Abstract. We consider ecological difference equations of the form X i t+1 = X i t Ai(Xt) where X i t is a vector of densities corresponding to the subpopulations of species i (e.g. subpopulations of different ages or living in different patches), Xt = (X 1 t , X 2 t , . . . , X m t ) is state of the entire community, and Ai(Xt) are matrices determining the update rule for species i. These equations are permanent if they are dissipative and the extinction set {X : i X i = 0} is repelling. If permanence persists under perturbations of the matrices Ai(X), the equations are robustly permanent. We provide sufficient and necessary conditions for robust permanence in terms of Lyapunov exponents for invariant measures supported by the extinction set. Applications to ecological and epidemiological models are given.

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This paper extends the work of Salceanu and Smith [12, 13] where Lyapunov exponents were used to obtain conditions for uniform persistence ina class of dissipative discrete-time dynamical systems on the positive orthant of R(m), generated by maps. Here a united approach is taken, for both discrete and continuous time, and the dissipativity assumption is relaxed. Sufficient conditions are given for compact subsets of an invariant part of the boundary of R(m+) to be robust uniform weak repellers. These conditions require Lyapunov exponents be positive on such sets. It is shown how this leads to robust uniform persistence. The results apply to the investigation of robust uniform persistence of the disease in host populations, as shown in an application.

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We analyse a discrete-time Ricker competition model with n competing species and give sufficient conditions, which depend on the competition coefficients only, for one species to survive (not necessarily at an equilibrium) and to drive all the other species to extinction. Our results complement and extend similar existing results from the literature. For the model reduced to three species (n = 3), we also investigate various scenarios under which all species coexist, in the sense that each species is robustly uniformly persistent. We provide a few numerical simulations to illustrate that coexistence does not necessarily mean convergence to the interior equilibrium, and that the interior dynamics can be quite complex.