Dispersal Does Not Increase Maximum Sustainable Yield in Multi-Patch Harvesting Models

We study a spatially structured fish population model described by an n-patch logistic system with harvesting and migration. Each patch supports a population with logistic growth and possibly heterogeneous fishing efforts. Using singular perturbation theory and Tikhonov’s theorem, we analyze the regime of large dispersal and derive a reduced model governing the total population. We characterize its equilibria and provide approximations of the full system dynamics. Our central theoretical result concerns the total maximum sustainable yield MSY. We prove that, for any migration rate epsilon, the MSY of the connected system satisfies the fundamental inequality: [Formula: see text] ([Formula: see text]) is less than or equal to the sum over i from 1 to n of ([Formula: see text]/4), where [Formula: see text] and [Formula: see text] denote the intrinsic growth rate and carrying capacity of patch i, respectively, this upper bound being precisely the sum of the MSYs of the isolated patches. Furthermore, for a given [Formula: see text], we show that equality holds if and only if the efforts [Formula: see text], which keep every patch at half of its carrying capacity ([Formula: see text] being the migration matrix), are all nonnegative. In particular, equality holds for every [Formula: see text] when K belongs to the kernel of [Formula: see text], whereas otherwise it holds only up to a finite threshold of the migration rate, beyond which dispersal strictly reduces the overall MSY; in the fast-dispersal limit, equality holds if and only if [Formula: see text] ker[Formula: see text]. Numerical simulations illustrate these theoretical findings and emphasize the impact of spatial structure on optimal harvesting strategies.