Asymptotic Behavior for a Competitive Model with Genetic Variation
Eduardo M. Munoz, James F. Selgrade
Source abstract
This paper is a study of a three-dimensional system of autonomous ordinary differential equations that models two competing populations with genetic variation in one population. The genetically varying population is diploid and diallelic. The per capita growth rate of each genotype is assumed to be a linear function of the population densities (Lotka–Volterra competition). On one allele frequency fixation plane the competition is assumed to exhibit stable coexistence and on the other, mutual exclusion. It is shown that the three-dimensional dynamics depends on the sign of a quadratic combination K of the competition parameters. Every solution orbit converges to an equilibrium solution regardless of the value of K. The asymptotic behavior for K positive and negative is analyzed, and the existence of heteroclinic orbits is established. The degenerate transition behavior when K equals zero is also discussed.
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