Uniqueness of Limit Cycles in a Rosenzweig–MacArthur Model with Prey Immigration

Abstract

Many natural predator and prey populations persist while their densities show sustained oscillations. Hence these populations must be regulated in such a way that the densities are kept away from the values where extinction is likely to occur. On the other hand, nonspatial simple predator-prey models show vigorous oscillations that can bring the populations to the brink of extinction or beyond. Predator-prey systems that are kept in the laboratory also tend to show fluctuations in densities that are severe enough to drive them to extinction. Since the amount of space that laboratory populations live in is small compared to that of natural populations, one is readily led to the hypothesis that spatial interactions must contribute to the regulation of natural predator-prey systems. In this paper, we construct a simplest type of spatially interacting populations by taking into account constant immigration of prey for a predator-prey model with a Holling type II functional response and derive necessary and sufficient conditions for both the uniqueness of limit cycles and the global asymptotic stability of a positive equilibrium. From these results, it is fully suggested (mathematically) that the prey immigration dampens the large fluctuations emerging in the predator-prey model and also stabilizes a positive equilibrium globally.