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Open AccessArticle

Carrying Capacity of a Population Diffusing in a Heterogeneous Environment

1
U.S. Geological Survey, Wetland and Aquatic Research Center, Gainesville, Florida, FL 32653, USA
2
Department of Environmental Science and Policy, University of California at Davis, Davis, CA 95616, USA
3
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
4
School of Science and Engineering, Chinese University of Hong Kong-Shenzhen, Shenzhen 518000, China
5
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 49; https://doi.org/10.3390/math8010049
Received: 7 November 2019 / Revised: 18 December 2019 / Accepted: 23 December 2019 / Published: 1 January 2020
(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)
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. View Full-Text
Keywords: carrying capacity; spatial heterogeneity; Pearl-Verhulst logistic model; reaction-diffusion model; energy constraints; total realized asymptotic population abundance; chemostat model carrying capacity; spatial heterogeneity; Pearl-Verhulst logistic model; reaction-diffusion model; energy constraints; total realized asymptotic population abundance; chemostat model
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DeAngelis, D.; Zhang, B.; Ni, W.-M.; Wang, Y. Carrying Capacity of a Population Diffusing in a Heterogeneous Environment. Mathematics 2020, 8, 49.

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