Dynamics of fear on a modified Leslie–Gower predator–prey system incorporating variable prey refuge

In the present article, we study the effect of anti-predator strategies of prey due to the induced fear by the predation on a modified Leslie-Gower predator-prey model incorporating variable prey refuge, where the predation rate of predators follows the Beddington-DeAngelis functional response. One introduces the refuge of prey, depending on the interaction of both species. We systematically demonstrate the various points of feasible equilibria with their stability, the existence of different bifurcations, and the conditions of the paradox of enrichment as well. We thoroughly investigate the impact of fear on the model and derive that an increased rate of fear can decrease the density of both species, leading to prey population extinction accordingly. Additionally, we explore the influence of prey refuge under varying levels of fear, observing that it can aid in the coexistence of both populations for moderate fear levels but may not prevent predator extinction for high fear levels. Finally, through numerical simulations, we have validated our theoretical findings. Our research provides insights into the role of safety zones for prey and the effects of fear on the delicate balance between predator and prey populations, offering implications that are significant for the management and conservation of ecosystems.