Increase of maximum sustainable yield for fishery in two patches with fast migration

We present a model of a fish population present on two patches connected by migrations. Fish grow logistically on each patch and are caught. We assume that migrations between the two sites are fast relative to local growth and fishing. Taking advantage of the time scales, we use methods of aggregation of variables to obtain a reduced model governing the total biomass of the fish population at a slow time scale. We are looking for the maximum sustainable yield (MSY) for the system of the two connected patches. We show that although the total equilibrium population may be greater than the sum of the carrying capacities on each isolated site, the total catch is always less than or equal to the sum of the catches on the isolated fishing sites. We then consider a prey–predator community of fish in the same environment. We assume that only the predator is caught and not its prey, growing logistically on each site. We study the Lotka–Volterra prey–predator model as well as the model with a type II Holling functional response. We show that the total catch at MSY of the system of connected sites can be greater than the sum of the captures on each isolated site. This result is obtained when a fishing site with a large prey carrying capacity and an average growth rate is connected to a site with a small carrying capacity but a large growth rate. Finally, we discuss fishery management methods on two fishing sites for the Lotka–Volterra model as well as the Holling type II model in the case of a prey refuge.