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On linear perturbations of the Ricker model.

Elena Braverman1, Damir Kinzebulatov

  • 1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB, Canada T2N 1N4. maelena@math.ucalgary.ca <maelena@math.ucalgary.ca>

Mathematical Biosciences
|June 27, 2006
PubMed
Summary

This study analyzes perturbed Ricker models, finding that stable equilibria in recruitment models ensure global stability. For high growth rates, population extinction or stable two-cycle periods emerge depending on perturbation type.

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Area of Science:

  • Population Dynamics
  • Mathematical Ecology
  • Theoretical Biology

Background:

  • Scramble competition models, such as the Ricker map, are fundamental in population dynamics.
  • Understanding the impact of perturbations (recruitment and harvesting) on model stability is crucial.
  • Previous work, including the Levin and May conjecture, provides a basis for stability analysis.

Purpose of the Study:

  • To investigate the stability properties of linearly perturbed discrete-time single-species models.
  • To analyze the effects of recruitment and harvesting perturbations on population dynamics.
  • To explore the transition to chaos and emergent behaviors like extinction and periodic solutions.

Main Methods:

  • Analysis of linearly perturbed discrete-time single-species scramble competition models.

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  • Examination of stability and bistability for time-independent and time-dependent parameters.
  • Study of models with recruitment, harvesting, and random discrete constant perturbations.

    • Main Results:

    • For recruitment models, local stability of the positive equilibrium implies global stability, supporting the Levin and May conjecture.
    • High intrinsic growth rates lead to population extinction (depletion) or stable two-cycle periods (immigration).
    • Perturbed Ricker models with time-dependent parameters exhibit extinction, persistence, and periodic solutions.

    Conclusions:

    • Linear perturbations significantly alter population dynamics, influencing stability and long-term outcomes.
    • The Ricker model's behavior under perturbation is complex, with distinct outcomes based on growth rates and perturbation types.
    • Results offer insights into ecological stability, chaos, and population persistence under environmental fluctuations.