Dynamics of discrete predator- prey system with fear effect and density dependent birth rate of the prey species

Debasis Mukherjee

doi:10.5206/mase/14496

Abstract

This paper analyses a discrete predator-prey system with fear effect and density dependent birth rate of the prey species. The fixed points of the system are determined and their stability is examined. The criterion for Neimark-Sacker bifurcation and flip bifurcation is developed. The chaotic orbit at an unstable fixed point can be stabilized by applying the state feedback control method. Numerically, we illustrate our analytical findings and observe the complex behaviour of the system that leads to stable state to chaotic one.

Highlights

In recent years, it is observed that the predator-prey interaction is governed by direct killing of prey by the predator, and the indirect effect such as fear caused by the predator
We have studied the qualitative behaviour of a discrete predator-prey model with fear effect and density dependent birth rate of the prey species
It is identified that the parameter d, the death rate of predator in the system is more relevant for the appearance of flip bifurcation and Neimark-Sacker bifurcation whenever it is varied in some appropriate interval

Summary

Introduction

It is observed that the predator-prey interaction is governed by direct killing of prey by the predator, and the indirect effect such as fear caused by the predator. In [11], the authors considered density dependent birth rate of the prey species and discussed the dynamical behaviour of the predator-prey system. Predator-prey system; fear effect; density dependent birth rate; bifurcation; chaos control. Agiza et al [12] discussed the dynamics of a discrete-time prey-predator model with Holling type II response function. They derived bifurcation diagrams, phase portraits and Lyapunov exponents for various system parameters. We propose a discrete predator-prey system with fear effect and density dependent birth rate of the prey species.

Discrete model

Fixed points and their nature

Chaos control

Numerical Simulation

Discussion