the field with rows spaced 50 cm apart and also 50 cm between plants within the rows. Two seeds per hill were planted in an area of approximately 50 m2. Based on the organic conditions, no fertilization was performed. Thinning was done 25 days after emergence, leaving one plant per hill. The other cultural practices were those normally adopted for cultivation in the region.Mass selection of populationsIn the first year, phenotypic studies were performed during the growing season and weak, diseased and underdeveloped plants were removed from the field before the flowering stage. Then plants with the same phenotype and the desired traits were selected and after harvesting, their seeds were mixed. This election cycle was repeated for 5 years. In the final year, the new mass selected population was compared in a pilot project with cultivated spinach in traits such as yield, resistance to wilt, cold and pests, diseases, and mineral contents. This variety before the certification in the related national organization is a candida cultivar. It is a developed population that will be evaluated in the session of the Iranian variety of introduction committee.The seeds of cultivated spinach (Spinacia oleracea L. |Varamin 88|) were prepared from the Research Institute of Modern Biological Techniques, University of Zanjan, Zanjan, Iran.Performing tests of stability, uniformity and differentiationTo assess morphologically and differentiate advanced uniformity in the studied population (Candida cultivar), the population was managed as a randomized complete block design with three replications over 2 years according to the instructions for spinach differentiation, uniformity, and stability (DUS Testing) of the International Union New Plant Cultivation (UPOV) and some morphological traits on plants or parts of plants. The studied traits included: cotyledon length, presence or absence of anthocyanin in petiole and veins, green color intensity, shrinkage, presence of lobes in the petiole, petiole state, petiole length, foil shape, foil edge shape, tip shape, and part of the length of the petiole, the time of flowering and the color of the seeds.Mineral analysesTo compare the mineral content of mass-selected population-medicinal spinach (MSP) with cultivated spinach (Spinacia oleracea L. var. Varamin 88), both plants were planted in pots and fields on similar conditions. In five leaves stage, plant samples were taken from both leaf and crown sections. The sampling method was such that after removing half a meter from the beginning and end of each plot (to remove the marginal effect) and also removing the two sidelines, five plants were harvested randomly for plant mineral analysis. Atomic absorption spectroscopy was used to determine the mineral content including iron (Fe), zinc (Z), manganese (Mn), and copper (Cu).The dried samples of root-crown and leave were stored, and later grounded and analyzed for iron (Fe), zinc (Z), manganese (Mn), and copper (Cu) in mass-selected variety (MSP) and cultivated spinach (CSP). Studied minerals were measured using atomic absorption spectrometry in the model of GBC AVANTA (GBC scientific equipment Ltd., Melbourne, Vic., Australia).Calibration of AAS was done using the working standard prepared from commercially available metal/mineral standard solutions (1000 μg/mL, Merck, Germany). The most appropriate wavelength, hollow cathode lamp current, gas mixture flow rate, slit width, and other AAS instrument parameters for metals/minerals were selected as given in the instrument user’s manual, and background correction was used during the determination of metals/minerals. Measurements were made within the linear range of working standards used for calibration15,16.The concentrations of all the minerals were expressed as mg/1000 g (ppm) dry weight of the sample. Each value is the mean of three replicate determination ± standard deviation.Scanning electron microscopy (SEM)For SEM studies, the seeds enveloping were removed and were acetolyzed in a 1:9 sulfuric acid-acetic anhydride solution. The seeds were vigorously shaken for 5 min. Then, they were left for 24–48 h in the solution. After this time, seeds were again shaken for 5 min and then washed.in distilled water by shaking for a further 5 min. The seeds were dried overnight and then were mounted on stubs and covered with Au–Pd by sputter coater model SC 7620. After coating, coated seeds were photographed with an LEO 1450 VP Scanning Electron Microscope. All photographs were taken in the Taban laboratory (Tehran, Iran).Statistical analysisThe statistical evaluation including: data transformation, analysis of variance and comparison of means were performed (SPSS software, Version 11.0). The experiment was structured following a randomized complete block design (RCBD) with three replications. Means comparisons were conducted using an ANOVA protected the least significant difference (LSD) test, with the ANOVA confidence levels of 0.95. Data were presented with their standard deviations (SD). More

Dispersal modelsIn this study, the dispersal model consists of two parts, namely, kernel and observation model (Fig. 1). The main purpose of the kernel was employed to estimate the proportion of pollen dispersed from location s′ to location s and calculate the expected number of CP grains. The observation model used the expected number of CP grains as a parameter and described the number of CP grains at location s (Ys) by a specific distribution in the following:$${Y}_{s}sim fleft(left.{y}_{s}right|{{varvec{theta}}}_{s}right),$$
(1)
where f indicates the probability density function (PDF) of the specific distribution. The θs is the parameter vector of the distribution. This study constructed eight different dispersal models combined with two observation models, two kernels, and two conditions of the field border (FB) effect (Table 1). The details of the kernels and observation models were described in the following subsections.Figure 1Graphical summary of the establishment of the dispersal model using ZIP distribution observation model as an example.Full size imageTable 1 List of dispersal models constructed in this study.Full size tableKernelsThe kernel indicates the probability when the pollen emitted at location s′ and would fall down at location s. It can be expressed as γ(s, s′), where s′ is the source location closest to location s. Numerous kernels have been used to describe various dispersal phenomena24. The output of the kernel represents the donor pollen density of location s. In order to calculate the expected number of CP grains, the donor pollen density is multiplied by the average total grain number described as follows:$${lambda }_{s}=Ktimes gamma left(s,{s}^{^{prime}}right),$$
