Abstract
A diffusive nutrient-microorganism model in a spatially heterogeneous environment with zero-flux boundary conditions is formulated. First, some fundamental properties of the related parabolic system are established. To be specific, some boundedness results are yielded by employing the comparison principle of parabolic equations, the theory of invariant regions and the method of Moser-Alikakos iteration. For the corresponding elliptic system, existence results of non-constant steady states are given by virtue of the Leray-Schauder degree theory. Profiles of the positive solutions are described when the diffusion rate of the nutrient is small or large, with the fixed diffusion rate of microorganisms. Different from the related previous models with spatial homogeneity, the obtained results suggest that such a diffusive nutrient-microorganism model in a spatially heterogeneous environment may be more realistic and perform more complex dynamical behaviors.
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