Evolutionary cycling in predator-prey interactions: Population dynamics and the Red Queen

This paper describes the coevolution of phenotypes in a community comprising a population of predators and of prey. It is shown that evolutionary cycling is a likely outcome of the process. The dynamical systems on which this description is based are constructed from microscopic stochastic birth and death events, together with a process of random mutation. Births and deaths are caused in part by phenotype-dependent interactions between predator and prey individuals and therefore generate natural selection. Three outcomes of evolution are demonstrated. A community may evolve to a state at which the predator becomes extinct, or to one at which the species coexist with constant phenotypic values, or the species may coexist with cyclic changes in phenotypic values. The last outcome corresponds to a Red Queen dynamic, in which the selection pressures arising from the predator-prey interaction cause the species to evolve without ever reaching an equilibrium phenotypic state. The Red Queen dynamic requires an intermediate harvesting efficiency of the prey by the predator and sufficiently high evolutionary rate constant of the prey, and is robust when the model is made stochastic and phenotypically polymorphic. A cyclic outcome lies outside the contemporary focus on evolutionary equilibria, and argues for an extension to a dynamical framework for describing the asymptotic states of evolution