Local Stability Analysis and Bifurcations of a Discrete-Time Host-Parasitoid Model

Tahmineh Azizi

doi:10.4236/ijmnta.2020.92002

Abstract

In this paper, we examine a discrete-time Host-Parasitoid model which is a non-dimensionalized Nicholson and Bailey model. Phase portraits are drawn for different ranges of parameters and display the complicated dynamics of this system. We conduct the bifurcation analysis with respect to intrinsic growth rate r and searching efficiency a. Many forms of complex dynamics such as chaos, periodic windows are observed. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for b≠a where a,b are searching efficiency. We study stable and unstable manifolds for different equilibrium points and coexistence of different attractors for this non-dimensionalize system. Without the parasitoid, the host population follows the dynamics of the Ricker model.

Highlights

More efficient computational models for numerical simulation can be created by discrete-time models and these models can present much more plentiful dynamic behaviors when they would be compared with the same type of continuous-time model [1] [2] [3]
We studied a discrete-time host-parasitoid model which is non-dimensionalized Nicholson-Bailey model
The model shows rich features and more complicated dynamics compared to the general Nicholson-Bailey model

Summary

Introduction

More efficient computational models for numerical simulation can be created by discrete-time models and these models can present much more plentiful dynamic behaviors when they would be compared with the same type of continuous-time model [1] [2] [3]. We observe various types of dynamics having stable fixed point, chaotic bands, periodic windows for the non-dimensionalized model. For some values of the parameters, this model has an asymptotically stable equilibrium with complex eigenvalues. When the parameter a increases, this asymptotically stable equilibrium loses its stability and rise to an invariant closed curve. The proof is straightforward using the fact that the positive fixed point of system is asymptotically stable if and only if tr ( J ) < 1+ det ( J ) < 2. For the case a = b , Taylor expansion of plant equation for r = 2 on the invariant manifold Pt = 0 has the condition of the period-doubling bifurcation correspond to Ricker dynamics. For a = 40 , when the parameter r is changed, the asymptotically stable fixed point loses its stability and to be an attracting closed invariant curve gradually. We study the different types of bifurcation such as Neimark-Sacker, saddle-node and period-doubling bifurcation

Solutions and Local Asymptotic Stability

Bifurcations

Result

Non-Unique Dynamics and Bistability

Conclusion