The Stability And Numerical Simulation Research On The Shallow Lake Ecosystem With Stochastic Excitation

The lake ecosystem is disturbed by various stochastic factors in the real environment. For more accurately describing the stability of the lake ecosystem, considering the effect of stochastic factors, we established the model of the lake ecosystem with stochastic excitation and studied the stochastic stability and stochastic Hopf bifurcation behavior of the lake ecosystem.(1) The dynamics property of the stochastic model. Considering the one-variable lake ecosystem model which is developed by Carpenter et al and the two-variable shallow lake ecosystem model of Scheffer et al, which describes the relation between macrophytes covers V and turbidity E in shallow lake, we established the stochastic model with the stochastic noise. The existence of the unique solution to the stochastic model is proved with Lyapunov analytical method. In addition, we established the asymptotic relationship between the solution of the stochastic model and the equilibrium point of the deterministic model by choosing suitable Lyapunov functions. The results show that stochastic model doesn't exist positive equilibrium point after adding the stochastic oise. But the solution of the stochastic model around a fixed point related to the balance point for random vibration in the sense of average time.(2) The shallow lake ecosystem with the multiplicative stochastic excitation. Considering the two-variable shallow lake ecosystem model of Scheffer et al, which describes the relation between macrophytes covers V and turbidity E in shallow lake, we established the stochastic model with the stochastic noise. The model of the shallow lake ecosystem with stochastic excitation was simplified with the stochastic averaging method and nonlinear dynamic theory. We studied the stochastic Hopf bifurcation behavior with the FPK method. Choosing linearly stable clear equilibrium state and linearly unstable equilibrium state as the initial states, selecting 500 individual sample paths and averaging over them, the stochastic model was numerically simulated by the Runge-Kutta numerical scheme of the stochastic differential equation. The results show that the stochastic factors can make the stability of the shallow lake ecosystem appear change; linearly stable equilibrium state becomeslinearly unstable equilibrium state; a pseudo-stochastic noise can make the shallow lake ecosystem appear regime shift and the bifurcation value shifts.(3) The shallow lake ecosystem with the external stochastic excitation and the multiplicative stochastic excitation. Considering the two-variable shallow lake ecosystem model of Scheffer et al, which describes the relation between macrophytes covers V and turbidity E in the shallow lake, we established the stochastic model with the stochastic noise. With the stochastic averaging method and nonlinear dynamic theory, the model of the shallow lake ecosystem with stochastic excitation was simplified. The stochastic Hopf bifurcation behavior with the FPK method was studied. The influence of bifurcation parameters on the system was also discussed and explained the influence. The results show that the stability of the original system appear change with the influence of stochastic factors; the original system could appear stochastic Hopf bifurcation with the probability meanings and the bifurcation location shifts with the increase of noise intensity. If the bifurcation position reaches the threshold which appears turbid state, it is likely to lead the transition between the different stable states and the shallow lake can switch from clear equilibrium state to turbid equilibrium state.