# American Institute of Mathematical Sciences

2016, 13(1): 1-18. doi: 10.3934/mbe.2016.13.1

## Competitive exclusion and coexistence in a two-strain pathogen model with diffusion

 1 Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010, United States 2 Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, United States

Received  December 2014 Revised  August 2015 Published  October 2015

We consider a two-strain pathogen model described by a system of reaction-diffusion equations. We define a basic reproduction number $R_0$ and show that when the model parameters are constant (spatially homogeneous), if $R_0 >1$ then one strain will outcompete the other strain and drive it to extinction, but if $R_0 \le 1$ then the disease-free equilibrium is globally attractive. When we assume that the diffusion rates are equal while the transmission and recovery rates are heterogeneous, then there are two possible outcomes under the condition $R_0 >1$: 1) Competitive exclusion where one strain dies out. 2) Coexistence between the two strains. Thus, spatial heterogeneity promotes coexistence.
Citation: Azmy S. Ackleh, Keng Deng, Yixiang Wu. Competitive exclusion and coexistence in a two-strain pathogen model with diffusion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 1-18. doi: 10.3934/mbe.2016.13.1
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