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

For citation:

Perevaryukha A. Y. Scenarios of the passage of the «population bottleneck» by an invasive species in the new model of population dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 5, pp. 63-80. DOI: 10.18500/0869-6632-2018-26-5-63-80

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 606)
Article type: 
519.6, 517.926

Scenarios of the passage of the «population bottleneck» by an invasive species in the new model of population dynamics

Perevaryukha A. Yu., St. Petersburg Institute for Informatics and Automation of RAS

Topic. The subject of the article is the expansion of the author’s research series in the direction of mathematical modeling of specific ecological situations and transitional regimes that arise in nonlinear population processes with complex internal regulation. Aim. The purpose of the article is to develop methods for modeling difficult-to-predict and abrupt changes in the ecology of communities of competing species. Such events occur after the invasion and adaptation of a species with a potentially high reproductive potential to a new area with favorable reproduction conditions. Relevance. The significance of the environmental problem we are considering based on the fact that when developing a rapid outbreak of insect, ordinary and easily mathematically formalized principles for regulating the efficiency of population reproduction do not work. In scenarios related to the manifestation of extreme population dynamics, traditional models of mathematical biology for describing asymptotic growth of populations or stable oscillatory regimes will not be applicable. Method. Activation of inclusion in the active counteraction for aggression is often significantly delayed, therefore, differential equations with delay are chosen by the mathematical description of transient situations. We believe in the development of new models that outbreaks of populations are a group of phenomena that are heterogeneous in their dynamic characteristics, stages and causes. The outbreaks of Hemiptera’s insects differ in terms of phases and duration from invasions of Lepidoptera pests. Variants of development and completion of outbreaks differ. Result. The main result of our work will be a model scenario based on modifications of differential equations with delay, when after invasive invasion the invading species passes through the «bottleneck mode» – of a critically small group of individuals, capable of further survival only under favorable conditions. We solved the problem of bottleneck in the population dynamics when considering in the modification of the active counteraction model of invasion, it is assumed that the undesirable species is able to substantially transform its new biotic environment. Discussion. One of the considered computational scenarios leads to the destruction of the cyclic regime that appeared in the equation, which reflects the elimination of the dangerous competitor species from the new habitat. Real biological processes of insect dynamics provide for several variants of final paths. An alternative version of the scenario obtained by us in the modification of the equation with independent withdrawal is the completion of the outbreak of the alien species with the formation of a stable small group, perhaps in the hidden shelter for the so-called «refugium mode».

