A model of competing species that exhibits zip bifurcation

Abstract

The purpose of this paper is to present a concrete model of competing population species that exhibits a phenomenon called zip bifurcation. The Zip Bifurcation was introduced by Farkas in 1984 for a three dimensional ODE prey-predator system describing a chemostat. We will study a three dimensional system of ordinary differential equations that model the competition of two predators species for one single prey species. The system is based on concrete trigonometric functions modeling the growth rate of the prey and the functional response of the predator. The model exhibits different kinds of behavior and shows examples of the so called "competitive exclusion principle," and the competition of one "r-strategist" and one "K-strategist." Additionally, in order to illustrate the zip bifurcation, we will present some numerical simulations for our model.

Zip and velcro bifurcations in competition models in ecology and economics

“…More recent work by Ferreira and Rao deals with zipbifurcation in a predator-prey model with diffusion [17], and in systems involving discrete delay [18,19] and crossdiffusion [20]. Zip bifurcations are also discussed by Escobar-Callejas et al [21], and by Echeverri et al [22].…”