Stochastic Hopf bifurcation analysis in a stochastic lagged logistic discrete-time system with Poisson distribution coefficient

In this paper, stochastic Hopf bifurcation behavior of a stochastic lagged logistic system is investigated. Firstly, the stochastic lagged logistic system with random parameter is approximately transformed as its equivalent deterministic system by the orthogonal polynomial approximation of discrete random function in the Hilbert spaces. Then, according to the bifurcation conditions of deterministic discrete system, Hopf bifurcation is existent in the equivalent deterministic system by mathematical analysis. Moreover, the direction and stability of its bifurcation is discussed by the normal form and center manifold theory. Finally, we verify the influence for the different random intensity on bifurcation critical value by numerical simulations and find that Hopf bifurcation phenomena under the influence of random intensity happen to drift.