### Abstract

The mathematical basis of a widely-known variance-mean power relationship of ecological populations was examined. It is shown that the log variance (S^{ 2})-log mean, (m) plot is virtually delimited by two lines log S^{ 2}=log n+2 log m and log S^{ 2}=log m, thus increasing the chance that a linear regression line can be successfully fitted, without a profoundly behavioural background. This makes difficult the task of interpreting a successful fit of the power law regression and its parameter b in a biologically meaningful manner. In comparison with the power law regression, Iwao's m^{ *}-m regression is structurally less constrained, i.e. has a wider spatial region in which data points can scatter. This suggests that a comparison between the two methods in terms of how good a fit is achieved for a particular data set is largely meaningless, since the power law regression may inherently produce a better fit due to its constrained spatial entity. Furthermore, it could be argued that a successful fit in Iwao's method, when found, is less taxed with mathematical arterfacts and perhaps more clearly linked to some biological mechanisms underlying spatial dispersion of populations.

Original language | English |
---|---|

Pages (from-to) | 43-48 |

Number of pages | 6 |

Journal | Researches on Population Ecology |

Volume | 37 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jun 1 1995 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Agricultural and Biological Sciences(all)

### Cite this

**On the mathematical basis of the variance-mean power relationship.** / Tokeshi, Mutsunori.

Research output: Contribution to journal › Article

*Researches on Population Ecology*, vol. 37, no. 1, pp. 43-48. https://doi.org/10.1007/BF02515760

}

TY - JOUR

T1 - On the mathematical basis of the variance-mean power relationship

AU - Tokeshi, Mutsunori

PY - 1995/6/1

Y1 - 1995/6/1

N2 - The mathematical basis of a widely-known variance-mean power relationship of ecological populations was examined. It is shown that the log variance (S 2)-log mean, (m) plot is virtually delimited by two lines log S 2=log n+2 log m and log S 2=log m, thus increasing the chance that a linear regression line can be successfully fitted, without a profoundly behavioural background. This makes difficult the task of interpreting a successful fit of the power law regression and its parameter b in a biologically meaningful manner. In comparison with the power law regression, Iwao's m *-m regression is structurally less constrained, i.e. has a wider spatial region in which data points can scatter. This suggests that a comparison between the two methods in terms of how good a fit is achieved for a particular data set is largely meaningless, since the power law regression may inherently produce a better fit due to its constrained spatial entity. Furthermore, it could be argued that a successful fit in Iwao's method, when found, is less taxed with mathematical arterfacts and perhaps more clearly linked to some biological mechanisms underlying spatial dispersion of populations.

AB - The mathematical basis of a widely-known variance-mean power relationship of ecological populations was examined. It is shown that the log variance (S 2)-log mean, (m) plot is virtually delimited by two lines log S 2=log n+2 log m and log S 2=log m, thus increasing the chance that a linear regression line can be successfully fitted, without a profoundly behavioural background. This makes difficult the task of interpreting a successful fit of the power law regression and its parameter b in a biologically meaningful manner. In comparison with the power law regression, Iwao's m *-m regression is structurally less constrained, i.e. has a wider spatial region in which data points can scatter. This suggests that a comparison between the two methods in terms of how good a fit is achieved for a particular data set is largely meaningless, since the power law regression may inherently produce a better fit due to its constrained spatial entity. Furthermore, it could be argued that a successful fit in Iwao's method, when found, is less taxed with mathematical arterfacts and perhaps more clearly linked to some biological mechanisms underlying spatial dispersion of populations.

UR - http://www.scopus.com/inward/record.url?scp=0028885365&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028885365&partnerID=8YFLogxK

U2 - 10.1007/BF02515760

DO - 10.1007/BF02515760

M3 - Article

AN - SCOPUS:0028885365

VL - 37

SP - 43

EP - 48

JO - Population Ecology

JF - Population Ecology

SN - 1438-3896

IS - 1

ER -