In the paper, we propose a novel stochastic population model with Markov chain and diffusion in a polluted environment. Under the condition that the diffusion coefficient satisfies the local Lipschitz condition, we prove the existence and uniqueness of invariant measure for the model. Moreover, we also discuss the existence and uniqueness of numerical invariance measure for stochastic population model under the discrete-time Euler-Maruyama scheme, and prove that numerical invariance measure converges to the invariance measure of the corresponding exact solution in the Wasserstein distance sense. Finally, we give the numerical simulation to show the correctness of the theoretical results.

中文翻译：

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污染环境下带马尔可夫链和扩散的随机种群模型不变测度的逼近

在本文中，我们提出了一种在污染环境下具有马尔可夫链和扩散的新型随机种群模型。在扩散系数满足局部Lipschitz条件的条件下，我们证明了该模型不变测度的存在性和唯一性。此外，我们还讨论了在离散时间Euler-Maruyama方案下随机种群模型的数值不变性度量的存在性和唯一性，并证明了数值不变性度量在Wasserstein距离意义上收敛于相应精确解的不变性度量。最后，我们给出了数值模拟，以证明理论结果的正确性。