Mathematical modeling of multi-site fisheries: Effects of heterogeneity on Maximum Sustainable Yield (MSY)

We present a short review of classical fishing models, the Schaefer, Gordon-Schaefer and variable price models. In the latter case, we show that the price dynamics dependent on supply and demand leads to a situation of bi-stability with the coexistence of a sustainable development equilibrium and an overexploitation equilibrium. We further consider a fishery consisting of N fishing sites connected by fish migrations. At each site, we assume the classic Schaefer model and fast migrations between the sites relative to local growth and fishing. Taking advantage of the time scales, we obtain a reduced model governing the total fish biomass at a slow time scale. We show that although the total equilibrium biomass may be greater than the sum of the carrying capacities on each isolated site, the Maximum Sustainable Yield (MSY) is always less than or equal to the sum of the MSY on the isolated fishing sites. We then consider a prey-predator fish community with type I functional response in the same environment. We assume that only the predator is caught and not its prey. We show that in this case due to connectivity the total catch at MSY can be greater than the sum of the captures on each isolated site. We show that heterogeneity between sites as well as asymmetry in migrations play an important role in the increase in MSY. Furthermore, an emergence phenomenon can be observed: even if none of the sites is viable for fishing, the entire multi-site fishery can be viable. Our study is extended to the prey-predator model with a type II Holling functional response.