Chaos Induced by Snap-Back Repeller in a Two Species Competitive Model

Tahmineh Azizi,Bacim Alali

doi:10.4236/ajcm.2020.102017

Abstract

In this paper, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform a local stability analysis for the fixed points and we will discuss about its persistence for boundary fixed points. This system inherits the dynamics of one-dimensional Ricker model such as cascade of period-doubling bifurcation, periodic windows and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and proving the existence of snap-back repeller. We use several dynamical systems tools to demonstrate the qualitative behaviors of the system.

Highlights

When we study the evolution of population dynamics, two major types of mathematical modelings can be used: the continuous-time dynamical systems and the discrete-time dynamical systems
We studied the complex dynamics of a two-species Ricker model which consists of four different biological parameters
We changed the model to a specific case with only three biological parameters and we discussed about the local stability of extinction and boundary fixed points of the system

Summary

Introduction

When we study the evolution of population dynamics, two major types of mathematical modelings can be used: the continuous-time dynamical systems and the discrete-time dynamical systems. The system is chaotic in the sense of Li-York [13] Marotto refined his theorem in 2005 and he explained that a fixed point z is called a repelling fixed point under differentiable function f : n → n if all eigenvalues of Df ( z ) exceed 1 in magnitude, but z is expanding only if f (x)− f (y) > s x − y where s > 1 , for all x, y sufficiently close to z with x ≠ y (for x, y ∈ Br′ ( z ) ). As Gardini et al discussed, in non-invertible maps homoclinic orbits may be associate with expanding fixed points and or expanding cycles They mentioned that in the neighborhood of such homoclinic orbits, there exists an invariant set on which the map is chaotic. We will numerically demonstrate the local and qualitative dynamics of the system using several dynamical system tools

The Two-Species Ricker Competitive Model and Its Local Dynamics

Persistence of the Species

Case 1

Case 2

Conclusion