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Numerical Study of Bifurcations Occurring at Fast Timescale in a Predator–Prey Model with Inertial Prey-Taxis

  • Yuri V. TyutyunovEmail author
  • Anna D. Zagrebneva
  • Vasiliy N. Govorukhin
  • Lyudmila I. Titova
Chapter
Part of the STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health book series (STEAM)

Abstract

Bifurcations occurring in a system of partial differential equations (PDEs) describing spatiotemporal dynamics of predator and prey populations with prey-taxis have been studied numerically. The model of the local kinetics of the system assumes logistic reproduction of the prey and a simplest Lotka–Volterra functional response of the predator. Since the model ignores relatively slow and rare demographic processes of birth and death in the population of predator, the predator abundance is kept constant under the considered zero-flux boundary conditions. The abundance of predator populations together with the predator taxis coefficient were used as bifurcation parameters in the numerical study that have been made with help of two qualitatively different techniques of discretization: the Bubnov–Galerkin method and grid method of lines. It has been shown that the considered simple model of prey-taxis in predator–prey system demonstrates complex bifurcation transitions leading to periodic, quasi-periodic and chaotic spatiotemporal dynamics.

Keywords

Population dynamics Bifurcations Numerical analysis Taxis Spatially heterogeneous periodic dynamics Quasi-periodic regime Spatiotemporal chaos Multistability 

Notes

Acknowledgments

The research was funded by the project 0259-2014-0004 (state reg.no. 01201363188) of SSC RAS “Development of GIS-based methods of modelling marine and terrestrial ecosystems” (Tyutyunov), by the basic part of the state assignment research, project 1.5169.2017/8.9 of the Southern Federal University “Fundamental and applied problems of mathematical modelling” (Titova), and RFBR grant 18-01-00453 “Multistable spatiotemporal scenarios for population models” (Tyutyunov, Zagrebneva, Govorukhin).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yuri V. Tyutyunov
    • 1
    • 2
    Email author
  • Anna D. Zagrebneva
    • 3
  • Vasiliy N. Govorukhin
    • 4
  • Lyudmila I. Titova
    • 4
  1. 1.Federal Research Center The Southern Scientific Centre of the Russian Academy of Sciences (SSC RAS)Rostov-on-DonRussia
  2. 2.Southern Federal UniversityRostov-on-DonRussia
  3. 3.Department of Computer and Computer-Based System Software, Faculty of IT Systems and TechnologiesDon State Technical UniversityRostov-on-DonRussia
  4. 4.Vorovich Institute of Mathematics, Mechanics and Computer SciencesSouthern Federal UniversityRostov-on-DonRussia

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