# Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates

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## Abstract

We study the competition of two species for a single resource in a chemostat. In the simplest space-homogeneous situation, it is known that only one species survives, namely the best competitor. In order to exhibit *coexistence* phenomena, where the two competitors are able to survive, we consider a space dependent situation: we assume that the two species and the resource follow a diffusion process in space, on top of the competition process. Besides, and in order to consider the most general case, we assume each population is associated with a *distinct* diffusion constant. This is a key difficulty in our analysis: the specific (and classical) case where all diffusion constants are equal, leads to a particular conservation law, which in turn allows to eliminate the resource in the equations, a fact that considerably simplifies the analysis and the qualitative phenomena. Using the global bifurcation theory, we prove that the underlying 2-species, stationary, diffusive, chemostat-like model, does possess *coexistence solutions*, where both species survive. On top of that, we identify the domain, in the space of the relevant bifurcation parameters, for which the system does have coexistence solutions.

## Keywords

Global bifurcation Elliptic systems Heterogeneous environment Coexistence Chemostat## Mathematics Subject Classification (2000)

35B32 35J61 35Q92 58J20 92D25## Notes

### Acknowledgments

We thanks the two anonymous referees for their careful reading and various comments which greatly improve the original manuscript. We thanks professor Y. Lagadeuc for bringing this problem to ours knowledge though valuable discussions and comments.

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