(2)
where λs and K indicate the expected number of CP grains at location s and the average number of grains per cob, respectively. The effect of the FB was introduced into the kernel to suit to the small-scale farming system in Asia. This study assumed that the relation between the pollen density at the first recipient row and the width of the FB displayed an exponential decrease25,26. To evaluate the improvement of the kernel with the FB effect, the kernels without the FB effect were also established in this study.The compound exponential kernel (γExpo) has been used in the previous pollen dispersal study27. Our study introduced the FB effect into this kernel. Therefore, the form of the compound exponential kernel can be expressed as follows:$$gamma_{{{text{Expo}}}} left( {s,s^{prime}} right) = left{ {begin{array}{*{20}l} {K_{e} exp left( { – a_{1} d^{*} left( {s,s^{prime}} right)} right)exp left( { – ksqrt {FB} } right),} \ {K_{e} exp left( { – a_{1} D – a_{2} left( {d^{*} left( {s,s^{prime}} right) – D} right)} right)exp left( { – ksqrt {FB} } right),} \ end{array} } right.begin{array}{*{20}l} {{text{if}},, d^{*} left( {s,s^{prime}} right) le D} \ {{text{if}} ,,d^{*} left( {s,s^{prime}} right) > D,} \ end{array}$$
(3)
where Ke, a1, a2, k, D are the parameters of the kernel. d*(s, s′) indicates the shortest distance between locations s′ and s in which the width of the FB has been subtracted. In the compound exponential kernel without the FB effect, the exponential term of the FB effect was removed and the d*(s, s′) was replaced directly by the shortest distance between s′ and s.The second kernel applied in this study was the modified Cauchy kernel (γCauchy) which was based on the PDF of the Cauchy distribution and the concept of compound distribution. The modified Cauchy kernel is represented as follows:$$gamma_{Cauchy} left( {s,s^{prime}} right) = left{ {begin{array}{*{20}l} {frac{2beta }{{pi left[ {beta^{2} + d^{*} left( {s,s^{prime}} right)^{2} } right]}}{text{exp}}left( { – ksqrt {FB} } right),} \ {frac{2beta }{{pi left[ {beta^{2} + D^{2} + c_{1} left( {d^{*} left( {s,s^{prime}} right) – D} right)^{2} } right]}}{text{exp}}left( { – ksqrt {FB} } right),} \ end{array} } right.begin{array}{*{20}l} {{text{if}} ,,d^{*} left( {s,s^{prime}} right) le D} \ {{text{if}} ,,d^{*} left( {s,s^{prime}} right) > D,} \ end{array}$$
(4)
where the β indicates the decline rate of the curve. Parameters of k and D are same as the compound exponential kernel. c1 indicates the relative slow decrease of pollen density at further distances. Similarly, in the modified Cauchy kernel without the FB effect, the term of the FB effect was removed and the d*(s, s′) was replaced directly by the shortest distance between s′ and s in which the row spacing (0.75 m) had been subtracted.Observation modelsBecause of the high proportions of zero value observations, the present study assumed that the CP grain count followed the zero-inflated Poisson (ZIP) distribution to account for zero-excess condition28. The ZIP distribution was first proposed by Lambert29, and several studies had applied the ZIP distribution to deal with the CP data27,30. The ZIP distribution consists of a Dirac distribution in zero and a Poisson distribution. Therefore, the distribution of CP grain count at location s (Ys) can be expressed as follows:$${Y}_{s}sim mathrm{ZIP}left(1-{q}_{s},{uplambda }_{s}right),$$
(5)
where qs indicates the probability of an observation following a Poisson distribution, and λs is the parameter of Poisson distribution calculated by Eq. (2). Furthermore, the parameter qs can be assumed to depend on the shortest distance between the recipient and donor plants. The border effect is also included in the estimation of qs because it is related to the distance effect. The relationship among distance, border, and the qs can be described using the following logistic function:$${q}_{s}=frac{1}{1+mathrm{exp}({b}_{1}-{b}_{2}{d}^{*}left(s,{s}^{^{prime}}right))},$$
(6)
where b1 and b2 are the parameters of the logistic function. The d*(s, s′) was the shortest distance between s′ and s in the version of dispersal models without the FB effect. The Poisson distribution was also used as an observation model for comparison with the ZIP observation model.Experimental and meteorological data collectionThe pollen dispersal data were collected from experiments performed in 2009 and 2010 at the geographic coordinates 23° 47′ N, 120° 26′ E, and an altitude of 20 m. These experiments were coded as 2009-1, 2009-2, and 2010-1, respectively. The experiment 2009-2 was divided into 2009-2A (without the FB) and 2009-2B (with the FB) based on the presence of the FB. The different layouts of the field experiments were designed to investigate the effect of the FB. Two commercial glutinous maize varieties, black pearl (purple grain) and Tainan No. 23 (white grain), were selected as the pollen donor and pollen recipient, respectively. The distance between the plants in a row was 25 cm, whereas the distance between the rows was 75 cm. The recipient plots consisted of 82 and 91 rows in 2009 and 2010 experiments, respectively.The CP rate was determined based on the differences in grain color on recipient cobs as a result of the xenia effect31. In the sampling framework, the whole field was divided into many grids and corn samples were collected from each grid in the whole field. The CP rate of each grid was calculated using the method presented in a previous study32 and defined as:$$mathrm{CP}left(%right)=left[sum_{i=1}^{n}{Cob}_{i}/left(ntimes Kright)right],$$
(7)

