Long time behavior of a degenerate stochastic system modeling the response of a population to its environmental perception

We study the asymptotics of a two-dimensional stochastic differential system with a degenerate diffusion matrix. This system describes the dynamics of a population where individuals contribute to the degradation of their environment through two different behaviors, responding more or less intensively to their environmental perception. We exploit the almost one-dimensional form of the dynamical system to compute explicitly the Freidlin-Wentzell action functional. This allows us to give conditions under which the small noise regime of the invariant measure is concentrated around the equilibria of the dynamical system having the smallest diffusion coefficient.