### 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) |
---|---|

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

### Cite this

**Carrying capacity of a spatially-structured population : Disentangling the effects of dispersal, growth parameters, habitat heterogeneity and habitat clustering.** / Van Dyken, James; Zhang, Bo.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Carrying capacity of a spatially-structured population

T2 - Disentangling the effects of dispersal, growth parameters, habitat heterogeneity and habitat clustering

AU - Van Dyken, James

AU - Zhang, Bo

PY - 2019/1/7

Y1 - 2019/1/7

N2 - 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, Ktotal, in an n-patch model that explicitly disentangle all of these factors are currently lacking. Therefore, here we derive Ktotal 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, Ktotal 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, βr,K. Strong dispersal increases Ktotal only if βr, K > 〈r〉/〈K〉 which requires a positive convex or negative concave association between r and K, and decreases Ktotal if βr, K < 〈r〉/〈K〉. Alternatively, weak dispersal increases Ktotal 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, Km), 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.

AB - 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, Ktotal, in an n-patch model that explicitly disentangle all of these factors are currently lacking. Therefore, here we derive Ktotal 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, Ktotal 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, βr,K. Strong dispersal increases Ktotal only if βr, K > 〈r〉/〈K〉 which requires a positive convex or negative concave association between r and K, and decreases Ktotal if βr, K < 〈r〉/〈K〉. Alternatively, weak dispersal increases Ktotal 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, Km), 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.

KW - Biomass yield

KW - Connectivity

KW - Dispersal

KW - Environmental heterogeneity

KW - Metapopulation

KW - Patchy habitat

KW - SLOSS

UR - http://www.scopus.com/inward/record.url?scp=85055040814&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055040814&partnerID=8YFLogxK

U2 - 10.1016/j.jtbi.2018.09.015

DO - 10.1016/j.jtbi.2018.09.015

M3 - Article

VL - 460

SP - 115

EP - 124

JO - Journal of Theoretical Biology

JF - Journal of Theoretical Biology

SN - 0022-5193

ER -