Bifurcation analysis of the prey-predator models incorporating herd behaviours

Abstract

In ecology, herd behaviour corresponds to the situation how a group of animals of the same species interacts in the time of danger. A herd mechanism offers more protection for preys against predators. But this mechanism can also provide defence from predators. To address this problem, we consider two ecological models of prey-predator interactions with the presence of different herd behaviours. While the first system represents a simple prey-predator model incorporating herd behaviour in prey populations only, the second one considers the herd behaviour in both prey and predator populations. We examine how different strategies of herd behaviours can determine the outcomes of species interactions. To do this, we employ stability analysis and numerical simulations to illustrate different dynamical behaviours of the two models, e.g. coexistence and extinction steady states and also limit cycles. By using bifurcation analysis, we observe the occurrences of Hopf bifurcation, transcritical bifurcation, limit point and limit point of cycles. These bifurcations can shape the overall dynamics of these two ecological systems.