Modified Holling Tanner diffusive and non-diffusive predator–prey models: The impact of prey refuge and fear effect

Abstract

The secondary consequences of predator species on prey species have substantial implications for population dynamics. A deeper comprehension of the dynamics between prey and predator can be achieved through the examination of indirect consequences. This work examines the dynamic behavior of a modified Holling-Tanner model. The interactions between the species are characterized by a functional response of the Beddington–DeAngelis type. Factors such as prey refuge, fear factor, disturbance intensity, and cross diffusion have been taken into account. The boundedness, feasibility of equilibrium points, their stability and restrictions for Hopf bifurcation of non-spatial model system are derived. The study explores the combined effects of prey refuge presence and fear factors on population dynamics. Furthermore, the investigation focuses on the stability of spatial self-diffusion and cross-diffusion model systems, as well as the specific conditions that lead to Turing instability. Ultimately, it has been shown that in the context of self-diffusion, a moderate level of fear promotes the survival of prey, whereas an excessive level of dread hinders the survival of prey. Concurrently, the mean density of prey exhibited a gradual decline as the refuge parameters increased. The spatial patterns of the population have also been investigated. As the mutual interference between prey populations intensifies, the spatial distribution of the prey population transitions from a clustered pattern to a combination of striped and clustered patterns, ultimately settling into a striped pattern. With the gradual growth of the half saturation constant, the prey population reached a state of uniform distribution. In the scenario of cross diffusion, when the prey is heavily impacted by the pursuit of predators, the fear effect, when appropriately used, did not have a significant impact on the survival of the prey. This work adds to the existing body of knowledge by revealing novel insights into the influence of indirect factors on the behavior of predator and prey populations.