where Cobi and n indicate ith cob and total number of cobs in the grid, respectively. K is the average grain number per cob. Meteorological data were collected from the meteorological station at geographic coordinates 23° 35′ N, 120° 27′ E, and an altitude of 20 m. The detailed experimental setup was described in our previous study33. The study complies with relevant institutional, national, and international guidelines and legislation.Statistical analysesAll statistical analyses were performed using SAS (Statistical Analysis System, version 9.4). The dispersal model parameters were estimated by two methods. First, the nonlinear model estimation was conducted by PROC NLMIXED to evaluate the fitting and predictive abilities of dispersal models. Then the dispersal models with the observation model performed better fitting ability were re-estimated using the Bayesian estimation method to assess the uncertainty by PROC MCMC. In the Bayesian method, the noninformative prior distribution was used to estimate all parameters (Supplementary Table S1). The iteration of Markov Chain was 500,000 times and the burn-in was set to 450,000 iterations. In order to reduce the autocorrelations in the chain, the thinned value was set to 25.The validation method used in this study was the threefold cross-validation for the results of both estimation methods. The data from three experiments were combined and randomly partitioned into three sub-datasets. To avoid the heterogeneity of the different field designs and distances among sub-datasets, the observations from the same field design and same distance were considered as a group, and then partitioned into three parts. Each sub-dataset contained one part of all groups. At each validation run, two sub-datasets were selected as the training set, and the remaining one was used for validation.The fitting ability of the dispersal models was evaluated based on two criteria, namely, Akaike information criterion (AIC), Deviance, and coefficient of determination (R2). The smaller values of AIC or deviance indicate a better fitting. The higher R2 value represents a better fitting performance. The correlation coefficient (r) between the predicted and actual CP rates was used to assess the predictive ability. The deviance information criterion (DIC) was used to evaluate the performance of dispersal model fitting for the Bayesian estimation. The criterion values calculated from three training and validation sets were averaged to assess the overall results. The uncertainty of the model parameter was quantified by the standard deviation (SD) of parameter posterior distribution. The 95% credible intervals of posterior predictive distribution constructed by the 2.5th and 97.5th percentiles of 200,000 samples generated from the posterior predictive distribution were used to assess the predictive uncertainty. Furthermore, to assess the zero-excess condition, the percentage of observed zero CP grain events was compared with the Poisson probability of the zero CP grain event. A zero-excess condition occurred if the observed percentage was higher than the Poisson probability34. More

Mathematical model 1—the relationship among the bodyweight of the initial WSSV-infected shrimp, number of deaths, and death time distributionThe experimental data show the time course of death for the infected shrimp satisfies the Laplacian distribution (Supplementary Tables 2–4). The relationship of the bodyweight of the initial infected shrimp number of deaths and death time distribution could be expressed by a mathematical model and the establishment of the mathematical model as shown below.Suppose that one dead shrimp could infect (n) healthy shrimp at the same day. These (n) infected shrimp do not die simultaneously but on different days (time course). The value of (n) is related to the weight of the dead shrimps—larger dead shrimp can infect more healthy shrimps of the same body weight. Our experimental results (Supplementary Tables 2–4) show the death time course for these (n) infected shrimp satisfies the Laplacian distribution, as follows:$$begin{array}{c}pleft(tright)=left{begin{array}{c}{b{{exp}}}left(-frac{left|t-aright|}{{c}_{1}}right),tle a\ {b{{exp}}}left(-frac{left|t-aright|}{{c}_{2}}right),t > aend{array}right.end{array}$$
(1)
where (a) is the peak time of number of dead shrimps, (b) is the maximal death percentage, ({c}_{1}) is related to the mortality increases of the infected shrimps, ({c}_{2}) is related to the mortality decreases of the infected shrimp, (p(t)) is the percentage of infected shrimp that die at time (t). The open bracket “{“ in formula (1) means the function is represented by two parallel expressions as described previously.Based on the Supplementary Tables 2–4, we can determine the value of (a), (b), ({c}_{1}), and ({c}_{2}) by the least square estimation method. As different weight corresponds to different distribution of death time, we can compute the relationship of weight of death shrimps with corresponding (a), (b), ({c}_{1}), and ({c}_{2}) (Supplementary Table 25).We found the relationship of (w) with (a), or (b), or ({c}_{1}) or ({c}_{2}) is quadratic (Eq. 2), with the data in Supplementary Table 25, we have$$begin{array}{c}left{begin{array}{c} a= -0.0918{w}^{2}+0.8772w+3.3449\ b=0.0029{w}^{2}-0.0369w+0.5849;;\ {c}_{1}=-0.0186{w}^{2}+0.1739w+0.7063\ {c}_{2}=0.0002{w}^{2}+0.0108w+1.0827;;,end{array}right.end{array}$$
(2)
Using Model 1, we can predict the effects of different body weights of dead WSSV-infected shrimp through the ingestion pathway of WSSV-infected dead shrimp on the WSSV transmission rate.Mathematical model 2—the dynamic changes of healthy, infected, and dead shrimp during WSSV transmissionWe derived and established Model 2 to simulate the WSS transmission dynamics in cultured shrimp. Using Model 2, we predicted the dynamic changes of three states (healthy, infected, and dead shrimps) in cultured shrimp as influenced by the WSS epidemic with the following:Now we can develop a model for the spread and break out of WSS. For any given weight (w) of shrimps, let ({s}_{h}(t)), ({s}_{i}(t)), and ({s}_{d}(t)) be the number of healthy shrimp, infected shrimp and dead shrimp respectively at time (t). Let (I(t)), (d(t)) be the number of daily infected shrimp, daily dead shrimp, respectively, at time (t).According to infection process, the decrement of healthy shrimp is caused by their infection, therefore we have (frac{d{s}_{h}}{{dt}}=-I(t)). The quantity change of infected shrimp includes the infection of healthy shrimp and the death of infected shrimp, we have (frac{d{s}_{i}}{{dt}}=I(t)-d(t)). The increment of dead shrimp is caused by the death of the infected shrimp; thus we have (frac{d{s}_{d}}{{dt}}=d(t)). We obtain the following system of ordinary differential equations:$$left{begin{array}{c}frac{d{s}_{h}}{{dt}}=-Ileft(tright)hfill\ ,frac{d{s}_{i}}{{dt}}=Ileft(tright)-dleft(tright)\ frac{d{s}_{d}}{{dt}}=d(t)hfillend{array}right.$$
