Sufficient Conditions for the Existence and Uniqueness of the Solution of the Dynamics of Biogeochemical Cycles in Coastal Systems Problem

Alexander Sukhinov, Yulia Belova, Alla Nikitina and Valentina Sidoryakina
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Alexander Sukhinov: Department of Mathematics and Informatics, Don State Technical University, 344000 Rostov-on-Don, Russia
Yulia Belova: Department of Mathematics and Informatics, Don State Technical University, 344000 Rostov-on-Don, Russia
Alla Nikitina: Department of Mathematics and Informatics, Don State Technical University, 344000 Rostov-on-Don, Russia
Valentina Sidoryakina: Department of Mathematics and Informatics, Don State Technical University, 344000 Rostov-on-Don, Russia

Mathematics, 2022, vol. 10, issue 12, 1-16

Abstract: The article considers a three-dimensional mathematical model of population dynamics based on a system of non-stationary parabolic advection-diffusion-reaction equations with lower derivatives describing the advective motion of the aquatic environment and non-linear source functions. In contrast to the previous authors’ works devoted to the description of this model and its numerical implementation, this article presents the results of an analytical study of the initial-boundary value problem corresponding to this model. For these purposes, the original initial-boundary value problem is linearized on a single time grid—for all nonlinear sources, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial—final data at each step of the time grid. For this chain of linearized problems, the existence and uniqueness of the solution of the initial-boundary value problem for the system of partial differential equations in the Hilbert space were researched. Numerical experiments were performed for model problems of the main types of phytoplankton populations in coastal systems under the influence of dynamically changing biotic and abiotic factors, the results of which are consistent with real physical experiments. The developed model, including the proposed model of biological kinetics, allows for the study of the productive and destructive processes of the shallow water body biocenosis to assess the state of the processes of reproduction of valuable and commercial fish participating in the food chain with selected species of summer phytoplankton.

Keywords: 3D biogeochemical mathematical model; three-population phytoplankton dynamics; linearization of nonlinear right side functions; Hilbert space; quadratic functional; sufficient conditions for the solution uniqueness; numerical experiments; software package (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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