A Coevolutionary Predator-Prey Model with Quantitative Characters

A new model for coevolution in generalized predator-prey systems is presented by incorporating quantitative characters relevant to predation in both prey and predator. Malthusian fitnesses are derived from ecological models, and they include interspecific frequency and density dependence. Both prey and predator characters are under stabilizing selection even without predation, and predation adds an additional linear selection component to both characters. The nonlinear system of differential equations is studied analytically by using local linearization near the equilibrium points. Parameters related to intrinsic growth and death rates and stabilizing selection determine whether there are zero, one, or two equilibria. Additive genetic variances do not have an effect on the equilibrium points, but genetic variability is crucial for determining their stability. Analysis of the linearized model shows that at most one equilibrium can be stable, and stability is achieved when additive genetic variance is high enough in both the prey and predator populations. The stability properties are illustrated by numerical examples of the full dynamics of the original nonlinear model.

[1]  R. Lande NATURAL SELECTION AND RANDOM GENETIC DRIFT IN PHENOTYPIC EVOLUTION , 1976, Evolution; international journal of organic evolution.

[2]  S. Levin,et al.  A Mathematical Model of Coevolving Populations , 1977, The American Naturalist.

[3]  R M May,et al.  Epidemiology and genetics in the coevolution of parasites and hosts , 1983, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[4]  R. A. Fisher,et al.  The Genetical Theory of Natural Selection , 1931 .

[5]  A. Lomnicki Evolution of the Herbivore-Plant, Predator-Prey, and Parasite-Host Systems: A Theoretical Model , 1974, The American Naturalist.

[6]  Sewall Wright,et al.  The analysis of variance and the correlations between relatives with respect to deviations from an optimum , 1935, Journal of Genetics.

[7]  C. M. Pease,et al.  ON THE EVOLUTIONARY REVERSAL OF COMPETITIVE DOMINANCE , 1984, Evolution; international journal of organic evolution.

[8]  R. Punnett,et al.  The Genetical Theory of Natural Selection , 1930, Nature.

[9]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[10]  W M Schaffer,et al.  Homage to the red queen. I. Coevolution of predators and their victims. , 1978, Theoretical population biology.

[11]  M. Kimura A stochastic model concerning the maintenance of genetic variability in quantitative characters. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[12]  W. E. Ritter AS TO THE CAUSES OF EVOLUTION. , 1923, Science.

[13]  David Pimentel,et al.  Animal Population Regulation by the Genetic Feed-Back Mechanism , 1961, The American Naturalist.

[14]  J. Endler Natural selection in the wild , 1987 .

[15]  D. Falconer,et al.  Introduction to Quantitative Genetics. , 1961 .

[16]  M. Kimura,et al.  An introduction to population genetics theory , 1971 .

[17]  C. Mode A GENERALIZED MODEL OF A HOST-PATHOGEN SYSTEM , 1961 .

[18]  S. Levin A Mathematical Analysis of the Genetic Feedback Mechanism , 1972, The American Naturalist.