(3)
where ({s}_{h}left(0right)={s}_{{h}_{0}}), ({s}_{i}left(0right)={s}_{{i}_{0}}), ({s}_{d}left(0right)={s}_{{d}_{0}}) are as the initial value, at (t=0).In the above system of ordinary differential equations, quantity (I(t)) can be expressed as follows$$begin{array}{c}Ileft(tright)={min }left{n{s}_{d}left(tright),{s}_{h}left(tright)-alpha {s}_{{h}_{0}}right}end{array},$$
(4)
(d(t)) can be expressed as$$begin{array}{c}dleft(tright)={int }_{0}^{T}{min }left{n{s}_{d}left(t-tau right),{s}_{h}left(t-tau right)-alpha {s}_{{h}_{0}}right}pleft(tau right)dtau end{array}$$
(5)
where (n) is the number of healthy shrimp infected by one dead shrimp on the first day. (p(tau )) is the death percentage of the (n) infected shrimp on the (tau) days, (T) is the longest survival time of infected shrimp.Now we explain how to set up the formulas (I(t)) and (d(t)). In the expression of (I(t)), (n{s}_{d}(t)) is the number of daily infected shrimp at time (t). But as the number of healthy shrimp decreases, there may not be as many as (n{s}_{d}(t)) healthy shrimp to be infected. Therefore, (I(t)) is the minimum of (n{s}_{d}(t)) and ({s}_{h}left(tright)-alpha {s}_{{h}_{0}}), where (alpha (0 , < , alpha, < , 1)) represents the percentage of healthy shrimp that may have resistance to viruses, (d(t)) is the number of shrimps infected from (0) to (t) die at time (t). We use this integral to express the number of shrimp die at time (t).To evaluate the performance of the model 2, we compare the simulated scenario and the biological experimental settings. Our experiments show the quantity change of dead shrimps and live shrimps with respect to time, which is consistent with the result of simulation (Supplementary Fig. 4).Mathematical model 3—use fish to control WSSWe established Model 3 for the prevention and control of WSS using fish. In Model 3, two parameters need to be determined before this model can be applied for evaluating the fish’s capability of WSS prevention and control. The two parameters are, (1) fish-feeding quantity of dead shrimp, and (2) fish-feeding ratio of dead shrimp over healthy shrimp. We obtained 1 kg grass carp’s feeding quantity of different body weights of shrimp and the feeding selectivity through experiments. The mathematical reasoning of Model 3 is as follows:To block the transmission of WSS, we apply fish to eat dead shrimp and infected shrimp. Let ({e}_{h}(t)), ({e}_{i}(t)), and ({e}_{d}(t)), respectively be the number of healthy shrimp, infected shrimp and dead shrimp eaten by fish daily at time (t), (f(t)) is the number of fish.The decrement of healthy shrimp is related to the number of infected healthy shrimp and the number of shrimp eaten by fish, as expressed in (frac{d{s}_{h}}{{dt}}=-I(t)-{e}_{h}(t)). Similarly, the dynamics of the infected shrimp is related to the number of infected healthy shrimp, the death number of infected shrimp, and the number of infected shrimp eaten by fish, as expressed in (frac{d{s}_{i}}{{dt}}=I(t)-d(t)-{e}_{i}(t)). The dynamics of dead shrimp is related to the death number of infected shrimp, and eaten by fish, as expressed in (frac{d{s}_{d}}{{dt}}=d(t)-{e}_{d}(t)). Combining the above formulae, we can write the model as follows:$$left{begin{array}{c}frac{d{s}_{h}}{{dt}}=-Ileft(tright)-{e}_{h}left(tright)quadhfill\ ,frac{d{s}_{i}}{{dt}}=Ileft(tright)-dleft(tright)-{e}_{i}left(tright)hfill\ frac{d{s}_{d}}{{dt}}=d(t)-{e}_{d}(t)hfillend{array}right.$$ (6) where ({s}_{h}left(0right)={s}_{{h}_{0}}), ({s}_{i}left(0right)={s}_{{i}_{0}}), ({s}_{d}left(0right)={s}_{{d}_{0}}) are as the initial value at (t=0). In the above model, (I(t)), (d(t)), ({e}_{h}(t)), ({e}_{i}(t)), and ({e}_{d}(t)) are respectively given as follows:$$left{begin{array}{c};, Ileft(tright)={min }left{n{s}_{d}left(tright),{s}_{h}left(tright)-alpha {s}_{{h}_{0}}right}hfill\ ;dleft(tright)={int }_{0}^{t}{min }left{n{s}_{d}left(t-tau right),{s}_{h}left(t-tau right)-alpha {s}_{{h}_{0}}right}pleft(tau right){exp }left{{int }_{t-tau }^{t}{{{{{rm{ln}}}}}}rleft(uright){du}right}dtau hfill\ {e}_{d}left(tright)={min }left{fleft(tright)cdot mcdot beta ,{s}_{d}left(tright)+dleft(tright)right}hfill\ ,{e}_{i}left(tright)={min }left{left(fleft(tright)cdot m-{e}_{d}left(tright)right)frac{{s}_{i}left(tright)+Ileft(tright)-dleft(tright)}{{s}_{i}left(tright)+{s}_{h}left(tright)-dleft(tright)},{s}_{i}left(tright)+Ileft(tright)-dleft(tright)right}hfill\ {e}_{h}left(tright)={min }left{fleft(tright)cdot m-{e}_{d}left(tright)-{e}_{i}left(tright),{s}_{h}left(tright)-Ileft(tright)right}hfill\ ;,rleft(tright)=1-frac{{e}_{i}left(tright)}{{s}_{i}left(tright)+Ileft(tright)-dleft(tright)}hfillend{array}right.$$ (7) where, (I(t)) is the same as in Eq. (4); for (d(t)), different from Eq. (5) is that we add an exponential item ({exp }left{{int }_{t-tau }^{t}{{{{{rm{ln}}}}}}rleft(uright){du}right}) to account for the infected shrimp that may be eaten by fish during the past (t) days. As for ({e}_{d}(t)) shown in Eq. (6), (m) is for that each fish eats (m) shrimps while (beta) accounts for a percentage of dead shrimp in (m) shrimp. In ({e}_{i}(t)), we introduce (frac{{s}_{i}left(tright)+Ileft(tright)-dleft(tright)}{{s}_{i}left(tright)+{s}_{h}left(tright)-dleft(tright)}) for the percentage of infected shrimp in live shrimp. ({e}_{h}(t)) accounts for the number of healthy shrimp eaten by fish. (r(t)) represents the percentage of infected shrimp not being eaten by fish. We performed the effects of 1 kg grass carps on shrimp with four different body weights. The simulated data agreed with the experimental results (Fig. 2c).The relationship among the bodyweight of one initial WSSV-infected shrimp, number of deaths, and death time distributionThree groups of 430 shrimp with a bodyweight of 1.98 ± 0.03, 6.13 ± 0.16, and 7.95 ± 0.13 g, respectively, were used. In each group, 30 shrimp were randomly selected and subjected to a two-step WSSV PCR assay. All the tested shrimp showed negative in the assay. The remaining 400 shrimps were divided equally and introduced to three experimental and one control ponds. All 12 aquariums (220 cm × 60 cm × 80 cm) were set up with a water volume of 0.5 m3 and a salinity of 8‰. Shrimp were quarantined for seven days before the experiment started. One piece of dead WSSV-infected shrimp was then introduced to each of the experimental aquariums. In addition, one