Bifurcation and chaotic behaviour in stochastic Rosenzweig-MacArthur prey-predator model with non-Gaussian stable L\'evy noise

TL;DR

This paper analyzes a stochastic prey-predator model driven by non-Gaussian Lévy noise, revealing bifurcations, stability conditions, and chaotic dynamics through analytical and numerical methods.

Contribution

It provides a detailed dynamical analysis of the stochastic Rosenzweig-MacArthur model with Lévy noise, including bifurcation and chaos characterization, which is novel.

Findings

Existence and stability of equilibrium points are characterized.

A transcritical bifurcation curve is identified.

Chaotic behaviour is observed under stochastic influence.

Abstract

We perform dynamical analysis on a stochastic Rosenzweig-MacArthur model driven by {\alpha}-stable L\'evy 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 behaviour in mutualistically interacting two-species prey-predator system subject to stochastic influence.