Dynamics of a population with two equal dominated species

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.