piece of frozen dead shrimp (WSSV-free) was introduced to the control aquarium. Shrimp were fed once a day with artificial feed that is 2% of their body weight. Shrimp feces were timely removed, and 50% of the water in the aquarium was exchanged every day. To prevent healthy shrimps from eating the moribund shrimp but not the initial dead WSSV-infected shrimp, shrimp were observed every 10 min to identify and remove moribund shrimp from the second day of the experiment. Moribund shrimp were identified as the ones having pleopod activity, but no response to glass rod agitation. The experiment was continued until three days after the appearance of the last moribund shrimp in each aquarium. Five pieces each of moribund and survived shrimps in each aquarium were subjected to a one-step WSSV PCR assay. All moribund shrimps showed WSSV-positive, while survived shrimps showed WSSV-negative. A mathematical model (Model 1) describing the relationship among the bodyweight of one initial WSSV-infected shrimp, number of deaths, and death time distribution was established based on the experimental results.The dynamic changes of live, infected, and dead shrimps during WSSV transmissionTo determine the changes in numbers of live and dead shrimp during WSSV transmission, 9 cement ponds (315 cm × 315 cm × 120 cm) were set up with a water volume of 5 m3 and salinity of 8‰. Regarding the stocking quantity of 7.5 × 105/ha in shrimp farming production, 750 healthy shrimp with an average body weight of 7.9 g were cultured in each of the nine ponds.To prepare the WSSV acute-infected shrimp, healthy shrimp were starved for 3 days, and then fed with parts of dead WSSV-infected shrimp that are 20% of their body weights twice a day. Five shrimp were randomly selected and subjected to a one-step WSSV PCR assay. If the tested shrimp showed WSSV positive in the assay. The rest of the shrimp in the aquarium was used as the WSSV acute-infected shrimp in the following experiments.Healthy shrimp were quarantined for seven days before the experiment started. Thirty WSSV acute-infected shrimp were then introduced in each pond. Shrimps were fed once a day with artificial feed that is 2% of their body weight. The numbers of survived shrimp were counted in three ponds on the 2nd, 4th, 8th day after WSSV infection, respectively. Five dead shrimps in each pond were subjected to a one-step WSSV PCR assay, showing WSSV-positive. Based on model 1, we established a mathematical model (Model 2) to describe the dynamic changes of healthy, infected, and dead shrimps during WSSV transmission.The dead shrimp ingestion rate of fishTo determine the dead shrimp ingestion rate of grass carp (Ctenopharyngodon idellus). Three cement ponds (315 cm × 315 cm × 120 cm) were set up with a water volume of 5 m3 and a salinity of 5‰. Three grass carps with an average body weight of 0.5 kg, 1 kg, and 1.5 kg were released in each of the three ponds, respectively. The fish were raised for four days and then fed with dead shrimps with an average weight of 5.3 g. In addition, to determine the dead shrimp ingestion rate of African sharptooth catfish (Clarias gariepinus). Four cement ponds (315 cm × 315 cm × 120 cm) were set up with a water volume of 5 m3 and salinity of 3‰. One African sharptooth catfish with bodyweight of 0.262, 0.496, 0.731, and 1.502 kg was released in each of the four ponds, respectively. The fish were raised for four days and then fed with dead shrimps with an average body weight of 6.2 g. Finally, to determine the dead shrimp ingestion rate of red drum (Sciaenops ocellatus). Three cement ponds (315 cm × 315 cm × 120 cm) were set up with water volume of 5 m3 and a salinity of 5‰. One red drum with a bodyweight of 0.590, 0.654, and 0.732 kg was released in each of the three ponds, respectively. The fish were raised for four days and then fed with dead shrimps with an average body weight of 3.9 g.During the five days of the experiment, dead shrimp that were not ingested by fish were exchanged with new dead shrimps every day. Additionally, the total body weight of dead shrimp ingested by fishes was calculated by subtracting the total body weight of dead shrimp that remained in the pond from the total body weight of dead shrimp put in the pond. The shrimp ingestion rate of fish is quantified by the daily ingestion rate (total body weight of ingested shrimps per day/total body weight of fishes). The daily ingestion rate of fish was calculated for 5 days.The healthy shrimp ingestion rate of fishTo determine the healthy shrimp ingestion rate of grass carp, three experimental ponds and one control pond (315 cm × 315 cm × 120 cm) were set up with a water volume of 5 m3 and salinity of 5‰. In total, 750 healthy shrimp with an average body weight of 5.3 g were cultured in each pond. One grass carp weighting 0.956, 1.013, and 1.050 kg was released in each of the experiment ponds, respectively. No fish was released in the control pond. Every two days, 50% of the water in each pond was changed. Live shrimp that remained in each pond were counted and weighted after 10 days of the experiment.To determine the healthy shrimp ingestion rate of African sharptooth catfish, one experimental pond and one control pond (315 cm × 315 cm × 120 cm) were set up with a water volume of 5 m3 and salinity of 3‰. In total, 750 healthy shrimp with an average body weight of 2.2 g were cultured in each pond. One African sharptooth fish weighting 1.050 kg was released in the experiment pond. No fish was released in the control pond. Every 2 days, 50% of the water in each pond was changed. Live shrimp that remained in each pond were counted and weighted after 10 days of the experiment.To determine the healthy shrimp ingestion rate of red drum, three experimental ponds and one control pond (315 cm × 315 cm × 120 cm) were set up with a water volume of 5 m3 and salinity of 5‰. In total, 750 healthy shrimp with an average body weight of 2.7 g were introduced in each pond. One red drum weighting 0.519, 0.554, and 0.595 kg was released in each of the experiment ponds, respectively. No fish was released in the control pond. Every two days, 50% of the water in each pond was changed. Live shrimp that remained in each pond were counted and weighted after 10 days of the experiment.The feeding selectivity of fish on dead, infected, and healthy shrimpsTo determine the feeding selectivity of grass carp on