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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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Agent-resource affiliation networksWe consider games involving populations of agents that extract from multiple common-pool sources (which term we use for nodes representing resources in accord with previous related work15,16). Agents’ access to sources is defined by bipartite networks, wherein a link between an agent and a source indicates that the agent can access that source. This access is determined by some exogenous factors and remains fixed in time. The set of agents affiliated with a particular source (s) is denoted as ({mathbf{A}}_{s}), while the set of sources affiliated with a particular agent (a) is denoted as ({mathbf{S}}_{a}). The degree of an agent (a) is denoted by (m(a)), and the degree of a source (s) by (n(s)).To explore the effects of network topology upon extraction dynamics and wealth distributions, we generate ensembles of ({10}^{3}) networks, each having (50) agents and (50) sources and sharing mean agent degree (langle mrangle =5) and mean source degree (langle nrangle =5). All networks thus share the same total numbers of agents, sources, and links, but differ in how these links are distributed among agents and sources. We generate 9 network ensembles, each generated to represent a particular combination of one of three types of degree heterogeneity in its source degree distribution (U: uniform-degree, L: low-heterogeneity, or H: high-heterogeneity) with one of three similar distributions of agent degree (u, l, or h39) (Supplementary Information S1.1). Degree histograms, averaged over each ensemble, provide a representative source degree distribution ({P}_{mathbf{S}}(n)) and agent degree distribution ({P}_{mathbf{A}}(m)) for each network type (Fig. 2a and b). It is worth noting that the results of the simulations depend primarily on the degree distributions of agents and sources rather than on the overall size of the networks used (Supplementary Information S3.1).Figure 2(a) Source degree distributions and (b) Agent degree distributions for 9 network ensembles, each representing a combination of a Uniform-degree (U), Low-heterogeneity (L), or High-heterogeneity (H) source degree distribution with a uniform-degree (u), low-heterogeneity (l), or high-heterogeneity (h) agent degree distribution. Ensemble mean time-averaged quantities from pure free adaptation dynamics: (c) Agent payoffs (f(a)) as a function of agent degree (m(a)); (d) Collective extraction (overrightarrow{q}(s)) as a function of source degree (n(s)); (e) Source quality (b(s)) as a function of source degree (n(s)); and (f) Period of oscillation (T(s)) as a function of source degree (n(s)). Means are computed from simulations on ({10}^{3}) networks of each type.Full size imageNetworked CPR extraction gameOn these networks, we simulate iterative games in which agents vary the extraction effort that they apply to their affiliated sources, altering the quality of these sources; in turn, these changes in source quality then influence how agents adapt their extraction levels in subsequent rounds. The extraction effort exerted by agent (a) upon its affiliated source (s) is denoted as (q(a,s)). The total effort exerted by an agent (a), its individual extraction, is denoted by (overleftarrow{q}left(aright)=sum_{sin {mathbf{S}}_{a}}q(a,s)). The total effort exerted upon source (s), or its collective extraction, is denoted by (overrightarrow{q}left(sright)=sum_{ain {mathbf{A}}_{s}}q(a,s)). The quality of a source (s) is quantified by the benefit (b(s)) per unit extraction effort applied that the source provides. The cost associated with extraction is given by a convex (quadratic) function of (overleftarrow{q}left(aright)), such that marginal costs increase with individual extraction15,16. In addition to modelling the increasing costs (i.e., diminishing returns) associated with the physical act of extraction itself, this could also reflect escalating, informal social penalties that result from increasing extraction (i.e., “graduated sanctions”1,40). The net payoff accumulated by an agent (a) in a game iteration is thus$$fleft( a right) = left[ {mathop sum limits_{{s in {mathbf{S}}_{a} }} qleft( {a,s} right) cdot bleft( s right)} right] – frac{gamma }{2}{ }mathop{q}limits^{leftarrow} left( a right)^{2} ,$$
(1)

where (gamma) is a positive cost parameter.Bistable model of CPR depletion and remediationSources are bistable, meaning that at any time they can occupy one of two states: (1) a viable state, during which the source provides a benefit of magnitude (alpha) in return for each unit of extraction effort, and (2) a depleted state, during which this benefit is reduced by (beta) ((0 vec{q}_{{text{D}}} left( s right)} \ {0, } & {{text{if }} chi_{t – 1} left( s right) = 1{text{ and }}vec{q}_{t} left( s right) le vec{q}_{R} left( s right)} \ {chi_{t – 1} left( s right),} & {text{otherwise }} \ end{array} } right.$$
(3)
