Global Hopf bifurcation and positive periodic solutions of a delayed diffusive predator–prey model with weak Allee effect for predator

• Published in: Advances in continuous and discrete models Vol. 2025; no. 1; p. 19
• Main Authors: Zhang, Xiaohui, Xu, Xiaofeng, Liu, Ming
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 27.01.2025; Springer Nature B.V
• Subjects: Analysis; Difference and Functional Equations; Differential equations; Diffusion models; Functional Analysis; Hopf bifurcation; Investigations; Liapunov functions; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Population decline; Predator-prey simulation; Predators; Steady state; Global Hopf bifurcation; Predator–prey; Diffusion; Weak Allee effect; Delay
• ISSN: 2731-4235; 1687-1839; 2731-4235; 1687-1847
• DOI: 10.1186/s13662-025-03883-2

In this paper, we investigate a delayed diffusive predator–prey model with weak Allee effect for predator. First, we discuss the existence and uniqueness of positive steady state and the local Hopf bifurcation. Next, we obtain the permanence and uniform boundedness of positive periodic solutions of the delayed reaction–diffusion system. The existence range of positive periodic solutions is further compressed by using iterative approach. For the system without delay, we prove the global attractiveness of unique positive steady state by using Lyapunov function, which ensures that the periods of positive periodic solutions are uniformly bounded. Then we obtain the global Hopf bifurcation and extended existence of positive periodic solutions by using the global Hopf bifurcation theorem of partial functional differential equation. Finally, the validity of the conclusion is verified through numerical simulation.