Bifurcation and complex dynamics of a discrete-time predator-prey system with simplified Monod-Haldane functional response

Sarker Md Sohel Rana

doi:10.1186/s13662-015-0680-7

Sarker Md Sohel Rana

Open Access

https://doi.org/10.1186/s13662-015-0680-7

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Journal: Advances in Difference Equations	Publication Date: Nov 5, 2015
Citations: 18	License type: CC BY 4.0

Affiliation: University of Dhaka

Abstract

In this paper, we investigate the dynamics of a discrete-time predator-prey system with simplified Monod-Haldane functional response. The existence and local stability of positive fixed point of the discrete dynamical system is analyzed algebraically. It is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of $\mathbb {R}^{2}_{+}$ by using bifurcation theory. Numerical simulation results not only show the consistence with the theoretical analysis but also display new and interesting dynamical behaviors, including phase portraits, period-11 orbits, attracting invariant circle, cascade of period-doubling bifurcation from period-11 leading to chaos, quasi-periodic orbits, and the sudden disappearance of the chaotic dynamics and attracting chaotic set. The Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors.

Highlights

It is well known the Lotka-Voltera predator-prey model is one of the fundamental population models; a predator-prey interaction has been described first by two pioneers Lotka [ ] and Voltera [ ] in two independent works
Rana Advances in Difference Equations (2015) 2015:345 there are few articles discussing the dynamical behaviors of predator-prey models, which include bifurcations and chaos phenomena for the discrete-time models
We rigorously prove that this discrete model possesses the flip bifurcation and the NS bifurcation, respectively

Summary

Introduction

It is well known the Lotka-Voltera predator-prey model is one of the fundamental population models; a predator-prey interaction has been described first by two pioneers Lotka [ ] and Voltera [ ] in two independent works. ( ) hold and c(δF ) = , system ( ) undergoes a flip bifurcation at the fixed point E (x∗, y∗) when the parameter δ varies in a small neighborhood of δF .

Results

Conclusion