The aim of this paper was to explore the impact of fear on the dynamics of prey and predator species. Specifically, we investigated a reaction-diffusion predator-prey model in which the prey was subjected to Beddington-DeAngelis type and the predator was subjected to modified Leslie-Gower type. First, we analyzed the existence and stability of equilibria of the nonspatial model, and further investigated the global stability and Hopf bifurcation at the unique positive equilibrium point. For the spatial model, we studied the local and global stability of the unique constant positive steady state solution and captured the existence of Turing instability, which depended on the diffusion rate ratio between the two species. Then, we demonstrated the existence of Hopf bifurcations and discussed the direction and stability of spatially homogeneous and inhomogeneous periodic solutions. Finally, the impact of fear and spatial diffusion on the dynamics of populations were probed by numerical simulations. Results revealed that spatial diffusion and fear both broaden the dynamical properties of this model, facilitating the emergence of periodic solutions and the formation of biodiversity.

In this paper, the permanence and extinction of a class of non-autonomous competitive population systems existing in an impulsively polluted environment are investigated. By establishing a time-varying differential equation model that couples population competition, pollutant dynamics, and discrete impulsive disturbances, we employ the comparison theorem of impulsive differential equations and analytical methods to derive sufficient criteria guaranteeing population persistence and global extinction. Furthermore, by constructing a Lyapunov function, we obtain conditions for the global attractivity of the system solutions. The theoretical analysis indicates that the ultimate fate of the population is jointly determined by the time-varying growth rates, competition intensities, period of impulsive disturbances, and pollutant toxicity levels. This study provides a quantitative theoretical basis for assessing the ecological risks of sporadic pollution events in competitive communities.