In mathematical biology there are classic models of population dynamics. These models are coupled ordinary differential equations. The rates of change being deterministic functions of the current population size. We consider a classic competing species model, but with the rates changed to the rate plus Gaussian white noise. We show that if the noise is not too large the stochastic version is ergodic. An explicit relation between the noise and original competing species parameters gives a sufficient condition for egrodicity. The methodology is based on an older idea of Bhattacharya from the 1970's. It involves finding functionals with certain monotone properties and then using a Markov's inequality to get upper or lower bounds on tail probabilities or expected hitting or return times, as needed. This is joint work with Zengjing Chen.
Groupe d’études et de recherche en analyse des décisions