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The current paper is concerned with the asymptotic dynamics of two speciescompetition systems with/without chemotaxis in heterogeneous media. In theprevious work \cite{ITBWS17a}, we find conditions on the parameters in suchsystems for the persistence of the two species and the existence of positivecoexistence states. In this paper, we find conditions on the parameters for theuniqueness and stability of positive coexistence states of such systems. Theestablished results are new even for the two species competition systemswithout chemotaxis but with space dependent coefficients.

Uniqueness and stability of coexistence states in two species models with/without chemotaxis on bounded heterogeneous environments