Stability, bifurcation, and chaos control of a novel discrete-time model involving Allee effect and cannibalism

Advances in Difference Equations

article

Shabbir, Muhammad Sajjad¹  Din, Qamar²  Ahmad, Khalil¹  Tassaddiq, Asifa³  Soori, Atif Hassan¹  Khan, Muhammad Asif² 

[1]Department of Mathematics, Air University

[2]Department of Mathematics, University of Poonch Rawalakot

[3]College of Computer and Information Sciences, Majmaah University

关键词: Allee effect;    Cannibalism;    Chaos control;    Neimark–Sacker bifurcation;    Period-doubling bifurcation;    Prey–predator system;    Stability;    Transcritical bifurcation;   

DOI  :  10.1186/s13662-020-02838-z

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

This paper is related to some dynamical aspects of a class of predator–prey interactions incorporating cannibalism and Allee effects for non-overlapping generations. Cannibalism has been frequently observed in natural populations, and it has an ability to alter the functional response concerning prey–predator interactions. On the other hand, from dynamical point of view cannibalism is considered as a procedure of stabilization or destabilization within predator–prey models. Taking into account the cannibalism in prey population and with addition of Allee effects, a new discrete-time system is proposed and studied in this paper. Moreover, existence of fixed points and their local dynamics are carried out. It is verified that the proposed model undergoes transcritical bifurcation about its trivial fixed point and period-doubling bifurcation around its boundary fixed point. Furthermore, it is also proved that the proposed system undergoes both period-doubling and Neimark–Sacker bifurcations (NSB) around its interior fixed point. Our study demonstrates that outbreaks of periodic nature may appear due to implementation of cannibalism in prey population, and these periodic oscillations are limited to prey density only without leaving an influence on predation. To restrain this periodic disturbance in prey population density, and other fluctuating and bifurcating behaviors of the model, various chaos control methods are applied. At the end, numerical simulations are presented to illustrate the effectiveness of our theoretical findings.

【 授权许可】

CC BY