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In this article, we study the near-optimal control of a class of stochastic vegetation-water model. The near-optimal control is one problem in which the density of vegetation and water is higher at the lowest cost. We have provided a priori estimates of the vegetation and water densities and obtained the sufficient and necessary conditions for the system's near-optimal control problem by applying the maximum condition of the Hamiltonian function and the Ekeland principle. A numerical simulation is presented to verify our theoretical results. KEYWORDSHamiltonian function, near-optimal, sufficient and necessary conditions, vegetation-water model MSC CLASSIFICATION 60H10 INTRODUCTIONSevere desertification has led to the extreme degradation of ecosystems, decline in living standards of residents, and large economic losses. These issues have aroused extensive scholarly attention in different branches, which has achieved some research results. 1-4 However, the restoration of degraded grasslands still faces great challenges. The majority of the research results obtained by ecologists are mainly experimental results, and there is a lack of quantitative analysis of the degraded grassland restoration problem. It is meaningful for scholars to formulate mathematical models based on experimental data and provide strategies for desertification restoration.Mathematical modeling is well-known as a tool for exploring the mechanisms of vegetation patterns. [5][6][7][8][9] To date, a variety of ecosystem models have been established. Ying 10 established a grassland plant-water coupling dynamics model and simulated the dynamic changes of plants and water under different precipitation parameters, regardless of the influence of the external environment. Suna 11 analyzed the trend of plant changes in semiarid areas, used complex network technology to from an early warning index for the oasis-desertization transition, and applied the indicators to the local positive feedback vegetation model. Liu et al 12 proposed a cross-diffusion vegetation system based on the reaction-diffusion equation and revealed that cross-diffusion is an important mechanism for dynamic vegetation changes.Note that the above research results assume that the system parameters are deterministic. However, in real life, the transmission coefficient may be affected by many random environment factors, such as temperature, wind, rain, and snow. 13 Stochastic fluctuation has been included in vegetation-water models. [14][15][16] For example, Zhang et al analyzed the dynamic evolution of a vegetation-water system disturbed by Gauss noise. 14 Guttal et al studied a simple ecological model in which mutations occur when control parameters change. 16 However, optimal control with stochastic vegetation-water models has not been widely studied.Math Meth Appl Sci. 2020;43:6043-6061.wileyonlinelibrary.com/journal/mma

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A class of time-varying delay impulsive reaction–diffusion tree–grass–water–nitrogen system driven by Lévy jump process is considered. First, we prove the existence and uniqueness of the global positive solution of the model by constructing the Lyapunov function. Secondly, several sufficient conditions for finite-time stability are given by using comparison theorem and mean impulse interval method. Finally, numerical simulations are carried out to verify the effectiveness of the theoretical analysis.

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