Continuous and discrete aspects of models of the dynamics of pasture ecosystems
A.N. SALUGIN
Federal Scientific Center for Agroecology, Integrated Land Reclamation and Protective Afforestation, Russian Academy of Sciences, Volgograd, Russia
Keywords: mathematical modeling, differential equations, Markov chains, stability, destabilization parameter, autonomous pulse processes
Abstract
The article presents methods of mathematical modeling in the study of the dynamics of phytocenoses of soil-plant systems in arid zones of the south of Russia. The models are developed on the basis of continuous and discrete formalisms. The results obtained in the form of analytical expressions in continuous models were used to identify the conditions for the existence of a stable state of pasture ecosystems by applying the destabilization parameter. Quantitative relationships between the permissible pasture load and the calculated destabilization parameter obtained from Markov chains were determined. The conditions for the stable functioning of pastures are discussed, demonstrating the capabilities of continuous models in the form of systems of ordinary differential equations and discrete ones: Markov chains and autonomous pulse processes. Under anthropogenic loads on pastures, Markov chains adequately reflect the nonlinear dynamics of pasture ecosystems and predict their final steady state. This state serves as the starting point for determining the destabilization parameter, which determines the degree of ecosystem deviation from equilibrium. It is shown that homogeneous Markov chains can be used to generate short-term forecasts and manage pasture forage reserves. Using autonomous pulse processes, a model of the dynamics of phytocenoses of natural pastures was developed. Computational experiments were conducted to determine the optimal model parameters corresponding to the steady-state regime and optimal pasture exploitation. The formalism of autonomous pulse processes was used to study the dynamics of interactions between pasture phytocenoses and to determine the optimal parameters of the binary interaction matrix when solving problems of pasture bioresource management. Mathematical modeling as a method for studying the dynamics of soil-plant ecosystems in the form of natural pastures is presented as one of the main research methods in the ecology of soil ecosystems.
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