Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise

This work focuses on population dynamics of two species described by Kolmogorov systems of competitive type under telegraph noise that is formulated as a continuous-time Markov chain with two states. Our main effort is on establishing the existence of an invariant (or a stationary) probability measure. In addition, the convergence in total variation of the instantaneous measure to the stationary measure is demonstrated under suitable conditions. Moreover, the Ω-limit set of a model in which each species is dominant in a state of the telegraph noise is examined in detail.