Non-smooth Dynamics Emerging from Predator-driven Discontinuous Prey Dispersal

In ecology, the refuge protection of the prey plays a significant role in the dynamics of the interactions between prey and predator. In this paper, we investigate the dynamics of a non-smooth prey-predator mathematical model characterized by density-dependent intermittent refuge protection of the prey. The model assumes the population density of the predator as an index for the prey to decide on when to avail or discontinue refuge protection, representing the level of apprehension of the prey by the predators. We apply Filippov's regularization approach to study the model and obtain the sliding segment of the system. We obtain the criterion for the existence of the regular or virtual equilibria, boundary equilibrium, tangent points, and pseudo-equilibria of the Filippov system. The conditions for the visibility (or invisibility) of the tangent points are derived. We investigate the regular or virtual equilibrium bifurcation, boundary-node bifurcation and pseudo-saddle-node bifurcation. Further, we examine the effects of dispersal delay on the Filippov system associated with prey vigilance in identifying the predator population density. We observe that the hysteresis in the Filippov system produces stable limit cycles around the predator population density threshold in some bounded region in the phase plane. Moreover, we find that the level of apprehension and vigilance of the prey play a significant role in their refuge-dispersion dynamics.

Predators risk injury too: the evolution of derring-do in a predator–prey foraging game

Derring-do is how aggressive a predator is in stalking and capturing prey. We model predator–prey interactions in which prey adjust vigilance behavior to mitigate risk of predation and predators their derring-do to manage risk of injury from capturing prey. High derring-do increases a predator's likelihood of capturing prey, but at higher risk of injury to itself. For fixed predator derring-do, prey increase vigilance in response to predator abundance, predator lethality, and predator encounter probability with prey and decrease vigilance with their own feeding rate; there is a humped-shaped relationship between prey vigilance and effectiveness of vigilance. For fixed prey vigilance, predators increase derring-do with the abundance of prey and predator lethality and decrease it with benefit of vigilance to prey and level of prey vigilance. When both prey and predator are behaviorally flexible, a predator–prey foraging game ensues whose solution represents an evolutionarily stable strategy (ESS). At the ESS, prey provide themselves with a public good as their vigilance causes predators to decrease derring-do. Conversely, predators have negative indirect effects on themselves as their derring-do causes prey to be more vigilant. These behavioral feedbacks create negative intra-specific interaction coefficients. Increasing the population size of prey (or predators) now has a direct negative effect on the prey (or predators). Both effects help stabilize predator–prey dynamics. Besides highlighting a common way by which predators may experience a food-safety tradeoff via dangerous prey, the model suggests why natural selection favors even small defensive measures by prey and hulky predators.