Global stability and period-doubling bifurcations of a discrete Kolmogorov predator-prey model with Ricker-type prey growth

TL;DR

This paper analyzes a discrete predator-prey model with Ricker prey growth, establishing conditions for stability, demonstrating period-doubling bifurcations, and revealing chaotic dynamics through numerical simulations.

Contribution

It provides the first comprehensive analysis of bifurcations and chaos in a discrete Kolmogorov predator-prey model with Ricker growth, including stability criteria and bifurcation conditions.

Findings

Existence and uniqueness of positive fixed point established.

Period-doubling bifurcations can lead to chaos.

Global stability criterion derived using geometric nullcline analysis.

Abstract

In this paper, we study the dynamics of a discrete Kolmogorov predator-prey model with Ricker-type prey growth. We give the sufficient and necessary condition to guarantee the existence and uniqueness of the positive fixed point. Using the center manifold theory, we prove that the period-doubling bifurcations can occur at the positive fixed point. Furthermore, our numerical simulations reveal that the model can exhibit cascades of period-doubling bifurcations leading to chaos, which is a significant difference from the behavior of continuous predator-prey models. Despite the complexities of the model dynamics, we are able to provide a criterion for the global stability of the positive fixed point by using a geometric analysis of the nullclines.