dead, infected, and healthy shrimp, one aquarium (220 cm × 60 cm × 80 cm) was set up with a water volume of 0.5 m3 and a salinity of 5‰. Grass carp weighting 1.58 kg was cultured in the aquarium for four days before the experiment started. The diseased shrimp infected with WSSV died within two days, which makes it hard to distinguish the initial dead shrimp from the ones that were died from diseased shrimp. The diseased shrimp had reduced activity, and the activity of shrimp was reduced after the endopods and exopods were removed. Thus, shrimp with endopods and exopods removed were utilized to resemble WSSV-infected shrimp. Thirty pieces each of dead, WSSV-infected (endopods and exopods removed), and healthy shrimps were introduced in the aquarium. The mean weight of shrimp used in the experiment is 3.5 g.To determine the feeding selectivity of African sharptooth catfish on dead, infected, and healthy shrimps, one aquarium (220 cm × 60 cm × 80 cm) was set up with a water volume of 0.5 m3 and salinity of 3‰. African sharptooth catfish with body weight of 1.03 kg was cultured in the aquarium for four days before the experiment started. Thirty pieces each of dead, WSSV-infected (endopods and exopods removed), and healthy shrimps were introduced in the aquarium. The mean weight of shrimps used in the experiment is 8.4 g.During the 9 days of the experiment, the dead, infected (endopods and exopods removed), and healthy shrimp that remained in the aquarium were counted and weighed every day. New shrimps were added to ensure there are 30 pieces each of dead, infected (endopods and exopods removed), and healthy shrimp in the aquarium. The daily total body weight of shrimp that were ingested by fish in each pond was calculated by subtracting the total body weight of shrimp that remained in the pond from the total weight of shrimp put in the pond. The shrimp ingestion rate of fish is quantified by the daily ingestion rate (total body weight of ingested shrimp per day/bodyweight of fish).The suitable bodyweight of grass carp for controlling WSSTo determine the suitable bodyweight of grass carp for controlling WSS, four experimental groups and two control groups were set up. Each group consisted of three ponds (315 cm × 315 cm × 120 cm). In total, 600 healthy and 3 WSSV-infected shrimp with an average body weight of 5 g were cultured in each pond of experimental groups. One grass carp with a bodyweight of 0.3, 0.5, 1.0, 1.5 kg was released in the ponds of each experimental group, respectively. In the positive control group, 600 healthy and 3 WSSV-infected shrimp with an average body weight of 5.0 g were cultured in each of the three ponds without introducing grass carp. In the negative control group, 600 healthy shrimp with an average body weight of 5.0 g were cocultured with one grass carp weighting 1.0 kg in each of the three ponds. The numbers of live shrimp were counted after ten days of the experiment. If there were dead shrimp in the ponds, they were subjected to a one-step WSSV PCR assay. All dead shrimps showed positive for WSSV infection.The suitable bodyweight of African sharptooth catfish for controlling WSSTo determine the suitable bodyweight of African sharptooth catfish for controlling WSS, four experimental groups and two control groups were set up. Each group consisted of three ponds (315 cm × 315 cm × 120 cm). In total, 600 healthy and WSSV carrying shrimp and 3 WSSV-infected shrimp with an average body weight of 1.5 g were cultured in each pond of experimental groups. The WSSV carrying shrimp were determined as the ones that showed positive in a two-step WSSV assay. One African sharptooth catfish with a bodyweight of 0.25, 0.5, 0.75, 1.5 kg was released in the ponds of each experimental group, respectively. In the positive control group, 600 healthy and 3 WSSV-infected shrimp with an average body weight of 1.5 g were cultured in each of the three ponds without introducing African sharptooth catfish. In the negative control group, 600 healthy shrimps with an average body weight of 1.5 g were cocultured with one African sharptooth catfish weighting 1.0 kg in each pond. The numbers of live shrimp were counted after ten days of the experiment. If there were dead shrimps in the ponds, they were subjected to a one-step WSSV PCR assay. All dead shrimps showed positive for WSSV infection.The capacity of grass carp for controlling WSSTo determine the capacity of grass carp for controlling WSS, the number of WSSV-infected shrimp that could be ingested by one grass carp weighting 1 kg was evaluated. Four groups of shrimp with different body weights (1.3 ± 0.1, 2.5 ± 0.2, 5.0 ± 0.3, 7.8 ± 0.5 g) were cocultured with 1-kg grass carp in the ponds.In 1.3 ± 0.1 g group, 750 healthy shrimp were cultured in each of the nine cement ponds (315 cm × 315 cm × 120 cm). Healthy shrimps were cultured with 3, 6, 9, 12, 15, 18, and 21 pieces of WSSV-infected shrimp in each of the seven experimental ponds, respectively. One grass carp weighting 1 kg was released in each of the seven ponds. Healthy shrimp were cultured with 3 WSSV-infected shrimps in one pond as a positive control. Additionally, healthy shrimps were cultured without WSSV-infected shrimp nor grass carp in one pond as a negative control. In 2.5 ± 0.2 g group, 750 healthy shrimp were cultured with 10, 20, 30, 40, 50, 60, and 70 pieces of WSSV-infected shrimp in each of the seven experimental ponds, respectively. One grass carp weighting 1 kg was released in each of the seven ponds. Healthy shrimp were cultured with 10 WSSV-infected shrimp in one pond as a positive control. Additionally, healthy shrimps were cultured without WSSV-infected shrimp nor grass carp in one pond as a negative control. In 5.0 ± 0.3 g group, 750 healthy shrimp were cultivated with 50, 70, 90, 110, 120, 130, and 140 pieces of WSSV-infected shrimp in each of the seven experimental ponds, respectively. One grass carp weighting 1 kg was released in each of the seven ponds. Healthy shrimp were cultured with 50 WSSV-infected shrimps in one pond as a positive control. Additionally, healthy shrimps were cultured without WSSV-infected shrimps nor grass carp in one pond as a negative control. In 7.8 ± 0.5 g group, 750 healthy shrimp were cultured with 30, 40, 50, or 60 pieces of WSSV-infected shrimps in four experimental ponds, respectively. One grass carp weighting 1 kg was released in each of the four ponds. Healthy