Résumé: In this paper we derive a model describing the dynamics of HIV-1 infection in tissue culture where the infection spreads directly from infected cells to healthy cells trough cell-to-cell contact. We assume that the infection rate between healthy and infected cells is a saturating function of cell concentration. Our analysis shows that if the basic reproduction number does not exceed unity then infected cells are cleared and the disease dies out. Otherwise, the infection is persistent with the existence of an infected equilibrium. Numerical simulations indicate that, depending on the fraction of cells surviving the incubation period, the solutions approach either an infected steady state or a periodic orbit.

Résumé: In this paper we consider a periodic non-autonomous competitive stage-structured system with infinite delay for the interaction between n species, the adult members of which are in competition. For each of the n species the model incorporates a time delay which represents the time from birth to maturity of that species. Infinite delay is introduced which denotes the influential effect of the entire past history of the system on the current competition interactions. We first prove by using the comparison principle that if the growth rates are sufficiently large then the solutions are uniformly permanent. Then by using Horn's fixed point Theorem, we show that the system with finite delay has a positive periodic solution. As a consequence of this result, we prove that even the system with infinite delay admits a positive periodic solution.

Résumé: In this paper we consider a nonautonomous periodic dispersal system which models the diffusion of a single species between n patches connected by discrete dispersal in a periodic environment. Using Gaines and Mawhin continuation theorem of coincidence degree (cf. [R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977]), we prove that the system has at least one positive periodic solution. With the help of an appropriately chosen Lyapunov functional it is proved that this periodic solution is globally attractive. We end the paper by some numerical simulations which illustrate the feasibility of our results.

Résumé: In this paper we consider a nonautonomous stage-structured competitive system of n-species population growth with distributed delays which takes into account the delayed feedback in both interspecific and intraspecific interactions. We obtain, by using the method of repeated replace, sufficient conditions for permanence and extinction of the species. The global attractivity of the unique positive equilibrium is proved in the autonomous case. Our results extend previous ones obtained by Liu et al. in [Nonlinear Anal. 51 (2002) 1347-1361; J. Math. Anal Appl. 274 (2002) 667-684].

Résumé: A generalization to n species of a system by Bass et al. (cf.[4]), which describes the self-organization of liver zones in a liver capillary in the case of two species, is proposed. We establish some hypotheses on the coefficient parameters of the system under which a part of the species is driven to extinction while the remaining ones are attracted by the non-trivial stationary solution.