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Mathematics 2018, 6(4), 59; https://doi.org/10.3390/math6040059

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

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK
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Received: 28 February 2018 / Revised: 5 April 2018 / Accepted: 13 April 2018 / Published: 17 April 2018
(This article belongs to the Special Issue Progress in Mathematical Ecology)
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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. View Full-Text
Keywords: animal movement; fragmented environment; critical size; extinction animal movement; fragmented environment; critical size; extinction
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Alharbi, W.; Petrovskii, S. Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics. Mathematics 2018, 6, 59.

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