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This research work successfully addresses the key issues related to the selection of optimal initial approximations depending on numerical parameters and available data in the context of reaction-diffusion. This approach significantly improves the accuracy and efficiency of monitoring and analyzing the dynamics of the reaction-diffusion process. In this study, it was shown that for the successful analysis and solution of complex nonlinear problems, it is effective to use the nonlinear splitting method and the method of standard equations. These methods have proven to be reliable tools for studying nonlinear phenomena in various systems. As a result of this study, a nonlinear splitting algorithm was justified, which is used to solve equations describing multicomponent cross-diffusion systems of a biological population. This algorithm provides an important tool for numerical simulation and analysis of such systems, providing more accurate and efficient results. The study confirms the significance of nonlinear splitting and reference equation methods in the context of nonlinear problems, and also provides a practical solution in the form of an algorithm for studying multicomponent cross-diffusion systems in biological populations. These results have potential for application in various fields of science and engineering, where modeling and numerical work with nonlinear differential equations is important. In addition, software packages have been created to help automate the processes of visualizing the development of solutions in time and space. These innovative tools provide the ability to conveniently observe and analyze the evolution of processes in various fields of science and engineering where complex dynamic systems, such as biological populations and others, are important. This work advances the understanding and modeling of complex reaction-diffusion processes and has potential applications in various disciplines of science and technology. [ABSTRACT FROM AUTHOR]

In this work, the properties of multicomponent cross-diffusion systems of a biological population with double nonlinearity and variable density are investigated. Using computer modeling, models have been created to analyze and simulate processes in such systems. The estimates necessary to solve the Cauchy problem in the context of these multicomponent cross-diffusion systems are obtained. The results of the study provide important data for understanding the dynamics of biological populations and may also be useful in the context of biological resource management and analysis of ecosystem interactions. This research has significant implications for the fields of ecology and biology and has the potential to advance our understanding of complex biological systems. [ABSTRACT FROM AUTHOR]

The work examines the solution of problems of a biological population taking into account reaction-diffusion. This research employs a technique to evaluate the qualitative properties of solutions to a differential equation system with the aim of addressing the challenge of selecting an optimal initial approximation for achieving efficient iterative convergence towards solving the Cauchy problem. The effectiveness of this approach is contingent on the numerical parameters and initial data of the system. Researchers utilize the asymptotic representation of the solution as an initial estimate, enabling them to conduct a numerical experiment and visualize the system's evolution as a function of parameter values. The study's findings indicate that the proposed nonlinear mathematical model aptly captures the dynamics of competitive biological populations characterized by double nonlinearity. Analysis of the derived solution estimates corroborates the preservation of confinement within a finite range and the system's source size. Thus, this work provides a complete, comprehensive picture of the process and can be a valuable contribution to the study of two-component competing systems in biological populations. [ABSTRACT FROM AUTHOR]