## 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.

Original language | English (US) |
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Article number | 49 |

Journal | Mathematics |

Volume | 8 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2020 |

### Bibliographical note

Funding Information:Funding: This research was funded by U.S. Geological Survey’s Greater Everglades Priority Ecosystem Science Grant GEPES ATLSS T5.17, Project Title KA30CTF.

Funding Information:

Acknowledgments: D.L.D. and B.Z. were supported by the USGS Greater Everglades Priority Ecosystems Science. B.Z. was also supported by the UC Davis Chancellor’s postdoc fellowship. The research of W.-M.N. and Y.W. were partially supported by NSF and Chinese NSF. We appreciate comments of Adrian Lam and Donald Schoolmaster on an earlier version of this manuscript.

Publisher Copyright:

© 2020 by the authors.

## Keywords

- Carrying capacity
- Chemostat model
- Energy constraints
- Pearl-verhulst logistic model
- Reaction-diffusion model
- Spatial heterogeneity
- Total realized asymptotic population abundance