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Interactions Obtained from Basic Mechanistic Principles: Prey Herds and Predators

Author

Listed:
  • Cecilia Berardo

    (Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland)

  • Iulia Martina Bulai

    (Department of Mathematics, Informatics and Economics, University of Basilicata, I-85100 Potenza, Italy
    GNCS Research Group, INdAM, I-00185 Rome, Italy)

  • Ezio Venturino

    (GNCS Research Group, INdAM, I-00185 Rome, Italy
    Department of Mathematics Giuseppe Peano, University of Torino, I-10100 Torino, Italy)

Abstract

We investigate four predator–prey Rosenzweig–MacArthur models in which the prey exhibit herd behaviour and only the individuals on the edge of the herd are subjected to the predators’ attacks. The key concept is the herding index, i.e., the parameter defining the characteristic shape of the herd. We derive the population equations from the individual state transitions using the mechanistic approach and time scale separation method. We consider one predator and one prey species, linear and hyperbolic responses and the occurrence of predators’ intraspecific competition. For all models, we study the equilibria and their stability and we give the bifurcation analysis. We use standard numerical methods and the software Xppaut to obtain the one-parameter and two-parameter bifurcation diagrams.

Suggested Citation

  • Cecilia Berardo & Iulia Martina Bulai & Ezio Venturino, 2021. "Interactions Obtained from Basic Mechanistic Principles: Prey Herds and Predators," Mathematics, MDPI, vol. 9(20), pages 1-18, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:20:p:2555-:d:654153
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    References listed on IDEAS

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    1. U. Dieckmann & M. Doebeli, 1999. "On the Origin of Species by Sympatric Speciation," Working Papers ir99013, International Institute for Applied Systems Analysis.
    2. Kumar, Sunil & Kumar, Ranbir & Cattani, Carlo & Samet, Bessem, 2020. "Chaotic behaviour of fractional predator-prey dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    3. Djilali, Salih, 2019. "Impact of prey herd shape on the predator-prey interaction," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 139-148.
    4. Bulai, Iulia Martina & Venturino, Ezio, 2017. "Shape effects on herd behavior in ecological interacting population models," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 40-55.
    5. Daniel I. Bolnick, 2001. "Intraspecific competition favours niche width expansion in Drosophila melanogaster," Nature, Nature, vol. 410(6827), pages 463-466, March.
    6. Ulf Dieckmann & Michael Doebeli, 1999. "On the origin of species by sympatric speciation," Nature, Nature, vol. 400(6742), pages 354-357, July.
    7. Bentout, Soufiane & Djilali, Salih & Kumar, Sunil, 2021. "Mathematical analysis of the influence of prey escaping from prey herd on three species fractional predator-prey interaction model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 572(C).
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    Cited by:

    1. Ezio Venturino, 2022. "Disease Spread among Hunted and Retaliating Herding Prey," Mathematics, MDPI, vol. 10(23), pages 1-21, November.

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