Carrying Capacity of a Population Diffusing in a Heterogeneous Environment

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.

Carrying Capacity of Spatially Distributed Metapopulations

“…We point out that carrying capacity, K, does not scale up in a simple way to give total population in heterogeneous space, but is affected by both the population's intrinsic growth rate and its dispersal rate. Despite the increasing mathematical work motivated by the new experimental and theoretical results [48,[50][51][52][53][54][55][56], more attention is still needed from ecologists. The concept of carrying capacity has long been central in the fields of wildlife management and conservation biology [57].…”

Individual heterogeneity and its importance for metapopulation dynamics

“…Spatial heterogeneity in demographic rates may alter and even reverse population-level predictions from mean-field theory (Holt 1985; Van Dyken and Zhang 2019; Deangelis et al 2020). When resource conditions are homogeneous among patches, as in our experiment, such a spatial variation in demography can only result from phenotypic structuring and self-sorting.…”

“…General metapopulation theory does not take into account that any local variation in dispersal and the emerging local population dynamics will promote phenotypic and genotypic sorting in the network, and thereby generate a spatial structuring in traits and relatedness. It is, however, known that spatial heterogeneity in demographic rates may alter and even reverse populationlevel predictions from mean-field theory[51]-[53] Such variation is typically assumed to be environmentally driven, but local phenotypic structuring may be at the basis of such spatial variation as well[27]. Our reshuffling treatment, where population densities and stage structure were maintained but not genetic (kin) and phenotypic (kind) structure[26] provided a unique opportunity to study the population demographic consequences of this self-organizing mechanism in metapopulations.…”

On the unboundedness of the ratio of species and resources for the diffusive logistic equation

“…Motivated by this fact, some optimization problems concerning (1) have been studied in the field of elliptic equations. We refer to [17,18,19] and [7,23,24,25] for the dependence of u d,m upon d > 0 (for fixed m) and m (for fixed d), respectively. See [9,10,11,12,13,14,21,22] for applications of information on u d,m to the dynamics of solutions to a class of diffusive Lotka-Volterra systems.…”