Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned on non-extinction

We are interested in the long time behavior of a two-type density-dependent biological population conditioned on non-extinction, in both cases of competition or weak cooperation between the two species. This population is described by a stochastic Lotka-Volterra system, obtained as limit of renormalized interacting birth and death processes. The weak cooperation assumption allows the system not to blow up. We study the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned on non-extinction. To this aim we generalize in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. The existence proof of a quasi-stationary distribution is reduced to the one for a d-dimensional Kolmogorov diffusion process under a symmetry assumption. The symmetry we need is satisfied under a local balance condition relying the ecological rates. A novelty is the outlined relation between the uniqueness of the quasi-stationary distribution and the ultracontractivity of the killed semi-group. By a comparison between the killing rates for the populations of each type and the one of the global population, we show that the quasi-stationary distribution can be either supported by individuals of one (the strongest one) type or supported by individuals of the two types. We thus highlight two different long time behaviors depending on the parameters of the model: either the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving, or there is a positive probability to have coexistence of the two species.