Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay
https://doi.org/10.18255/1818-1015-2020-3-344-355
Abstract
The logistic equation with delay or Hutchinson’s equation is one of the fundamental equations of population dynamics and is widely used in problems of mathematical ecology. We consider a family of mappings built for this equation based on central separated differences. Such difference schemes are usually used in the numerical simulation of this problem. The presented mappings themselves can serve as models of population dynamics; therefore, their study is of considerable interest. We compare the properties of the trajectories of these mappings and the original equation with delay. It is shown that the behavior of the solutions of the mappings constructed on the basis of the central separated differences does not preserve, even with a sufficiently small value of the time step, the basic dynamic properties of the logistic equation with delay. In particular, this map does not have a stable invariant curve bifurcating under the oscillatory loss of stability of a nonzero equilibrium state. This curve corresponds in such mappings to the stable limit cycle of the original continuous equation. Thus, it is shown that such a difference scheme cannot be used for numerical modeling of the logistic equation with delay.
Supporting agencies: The author were supported by the Russian Foundation for Basic Research (project no. 18-29-10043)
About the Authors
Sergey D. Glyzin
Centre of Integrable Systems, P. G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
Head of the Department of Computer Networks, professor.
Sovetskaya str., 14, Yaroslavl, 150003
Sergey A. Kashchenko
Centre of Integrable Systems, P. G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
First Vice-Rector, professor.
Sovetskaya str., 14, Yaroslavl, 150003
Anna O. Tolbey
Centre of Integrable Systems, P. G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
Associate Professor, PhD.
Sovetskaya str., 14, Yaroslavl, 150003
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Review
For citations:
Glyzin S.D., Kashchenko S.A., Tolbey A.O. Features of the Algorithmic Implementation of Difference Analogues of the Logistic Equation with Delay. Modeling and Analysis of Information Systems. 2020;27(3):344-355. (In Russ.) https://doi.org/10.18255/1818-1015-2020-3-344-355
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