Dynamics of a population with two equal dominated species

TL;DR

This paper analyzes the complex dynamics of a two-species population model, revealing fixed points, chaos, and invariant sets, with implications for understanding population stability and fluctuations.

Contribution

It introduces a novel one-dimensional piecewise-continuous model for two equal dominated species and characterizes its fixed points, periodic points, and chaotic behavior.

Findings

All periodic points are repelling.

The system exhibits chaos with non-negative Lyapunov exponents.

An invariant set attracts most trajectories over time.

Abstract

We consider a population with two equal dominated species, dynamics of which is defined by one-dimensional piecewise-continuous, two parametric functions. It is shown that for any non-zero parameters this function has two fixed points and several periodic points. We prove that all periodic (in particular fixed) points are repelling, and find an invariant set which asymptotically involves the trajectories of any initial point except fixed and periodic ones. We showed that the orbits are unstable and chaotic because Lyapunov exponent is non-negative. The limit sets analyzed by bifurcation diagrams. We give biological interpretations of our results.