In this paper, we study a system of partial differential equations describing the evolution of a population under chemotactic effects with non-local reaction terms. We consider an external application of chemoattractant in the system and study the cases of one and two populations in competition. By introducing global competitive/cooperative factors in terms of the total mass of the populations, we obtain, for a range of parameters, that any solution with positive and bounded initial data converges to a spatially homogeneous state with positive components. The proofs rely on the maximum principle for spatially homogeneous sub-and super-solutions.

No abstract

No abstract

We study the behavior of two biological populations "w" 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 satisfles an ODE, more precisely, , t >0, w¡ = h(u, v, w), x e Q, f > 0, under appropriate boundary and initial conditions in an «-dimensional open and bounded domain Q. We consider the cases of positive chemo-sensitivities, not necessarily constant elements. The chemical production function h increases as the concentration of the species "w" and ' V increases. We flrst 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, Xi an d the size of fj,¡. Finally, some examples of the theoretical results are presented for particular functions h and xi •

In this paper we consider a system of three parabolic equations modeling the behavior of two biological species moving attracted by a chemical factor. The chemical substance verifies a parabolic equation with slow diffusion. The system contains second order terms in the first two equations modeling the chemotactic effects. We apply an iterative method to obtain the global existence of solutions using that the total mass of the biological species is conserved. The stability of the homogeneous steady states is studied by using an energy method. A final example is presented to illustrate the theoretical results.

In this paper we prove a discrete version of the classical Ingham inequality for nonharmonic Fourier series whose exponents satisfy a gap condition. Time integrals are replaced by discrete sums on a discrete mesh. We prove that, as the mesh becomes finer and finer, the limit of the discrete Ingham inequality is the classical continuous one. This analysis is partially motivated by control-theoretical issues. As an application we analyze the control/observation properties of numerical approximation schemes of the 1-d wave equation. The discrete Ingham inequality provides observability and controllability results which are uniform with respect to the mesh-size in suitable classes of numerical solutions in which the high frequency components have been filtered. We also discuss the optimality of these results in connection with the dispersion diagrams of the numerical schemes.

No abstract

In this paper we consider a general system of reaction-diffusion equations and introduce a comparison method to obtain qualitative properties of its solutions. The comparison method is applied to study the stability of homogeneous steady states and the asymptotic behavior of the solutions of different systems with a chemotactic term. The theoretical results obtained are slightly modified to be applied to the problems where the systems are coupled in the differentiated terms and / or contain nonlocal terms. We obtain results concerning the global stability of the steady states by comparison with solutions of Ordinary Differential Equations.(1) 2010 Mathematics Subject Classification. Primary: 35B35, 35B40; Secondary: 35B51.