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We consider general models of coupled reaction-diffusion systems for interacting variants of a species. When the total population becomes large with intensive competition, we prove that the frequency (i.e., proportion) of a variant can be approached by the solution of a single reaction-diffusion equation, through a singular limit method and a relative compactness argument. Applying this result, we retrieve the classical bistable equation for Wolbachia's spread into an arthropod population from a system modeling interaction between infected and uninfected individuals.

Reduction to a Single Closed Equation for 2-by-2 Reaction-Diffusion Systems of Lotka--Volterra Type