In the results that follow, we focus upon a uniform capacity scenario, wherein all sources share identical threshold values (vec{q}_{{text{D}}} left( s right) equiv vec{q}_{{text{D}}}) and (vec{q}_{{text{R}}} left( s right) equiv vec{q}_{{text{R}}} left( s right)). An alternative degree-proportional capacity scenario, in which threshold values increase with source degree, is discussed in the Supplementary Information (S3.4.2).Free adaptationUnder the free adaptation strategy, an agent updates its extraction levels independently at each of its affiliated sources depending on the state of each (Fig. 1b). As in the replicator rule often applied in networked evolutionary game models17,41,42, the rate at which an agent adapts its extraction levels within a time interval ({Delta }t) is proportional to the marginal payoff that the agent expects to attain thereby:$$frac{{{Delta }qleft( {a,s} right)}}{{{Delta }t}} = kfrac{partial fleft( a right)}{{partial qleft( {a,s} right)}},$$
(4)
where (k) is a rate constant. So, each extraction level (qleft( {a,s} right)) is updated according to$$q_{t + 1} left( {a,s} right) = q_{t} left( {a,s} right) + kleft[ {alpha – beta chi_{t} left( s right) – gamma {mathop{q}limits^{leftarrow}}_{t} left( a right)} right].$$
(5)
The higher an agent’s individual extraction (overleftarrow{q}(a)), the more slowly it will increase its extraction from viable sources, and the more rapidly it will reduce its extraction from depleted sources.Uniform adaptationWhen applying the uniform adaptation strategy, an agent adjusts each of its extraction levels by the same magnitude (Delta qleft(a,sright)equivDelta overleftarrow{q}(a)/mleft(aright)) (Fig. 1c). Assuming again that the rate at which an agent enacts this update is proportional to the associated marginal payoff, an agent adapts its extraction levels at all of its affiliated sources (s) by$$q_{t + 1} left( {a,s} right) = q_{t} left( {a,s} right) + kleft[ {alpha – beta overline{chi }left( a right) – gamma {mathop{q}limits^{leftarrow}}_{t} left( a right)} right],$$
(6)
where (overline{chi }left( a right) = left[ {mathop sum nolimits_{{s^{prime} in {mathbf{S}}_{a} }} chi left( {s^{prime}} right)} right]/mleft( a right)) is the mean state of the agent’s affiliated sources.ReallocationWhen practicing reallocation, an agent shifts an increment of extraction effort from a depleted source to a viable source such that its overall individual extraction (mathop{q}limits^{leftarrow} left( a right)) remains unchanged (Fig. 1d). The agent thus randomly selects one depleted source (s_{{text{D}}} in {mathbf{S}}_{a}) and one viable source (s_{{text{V}}} in {mathbf{S}}_{a}), if available. Since the marginal payoff per unit reallocated is (beta), updates its extraction levels such that$$q_{t + 1} left( {a,s} right) = left{ {begin{array}{*{20}c} {q_{t} left( {a,s} right) – kbeta , } & {{text{if}} s = s_{{text{D}}} } \ {q_{t} left( {a,s} right) + kbeta , } & {{text{if}} s = s_{{text{V}}} } \ {q_{t} left( {a,s} right),} & {text{otherwise }} \ end{array} } right.$$
(7)

When an agent’s affiliated sources all share the same quality value, no such reallocation is possible, and so the agent retains its present extraction levels: (q_{t + 1} left( {a,s} right) = q_{t} left( {a,s} right)) for all (s in {mathbf{S}}_{a}).Mixed strategiesAn agent’s adaptation strategy (({p}_{0},{p}_{updownarrow },{p}_{leftrightarrow })) comprises the probabilities that it will practice each of these update rules in any given round: its free adaptation propensity (({p}_{0})), its uniform adaptation propensity (({p}_{updownarrow })), and its reallocation propensity (({p}_{leftrightarrow })). An agent’s choice of a particular update rule is thus based only on its own innate inclinations, but the rate at which it enacts the selected rule is influenced by current resource conditions. We first simulate dynamics in which the same adaptation strategy is shared by all members of a population throughout the entire course of a simulation. We then consider games in which agents’ individual adaptation strategies are each allowed to independently evolve under generalized reinforcement learning38,43 (Supplementary Information S1.3.4). That is, after enacting a chosen update rule in an iteration (t), each agent (a) observes the payoff change (Delta {f}_{t}left(aright)={f}_{t}left(aright)-{f}_{t-1}(a)). If (Delta {f}_{t}left(aright) >0), then the agent’s relative propensity to practice this update rule in subsequent rounds is increased. If the agent’s payoffs decreased ((Delta {f}_{t}left(aright)0) or remediation threshold ({overrightarrow{q}}_{mathrm{R}}left(sright)). That is, all resource depletion events are assumed to be extreme enough to motivate agents to continuously “self-regulate” by remediating depleted sources (see Supplementary Information S5 for a more thorough discussion of these parameter settings).In simulations where reinforcement learning is applied, all agents are initialized with ({p}_{updownarrow }={p}_{leftrightarrow }=.333). For pure free adaptation simulations (({p}_{0}=1)), initial extraction levels were randomized (({q}_{t=0}left(a,sright)in [0,frac{{overrightarrow{q}}_{mathrm{D}}left(sright)}{nleft(sright)}])). All other simulations (({p}_{0} More