  1. Perevaryukha A.Yu. Transition from relaxation oscillations to pseudoperiodic trajectory in the new model of population dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, no. 2, pp. 51–62 (in Russian).
  2. Reshetnikov Yu.S., Tereshchenkov V.G. Quantitative level of research in fish ecology and errors associated with it. Russian Journal of Ecology, 2017, vol. 48, pp. 233–239.
  3. Halls A. Dynamics of river fish populations in response to hydrological conditions: A simulation study. River Research and Applications, 2004, vol. 20, pp. 985–1000.
  4. Slynko Yu.V., Dgebuadze U.U., Novitskiy U.A., Kchristov A.O. Invasions of alien fishes in the basins of the largest rivers of the Ponto-Caspian Basin: Composition, vectors, invasion routes, and rates. Russian Journal of Biological Invasions, 2011, no. 1, pp. 74–89.
  5. Daunys D. Impact of the zebra mussel Dreissena polymorpha invasion on the budget of suspended material in a shallow lagoon ecosystem. Helgoland Marine Research, 2006, vol. 60, pp. 113–121.
  6. Gilg O., Sittler T. Climate change and cyclic predator–prey population dynamics in the high Arctic. Global Change Biology, 2009, vol. 15, pp. 2634–2652.
  7. Odum H.T Systems Ecology. New York, Wiley, 1983, p. 644.
  8. URL:
  9. Rath D. Amlinger L. The CRISPR-Cas immune system: Biology, mechanisms and applications. Biochimie, 2015, vol. 117, pp. 119–128.
  10. Hutchinson G.E. Circular causal systems in ecology. Ann. New York Acad. Sci., 1948, vol. 50, pp. 221–246.
  11. Shirsat N. Revisiting Verhulst and Monod models: Analysis of batch and fed-batch cultures. Cytotechnology, 2015, Vol. 67, pp. 515–530.
  12. Hadeler K.P. Where to put delays in population models, in particular in the neutral case. Canadian Applied Mathematics Quarterly, 2003, vol. 11, no. 2, pp. 150–173.
  13. Birch D., Colin T. A New generalized logistic sigmoid growth equation compared with the Richard’s growth equation. Annals of Botany, 1999, vol. 83, no. 6, pp. 713–723.
  14. Bazykin A.D. Nonlinear Dynamics of Interacting Populations. London, WSP, 1998, 198 p.
  15. Hutchinson G.E. An Introduction to Population Ecology. New Haven, Yale University Press, 1978, 234 p.
  16. Balanova Z., Ruan H.G.E. Hutchinson’s delay logistic system with symmetries and spatial diffusion. Nonlinear Analysis: Real World Applications, 2008, vol. 9, pp. 154–182.
  17. Jones G.S. The existence of periodic solutions of f′(x) = αf(x(t − 1))1 + f(x). J. Math. Anal. Appl., 1962, vol. 5, pp. 435–450.
  18. Kashchenko S.A. Dynamics of the logistic equation with delay. Mathematical Notes, 2015, vol. 98, no. 2, pp. 98–110.
  19. Sakanoue S. Extended logistic model for growth of single-species populations. Ecological Modeling, 2007, vol. 2005, no. 1, pp. 159–168. 
  20. Gopalsamy K., Kulenovic M., Ladas G. Time lags in a «food-limited» population model. Applicable Analysis, 1988, vol. 31, no. 3, pp. 225–237.
  21. Kolesov A., Mishchenko E., Kolesov Yu. A modification of Hutchinson’s equation. Computational Mathematics and Mathematical Physics, 2010, no. 12, pp. 1990– 2002.
  22. Perevaryukha A.Yu. Comparative modeling of two special scenarios of extreme dynamics in forest ecosystems: Psillides in Australia and spruce budworm moth in Canada. Journal of Automation and Information Sciences, 2018, no. 5, pp. 22–33.
  23. Hutchings J.A. Renaissance of a caveat: Allee effects in marine fish. ICES Journal of Marine Science, 2014, vol. 71, no. 8, pp. 2152–2157.
  24. Roughgarden J. Why fisheries collapse and what to do about it. Proc. Natl. Acad. Sci. USA, 1996, vol. 93, pp. 5078–5083.
  25. Perevaryukha A.Yu. Computer modeling of sturgeon population of the Caspian sea with two types of aperiodic oscillations. Radio Electronics Computer Science Control, 2015, vol. 1, no. 1, pp. 26–32.
  26. Pushkin С.V. Introduction of striped leaf-eating insect Zygoramma suturalis (Coleoptera, Chrysomelidae) in Stavropol territory. Russian Journal of Biological Invasions, 2008, no. 1, pp. 42–44.
  27. Gause G.F. The Struggle for Existence. Baltimore, Williams & Wilkins, 1934, 163 p.
  28. Meyerhans А. Virus-host interactions URL:
  29. Ludwig D., Jones D., Holling S. Qualitative analysis of insect outbreak systems: The spruce budworm and forest. The Journal of Animal Ecology, 1978, vol. 47, no. 1, pp. 315–332.
  30. Wan H. Biology and natural enemies of Cydalima perspectalis in Asia: Is there biological control potential in Europe? Journal of Applied Entomology, 2014, vol. 138, no. 10, pp. 715–722.
  31. Verner J., Zennaro M. Continuous explicit Runge–Kutta methods of order 5. Mathematics of computation, 1995, vol. 64, pp. 1123–1146.
  32. Baker T., Paul H. Computing stability regions Runge–Kutta methods for delay differential equations. IMA Journal of Numerical Analysis, 1994, vol. 14, no. 4, pp. 347–362.
Short text (in English):
(downloads: 115)