How predator evolution to resist lethal or sublethal toxicant effects impact the dynamics of a discrete-time predator-prey system

We extend the predator-prey model developed by Ackleh et al. [Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics. J. Differ. Equ. Appl. 2019;25:1568-1603] to incorporate the evolution of a predator's resistance to toxicant effects. We consider three cases: (1) lethal effects, where the toxicant directly influences the predator's survival; (2) sublethal effects, where the toxicant impacts the predator's fecundity, and (3) mixed effects, where the toxicant impacts both vital rates. For the first two cases, we derive conditions for existence and stability of model equilibria and for system persistence. These cases are also analyzed numerically to further understand the system dynamics. Overall, we find that evolution of a predator to resist a toxicant may allow for predator survival when otherwise it would have faced extinction. However, evolution in response to lethal effects can generate multiple boundary equilibria, leading to alternative stable states. When this occurs, evolution in response to a toxicant may result in the extinction of the predator while, without evolution, the predator survives.