Chaotic attractors in Atkinson–Allen model of four competing species

Abstract

We study the occurrence of chaos in the Atkinson–Allen model of four competing species, which plays the role as a discrete-time Lotka–Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark–Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson–Allen dynamics can lead to chaos.

Keywords

• atkinson–allen model
• carrying simplex
• neimark–sacker bifurcation
• quasiperiod-doubling bifurcation
• chaotic attractor
• invasion