Engineering Technologies and Systems - 02. Shamanaev P. A., Yazovtseva O. S. Studying the Equilibrium State Stability of the Biocenosis Dynamics System under the Conditions of Interspecies Interaction.

UDK 519.87:581.55

DOI: 10.15507/0236-2910.028.201803.321-332

Studying the Equilibrium State Stability of the Biocenosis Dynamics System under the Conditions of Interspecies Interaction

Pavel A. Shamanaev
Associate Professor, Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Physics and Mathematics), ResearcherID: J-6591-2018, ORCID: http://orcid.org/0000-0002-0135-317X, This email address is being protected from spambots. You need JavaScript enabled to view it.

Olga S. Yazovtseva
Postgraduate Student, Chair of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ResearcherID: J-6507-2018, ORCID: http://orcid.org/0000-0001-8075-4491, This email address is being protected from spambots. You need JavaScript enabled to view it.

Introduction. The article considers the problem of stability of the mathematical model of the trivial equilibrium. The model describes the biocenosis dynamics with the predatorprey type interspecific interaction, which is a nonlinear system of ordinary differential equations with perturbations in the form of vector polynoms. The examined system is considered provided that the birth rate of biological species does not exceed mortality rate.
Materials and Methods. The article states the sufficient conditions for asymptomatic stability. The proof is based on the construction of an operator equation in a Banach space, which connects the solution of the nonlinear system and its linear approximation. The existence of the operator equation solution is proved through using the Schauder fixed point principle. It is shown that there is Brauer local asymptotic equivalence between the solutions of the investigated system and its linear approximation and the differences between the components of the solutions of the nonlinear system and its linear approximation tends to zero evenly with respect to the initial values.
Results. As a case in point, the authors consider the model of the predator-prey type in the case when two species feed on the third one. The conditions for stability and asymptotic stability for a part of the variables of the trivial equilibrium of the abundance dynamics of two predator populations and one prey population under different fertility rates of biological species are given. The graphs of a number of populations with different vaues of the difference between the birth rate and the mortality rate of partucular species are constructed.
Conclusions. Depending on the difference between fertility and mortality of biological species, the population dynamics of two populations of “predators” and one population of “preys” is analyzed over time.

Keywords: predator-prey model, stability in the respect to a part of variables, nonlinear ordinary differential equations, local asymptotic equivalence by Brauer, Schauder fixed point principle

For citation: Shamanaev P. A., Yazovtseva O. S. Studying the Equilibrium State Stability of the Biocenosis Dynamics System under the Conditions of Interspecies Interaction. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(3):321–332. DOI: https://doi.org/10.15507/0236-2910.028.201803.321-332

Authors’ contribution: П. P. A. Shamanaev – formulating the problem, prooving the theorem; O. S. Yazovtseva – reviewing the relevant literature, presenting and analysis of the research results.

All authors have read and approved the final version of the paper

Received 14.05.2018; revised 29.06.2018; published online 20.09.2018

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