Bifurcation and chaos analysis for a discrete ecological developmental systems.

Xiao-Wei Jiang,Xian-He Zhang,Huaicheng Yan,Ming Chi,Chaoyang Chen

doi:10.1007/s11071-021-06474-4

Abstract

This work concentrates on the dynamic analysis including bifurcation and chaos of a discrete ecological developmental systems. Specifically, it is a prey–predator–scavenger (PPS) system, which is derived by Euler discretization method. By choosing the step size h as a bifurcation parameter, we determine the set consists of all system’s parameters, in which the system can undergo flip bifurcation (FB) and Neimark–Sacker bifurcation (NSB). The theoretical results are verified by some numerical simulations. It is shown that the discrete systems exhibit more interesting behaviors, including the chaotic sets, quasi-periodic orbits, and the cascade of period-doubling bifurcation in orbits of periods 2, 4, 8, 16. Finally, corresponding to the two bifurcation behaviors discussed, the maximum Lyapunov exponent is numerically calculated, which further verifies the rich dynamic characteristics of the discrete system.

Highlights

In the last decades, the study on revealing the relationship between complex dynamical behavior, such as bifurcation and chaos, and systems’ parameters has never been stopped, especially for ecological developmental system
Ecological developmental system refers to the interaction, mutual restriction, and continuous evolution between creatures and environment in a certain space of nature, so as to achieve a dynamic balance and relatively stable integrated whole
This paper has analyzed the dynamics of the discrete PPS model which are discussed

Summary

Introduction

The study on revealing the relationship between complex dynamical behavior, such as bifurcation and chaos, and systems’ parameters has never been stopped, especially for ecological developmental system. It is not difficult to understand that food and energy are of great significance for all species, such as disease spread model including SARSCoV-2 [13] It is an interesting research direction by combining the multi-species food web systems with the harvesting [14,15,16]. In this work, by considering the effects of harvesting and toxicant on prey, predator, and scavenger, we propose and discuss a PPS food web model. The point of view of experiment or calculation Another reason is that different from the continuous case, the discrete model can produce more abundant dynamical behaviors [18,19].

Existence and stability of positive equilibrium

Bifurcation behavior analysis

Illustrative example

Conclusions