BIFURCATION ANALYSIS IN A PREDATOR-PREY MODEL WITH AN ALLEE EFFECT AND A DELAYED MECHANISM*

doi:10.1007/s10473-023-0324-z

Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (3): 1415-1438.doi: 10.1007/s10473-023-0324-z

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BIFURCATION ANALYSIS IN A PREDATOR-PREY MODEL WITH AN ALLEE EFFECT AND A DELAYED MECHANISM^*

Danyang LI¹, Hua LIU^1,†, Haotian ZHANG¹, Ming MA¹, Yong YE², Yumei WEI³

1. School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730000, China;
2. School of Science, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China;
3. Experimental Teaching Department, Northwest Minzu University, Lanzhou 730030, China;

Received:2022-02-11 Revised:2022-08-25 Online:2023-06-25 Published:2023-06-06
Contact: ^† Hua LIU, E-mail: 7783360@qq.com
About author:Danyang LI, E-mail: 1964757049@qq.com; Haotian ZHANG, E-mail: 845420039@qq.com; Ming MA, E-mail: 1020841601@qq.com; Yong YE, E-mail:13339239813@163.com; Yumei WEI, E-mail: 649118046@qq.com
Supported by:
Gansu Science and Technology Fund (20JR5RA512), the Research Fund for Humanities and Social Sciences of the Ministry of Education (20XJAZH006), the Fundamental Research Funds for the Central Universities (31920220066), the Gansu Provincial Education Department: Outstanding Postgraduate Innovation Star Project (2023CXZX-196), the Leading Talents Project of State Ethnic Affairs Commission of China and the Innovation Team of Intelligent Computing and Dynamical System Analysis and Application of Northwest Minzu University.

Abstract

Abstract: Regarding delay-induced predator-prey models, much research has been done on delayed destabilization, but whether delays are stabilizing or destabilizing is a subtle issue. In this study, we investigate predator-prey dynamics affected by both delays and the Allee effect. We analyze the consequences of delays in different feedback mechanisms. The existence of a Hopf bifurcation is studied, and we calculate the value of the delay that leads to the Hopf bifurcation. Furthermore, applying the normal form theory and a center manifold theorem, we consider the direction and stability of the Hopf bifurcation. Finally, we present numerical experiments that validate our theoretical analysis. Interestingly, depending on the chosen delay mechanism, we find that delays are not necessarily destabilizing. The Allee effect generally increases the stability of the equilibrium, and when the Allee effect involves a delay term, the stabilization effect is more pronounced.

Key words: delays, Allee effect, Hopf bifurcation, stability

Danyang LI, Hua LIU, Haotian ZHANG, Ming MA, Yong YE, Yumei WEI. BIFURCATION ANALYSIS IN A PREDATOR-PREY MODEL WITH AN ALLEE EFFECT AND A DELAYED MECHANISM^*[J].Acta mathematica scientia,Series B, 2023, 43(3): 1415-1438.

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BIFURCATION ANALYSIS IN A PREDATOR-PREY MODEL WITH AN ALLEE EFFECT AND A DELAYED MECHANISM^*

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