shrimp were cultured with 30 WSSV-infected shrimps in one pond as a positive control. In addition, healthy shrimps were cultured without WSSV-infected shrimps nor grass carp in one pond as a negative control.In all the ponds, shrimp were fed with artificial feed that is 2% of their body weight. And 50% of the water was changed every day. The numbers of the remaining live shrimp were counted after 15 days of the experiment. A mathematical model (Model 3) was established based on the relationship of healthy shrimp, infected shrimp, dead shrimp, and fish.Determine the numbers of grass carp and African sharptooth catfish required for controlling WSS in L. vanmamei cultivationThe number of grass carp required for controlling WSS in shrimp production was determined in Pinggang Aquaculture Base, Yangjiang, China in 2010. Forty ponds (0.34 ± 0.04 ha/pond) were divided into eight groups; each group consisted of 5 ponds. We cultured 675,000/ha of shrimp in the ponds. Shrimp were cultured for 20 days before 45, 150, 225, 300, 450, 600, 750/ha of grass carp with an average body weight of 1.0 kg were released in the ponds of group 2 to group 8. Shrimp were cultured without fish in the ponds of group 1. These 40 ponds were managed by using the same farming method. If the WSS outbreak occurred, shrimps were harvested immediately; if not, shrimps were harvested after 110 days of cultivation.The number of African sharptooth catfish required for controlling WSS in shrimp production was determined in Pinggang Aquaculture Base, Yangjiang, China in 2010. Thirty-five ponds (0.37 ± 0.06 ha/pond) were divided into seven groups; each group consisted of 5 ponds. We cultured 675,000/ha of shrimp in the ponds. Shrimp were cultured for 10 days before 150, 300, 450, 600, 750, 900/ha of African sharptooth catfish with an average body weight of 1.0 kg were released in the ponds of group 2 to group 7. Shrimp were cultured without fish in the ponds of group 1. These 35 ponds were managed by using the same farming method. If the WSS outbreak occurred, shrimps were harvested immediately; if not, shrimps were harvested after 110 days of cultivation.Validation of coculturing shrimp and grass carp for controlling WSS in L. vanmamei farmingIn 2011, the polyculture system of coculturing L. vanmamei and grass carps was validated at a farm in Maoming, Guangdong Province, China (Farm 1). Forty-six farm ponds (17.33 ha) were divided into zone A and zone B. Zone A consisted of 18 ponds with a total area of 6.03 ha, and zone B consisted of 28 ponds with a total area of 11.30 ha. The stocking quantity of shrimp in the ponds of zone A is 900,000/ha. Shrimp were cultured in the ponds for 20 days before releasing grass carps with an average body weight of 1.0 kg. The stocking quantity of fish is 317–450/ha. Shrimp were cultured without fish in the ponds of zone B, and the stocking quantity of shrimp is 900,000/ha. In 2012, we switched zones A and B, cultivating shrimp with grass carp in zone B but without fish in zone A. The stocking quantities of shrimp and fish were the same as in 2011. If a WSS outbreak occurred, shrimps were harvested immediately; if not, shrimps were harvested after 110 days of cultivation, and yields were measured.Validation of coculturing shrimp and African sharptooth catfish for controlling WSS in L. vanmamei farmingIn 2011, the polyculture system of coculturing L. vanmamei and African sharptooth catfish was validated at a farm in Qinzhou, Guangxi Province, China (Farm 2). Ninety-five farm ponds (88.2 ha) were divided into zone A and zone B. Zone A consisted of 38 ponds with a total area of 21.2 ha, and zone B consisted of 57 ponds with a total area of 67.0 ha. The stocking quantity of shrimp in the ponds of zone A is 750,000/ha. Shrimp were cultured in the ponds for 10 days before releasing African sharptooth catfish with an average body weight of 0.5 kg. The stocking quantity of fish is 525–750/ha. Shrimp were cultured without fish in the ponds of zone B, and the stocking quantity of shrimp is 750,000/ha. In 2012, we split zone B into zones B1 and B2. Shrimp were cultivated with catfish in 38 ponds of zone A and 25 ponds (27.00 ha) of zone B1, while shrimp were cultivated without fish in 32 ponds (40.00 ha) of zone B2. The stocking quantities of shrimp and fish were the same as in 2011. If WSS outbreak occurred, shrimps were harvested immediately; if not, shrimps were harvested after 110 days of cultivation, and yields were measured.Long-term validation of coculturing shrimp and fish for controlling WSS in L. vanmamei cultivationWe tested the effectiveness of using fish for controlling WSS in shrimp production at a farm in Maoming, Guangdong Province, China (Farm 1) from 2013 to 2019. In 2013, shrimp were co-cultured with African sharptooth catfish of body weight ranging from 0.5 to 0.6 kg in 13 ponds (3.73 ha). The stocking quantity of shrimp in these ponds ranges from 878,788/ha to 1,230,769/ha. And shrimp were co-cultured with grass carp of body weight ranges from 0.7 kg to 1.0 kg and African sharptooth catfish of body weight ranges from 0.5 kg to 0.6 kg in 10 ponds (3.7 ha). The stocking quantity of shrimp in these ponds ranges from 909,091/ha to 1,212,121/ha. Additionally, shrimp were cultured without fish in 11 ponds (3.63 ha). The stocking quantity of shrimp in these ponds ranges from 878,788/ha to 969,697/ha. If WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 110 days of cultivation.In 2014, shrimp were cocultured with grass carp of body weight ranging from 0.7 to 1.0 kg in 8 ponds (2.76 ha). The stocking quantity of shrimp in these ponds ranges from 833,333/ha to 1,060,606/ha. And shrimp were co-cultured with grass carp of body weight ranges from 0.7 to 1.0 kg and African sharptooth catfish of body weight ranges from 0.5 to 0.6 kg in 12 ponds (4.03 ha). The stocking quantity of shrimp in these ponds ranges from 825,000/ha to 1,060,606/ha. Additionally, shrimp were cultured without fish in 5 ponds. The stocking quantity of shrimp in these ponds was 1,060,606/ha. If a WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 110 days of cultivation.In 2015, shrimp were co-cultured with grass carp of body weight ranging from 0.7 to 1.0 kg in 19 ponds (7.4 ha). The stocking quantity of shrimp in these ponds ranges from 746,269 to 1,538,462/ha. In addition, shrimp were cultured without fish in 10 ponds (3.8 ha). The stocking