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Competitive Exclusion and Coexistence in a Two-Strain Pathogen Model with Diffusion

We consider a two-strain pathogen model described by a system of reaction-diffusion equations. We define a basic reproduction number and show that when the model parameters are constant (spatially homogenous), if  then one strain will outcompete the other strain and drive it to extinction, but if then the disease-free equilibrium is globally attractive. When we assume that the transmission and recovery rates are heterogenous, then there are two possible outcomes under the condition : 1) Competitive exclusion where one strain dies out.  2) Coexistence between the two strains. Thus, spatial heterogeneity promotes coexistence.