SOLUTION OF BIOLOGICAL POPULATION TASK BY TAKING INTO ACCOUNT THE REACTION-DIFFUSION

Аннотация

The work examines the solution of problems of a biological population taking into account reaction-diffusion. This research work uses a method for analyzing the qualitative characteristics of solutions to a system of differential equations in order to solve the problem of choosing the optimal initial approximation for effective iterative convergence to the solution of the Cauchy problem. The effectiveness of this method depends on the numerical parameters and initial data of the system. Researchers use the asymptotic representation of the solution as an initial approximation, which allows them to conduct a numerical experiment and visualize the evolution of the system depending on the parameter values. The results of the study show that the proposed nonlinear mathematical model successfully reflects the process of competing biological populations with double nonlinearity. Analysis of the obtained solution estimates confirms the preservation of localization in a finite range and the size of the source in the system. 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. 1 INTRODUCTION species, studying predator-prey interactions or competition between species in ecosystems. Cross-diffusion, or interactions between different species or subpopulations in Research on the development and application of biological systems, is one of the key characteristics of nonlinear mathematical models is carried out in many nonlinear biological models. Research into this phenomenon developed countries, including the USA, Japan, Spain, can be important for understanding ecological processes, Germany, Great Britain, France, the Russian Federation, including the spread of disease, species migration, and Uzbekistan and many others. Nonlinear mathematical models population dynamics. The development of methods and play an important role in various fields of science and algorithms for solving nonlinear problems in biological industry, such as physics, biology, economics, engineering, populations is an important area of research. These methods artificial intelligence, climatology and others. These models may include numerical algorithms for solving systems of are used to analyze complex systems that cannot always be differential equations, optimizing model parameters, and described by linear equations. For example, in medical analyzing the sensitivity of models to changes in parameters. research, nonlinear models can be used to study disease Given the complexity and diversity of biological systems, dynamics, in economics to analyze market trends and forecast further development of nonlinear models and methods for economic development, and in climatology to model changes their solution will be important for science and practical in climate. Research in this area may include the development applications in biology and ecology [6,7]. of new methods and algorithms for analyzing and solving As a result of research on numerical modeling of cross- nonlinear mathematical models, as well as their application to diffusion systems carried out around the world, a number of specific applied problems. It is important to note that with scientific results have been obtained and represent important increasing computing power and data availability, interest in contributions to the field of mathematical modeling and nonlinear models continues to grow, and they play a key role understanding of complex processes in nature. The proof of in modern research and development [1-5]. the Turing bifurcation in a one-dimensional stationary model Mathematical modeling of nonlinear processes is of is an important result in the theory of nonlinear diffusion fundamental importance in modern science, and this is also systems. This type of bifurcation can lead to the formation of true in biology and ecology, where it is important to structural patterns in a system, such as the formation of understand and predict the complex interactions between patterns and structures, and has wide applications in biology biological populations and their environment. Modeling and chemistry. The study of the conditions for the existence biological populations using nonlinear models can help in the and uniqueness of solutions in diffusion systems is of great study of population dynamics and in the development of importance for substantiating the mathematical correctness of natural resource management and conservation strategies. For models and their applicability in real problems. Studying the example, nonlinear models can be useful for analyzing the stability of stationary solutions and diffusion instability helps effects of climate change on the distribution and behavior of determine which structures and solutions can be stable in the parameters may correspond to the diffusion rate and the initial long term. The Maxwell-Stephan system of equations concentrati