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Carrying Capacity of a Population Diffusing in a Heterogeneous Environment

Author

Listed:
  • D.L. DeAngelis

    (U.S. Geological Survey, Wetland and Aquatic Research Center, Gainesville, Florida, FL 32653, USA)

  • Bo Zhang

    (Department of Environmental Science and Policy, University of California at Davis, Davis, CA 95616, USA)

  • Wei-Ming Ni

    (School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
    School of Science and Engineering, Chinese University of Hong Kong-Shenzhen, Shenzhen 518000, China)

  • Yuanshi Wang

    (School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China)

Abstract

The carrying capacity of the environment for a population is one of the key concepts in ecology and it is incorporated in the growth term of reaction-diffusion equations describing populations in space. Analysis of reaction-diffusion models of populations in heterogeneous space have shown that, when the maximum growth rate and carrying capacity in a logistic growth function vary in space, conditions exist for which the total population size at equilibrium (i) exceeds the total population that which would occur in the absence of diffusion and (ii) exceeds that which would occur if the system were homogeneous and the total carrying capacity, computed as the integral over the local carrying capacities, was the same in the heterogeneous and homogeneous cases. We review here work over the past few years that has explained these apparently counter-intuitive results in terms of the way input of energy or another limiting resource (e.g., a nutrient) varies across the system. We report on both mathematical analysis and laboratory experiments confirming that total population size in a heterogeneous system with diffusion can exceed that in the system without diffusion. We further report, however, that when the resource of the population in question is explicitly modeled as a coupled variable, as in a reaction-diffusion chemostat model rather than a model with logistic growth, the total population in the heterogeneous system with diffusion cannot exceed the total population size in the corresponding homogeneous system in which the total carrying capacities are the same.

Suggested Citation

  • D.L. DeAngelis & Bo Zhang & Wei-Ming Ni & Yuanshi Wang, 2020. "Carrying Capacity of a Population Diffusing in a Heterogeneous Environment," Mathematics, MDPI, vol. 8(1), pages 1-12, January.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:49-:d:304200
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    References listed on IDEAS

    as
    1. Wang, Yuanshi & DeAngelis, Donald L., 2019. "Energetic constraints and the paradox of a diffusing population in a heterogeneous environment," Theoretical Population Biology, Elsevier, vol. 125(C), pages 30-37.
    2. DeAngelis, Donald L. & Trexler, Joel C. & Cosner, Chris & Obaza, Adam & Jopp, Fred, 2010. "Fish population dynamics in a seasonally varying wetland," Ecological Modelling, Elsevier, vol. 221(8), pages 1131-1137.
    3. Arditi, Roger & Lobry, Claude & Sari, Tewfik, 2015. "Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation," Theoretical Population Biology, Elsevier, vol. 106(C), pages 45-59.
    4. Ang, Tau Keong & Safuan, Hamizah M., 2019. "Harvesting in a toxicated intraguild predator–prey fishery model with variable carrying capacity," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 158-168.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Alexander Korotkov & Sergei Petrovskii, 2023. "Extinctions in a Metapopulation with Nonlinear Dispersal Coupling," Mathematics, MDPI, vol. 11(20), pages 1-22, October.

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