Frequency‐dependent evolution in a predator–prey system

Abstract

AbstractWe present a discrete‐time predator–prey model in which the prey population is assumed to evolve in response to a toxicant. We incorporate frequency‐dependent selection into the prey evolution, assuming that an individual's susceptibility to predation depends on both the trait it possesses and the traits of others in the population. When frequency‐dependent selection is symmetric, we show that the trait equation is unable to track changes in the fitness landscape, that is, the fitness landscape may change while the trait continues to evolve to the same trait value. As a result, evolution may produce unfit prey populations. Meanwhile, we show that asymmetric frequency‐dependence may have a destabilizing effect on the system, resulting in a closed invariant curve via a Neimark–Sacker bifurcation.Recommendations for Resource Managers When considering the impact that a toxicant or other disturbance may have on a system, it is important to consider the impact duration of the toxicant and whether any impacted species have the potential to evolve to reduce the impact of the toxicant. Evolution of toxicant resistance may change system dynamics in a number of ways, such as enabling species persistence at higher toxicant levels. However, here we show that frequency‐dependent evolution has the potential to destabilize system dynamics, resulting in stable closed invariant curves. Destabilization of system dynamics may have important consequences for species persistence, as periods of low abundance may increase the possibility of species extinction.