Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant

We study the behavior of two biological populations u and v attracted by the same chemical substance whose behavior is described in terms of second order parabolic equations. The model considers a logistic growth of the species and the interactions between them are relegated to the chemoattractant production The system is completed with a third equation modeling the evolution of chemical. We assume that the chemical w is a non-diffusive substance and satisfies an ODE, more precisely, {u(t) = Delta u - del . (u chi(1)(w)del w) + mu(1)u(1-u), x is an element of Omega, t > 0, v(t) = Delta v - del . (v chi(2)(w)del w)) + mu(2)v(1-v), x is an element of Omega, t > 0, w(t) = h(u, v, w), x is an element of Omega, t > 0, under appropriate boundary and initial conditions in an n-dimensional open and bounded domain Omega. We consider the cases of positive chemo-sensitivities, not necessarily constant elements. The chemical production function h increases as the concentration of the species u and v increases. We first study the global existence and uniform boundedness of the solutions by using an iterative approach. The asymptotic stability of the homogeneous steady state is a consequence of the growth of h, chi(i) and the size of mu(i). Finally, some examples of the theoretical results are presented for particular functions h and chi(i). (C) 2014 Elsevier Inc. All rights reserved.