Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 116, pp. 1-19.

Title: Markov semigroup approach to the analysis of a nonlinear
       stochastic plant disease model

Authors: Haokun Qi     (Shandong Univ. of Science and Tech., Qingdao, China)
         Xinzhu Meng   (Shandong Univ. of Science and Tech., Qingdao, China)
         Zhengbo Chang (Shandong Univ. of Science and Tech., Qingdao, China)

Abstract:
 In this article, we consider a stochastic plant disease model with logistic growth
 and saturated incidence rate. We analyze long-term behaviors of densities of
 the distributions of the solution. On the basis of the theory of Markov semigroup,
 we obtain the existence of asymptotically stable stationary distribution density
 of the stochastic system. We demonstrate that the densities can converge in L^1
 to an invariant density under appropriate conditions. Moreover, we obtain the
 sufficient conditions for extinction of the disease.
 Also, we present a series of numerical simulations to illustrate our
 theoretical results.

Submitted March 4, 2019. Published October 18, 2019.
Math Subject Classifications: 34D05, 60H10, 92B05.
Key Words: Plant disease model; Markov semigroup; stationary distribution;
           extinction.