Carrying Capacity of a Population Diffusing in a Heterogeneous Environment

D.L Deangelis,Yuanshi Wang,Wei-Ming Ni,Bo Zhang

doi:10.3390/math8010049

D.L Deangelis, Yuanshi Wang + Show 2 more

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https://doi.org/10.3390/math8010049

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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.

Highlights

Partial differential equations have long been used in spatial ecology, usually in the form of reaction-diffusion equations, to describe such phenomena as spatial pattern formation, the spread of populations in space, and the effects of spatial heterogeneity on populations
The first result of the logistic reaction-diffusion model is that if a consumer population exists in an environment in which an exploited renewable resource input is heterogeneously distributed, and there is a positive relationship between growth rate and carrying capacity, the total steady state of a diffusing population (TRAPA) can attain a greater abundance than the non-diffusing population
We reviewed two important and related results that have been proved for the reaction-diffusion equation of a population of the type (3) or (8) in a spatially heterogeneous environment when the growth rate, r(x) and carrying capacity K(x) are positively related (which is automatically true for (3))

Summary

Introduction

Partial differential equations have long been used in spatial ecology, usually in the form of reaction-diffusion equations, to describe such phenomena as spatial pattern formation (e.g., references [1,2,3]), the spread of populations in space (e.g., references [4,5]), and the effects of spatial heterogeneity on populations (e.g., reference [6]). The need to extend the logistic population model to space and to include capacity for organism movement was realized [21,22] To extend this model to populations over a spatial area such as a landscape or region partial differential equations are useful tools. Using a simplified form of Equation (2), in which maximum growth rate and carrying capacity are combined into one spatially varying parameter, g(x), Lou [23] (see references [24,25]) solved for the total population size of a population diffusing in a heterogeneous spatial region described by the equation: du. In Equation (3) is due to the fact that g(x) is both the maximum growth rate and carrying capacity, which results in these being positively correlated. The parameters r(x) and K(x) may not necessarily be positively correlated, so the increase in TRAPA with diffusion in heterogeneous space is not a general phenomenon, we will see later that a positive correlation is likely

Analogy and Insight from Spatially Discretized Model

Analysis and Experiment

Discussion

Conclusions