Prey herd behavior modeled by a generic non-differentiable functional response

Abstract

Over the past decade, many works have studied an antipredator behavior (APB) namedprey herd behavior. Analyzes have been conducted by modifying the classical predator consumption rate to be dependent only on the prey population size assuming thesquare root functional response. This work focuses analyzing the dynamics of a Gause-type predator-prey model considering that social behavior of prey. However, we model this phenomenon using a Holling type II non-differentiable rational functional response, which is more general than that mentioned above. The studied model exhibits richer dynamics than those with differentiable functional responses, and one the main consequences of including this type of function is the existence of initial values for which the extinction of prey occurs within a finite time for all parameter conditions, which is a direct consequence of the non-uniqueness of the solutions over the vertical axes and of the existence of a separatrix curve dividing the phase plane. A discussion on what represents a well-posed problem from both the mathematical and the ecological points of view is presented. Additionally, the differences in other social behaviors of the prey are also established. Numerical simulations are provided to validate the mathematical results.