Global existence and boundedness of classical solutions to a forager–exploiter model with volume-filling effects

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This work is concerned with a chemotactic model for the dynamics of social interactions between two species — foragers u and exploiters v, as well as the dynamics of food resources w consumed by these two species. The foragers search for food directly, while the exploiters head for food by following the foragers. Specifically, the parabolic system in a smoothly bounded convex n-dimensional domain Ω, ut=Δu−∇⋅(S1(u)∇w),x∈Ω,t>0,vt=Δv−∇⋅(S2(v)∇u),x∈Ω,t>0,wt=dΔw−λ(u+v)w−μw+r(x,t),x∈Ω,t>0,is considered for the constants d,λ,μ>0 and r∈C0(Ω×[0,∞)) with a uniform bound. Volume-filling effects account for a simple version by taking S1(u)=u(1−u),S2(v)=v(1−v).We prove the global existence and boundedness of the unique classical solution to this forager–exploiter model associated with no-flux boundary conditions under the mild assumption that the initial data u0, v0, w0 satisfy 0≤u0,v0≤1 and w0≥0.