Stability and Hopf bifurcation in a resources-based and carrying-capacity-driven diffusion model

The dynamics of a delay system with logistic growth is investigated. Considering the various diffusion strategies, the resources-based and carrying-capacity-driven diffusion is introduced into the model. Under certain conditions, we discuss the existence and expression of the spatial inhomogeneous steady-state solution of the system. For the case without delay, the existence, permanence and stability of the solutions are studied, especially the stability of spatially inhomogeneous steady-state solutions. When the time delay increases, the Hopf bifurcation caused by the delay is explored, and the direction of the bifurcation and the stability of the bifurcating spatially inhomogeneous periodic solution are further studied. The discussion in this paper emphasizes the importance of resources and spatial carrying capacity for species survival. The research results show that species will spread to places with abundant resources and strong spatial carrying capacity, which is conducive to species survival, however, excessive diffusion will lead to population extinction.