On Periodic Regimes Triggered by Herd Behaviour in Population Systems

Abstract

Different response functions have been proposed to model predator–prey interactions. In particular, Lotka–Volterra models work with the mass action law, resulting in a Holling type I response function. More recently, authors have proposed a term proportional to the square root of the prey population, in order to model herd behaviour and group defense. We present a model in which the response function is defined piecewisely: below a certain threshold (populations too small to display group defense) we have a Lotka–Volterra type interaction and above it we have herd behaviour type response. The model is analysed using standard techniques and also complementary techniques designed specifically for piecewise systems. Both stability of equilibria and bifurcations are investigated. In particular, we were able to prove that both supercritical and subcritical Hopf bifurcations occur, one of those leading to the existence of two limit cycles (one stable and the other unstable). We conclude that the proposed model displays novel behaviour in comparison to previous models and serves as a coherent tool to model predator–prey interactions.