Exponential ergodicity for population dynamics driven by α-stable processes

In this paper, we consider the stochastic Lotka-Volterra model driven by spectrally positive stable processes. We show that if the coefficients of the noise are small, then this kind of pure jump stochastic dynamic has a unique stationary distribution. Besides, we prove that the rate of the transition semigroup convergence to the stationary distribution in the total variation distance is exponential. However, if the noise is sufficiently large, then this stochastic dynamic will become extinct with probability one. Computer simulations are presented to illustrate our theory. To the best of our knowledge, it is the first result to give the exponential ergodicity for population dynamics driven by spectrally positive α-stable processes.