Stable periodic solutions for two species, density dependent coevolution
This paper studies a four dimensional system of time-autonomous ordinary differential equations which models the interaction of two diploid, diallelic populations with overlapping generations. The variables are two population densities and an allele frequency in each of the populations. For single species models, the existence of periodic solutions requires that the genotype fitness functions be both frequency and density dependent. But, for two species exhibiting a predator-prey interaction, two examples are presented where there exists asymptotically stable cycles with fitness functions only density dependent. In the first example, the Hopf bifurcation theorem is used on a two parameter, polynomial vector field. The second example has a Michaelis- Menten or Holling term for the interaction between predator and prey; and, for this example, the existence and uniqueness of limit cycles for a wide range of parameter values has been established in the literature.
four dimensional system of time-autonomous ordinary differential equations
interaction of two diploid, diallelic populations
scientific article; zbMATH DE number 3891153