STABILITY OF EVOLUTIONARY SYSTEMS

Boykov Il'ya Vladimirovich, Doctor of physical and mathematical sciences, professor, head of sub-departmentof higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), boikov@pnzgu.ru
Zakharova Yuliya Fridrikhovna, Candidate of physical and mathematical sciences, associate professor, sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), math@pnzgu.ru
Dmitrieva Alla Arkad'evna, Senior lecturer, sub-department of higher and applied mathematics, Penza State
University (40 Krasnaya street, Penza, Russia), math@pnzgu.ru

518.5

Background. Recently the evolutionary systems have gained growing significance in various fields of science and technology. A crucial example of the evolutionary systems are the various sectors of economy, separate enterprises, computing centers and networks thereof, human organism, cells, organism’s systems, various populations. Thereby it is topical to research dynamic processes progressing in the evolutionary systems and, first of all, to research the stability of the system itself. In the article, by the example of models of interaction of the environment with pollution and the models of immunology, the authors research the stability of the evolutionary systems, described by Lotka Volterra equations. The article describes the application of therapy in the base model of immunology.
Materials and methods. The researchers use the modification of Lyapunov first method, intended for research of stability of nonautonomous differential equation systems. For this purpose the authors build a family of linear operators and determine the stability of differential equation system by signs of operators’ logarithmical norms.
Results. The researchers obtained the criteria of stability and asymptotic stability according to Lyapunov of the fixed points in the model of interaction of the environment with pollution. The article adduces a qualitative research of a number of models of immunology. The authors investigated the application of therapy in the base model of immunology.
Conclusions. The suggested method may be used in research of a wide class of evolutionary systems.

evolutionary systems, dynamic process, stability, Lotka Volterra equa-tions, models of immunology.

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