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Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models

Author

Listed:
  • Vitaly G. Il’ichev

    (Southern Scientific Centre of the Russian Academy of Sciences, 344006 Rostov-on-Don, Russia)

  • Dmitry B. Rokhlin

    (Institute of Mathematics, Mechanics and Computer Sciences, Regional Scientific and Educational Mathematical Center, Southern Federal University, 344090 Rostov-on-Don, Russia)

Abstract

We demonstrate a basic technique for simplifying time-periodic competition models, which is based on the utilization of periodic delta functions as population growth rates. We show that the Poincare mapping splits into a sequence of one-dimensional mappings. The study of the corresponding stable equilibria allows us to make conclusions concerning the coexistence and selection of the family of competitors. In particular, in “all vs. all” systems, for one of the populations to dominate, it is enough to surpass the others with a certain margin, and the correspondent stock constant does not depend on the number of competitors. We present paradoxical examples, where (1) a low-productive population can displace a highly productive one, (2) the displacement is non-transitive, (3) the coexistence is non-transitive. We also show how the delta functions can be utilized for the analysis of a “predator–prey” system.

Suggested Citation

  • Vitaly G. Il’ichev & Dmitry B. Rokhlin, 2023. "Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models," Mathematics, MDPI, vol. 12(1), pages 1-12, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:125-:d:1310465
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    References listed on IDEAS

    as
    1. Yuri V. Tyutyunov & Anna D. Zagrebneva & Andrey I. Azovsky, 2020. "Spatiotemporal Pattern Formation in a Prey-Predator System: The Case Study of Short-Term Interactions Between Diatom Microalgae and Microcrustaceans," Mathematics, MDPI, vol. 8(7), pages 1-15, July.
    2. Chunxiang Li & Fangshu Hui & Fangfei Li, 2023. "Stability of Differential Systems with Impulsive Effects," Mathematics, MDPI, vol. 11(20), pages 1-23, October.
    3. Vitaly G. Il’ichev & Dmitry B. Rokhlin, 2022. "Internal Prices and Optimal Exploitation of Natural Resources," Mathematics, MDPI, vol. 10(11), pages 1-14, May.
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