ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Kashchenko I. S., Kashchenko S. A. Dynamics of equation with two delays modelling the number of population. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 2, pp. 21-38. DOI: 10.18500/0869-6632-2019-27-2-21-38

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2019-27-2-21-38

Dynamics of equation with two delays modelling the number of population

Autors: 
Kashchenko I. S., P. G. Demidov Yaroslavl State University
Kashchenko Sergej Aleksandrovich, P. G. Demidov Yaroslavl State University
Abstract: 

Issue. The paper investigates the behavior of solutions of a logistic equation with two delays from some neighborhood of the equilibrium state with a large value of the coefficient of linear growth. Such problems arise in modeling the population size taking into account the age structure, as a model of the number of insets, etc. Innovation. It is shown that the critical cases arising in the problem of the stability of an equilibrium state have infinite dimension: an infinitely large number of roots of the characteristic equation tend to the imaginary axis. In addition, in a number of studied situations, an additional degeneracy arises that significantly affects the structure of solutions. Investigation methods. To study the behavior of solutions in close to critical cases, an asymptotic method has been developed. With its help, special nonlinear equations – quasi-normal forms – whose solutions provide asymptotic approximations of solutions to the original problem. Results. It is shown that in critical cases the behavior of the solutions of the original singularly perturbed problem is determined by the dynamics of the quasi-normal form. The asymptotic formulas connecting their solutions are given. Complex parabolic Ginzburg–Landau equation can serve as a quasi-normal form, and for some degenerations, equations with one (possibly large) delay or the generalized Korteweg–de Vries equation. These tasks either do not contain a small parameter, or depend on it regularly. Conclusions. The behavior of solutions of a singularly perturbed logistic equation with two delays is studied. Critical cases are found and bifurcations are investigated. It is shown that the system under study has such dynamic effects as multistability and hypermultistability, as well as an infinite process of direct and inverse bifurcations as the small parameter tends to zero.

Acknowledgements. The reported study was funded by RFBR according to the research project No 18-29-10043.

Key words: 
equation with delay
two delays
small parameter
normal form
number of population
Ginzburg–Landau equation
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Received: 
18.12.2018
Accepted: 
01.03.2019
Published: 
24.04.2019
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