Application Of The Conditional Nonlinear Optimal Perturbations Method In Sensitivity Analysis Of The Lake Eutrophication

The Lake eutrophication problem has been one of most serious environmental problems which are concerned by government and public.Lake eutrophication forms a very complex ecological community as all open nonlinear system in which a variety of complex physical, chemical and biological reactions occur.In this paper, based on the dynamical model for nutrient cycling in shallow lakes, a new approach of conditional nonlinear optimal perturbation (CNOP) is adopted to investigate the instability and sensitivity of the lake ecosystem to finite-amplitude perturbation.The main researches of this paper are concluded as follows:(1) The impact of the nutrient loading rate c on this lake ecosystem has been studied in this paper.For c less than the bifurcation point c₁ , OES is nonlinearly stable,no matter how large the initial finite-amplitude perturbation is.This implies that OES has non-zero resilience, human activities and natural factors of instantaneous or short type can not make the ecosystem experience a regime shift.When c is lager than the bifurcation point c₂ , EES is also nonlinearly stable, which means that once the lake eutrophication occur, it is difficult for this lake to come back to the clear state.In case of c between c₁ and c₂ , the lake ecosystem is fragile, OES (EES) is linearly stable but nonlinearly unstable, which indicates that a large enough initial finite-amplitude perturbation can induce a transition from OES (EES) to EES (OES).Under the same constraint condition, the evolution of corresponding finite amplitude perturbation represented CNOP-I for different c is different.With the approach of CNOP-I, we determine the threshold, the critical valueδ_c , which is the smallest magnitude of the finite-amplitude perturbation that induces a transition from OES (EES) to EES (OES).(2) By numerical simulation, under the given constraint condition, the perturbations of control parameterc which have the largest effect on the evolution of lake ecosystem are given in this paper.The numerical results demonstrate the optimal perturbation of control parameter c, named CNOP-P, have the largest effect on lake ecosystem reaching to other equilibrium states.For c less than the bifurcation pointc₂ , if the constraint condition of perturbation is lager than | c₂ - c|, then the original clear lake ecosystems are prone to form eutrophication states.While in the case of c lager than the bifurcation point c₁ , if the constraint condition of perturbation is lager than | c₁ - c|, the lake in the state of eutrophication can returned to the clear state.