quantity of shrimp in these ponds ranges from 750,000 to 909,091/ha. If a WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 110 days of cultivation.In 2016, shrimp were co-cultured with grass carp of body weight ranging from 0.7 to 1.0 kg in 19 ponds (8.11 ha). The stocking quantity of shrimp in these ponds ranges from 488,372/ha to 636,364/ha. Additionally, shrimp were cultured without fish in 8 ponds (2.84 ha). The stocking quantity of shrimp in these ponds ranges from 543,478/ha to 636,364/ha. If a WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 110 days of cultivation.In 2017, shrimp were cocultured with grass carp of body weight ranging from 0.7 to 1.0 kg in 6 ponds (1.56 ha). The stocking quantity of shrimp in these ponds was 961,538/ha. And shrimps were co-cultured with grass carp of body weight ranging from 0.7 kg to 1.0 kg and African sharptooth catfish of body weight ranges from 0.5 to 0.6 kg in 12 ponds (3.96 ha). The stocking quantity of shrimp in these ponds ranges from 848,485/ha to 909,091/ha. Additionally, shrimp were cultured without fish in 9 ponds (2.76 ha). The stocking quantity of shrimp in these ponds ranges from 848,485/ha to 961,538/ha. If a WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 110 days of cultivation.In 2018, shrimp were cocultured with grass carp of body weight ranging from 0.7g to 1.0 kg in 22 ponds (9.24 ha). The stocking quantity of shrimp in these ponds ranges from 454,545/ha to 869,565/ha. Additionally, shrimp were cultured without fish in 9 ponds (3.36 ha). The stocking quantity of shrimp in these ponds ranges from 695,652/ha to 861,111/ha. If a WSS outbreak occurred, shrimp were harvested; if not, shrimp were harvested after 110 days of cultivation.In 2019, shrimp were cocultured with grass carp of body weight ranging from 0.7 to 1.0 kg in 30 ponds (11.31 ha). The stocking quantity of shrimp in these ponds ranges from 652,174/ha to 1,000,000/ha. Additionally, shrimp were cultured without fish in 10 ponds (3.57 ha). The stocking quantity of shrimp in these ponds ranges from 666,667/ha to 1,000,000/ha. If a WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 110 days of cultivation.Validation of coculturing shrimp and brown-marbled grouper for controlling WSS in P. monodon farmingIn 2013, the polyculture system of coculturing P. monodon and brown-marbled grouper was validated at a farm in Changjiang, Hainan Province, China (Farm 3). We cultured 6 × 105/ha of non-SPF shrimp in 6 ponds (1.60 ha) for 30 days and then introduced 600~750/ha of brown-marbled grouper with an average body weight of 0.1 kg. Shrimp were also cultured without fish in 3 ponds (0.8 ha). The stocking quantity of shrimp in these ponds is 6 × 105/ha. If a WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 150 days of cultivation, and yields were measured.In 2014, we cultured 6 × 105/ha of non-SPF shrimp in 6 ponds (1.60 ha) for 30 days and then introduced 600–750/ha of brown-marbled grouper with an average body weight of 0.1 kg. Shrimps were also cultured without fish in 3 ponds (0.8 ha). The stocking quantity of shrimp in these ponds is 6 × 105/ha. If a WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 150 days of cultivation, and yields were measured.Validation of coculturing shrimp and branded gobies for controlling WSS in M. japonica farmingIn 2013, the polyculture system of coculturing M. japonica and branded gobies was validated at a farm in Qingdao, Shandong Province, China (Farm 4). We cultured 1.5 × 105/ha of non-SPF shrimp in 10 ponds (13.40 ha) for 30 days and then introduced 750~900/ha of branded gobies with an average body weight of 0.05 kg. Shrimp were also cultured without fish in 5 ponds (6.70 ha). The stocking quantity of shrimp in these ponds is 1.5 × 105/ha. If a WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 100 days of cultivation, and yields were measured.In 2014, we cultured 1.5 × 105/ha of non-SPF shrimp in 10 ponds (13.40 ha) for 30 days and then introduced 750~900/ha of branded gobies with an average body weight of 0.1 kg. Shrimp were also cultured without fish in 5 ponds (6.70 ha). The stocking quantity of shrimp in these ponds is 1.5 × 105/ha. If a WSS outbreak occurred, shrimp were harvested immediately; if not, shrimp were harvested after 100 days of cultivation, and yields were measured.Promotion of the polyculture system at a farmers’ association in Nansha, ChinaWhen we promoted the polyculture system at the farmers’ association in 2015, only 6 farmers decided to adopt the system, as most of the farmers worried that fish would ingest healthy shrimp. Each of the 6 farmers introduced 225,000, 360,000, and 360,000 P.monodon postlarva to his/her earthen pond (3 ha) on March 28, May 8, and June 15, respectively. And 1350 grass carps with an average body weight of 1 kg were released in the ponds on April 30. These farmers harvested shrimp from May to November, and grass carp on December 14. The yields of shrimp and fish of these six ponds were recoded. The other farmers in the association introduced 225,000 and 360,000 of P.monodon postlarva to their ponds (3 ha) on March 28 and May 8, respectively. WSS outbreaks occur in their ponds from May 15 to May 23. Therefore, these farmers only harvested shrimp in May. Six ponds were randomly selected, and the yields of these ponds were recorded.Promotion of the polyculture system at a farmers’ association in Tanghai, ChinaFarmers at the farmers’ association used to culture 1500/ha of F. chinensis in earthen pond (5 ha) before the promotion of the polyculture system in 2015. The yields of 10 randomly selected ponds in 2014 were recorded. In 2015, farmers at the association started to culture 8,000/ha of F. chinensis in their ponds. The shrimp were cultured 20 days before 800/ha of branded gobies with an average body weight of 0.05 kg were released in the ponds. Branded gobies were cultivated for 15 days before introducing to the ponds. Shrimps were harvested after 120 days of cultivation. The yields of ten randomly selected ponds were recorded.Statistics and reproducibilityAlpha levels of 0.05 were regarded as statistically significant throughout the study. Three replicates were set up for each experiment to confirm the reproducibility of the data. All data are reported as the mean ± standard errors.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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