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

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. Two major types of mathematical modelings can be used:
the continuous-time dynamical systems and the discrete-time dynamical systems. 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. Finally, the study used bifurcation diagram to demonstrate the interesting dynamics of new
system and the role of biological parameters r and k in appearance of different types of complicated
dynamics. The new system has the same number of fixed points as the first system and the
bifurcation analysis displayed the same qualitative dynamics for both species as we expected.