Stochastic bifurcation and Break-out of dynamic balance of predator-prey system with Markov switching

Abstract

This paper investigates Markov-switching-induced stochastic P-bifurcation and the dynamical balance in a predator-prey system. Based on the limit averaging, a probability-weighted system is established to approximate the original stochastic switching ecosystem, then the stationary distribution and first-passage problems are theoretically addressed. Stochastic bifurcations are discussed through a qualitative change of the stationary probability density, which indicates that the transition rate matrix of Markov switching can be treated as a bifurcation parameter. They also imply that the stationary distribution is more sensitive to the transition rate λ12. Besides, by the first-passage theory, the dynamic balance is explored to elucidate the mechanisms underlying species coexistence. Astonishingly, the Markov switching may maintain or even improve the dynamic stability of species coexistence. Biologically, apart from human destruction, the species are even more robust and capable of adapting to general environmental disturbances. The utility and the accuracy of the theoretical analysis are demonstrated by direct numerical simulations.