  We are interested in the long time behavior of a two-type density-dependent
biological population conditioned to 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 to 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.
