Impact of Fear on Searching Efficiency of Prey: A Prey–Predator Model with Weak Allee Effect

Abstract

Reduced population growth at low density has important implications for conservation, colonization success, and wildlife management. In this context, the Allee effect, i.e. the positive relationship between per capita growth rate and biomass of small population density, is a crucial biological phenomenon since it is directly related to population extinction. The present paper deals with a two-species interacting model with a predator–prey relationship, where the prey population experiences the mate-finding Allee effect caused by the predator. We assume that the searching efficiency of prey individuals decreases linearly with predator density due to predation fear and investigate how predation intensity affects predator–prey dynamics. Moreover, we consider the Monod–Haldane type functional response for predator–prey interactions, which shows group defense of prey against the predator. We provide detailed mathematical analyses, including the positivity and boundedness of solutions, all biologically feasible equilibria, and their local and global stabilities. From our detailed mathematical analyses, we observe that when the carrying capacity of prey is low, at most one interior equilibrium exists, and system dynamics is simple compared to the case with high carrying capacity in which multiple coexistence equilibria may exist. We discuss three codimension-one bifurcations mathematically, e.g. Hopf bifurcation, transcritical bifurcation, saddle-node bifurcation. We notice bistability in the system when there are two interior equilibria with high carrying capacity. However, a unique attractor exists when there is only a single interior equilibrium and both populations persist. We perform extensive numerical simulations by varying two parameters simultaneously and explore how the system dynamics become complex when carrying capacity is high compared to low carrying capacity. Moreover, we discuss other important biological phenomena, e.g. the paradox of enrichment, bubbling phenomenon, etc.