Ergodicity and exponential ergodicity of generalized stochastic Gilpin–Ayala model driven by α-stable process

In this paper, we develop a generalized stochastic Gilpin–Ayala competition model that, by employing an α-stable process, moves beyond the Brownian motion framework to better capture large-jump environmental noise, and, for the first time, incorporates a nonlinear diffusion term to realistically reflect the population–disaster impact relationship. The existence of a local solution is established using the interlacing method, while the existence and uniqueness of a global positive solution are rigorously proved via a Lyapunov function, which reveals the suppressive effect of noise on population explosion. By constructing a suitable function V satisfying Khasminskii’s lemma, we derive sufficient conditions for ergodicity and exponential ergodicity, which reduce to known results in degenerate cases. The findings generalize existing theories. Furthermore, we show that the solution is stochastically ultimately bounded: as t→∞, the population size converges to a bounded region with arbitrarily high probability.