Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation

This on the one hand shows that in comparison to the well-understood simple nutrient taxis system obtained on letting ρ=μ=0 and removing the proliferation term +uv in (⋆), the introduction of the latter does not substantially affect the tendency of the system to support relaxation into spatial homogeneity. Apart from that, (0.1) complements previous findings on convergence to equilibria in prey-taxis systems of type (⋆), which seem to exclusively concentrate on cases in which unlike in (0.1) the constant limit functions are a priori known, and in which thereby the corresponding asymptotic analysis is apparently simplified to a considerable extent.