Stochastic Bifurcations in the Plankton-fish System

Abstract

Since each creature interacts with each other through a food chain, an analysis of the food chain is a crucial problem in an ecosystem. In this paper, we consider the plankton-fish system consisting of fish, zooplankton and phytoplankton, and study the influence of the random noise on the stability of the plankton system under fish school by numerical simulations. The qualitative change of the solutions of differential equations by variation of a system parameter is generally called a bifurcation. This paper is concerned with the bifurcation analysis of a stochastic plankton-fish system. As major analytical methods for the stochastic bifurcation, the D-bifurcation (Dynamical bifurcation) and the P-bifurcation (Phenomenological bifurcation) approaches are cited. The D-bifurcation considers the stability of invariant measures, while the P-bifurcation is characterized with a qualitative change of the stationary probability distribution, e.g., a transition from unimodal to bimodal distribution. By the numerical simulations, we show that not the D-bifurcation but the P-bifurcation occurs in the stochastic plankton-fish system considered here.