Persistence and stability analysis of discrete-time predator–prey models: a study of population and evolutionary dynamics

Prolonged exposure to disturbances, such as toxicants, has the potential to result in evolutionary responses in a population. These evolutionary responses may impact not just that species but also other species with which the population interacts. In this paper, we develop a discrete-time predator–prey model and establish conditions for the existence and stability of equilibria, as well as conditions for the persistence of the prey and predator populations. We then extend this model to an evolutionary model which couples the population dynamics with the dynamics of an evolving phenotypic trait that provides a measure for the level of toxicant resistance developed by the prey. We study the persistence dynamics of this model and establish conditions for when the prey evolves toxicant resistance. We show that the evolution of toxicant resistance may allow both the predator and prey to persist when, without the evolution, both may go extinct.