Diffusively coupled Allee effect on heterogeneous and homogeneous graphs

Abstract

We study how the Allee effect with diffusion changes depending on the graph topology. We present the metapopulation dynamic model on heterogeneous and homogeneous graphs for the Allee effect of mobile individuals. We consider the small graphs with three and four nodes and the star and complete graphs with N nodes. A subpopulation (patch) is represented by a node on a graph. A link represents a migration path between subpopulations. Mobile individuals move by random walk through a link between nodes. Individuals grow or die out according to Allee population dynamics within each subpopulation. The population dynamics in the metapopulation model are presented by reaction–diffusion equations. To evaluate the population size of each subpopulation (node), we obtain the solutions of the reaction–diffusion equations numerically for the small graphs, star graphs, and complete graphs. The equilibrium points of the Allee metapopulation model depend highly on the network topology. The dependence of the critical Allee parameter on the number of nodes is derived for star and complete graphs.