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    Spreading Speed, Traveling Waves, and Minimal Domain Size in Impulsive Reaction–Diffusion Models

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    Type
    Article
    Authors
    Lewis, Mark A.
    Li, Bingtuan
    KAUST Grant Number
    KUK-CI013-04
    Date
    2012-08-15
    Online Publication Date
    2012-08-15
    Print Publication Date
    2012-10
    Permanent link to this record
    http://hdl.handle.net/10754/599701
    
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    Abstract
    How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results. © 2012 Society for Mathematical Biology.
    Citation
    Lewis MA, Li B (2012) Spreading Speed, Traveling Waves, and Minimal Domain Size in Impulsive Reaction–Diffusion Models. Bull Math Biol 74: 2383–2402. Available: http://dx.doi.org/10.1007/s11538-012-9757-6.
    Sponsors
    The authors thank two anonymous reviewers for their helpful comments and suggestions. This research was supported by NSERC Discovery and Accelerator grants and by a Canada Research Chair. M. A. L. gratefully acknowledges a Research Fellowship from Oxford Centre for Collaborative and Applied Mathematics, supported by Award No KUK-CI013-04 made by King Abdullah University of Science and Technology (KAUST).This research was partially supported by the National Science Foundation under Grant DMS-616445 and Grant DMS-1225693.
    Publisher
    Springer Nature
    Journal
    Bulletin of Mathematical Biology
    DOI
    10.1007/s11538-012-9757-6
    PubMed ID
    22893042
    ae974a485f413a2113503eed53cd6c53
    10.1007/s11538-012-9757-6
    Scopus Count
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