Analysis of the impact of fear in the presence of additional food and prey refuge with nonlocal predator–prey models

There are many positive and negative factors present in the predator–prey interaction which affect the net growth of the species. Fear of predation is one such factor that creates psychological stress in a prey species, which causes a negative impact on their overall growth. This work considers a predator–prey model where the prey species faces a reduction in their growth out of fear, and the predator has an alternative food source that helps the prey to hide in a safer place. As an extension into the nonlocal spatio-temporal model, a nonlocal term is considered in the prey growth to incorporate a fear-effect range around their spatial location. Linear stability analysis helps to analyse the temporal model and produces a wide range of interesting results, including the presence of a certain amount of fear or even prey refuge, which helps in population coexistence. Furthermore, the numerical simulations of the local and nonlocal spatio-temporal models show different types of spatial–temporal patterns, such as Turing and non-Turing patterns. Nevertheless, an increase in fear level reduces the range of the Turing domain for the local model, whereas the opposite happens when the range of nonlocal interaction is increased.