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Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

Author

Listed:
  • Weam Alharbi

    (Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK)

  • Sergei Petrovskii

    (Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK)

Abstract

A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that causes species extinction worldwide, we consider the reaction–telegraph equation (i.e., telegraph equation combined with the population growth) on a bounded domain with the goal to establish the conditions of species survival. We first show analytically that, in the case of linear growth, the expression for the domain’s critical size coincides with the critical size of the corresponding reaction–diffusion model. We then consider two biologically relevant cases of nonlinear growth, i.e., the logistic growth and the growth with a strong Allee effect. Using extensive numerical simulations, we show that in both cases the critical domain size of the reaction–telegraph equation is larger than the critical domain size of the reaction–diffusion equation. Finally, we discuss possible modifications of the model in order to enhance the positivity of its solutions.

Suggested Citation

  • Weam Alharbi & Sergei Petrovskii, 2018. "Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics," Mathematics, MDPI, vol. 6(4), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:4:p:59-:d:141488
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    References listed on IDEAS

    as
    1. Jacek Banasiak & Janusz R. Mika, 1998. "Singularly perturbed telegraph equations with applications in the random walk theory," International Journal of Stochastic Analysis, Hindawi, vol. 11, pages 1-20, January.
    2. E. Ahmed & H. A. Abdusalam & E. S. Fahmy, 2001. "On Telegraph Reaction Diffusion And Coupled Map Lattice In Some Biological Systems," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 12(05), pages 717-726.
    3. Antonio Di Crescenzo & Barbara Martinucci & Shelemyahu Zacks, 2018. "Telegraph Process with Elastic Boundary at the Origin," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 333-352, March.
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    Cited by:

    1. Antonio Di Crescenzo & Paola Paraggio, 2019. "Logistic Growth Described by Birth-Death and Diffusion Processes," Mathematics, MDPI, vol. 7(6), pages 1-28, May.
    2. Nikita Ratanov & Mikhail Turov, 2023. "On Local Time for Telegraph Processes," Mathematics, MDPI, vol. 11(4), pages 1-12, February.
    3. Alqubori, Omar & Petrovskii, Sergei, 2022. "Analysis of simulated trap counts arising from correlated and biased random walks," Ecological Modelling, Elsevier, vol. 470(C).

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