The Nonlinear Stability Research On The Shallow Lake Ecological Degradation And Restoration

There exists the serious degradation in the shallow lake ecosystem, while it is more significant to explore the dynamic process in order to study the recovery mechanism of the shallow lake ecosystem. In this paper, the conditional nonlinear optimal perturbations method and the nonlinear stochastic dynamic theory are adopted to explore the stability and sensitivity of the ecosystem. The main works are as follows:(1) For the determinate ecosystem, when we ignore the external stochastic disturbance, i.e., the intensity of the external noise is 0, we use the two-variable model of Scheffer et al., which describes the relation between macrophytes covers V and turbidity E in shallow lakes. Considering the errors in the initial field and the parameter in the model and using the CNOP method proposed by M. Mu, we discuss the instability and sensitivity of the ecosystem to the finite-amplitude perturbations related to the initial condition and the parameter condition. Results show that the linearly stable clear(turbid) water states can be nonlinearly unstable with the finite amplitude perturbations. The results also support the viewpoint of Scheffer et al., whose emphasis is that the facilitation interactions between the submerged macrophytes and the water transparency are the main trigger for an occasional shift from a turbid to a clear state. Also, by the comparison with CNOP-I, CNOP-P, CNOP and(CNOP-I, CNOP-P), results demonstrate that CNOP, which is not a simple combination of CNOP-I and CNOP-P, could induce the shallow lake ecosystem the largest departure from the same ground state rather than CNOP-I, CNOP-P and(CNOP-I, CNOP-P).(2) For the one-variable lake ecosystem model which is developed by Carpenter et al., we consider that the model is disturbed by the stochastic factors and we simplify the external disturbance the Gaussian white noise. In this paper, by establishing the model of the lake eutrophication stochastic ecosystem, we studied the bifurcation characteristics with the FPK method based on the FPK equation and the maximumLyapunov exponent method. Also, with the fourth-order Runge-Kutta numerical scheme of the stochastic differential equation, choosing 500 individual sample paths and averaging over them, we discussed the stability and the regime shift of the ecosystem by analyzing the evolution of the ecological variable. Further more, considering that the model is disturbed by the multiplicative noise, we discuss that how the noise intensity and the nutrient loading rate affect the shallow lake ecosystem.(3) For the two-variable model of Scheffer et al., we established the stochastic model which is subjected to the external stochastic excitation and explore the bifurcation characteristics with the FPK method based on the Stratonovich-Khasminiskii stochastic average principle. Also, we study the effect of the noise intensity on the ecosystem.