Lyapunov functions for Lotka-Volterra predator-prey models with optimal foraging behavior

The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka-Volterra predator-prey systems with optimal foraging behavior. We examine a predator-prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka-Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations.