具有跳跃的随机蚊子种群模型的不变测度

Invariant Measure of Random Mosquito Population Model with Jumping

摘要: 本文主要研究具有马尔可夫链的随机蚊子种群模型的不变测度。首先，巧妙构造李雅普诺夫函数，利用伊藤定理、比较定理，证明了随机蚊子种群模型存在唯一的全局连续正解。其次，如果λ≤0，不育蚊子种群会灭绝，而野生蚊子种群的分布弱收敛于唯一不变概率测度；如果λ > 0，则系统具有不变概率测度，解过程的转移概率收敛于不变测度。最后证明了随机过程的转移概率收敛到其不变测度的指数收敛速度。

Abstract: This paper mainly studies the invariant measures of the random mosquito population model with Markov chains. First, the Lyapunov function is cleverly constructed, and the Itô theorem and the comparison theorem are used to prove that the random mosquito population model has a unique global continuous positive solution. Second, if λ≤0, the sterile mosquito population will be extinct, and the distribution of the wild mosquito population weakly converges to the only constant probability measure; if λ > 0, then the system has an invariant probability measure, and the transition probability of the solution process converges to an invariant measure. Finally, it is proved that the transition probability of a stochastic process converges to the exponential convergence rate of its invariant measure.