Bifurcation and chaos in a host-parasitoid model with a lower bound for the host

Xijuan Liu,Yandong Chu,Yun Liu

doi:10.1186/s13662-018-1476-3

Xijuan Liu, Yandong Chu + Show 1 more

Open Access

https://doi.org/10.1186/s13662-018-1476-3

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Abstract

In this paper, a discrete-time biological model and its dynamical behaviors are studied in detail. The existence and stability of the equilibria of the model are qualitatively discussed. More precisely, the conditions for the existence of a flip bifurcation and a Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Numerical simulations are presented not only to validate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors. We also analyze the dynamic characteristics of the system in a two-dimensional parameter space. Numerical results indicate that we can more clearly and directly observe the chaotic phenomenon, period-doubling and period-adding, and the optimal parameters matching interval can also be found easily.

Highlights

A host-parasitoid model with a lower bound for the host is given by [1] as follows: ⎧ ⎨H(t + 1) = H (t) exp[ r(1– H(t) k )(H (t)–c)H(t)+m abP(t) 1+aH(t)
Our objective is to study systematically the existence conditions of the flip bifurcation and the Neimark-Sacker bifurcation, which will be derived using the center manifold theorem and the bifurcation theory
These findings prove that there are possibilities for periodic and chaotic motions to exist in the parameter space

Summary

Introduction

A host-parasitoid model with a lower bound for the host is given by [1] as follows:. where H(t) is the host population size in generation t, P(t) is the parasitoid population size in generation t, r is the intrinsic growth rate. k is the carrying capacity of the environment and c is the lower bound for the host. Our objective is to study systematically the existence conditions of the flip bifurcation and the Neimark-Sacker bifurcation, which will be derived using the center manifold theorem and the bifurcation theory (see [14, 15]) The effectiveness of these theoretical analyses is determined by bifurcation diagrams with one control parameter. These findings prove that there are possibilities for periodic and chaotic motions to exist in the parameter space. We have the following theorem on the stability of a positive fixed point of system (1.2). We will investigate the flip bifurcation of E1 if parameters vary in a small vicinity of FB and the NeimarkSacker bifurcation of E1 if parameters lie in a small scope of HB

Flip bifurcation

N-S bifurcation

Conclusion