Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics
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...
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MDPI AG
2018-04-01
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| Series: | Mathematics |
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| Online Access: | http://www.mdpi.com/2227-7390/6/4/59 |
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| author | Weam Alharbi Sergei Petrovskii |
| author_facet | Weam Alharbi Sergei Petrovskii |
| author_sort | Weam Alharbi |
| collection | DOAJ |
| description | 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. |
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| language | English |
| last_indexed | 2024-12-11T06:20:03Z |
| publishDate | 2018-04-01 |
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| spelling | doaj.art-160656b27f594e0c8a3c5331a06442732022-12-22T01:17:51ZengMDPI AGMathematics2227-73902018-04-01645910.3390/math6040059math6040059Critical Domain Problem for the Reaction–Telegraph Equation Model of Population DynamicsWeam Alharbi0Sergei Petrovskii1Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UKDepartment of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UKA 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.http://www.mdpi.com/2227-7390/6/4/59animal movementfragmented environmentcritical sizeextinction |
| spellingShingle | Weam Alharbi Sergei Petrovskii Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics Mathematics animal movement fragmented environment critical size extinction |
| title | Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics |
| title_full | Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics |
| title_fullStr | Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics |
| title_full_unstemmed | Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics |
| title_short | Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics |
| title_sort | critical domain problem for the reaction telegraph equation model of population dynamics |
| topic | animal movement fragmented environment critical size extinction |
| url | http://www.mdpi.com/2227-7390/6/4/59 |
| work_keys_str_mv | AT weamalharbi criticaldomainproblemforthereactiontelegraphequationmodelofpopulationdynamics AT sergeipetrovskii criticaldomainproblemforthereactiontelegraphequationmodelofpopulationdynamics |