Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 39. no. 2. pp. 91–102. ISSN 2079-6641

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MSC 37N25, 92C42, 81T80 

Research Article

Scenario of the invasive process in the modification of Bazykins population equation with delayed regulation and high reproductive potential

A.Yu. Perevaryukha

St. Petersburg Federal Research Center of the Russian Academy of Sciences.


The paper discusses modeling of the variant of the development of a rapid invasive process in competitive biosystems. The emergence of dangerous alien species leads to extreme phenomena in the dynamics of populations. Invasions generate a phase of active spread of the alien species, but outbreaks are often followed by a phase of sharp depression. Changes in the process are associated with active resistance, which has a delayed activation time interval and a threshold level of maximizing the impact J . For the mathematical formalization of the successively following stages of the outbreak/crisis, equations with a deviating argument are used. In a variant of the equation with a delayed tuning of the biotic reaction \dot x = rf(x(t-\tau)) - \mathfrak{F}(x^m(t-\nu); J) a variant of the passage of the crisis that occurs it is in the phase of rapid growth until a balance is reached with the resources of the environment. Due to the threshold feedback, the competitive pressure after a deep crisis is weakened and the invasive population goes into a mode of damped oscillations. The asymptotic level of equilibrium in the scenario with a crisis turns out to be much less than the theoretically permissible limiting level of abundance for an alien species in a given environment. The new Equation also has an interpretation to describe the weakening development of the immune response in a situation of chronicity of the infectious process.

Key words: modeling of extreme events, threshold effects, equations with delay, nonlinear ecological regulation.

DOI: 10.26117/2079-6641-2022-39-2-91–102

Original article submitted: 30.04.2022

Revision submitted: 02.06.2022

For citation. Perevaryukha A.Yu. Scenario of the invasive process in the modification of Bazykins population equation with delayed regulation and high reproductive potential. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 39: 2, 91–102. DOI: 10.26117/2079-6641-2022-39-2-91–102

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and responsibility. The author contributed to the writing of the article and is solely responsible for submitting the final version of the article to the press. The final version of the manuscript was approved by the author.

The content is published under the terms of the Creative Commons Attribution 4.0 International License (

© Perevaryukha A.Yu., 2022


  1. Simberloff D., Gibbons L. Now you see them, Now you don’t! – Population crashes of established introduced species, Biological Invasions, 2004, vol. 6, no. 2, pp. 116-172
  2. Peleg M., Corradini M. G., Normand M. D. The logistic (Verhulst) model for sigmoid microbial growth curves revisited, Food Research International, 2007, vol. 40, no. 7, pp 808-818
  3. Kurkin A. A., Kurkina O. E., Pelinovsky E. N. Logistic models of the spread of epidemics, Proceedings of NSTU im. R.E. Alekseeva, 2020, vol. 2, no. 2. pp. 9-18 (In Russian).
  4. Hutchinson G. E. Circular causal systems in ecology, Ann. New York Acad. Sci., 1948, vol. 50, no. 2, pp. 221-246
  5. Glyzin S. D., Kolesov A. Y., Rozov N. K. A self-symmetric cycle in a system of two diffusely connected Hutchinson’s equations, Sbornik: Mathematics, 2019, vol. 210, no. 2, pp. 184-233
  6. Kaschenko S. A., Loginov D. O. About global stable of solutions of logistic equation with delay, Journal of Physics: Conf. Series, 2017. vol. 937, no. 2, pp. 120-139
  7. Gopalsamy K. Persistence and Global Stability in a Population Model, Journal of Mathematical Analysis and Applications, 1998, vol. 224, no. 3, pp. 59-80
  8. Khokhlov A. D. Population survival conditions in Nicholson’s models with delay, Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2010, vol. 30, no. 3, pp. 29–32
  9. Hale J. Persistence in infinite dimensional systems, SIAM J. Math. Anal, 1989, vol. 20, no. 4, pp. 388-395
  10. Glyzin D. S., Kashchenko S. A., Polstyanov A. S. Spatially inhomogeneous periodic solutions of the Hutchinson equation with distributed saturation, Modeling and analysis of information systems, 2011, vol. 7, no. 1, pp. 37-45
  11. Kolesov A. Y., Mishchenko E. F., Rozov N. K. A modification of Hutchinson’s equation, Computational Mathematics and Mathematical Physics, 2010, vol. 50, no. 12, pp. 1990-2002
  12. Bazykin A. D., Aponina E. A.Model of an ecosystem of three trophic levels taking into account the existence of a lower critical density of the producer population, Problemy ekologicheskogo monitoringa i modelirovaniya ekosistem, 1981, vol. 4, no. 2, pp. 186-203.
  13. Buck J., Hechinger R. Host density increases parasite recruitment but decreases host risk in a snail-trematode system, Ecology, 2017, vol. 98, no. 8, pp. 2029-2038
  14. Perevaryukha A.Yu. An iterative continuous-event model of the population outbreak of a phytophagous hemipteran, Biophysics, 2016, vol. 61, no. 2, pp. 334-341
  15. Aarde V., Whyte T., Pimm L. Culling and the dynamics of the Kruger National Park African elephant population, Animal Conservation, 1999, vol. 2, no. 4, pp. 287-294
  16. Borisova T.Yu., Solovieva I. V. Problematic aspects of modeling population processes and criteria for their agreement, Mathematical machines and systems, 2017, vol. 1, no. 1, pp. 71-81 (In Russian).
  17. Nikitina A. V. Optimal control of sustainable development in the biological rehabilitation of the Azov Sea, Math. Mod. Comp. Simul. 2017, vol. 2, no. 1, pp. 101-107.

Perevaryukha Andrey Yurievich – PhD (Tech.), senior reseacher, Saint Petersburg Federal Research Center of the Russian Academy of Sciences, senior reseacher, Saint Petersburg, Russian, ORCID 0000-0002-1049-0096.