Bifurcation and chaotic behavior in stochastic Rosenzweig–MacArthur prey–predator model with non-Gaussian stable Lévy noise

Abstract

We perform dynamical analysis on a stochastic Rosenzweig–MacArthur model driven by α-stable Lévy motion. We analyze the existence of the equilibrium points, and provide a clear illustration of their stability. It is shown that the nonlinear model has at most three equilibrium points. If the coexistence equilibrium exists, it is asymptotically stable attracting all nearby trajectories. The phase portraits are drawn to gain useful insights into the dynamical underpinnings of prey–predator interaction. Specifically, we present a transcritical bifurcation curve at which system bifurcates. The stationary probability density is characterized by the non-local Fokker–Planck equation and confirmed by some numerical simulations. By applying Monte Carlo method and using statistical data, we plot a substantial number of simulated trajectories for stochastic system as parameter varies. For initial conditions that are arbitrarily close to the origin, parameter changes in noise terms can lead to significantly different future paths or trajectories with variations, which reflect chaotic behavior in mutualistically interacting two-species prey–predator system subject to stochastic influence.