Investigation in general of the behaviour of the trajectories of a predator – prey system

Authors

  • Adam D. Ushkho Adyghe State University, 208 Pervomayskaya Street, Maykop 385000, Russia
  • Damir S. Ushkho Adyghe State University, 208 Pervomayskaya Street, Maykop 385000, Russia

Keywords:

A. N. Kolmogorov, predator – prey system, global phase portrait, Poincare disk, equilibrium states, limit cycle
Supporting Agencies
The authors are grateful to professor V. B. Tlyachev for useful comments, as well as to the Adygheya Republic Physical Society for partial financial support (grant No. 2022/01).

Abstract

By the methods of the classical qualitative theory of dynamical systems on the plane, the problem of constructing a phase portrait of Kolmogorov’s predator – prey system has been solved in general. Possible topological structures of this model are considered for six cases of coefficient conditions with positive values of three parameters. The phase portraits in the Poincare disk are constructed by dividing the set of values of one of the parameters into intervals. The values of this parameter are found at which the self-oscillation mode is possible in the system. It is shown that a weak focus of order 1 (multiplicity 1) is not surrounded by closed trajectories. Based on the analysis of the location of the main isoclines of the system on the entire phase plane, exclusively geometrically, the topological structure of a complex equilibrium state at infinity is established without relying on known analytical (more time-consuming) methods.

Author Biographies

  • Adam D. Ushkho, Adyghe State University, 208 Pervomayskaya Street, Maykop 385000, Russia

    PhD (physics and mathematics), docent; head of the department of theoretical physics, faculty of engineering and physics

  • Damir S. Ushkho, Adyghe State University, 208 Pervomayskaya Street, Maykop 385000, Russia

    PhD (physics and mathematics), docent; associate professor at the department of mathematical analysis and methods of teaching mathematics, faculty of mathematics and computer science

References

  1. Diz-Pita É, Llibre J, Otero-Espinar MV. Global phase portraits of a predator – prey system. Electronic Journal of Qualitative Theory of Differential Equations. 2022;16:1–13. DOI: 10.14232/ejqtde.2022.1.16.
  2. Bautin NN, Leontovich EA. Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti [Methods and techniques of the qualitative study of dynamical systems on the plane]. Moscow: Nauka; 1976. 496 p. (Spravochnaya matematicheskaya biblioteka). Russian.
  3. Otrokov NF. Analiticheskie integraly i predel’nye tsikly [Analytical integrals and limit cycles]. Gorky: Volgo-Vyatskoe knizhnoe izdatel’stvo; 1972. 216 p. Russian.
  4. Andronov AA, Leontovich EA, Gordon II, Mayer AG. Kachestvennaya teoriya dinamicheskikh sistem vtorogo poryadka [Qualitative theory of dynamical systems of the second order]. Moscow: Nauka; 1966. 568 p. Russian.
  5. Andronov AA, Leontovich EA, Gordon II, Mayer AG. Teoriya bifurkatsii dinamicheskikh sistem na ploskosti [Theory of bifurcations of dynamical systems on a plane]. Moscow: Nauka; 1967. 488 p. Russian.
  6. Poincare A. O krivykh, opredelyaemykh differentsial’nymi uravneniyami [On curves defined by differential equations]. Leontovich E, Mayer A, translators; Andronov AA, editor. Moscow: OGIZ; 1947. 392 p. Co-published by the Gosudarstvennoe izdatel’stvo tekhniko-teoreticheskoi literatury (Klassiki estestvoznaniya. Matematika. Mekhanika. Fizika. Astronomiya). Russian.
  7. Frommer M. [Integral curves of an ordinary differential equation of the first order in the vicinity of a singular point having a rational character]. Uspekhi matematicheskikh nauk. 1941;9:212–253. Russian.
  8. Liou Lii-Perng, Cheng Kuo-Shung. On the uniqueness of a limit cycle for a predator – prey system. SIAM Journal on Mathematical Analysis. 1988;19(4):867–878. DOI: 10.1137/0519060.
  9. Sadovskii AP, Shcheglova TV. [Centers of a cubic system with eleven parameters]. Vestnik BGU. Seriya 1, Fizika. Matematika. Informatika. 2011;1:71–75. Russian.
  10. Sidorenko IN. [Limit cycles of normal size of some classes of quadratic systems on the plane]. Vestnik BGU. Seriya 1, Fizika. Matematika. Informatika. 2008;3:63–68. Russian.

Downloads

Additional Files

Published

2023-03-23

Issue

No. 1 (2023): Journal of the Belarusian State University. Mathematics and Informatics

Section

Differential Equations and Optimal Control

License

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)

How to Cite

[1]
Ushkho, A.D. and Ushkho, D.S. 2023. Investigation in general of the behaviour of the trajectories of a predator – prey system. Journal of the Belarusian State University. Mathematics and Informatics. 1 (Mar. 2023), 20–30. DOI:https://doi.org/10.33581/2520-6508-2023-1-20-30.