### Abstract

Carrying capacity, K, is a fundamental quantity in theoretical and applied ecology. When populations are distributed over space, carrying capacity becomes a complicated function of local, global and nearby environments, dispersal rate, and the relationship between population growth parameters, e.g., r and K. Expressions for the total carrying capacity, K_{total}, in an n-patch model that explicitly disentangle all of these factors are currently lacking. Therefore, here we derive K_{total} for a linear spatial array of n habitat patches with logistic growth and strong or weak random dispersal of individuals between adjacent patches. With strong dispersal, K_{total} depends on the mean r and K over all patches (〈r〉 and 〈K〉), the among-patch variance in K, and the linear regression coefficient of r on K, β_{r,K}. Strong dispersal increases K_{total} only if β_{r, K} > 〈r〉/〈K〉 which requires a positive convex or negative concave association between r and K, and decreases K_{total} if β_{r, K} < 〈r〉/〈K〉. Alternatively, weak dispersal increases K_{total} only if the within-patch covariance of r and K, cov(r, K) is greater than the spatial covariance between r and K, cov(r, K_{m}), defined as the average covariance between r in a focal patch and K in neighboring patches. Unlike the strong dispersal limit, this condition depends not only on the magnitude of environmental heterogeneity, but explicitly on the spatial distribution of heterogeneity (i.e., habitat clustering). This work clarifies how the interaction between dispersal, habitat heterogeneity, and population growth parameters shape carrying capacity in spatial populations, with implications for species management, conservation and evolution.

Original language | English (US) |
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Pages (from-to) | 115-124 |

Number of pages | 10 |

Journal | Journal of Theoretical Biology |

Volume | 460 |

DOIs | |

State | Published - Jan 7 2019 |

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### Keywords

- Biomass yield
- Connectivity
- Dispersal
- Environmental heterogeneity
- Metapopulation
- Patchy habitat
- SLOSS

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Biochemistry, Genetics and Molecular Biology(all)
- Immunology and Microbiology(all)
- Agricultural and Biological Sciences(all)
- Applied Mathematics