Abstract
We study the local dynamics and bifurcation analysis of a discrete-time modified Nicholson–Bailey model in the closed first quadrant R+2. It is proved that model has two boundary equilibria: O0,0,Aζ1−1/ζ2,0, and a unique positive equilibrium Brer/er−1,r under certain parametric conditions. We study the local dynamics along their topological types by imposing method of Linearization. It is proved that fold bifurcation occurs about the boundary equilibria: O0,0,Aζ1−1/ζ2,0. It is also proved that model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium Brer/er−1,r and meanwhile stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasi-periodic oscillations between host and parasitoid populations. Some simulations are presented to verify theoretical results. Finally, bifurcation diagrams and corresponding maximum Lyapunov exponents are presented for the under consideration model.
Highlights
/ − 1, and stable invariant closed curve appears
Where and represent the population size of the host and parasite in successive generations and + 1respectively. e parameter is the host nite rate of increase in the absence of parasites, is the biomass conversion constant and is the function de ning the fractional survival of hosts from parasitism. e simplest version of this model is that of Nicholson, and Nicholson and Bailey who explored in depth a model in which the proportion of hosts escaping parasitism is given by the zero term of the Poisson distribution [1,2,3]:
Further in 2015, Khan and Qureshi [5] have investigated the dynamics of the following modi ed Nicholson–Bailey model:
Summary
E rest of the paper is organized as follows: Section 2 deals with the study of existence of equilibria of the model (8). We study the existence of equilibria of the model (8) in R2+. Discrete-time model (8) has at least two boundary equilibria and the unique positive equilibrium point in R2+. (i) For all parametric values 1 and 2, model (8) has a unique equilibrium point: (0, 0);. / − 1 , has a unique positive equilibrium point of (8). / − 1 , of is Section deals with the study of Neimark–Sacker bifurcation of the model (8) about.
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