On the dynamics of evolutionary competition models

As exemplified by classic Lotka-Volterra theory, there are several canonical outcomes possible to a two species (interference) competitive interaction: coexistence, initial condition-dependent competitive exclusion of one species, or the global exclusion of one species. Evolutionary versions of Lotka-Volterra dynamics have been investigated in order to see the role that evolutionary adaptation can play in influencing the competitive outcome. For the most part, however, these investigations have been carried out by numerical simulations. In this paper, we provide some rigorous mathematical criteria concerning the outcome of a competition between two species x₁ and x₂ when evolution is taken into account. Motivated by two classic experiments in which the outcome of two competing beetle species was observed to change due to phenotypic or genetic changes in one species, we consider the case when only the species x₂ can evolutionarily adapt. Using methods from persistence theory, we obtain criteria under which x₂ will persist and criteria under which x₁ will persist.