The multi-patch logistic equation with asymmetric migration

TL;DR

This paper studies a multi-patch logistic model with asymmetric migration, analyzing how migration affects total population dynamics and equilibrium, including perfect mixing and specific case conditions.

Contribution

It introduces a multi-patch logistic model with asymmetric migration and characterizes population behavior under perfect mixing and specific conditions.

Findings

Total population follows a modified logistic equation under perfect mixing.

Fragmentation and asymmetry can alter total equilibrium population relative to capacities.

Numerical evidence of critical migration rates affecting total equilibrium in three-patch models.

Abstract

This paper considers a multi-patch model, where each patch follows a logistic law, and patches are coupled by asymmetrical migration terms. First, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic equation with a carrying capacity which in general is different from the sum of the n carrying capacities, and depends on the migration terms. Second, we determine, in some particular cases, the conditions under which fragmentation and asymmetrical migration can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities. Finally, for the three-patch model, we show numerically the existence of at least three critical values of the migration rate for which the total equilibrium population equals the sum of the carrying capacities.