Analysis of Coexistence and Extinction in a Two-Species Competition Model

Abstract

We study a competition model of two competing species in population biology having exponential and rational growth functions described by Alexander et al. [1992]. They observed that, for some choice of parameters, the competition model has a chaotic attractor [Formula: see text] for which the basin of attraction is riddled. Here, we give a detailed analysis to illustrate what happens when the normal parameter in this model changes. In fact, by varying the normal parameter, we discuss how the geometry of the basin of attraction of [Formula: see text], the region of coexistence or extinction, changes and illustrate the transitions between the set [Formula: see text] being an asymptotically stable attractor (extinction of rational species), a locally riddled basin attractor and a normally repelling chaotic saddle (extinction of exponential species). Additionally, we show that the riddling and the blowout bifurcation occur. Numerical simulations are presented graphically to confirm the validity of our results. In particular, we verify the occurrence of synchronization for some values of parameters. Finally, we apply the uncertainty exponent and the stability index to quantify the degree of riddling basin. Our observation indicates that the stability index is positive for Lebesgue for almost all points whenever the riddling occurs.