Complex Behaviour in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species

TL;DR

This paper models the complex dynamics of two interacting species using a coupled logistic map, revealing various behaviors from extinction to chaos as the mutual benefit parameter varies.

Contribution

Introduces a symmetrical cubic coupled logistic model for symbiotic species interaction, analyzing its diverse dynamical regimes and bifurcation structures.

Findings

Species extinction at low mutual benefit

Stable synchronization or periodic oscillations at moderate benefit

Chaotic dynamics or extinction at high benefit

Abstract

A symmetrical cubic discrete coupled logistic equation is proposed to model the symbiotic interaction of two isolated species. The coupling depends on the population size of both species and on a positive constant $\lambda$, named the mutual benefit. Different dynamical regimes are obtained when the mutual benefit is modified. For small $\lambda$, the species become extinct. For increasing $\lambda$, the system stabilizes in a synchronized state or oscillates in a 2 periodic orbit. For the greatest permitted values of $\lambda$, the dynamics evolves into a quasiperiodic, into a chaotic scenario or into extinction. The basins for these regimes are visualized as coloured figures on the plane. These patterns suffer different change as consequence of basins' bifurcations. The use of the critical curves let us to determine the influence of the zones with different number of first rank preimages in those bifurcation mechanisms.