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Features of the Algorithmic Implementation of Difference Analogs of the Delayed Logistic Equation

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Abstract

The logistic equation with delay, or the Hutchinson equation, is one of the fundamental equations of population dynamics and is widely used in problems of mathematical ecology. A family of mappings built for this equation based on central separated differences is considered. Such finite-difference schemes are usually applied 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. The properties of the trajectories of these mappings and the original delayed equation are compared. 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 delayed logistic equation. In particular, this map has no 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 demonstrated that such a finite-difference scheme cannot be used for numerical modeling of the logistic equation with delay.

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Funding

The work is supported by the Russian Foundation for Basic Research, project no. 18-29-10043.

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Authors and Affiliations

  1. Center of Integrable Systems, Demidov Yaroslavl State University, 150003, Yaroslavl, Russia

    S. D. Glyzin, S. A. Kashchenko & A. O. Tolbey

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  1. S. D. Glyzin
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  2. S. A. Kashchenko
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  3. A. O. Tolbey
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Correspondence to S. D. Glyzin, S. A. Kashchenko or A. O. Tolbey.

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The authors declare that they have no conflicts of interest.

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Translated by E. Oborin

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Glyzin, S.D., Kashchenko, S.A. & Tolbey, A.O. Features of the Algorithmic Implementation of Difference Analogs of the Delayed Logistic Equation. Aut. Control Comp. Sci. 55, 723–730 (2021). https://doi.org/10.3103/S014641162107004X

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  • DOI: https://doi.org/10.3103/S014641162107004X

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