Nonlinear diffusion in multi-patch logistic model

Abstract

We examine a multi-patch model of a population connected by nonlinear asymmetrical migration, where the population grows logistically on each patch. Utilizing the theory of cooperative differential systems, we prove the global stability of the model. In cases of perfect mixing, where migration rates approach infinity, the total population follows a logistic law with a carrying capacity that is distinct from the sum of carrying capacities and is influenced by migration terms. Furthermore, we establish conditions under which fragmentation and nonlinear asymmetrical migration can lead to a total equilibrium population that is either greater or smaller than the sum of carrying capacities. Finally, for the two-patch model, we classify the model parameter space to determine if nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities.

Keywords: Logistic equation; Nonlinear diffusion; Perfect mixing; Population dynamics; Slow-fast systems; Tikhonov’s theorem.