Hopf bifurcation and periodic solution of a delayed predator-prey-mutualist system

Advances in Difference Equations

Jianwen Jia¹  Liping Li¹ 

[1] School of Mathematical and Computer Science, Shanxi Normal University, Linfen, P.R. China

关键词: predator-prey-mutualist system;    digestion delay;    stability;    Hopf bifurcation;    periodic solution;   

DOI  :  10.1186/s13662-016-0907-2

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, we study a predator-prey-mutualist system with digestion delay. First, we calculate the threshold value of delay and prove that the positive equilibrium is locally asymptotically stable when the delay is less than the threshold value and the system undergoes a Hopf bifurcation at the positive equilibrium when the delay is equal to the threshold value. Second, by applying the normal form method and center manifold theorem, we investigate the properties of Hopf bifurcation, such as the direction and stability. Finally, some numerical simulations are carried out to verify the main theoretical conclusions.

【 授权许可】

CC BY