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In this paper, we consider a diffusive predator–prey model with spatial memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a fractional-order two delays neural network with ring-hub structure is investigated. Firstly, the stability and the existence of Hopf bifurcation of proposed system are obtained by taking the sum of two delays as the bifurcation parameter. Furthermore, a parameters delay feedback controller is introduced to control successfully Hopf bifurcation. The novelty of this paper is that the characteristic equation corresponding to system has two time delays and the parameters depend on one of them. Selecting two time delays as the bifurcation parameters simultaneously, stability switching curves in (τ 1 , τ 2) plane and crossing direction are obtained. Sufficient criteria for the stability and the existence of Hopf bifurcation of controlled system are given. Ultimately, numerical simulation shows that parameters delay feedback controller can effectively control Hopf bifurcation of system. [ABSTRACT FROM AUTHOR]

This study is devoted to considering dynamic properties of an environmental protection expenditures model for protected areas (PAs) with two delays. It presents an improved understanding of relationship among tourism, environment, and capital stock in PAs and provides a sustainable strategy to keep the balance between them. The model considers the interactions among environmental resource E$$ E $$, visitors V$$ V $$, and capital stock K$$ K $$ in PAs. The tourism income is used partly to increase the capital stock and partly to protect the environment. In this paper, time delays τ1$$ {\tau}_1 $$ and τ2$$ {\tau}_2 $$ are introduced into the system. The technique of stability crossing curves for two delays is proposed to analyze the occurrence of Hopf bifurcation. Numerical simulations are performed for understanding the analytical results. The results show that under some parameter conditions, the share η$$ \eta $$ of total revenue generated by tourism user fees for environmental protection is related to delays τ1$$ {\tau}_1 $$ and τ2$$ {\tau}_2 $$. Moreover, the environment and the capital stock can be maintained persistently if time delay τ1$$ {\tau}_1 $$ is not too large. [ABSTRACT FROM AUTHOR]