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

For citation:

Perevaryukha A. Y. Transition from relaxation oscillations to pseudoperiodic trajectory in the new model of population dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, iss. 2, pp. 51-62. DOI: 10.18500/0869-6632-2017-25-2-51-62

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: 161)
Article type: 
519.6, 517.926

Transition from relaxation oscillations to pseudoperiodic trajectory in the new model of population dynamics

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

In this article we consider simulation of appearance oscillations in abundance of biological species and concomitant abrupt changes in the population development process. The inductors for the appearance of many long-period fluctuations, such as herring cycles in the Pacific Ocean, have not yet been fully established. Changes in population cycles can take extreme nature and violate prevailing uniformity of biological diversity, even evoke significant impacts on ecosystems. Certain variation of abundance outbreaks can grow long enough in the form of series of peaks and lead to temporary habitat degradation. Spontaneous oscillations can occur in isolated laboratory populations without any external influence, while maintaining permanent experimental conditions. For the mathematical explanation of the internal mechanisms for the appearance of oscillatory regimes are used differential equations with delay. Widely known several such models Nicholson, Cushing, and Hutchinson equations – greater detail studied. Models show the occurrence of complicated forms of cycles after the Andronov–Hopf bifurcation. Features of relaxation oscillations in these models, like huge amplitude peaks at unreal depth lows, very few correspond to ecological reality. Burst to peak values of a population usually start after overcoming some threshold when suddenly action of mechanisms of regulation abruptly reduced. Modification of equation with delay remains an actual task for analysis of particular scenarios that are observed in reality for the population, such as the specific cases or insect outbreaks. We have proposed the modification of the Hutchinson model to describe the rare but critical for the condition of the forest scenario of population dynamics – the outbreak in the form of a series of peaks, but without intermediate minimums approaching to zero. Any outbreak of is a transient regime of existence of species within the ecosystem. Further, we have developed a basically different model, which involve one of the observed scenarios for completion of such process and reviewing the traditional interpretation of the parameter threshold capacity of environment. Relaxation oscillations occurring after the bifurcation in the new equation with increasing amplitude do not form a stable orbital cycle, but they are rapidly moving in infinitely increasing pseudo-periodic trajectory. The loss of properties of being dissipative trajectory is indicates fatal consequences for the existence of species in the local natural environment. Model with abnormal transition has practical significance in the description for the situations of invasion of alien species in isolated ecosystems and breeding of pests in agricultural areas already adapt to toxic chemicals. In the field of system ecology scenario of uncontrolled of resource exhaustion historically associated with the problem of Easter Island.

  1. Verhulst P. Recherches mathematiques sur la loi d’accroissement de la population. Nouveaux memoires de l’academie royale des sciences et belles-letters de Bruxelles. 1845. Vol.18. P. 1–38.
  2. Bacaer N. A Short History of Mathematical Population Dynamics. London, Springer-Verlag, 2011.
  3. Bazykin A.D. Theoretical and mathematical ecology: dangerous boundaries and criteria of approaсh them. Mathematics and Modelling. Nova Sci. Publishers, N.-Y., 1993. P. 321–328.
  4. Nicholson A. An outline of the dynamics of animal populations. Australian Journal of Zoology. 1954. Vol. 2. Issue 1. P. 9–65.
  5. Hutchinson G. An Introduction to Population Ecology. New Haven, Yale University Press, 1978.
  6. Kashchenko S.A. Dynamics of the logistic equation with delay. Mathematical Notes. 2015. Vol. 98. P. 98–110.
  7. Kolesov A.Yu., Kolesov Yu.S. Relaxation Oscillations in Mathematical Models of Ecology. Rhode Island, AMS, 1993.
  8. Kolesov A.Yu., Mishchenko E.F., Rozov N.K. A modification of Hutchinson’s equation. Computational Mathematics and Mathematical Physics. 2010. Vol. 50. Issue 12. P. 1990–2002.
  9. Gopalsamy K., Kulenovic M., Ladas G. Time lags in a «food-limited» population madel. Applicable Analysis. 1988. Vol. 31. Issue 3. P. 225–237.
  10. Ladas G., Qian C. Oscillation and global stability in a delay logistic equation. Dynamics and Stability of Systems. 1994. Vol. 9. Issue 2. P. 153–162.
  11. Ruan S. Delay Differential Equations in Single Species Dynamics. In the book: Delay Differential Equations and Applications. Springer, Berlin, 2006. P. 477–517.
  12. Lin X., Wang H. Stability analysis of delay differential equations with two discrete delays. Canadian applied mathematics quarterly. 2012. Vol. 20. Issue 4. P. 519–533.
  13. Perevaryukha A.Y. A model of development of a spontaneous outbreak of an insect with aperiodic dynamics. Entomological Review. 2015.Vol. 95. Issue 3. P. 397–405.
  14. Cooke B., Nealis V., Regniere J. Insect defoliators as periodic disturbances in northern forest ecosystems. Plant Disturbance Ecology: The Process and the Response. Elsevier, Burlington, 2007. P. 487–525.
  15. Gray D.R. Historical spruce budworm defoliation records adjusted for insecticide protection in New Brunswick. Journal of the Acadian Entomological Society. 2007. Vol. 115. Issue 1. P. 1–6.
Short text (in English):
(downloads: 125)