On the effect of density-dependent dispersal on the global dynamics of population

In this work, we study a multi-patch model, where the patches are coupled by nonlinear asymmetrical migration terms, and each patch follows a logistic law. First, used the theory of a cooperative differential system, we prove the global stability of the model. Next, in the case of perfect mixing, i.e. when the migration rate tends to infinity, we compute the limit of the total equilibrium population, which in general is different from the sum of the $n$ carrying capacities, and depends on the migration terms, carrying capacities and growth rates. Second, we determine the conditions under which fragmentation and nonlinear asymmetrical migration can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities. We end by considering the case of two patches. We give the explicit formula of the total equilibrium population, and we compare with the sum of two carrying capacities.

Keywords

nonlinear diffusion, migration rate, logistic growth, global stability, perfect mixing

2010 Mathematics Subject Classification

Primary 37N25, 92D25. Secondary 34D15, 34D23.

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Received 17 May 2023

Accepted 21 August 2023

Published 15 August 2024