From local bi- and quadro-stability to space-time inhomogeneity: a review of mathematical models and meaningful conclusions

Frisman E.Y., Kulakov M.P.
 pdf (3951K)

Bistability is a fundamental property of nonlinear systems and is found in many applied and theoretical studies of biological systems (populations and communities). In the simplest case it is expressed in the coexistence of diametrically opposed alternative stable equilibrium states of the system, and which of them will be achieved depends on the initial conditions. Bistability in simple models can lead to quad-stability as models become more complex, for example, when adding genetic, age and spatial structure. This occurs in different models from completely different subject area and leads to very interesting, often counterintuitive conclusions. In this article, we review such situations. The paper deals with bifurcations leading to bi- and quad-stability in mathematical models of the following biological objects. The first one is the system of two populations coupled by migration and under the action of natural selection, in which all genetic diversity is associated with a single diallelic locus with a significant difference in fitness for homo- and heterozygotes. The second is the system of two limited populations described by the Bazykin model or the Ricker model and coupled by migration. The third is a population with two age stages and density-dependent regulation of birth rate which is determined either only by population density, or additionally depends on the genetic structure of adjacent generations. We found that all these models have similar scenarios for the birth of equilibrium states that correspond to the formation of spatiotemporal inhomogeneity or to the differentiation by phenotypes of individuals from different age stages. Such inhomogeneity is a consequence of local bistability and appears as a result of a combination of pitchfork bifurcation (period doubling) and saddle-node bifurcation.

Keywords: population, dynamics, age structure, migration, genetic divergence, bistability, bifurcations
Citation in English: Frisman E.Y., Kulakov M.P. From local bi- and quadro-stability to space-time inhomogeneity: a review of mathematical models and meaningful conclusions // Computer Research and Modeling, 2023, vol. 15, no. 1, pp. 75-109
Citation in English: Frisman E.Y., Kulakov M.P. From local bi- and quadro-stability to space-time inhomogeneity: a review of mathematical models and meaningful conclusions // Computer Research and Modeling, 2023, vol. 15, no. 1, pp. 75-109
DOI: 10.20537/2076-7633-2023-15-1-75-109
URL: http://crm-en.ics.org.ru/journal/article/3290/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Indexed in Scopus

Full-text version of the journal is also available on the web site of the scientific electronic library eLIBRARY.RU

The journal is included in the Russian Science Citation Index

The journal is included in the RSCI

International Interdisciplinary Conference "Mathematics. Computing. Education"

Copyright © 2009–2024 Institute of Computer Science