The multi-patch logistic equation

Abstract

<p style='text-indent:20px;'>The paper considers a <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-patch model with migration terms, where each patch follows a logistic law. First, we give some properties of the total equilibrium population. In some particular cases, we determine the conditions under which fragmentation and migration can lead to a total equilibrium population which might be greater or smaller than the sum of the <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula> carrying capacities. Second, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic law with a carrying capacity which in general is different from the sum of the <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula> carrying capacities. Finally, for the three-patch model we show numerically that the increase in number of patches from two to three gives a new behavior for the dynamics of the total equilibrium population as a function of the migration rate.</p>