Mathematical Models and Degradation Forecast of Arid Soil–Plant Systems

Mathematical modeling is considered as a method for studying the dynamics of grassland soil and plant systems in arid zones of Russia. The evolutionary development of grassland phytocenoses was modeled using the principles of nonequilibrium thermodynamics based on continuous and discrete mathematical formalisms. Examples are given, and problems of sustainable development of such systems under conditions of anthropogenic load and climate change are discussed. This study demonstrates new methodological capabilities of mathematical models of different types: in the form of systems of ordinary differential equations and discrete Markov chains. Prediction of degradation processes occurring in grasslands using these models has been studied in a comparative aspect. Differential models with constant and variable coefficients showed different results due to the nonlinearity of succession dynamics. The model with constant coefficients was refined by introducing time-dependent coefficients. The stability of the functioning of grassland ecosystems is discussed from the point of view of the formal parametric stability of solutions for a system of ordinary differential equations. Discrete modeling using the Markov chain technique showed that the degradation of soil and plant systems when the animal load is exceeded is described by a heterogeneous Markov process. Homogeneous Markov chains give adequate forecasts on one observation time interval. Prediction of the final states in the homogeneous approximation revealed that the observed nonlinearity in the dynamics of phytocenoses was caused by changes in the rate of development, ultimately leading to heterogeneity of the Markov chain. The issues of modeling nonlinear processes in the ecology of grassland soil and plant systems in the context of heterogeneous Markov processes are discussed.