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Pattern Formation and Bistability in a Generalist Predator-Prey Model

Author

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  • Vagner Weide Rodrigues

    (Instituto Federal do Rio Grande do Sul–Campus Bento Gonçalves, Bento Gonçalves RS 95700-206, Brazil
    These authors contributed equally to this work.)

  • Diomar Cristina Mistro

    (Departamento de Matemática, Universidade Federal de Santa Maria, Santa Maria RS 97105-900, Brazil
    These authors contributed equally to this work.)

  • Luiz Alberto Díaz Rodrigues

    (Departamento de Matemática, Universidade Federal de Santa Maria, Santa Maria RS 97105-900, Brazil
    These authors contributed equally to this work.)

Abstract

Generalist predators have several food sources and do not depend on one prey species to survive. There has been considerable attention paid by modellers to generalist predator-prey interactions in recent years. Erbach and collaborators in 2013 found a complex dynamics with bistability, limit-cycles and bifurcations in a generalist predator-prey system. In this paper we explore the spatio-temporal dynamics of a reaction-diffusion PDE model for the generalist predator-prey dynamics analyzed by Erbach and colleagues. In particular, we study the Turing and Turing-Hopf pattern formation with special attention to the regime of bistability exhibited by the local model. We derive the conditions for Turing instability and find the region of parameters for which Turing and/or Turing-Hopf instability are possible. By means of numerical simulations, we present the main types of patterns observed for parameters in the Turing domain. In the Turing-Hopf range of the parameters, we observed either stable patterns or homogeneous periodic distributions. Our findings reveal that movement can break the effect of hysteresis observed in the local dynamics, what can have important implication in pest management and species conservation.

Suggested Citation

  • Vagner Weide Rodrigues & Diomar Cristina Mistro & Luiz Alberto Díaz Rodrigues, 2019. "Pattern Formation and Bistability in a Generalist Predator-Prey Model," Mathematics, MDPI, vol. 8(1), pages 1-18, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2019:i:1:p:20-:d:300207
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    References listed on IDEAS

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    1. Marten Scheffer & Steve Carpenter & Jonathan A. Foley & Carl Folke & Brian Walker, 2001. "Catastrophic shifts in ecosystems," Nature, Nature, vol. 413(6856), pages 591-596, October.
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    1. Rana, Sourav & Bhattacharya, Sabyasachi & Samanta, Sudip, 2022. "Spatiotemporal dynamics of Leslie–Gower predator–prey model with Allee effect on both populations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 32-49.
    2. Mondal, Bapin & Thirthar, Ashraf Adnan & Sk, Nazmul & Alqudah, Manar A. & Abdeljawad, Thabet, 2024. "Complex dynamics in a two species system with Crowley–Martin response function: Role of cooperation, additional food and seasonal perturbations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 415-434.

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