Local dynamical properties and supercritical N-S bifurcation of a discrete-time host-parasitoid model with Allee effect

Abstract

We explore thelocal dynamical properties and supercritical N-S bifurcation of the following Beddington model with Allee effect in R2+: xt+1= xt exp(r(1- xt)- yt), yt+1=m xt (1-exp(- yt)) y t/(B+ yt), where xt (respectively yt) denotes densities of host (respectively parasitoid) at time t, r and m respectively denotes number of eggs laid by host and parasitoid which survive through larvae, pupae, and adult stages, and B is constant. More specifically, we explored that model has three equilibria namely the trivial, boundary and positive equilibrium point. We studied the local dynamics along with topological classification about equilibria of the under consideration model. We also explored the existence of bifurcation about equilibria of the model. It is proved about boundary equilibrium point parasitoidgoes to extinction whilehost population undergoes a flip bifurcation to chaos by taking r as bifurcation parameter. It is explored that aboutpositive equilibrium point, model undergoes N-S bifurcation and in meantime invariant closed curve appears. In the perspective of the biology, these curves correspond to periodic or quasi-periodic oscillations between host and parasitoid populations. Finally theoretical results are verified numerically.

Keywords

• beddington model
• stability and bifurcation
• allee effect
• numerical simulation