| Contents |
| We study the behavior of two biological populations ``$u$" and ``$v$" attracted by the same chemical substance which 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 chemoatractant 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, %the system is presented by
$$
\left\{ \begin{array}{l}
u_t= \Delta u - \nabla \cdot (u\chi_1(w) \nabla w) + \mu_1u(1-u), \qquad x\in\Omega, \ t>0, \\[1mm]
v_t= \Delta v - \nabla \cdot (v\chi_2(w) \nabla w) +\mu_2v(1 -v), \qquad x\in\Omega, \ t>0, \\[1mm]
w_t= % \varepsilon \Delta w+
h(u,v,w), \qquad\qquad x\in\Omega, \ t>0,
\end{array} \right.
$$
under appropriate boundary and initial conditions in a $n$-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 $\chi_i$. |
|