Qualitative properties of the task of biological population task with double nonlinear diffusion.

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]