ISSUE 180, article 7


Kibern. vyčisl. teh., 2015, Issue 179, pp 83-92.

Chernyshenko Sergei V., Dr (Biology), PhD (Phys. and Math.), Prof., Head of the Department of Applied Mathematics and Social Informatics of Khmelnytsky National University, st. Institutskaya, 11, Khmelnitsky, 29016, Ukraine, e-mail:

Ruzich Roman V., Assistant of the Department of Applied Mathematics and Social Informatics of Khmelnitsky National University, st. Institutskaya, 11, Khmelnitsky, 29016, Ukraine, e-mail:


Introduction. A long-term ecological successions are considered as step-by-step process. The continuous model (model of open Eigen’s hypercycle) is used to describe this process.

The purpose of the paper is to investigate non-linear properties of the system, which define discrete processes that occur in the one.

Results. The multi-dimension case of the model of open Eigen’s hypercycle has been analyzed. It is shown that in many cases the consideration of dynamics of the -dimensional system can be simplified by partial reduction to -dimensional cases.

It is mathematically shown that evolution of system, which is described by the -dimensional model of open Eigen’s hypercycle has, as maximum, stages. Presence and duration of each stage are determined by the size of the ecological niche, as a characteristics of the environment. As an example: if the niche is very small (), there is only one association in the stable state of the ecosystem.

Conclusion. It is shown that the continuous model can describe discrete processes of sucessions. The quasi-discrete dynamics of the system is explained by its bifurcation properties, produced step-by-step changing of the system structure.

Keywords: succession, discrete process, continuous model, Eigen’s hypercycle, bifurcation.

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Received 17.03.2015