Asymptotic behavior in a doubly tactic resource consumption model with proliferation

This paper is concerned with the doubly tactic model {ut=Δu-χu∇·(u∇w)+uw,x∈Ω,t>0,vt=Δv-χv∇·(v∇u)+vw,x∈Ω,t>0,wt=Δw-λ(u+v)w-μw,x∈Ω,t>0,in a smoothly bounded domain Ω ⊂ R^N (N≥ 1) with positive parameters χ_u, χ_v, λ and nonnegative parameter μ, for the spatiotemporal evolution of forager–exploiter groups u and v, which simultaneously consume a common nutrient w and proliferate. It is shown that for all suitably regular small initial data, the corresponding Neumann initial-boundary value problem possesses a globally classical solution, which approaches spatially homogeneous profiles at an exponential rate.