Role of nonlinear interaction between water and plant in stability analysis of nonspatial plants

Abstract. In this study, a theoretical ecosystem model is applied to discuss a stability of plant using linear and nonlinear methods. Two common linear methods are employed to analyze a linear stability of plant through judging the positive or negative of eigenvalues (Lyapunov method), and solving a linear singular vector (LSV). To explore the nonlinear stability of plant, a conditional nonlinear optimal perturbation (CNOP) approach is used. The CNOP, which is a type of initial perturbation, could cause the nonlinearly most unstable for an equilibrium state. The CNOP is a nonlinear development of the LSV which is the rapidest initial perturbation with a linear framework. The numerical results show that two linear stable equilibrium states (plant and desert) with the linear methods are nonlinear unstable with the CNOP method. When there is large enough magnitude of initial perturbation, the linear stable plant (desert) equilibrium state will be evolved into desert (plant) equilibrium state using the CNOP-type initial perturbation. This character disappears using the LSV-type initial perturbation. The above results are effective for two types of plant, namely grasslands and trees. Through analyzing the nonlinear dynamics of the theoretical model, it is found that the nonlinear interaction between plant and water play more important role to a transitions between two equilibriums states than the evaporation and the plant losses expressed by linear terms in the theoretical model. The findings could be exhibited by using the nonlinear method (the CNOP method), but fail by using the linear methods.