Dynamics of a diffusive predator-prey system with fear effect in advective environments

We explore a diffusive predator-prey system that incorporates the fear effect in advective environments. Firstly, we analyze the eigenvalue problem and the adjoint operator, considering Constant-Flux and Dirichlet (CF/D) boundary conditions, as well as Free-Flow (FF) boundary conditions. Our investigation focuses on determining the direction and stability of spatial Hopf bifurcation, with the generation delay $\tau$ serving as the bifurcation parameter. Additionally, we examine the influence of both linear and Holling-II functional responses on the dynamics of the model. Through these analyses, we aim to gain a better understanding of the intricate relationship between advection, predation, and prey response in this system

Existence and stability of periodic solutions for a delayed prey–predator model with diffusion effects

Existence and stability of spatially periodic solutions for a delay prey-predator diffusion system are concerned in this work. We obtain that the system can generate the spatially nonhomogeneous periodic solutions when the diffusive rates are suitably small. This result demonstrates that the diffusion plays an important role on deriving the complex spatiotemporal dynamics. Meanwhile, the stability of the spatially periodic solutions is also studied. Finally, in order to verify our theoretical results, some numerical simulations are also included

Stability and Hopf bifurcation of a diffusive Gompertz population model with nonlocal delay effect

In this paper, we investigate the dynamics of a diffusive Gompertz population model with nonlocal delay effect and Dirichlet boundary condition. The stability of the positive spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcations with the change of the time delay are discussed by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. Then we derive the stability and bifurcation direction of Hopf bifurcating periodic orbits by using the normal form theory and the center manifold reduction. Finally, we give some numerical simulations

Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented