Theoretical Ecology

, Volume 8, Issue 2, pp 175–186 | Cite as

Mean and variance of population density and temporal Taylor’s law in stochastic stage-structured density-dependent models of exploited fish populations

  • Masami FujiwaraEmail author
  • Joel E. Cohen


How does fishing affect the mean and variance of population density in the presence of environmental fluctuations? Several recent authors have suggested that an increasing ratio of standard deviation to mean (coefficient of variation, or CV) in population density indicates declining population stability. We investigated the relationship between the mean and variance of population density in stochastic, density-dependent, stage-structured fish population models. Our models included either compensatory or overcompensatory density dependence affecting either fertility or juvenile survival. Environmental stochasticity affected either juvenile survival (when density dependence affected fertility) or fertility (when density dependence affected juvenile survival). The mean and variance of population density were compared as fishing mortality changed. In some cases, the relationship between the natural logarithms of mean and variance is linear under some parameters (life history strategy) of some models (the type of density dependence and the timing of density dependence and stochasticity), supporting Taylor’s law. In other cases, the relationship can be non-linear, especially when density dependence is overcompensatory, and depends on the stage observed. For example, the variance of adult density may increase with its mean while the variance of juvenile density of the same population may decline, or vice versa. The sequence in which individuals experience stochasticity and density dependence matters because density dependence can attenuate or magnify the fluctuation. In conclusion, the use of the CV as a proxy for population instability is not appropriate, and the CV of population density has to be interpreted carefully for other purposes.


Coefficient of variation Density dependence Environmental stochasticity Fluctuation scaling Stage-structured population model Taylor’s law 



We thank A. Hastings and an anonymous reviewer for valuable feedback on a previous version of this manuscript. This project was developed during The Keyfitz Centennial Symposium on Mathematical Demography organized by John Bongaarts, Hal Caswell, Noreen Goldman, Josh Goldstein, Ron Lee, and Shripad Tuljapurkar in 2013. MF was funded in part by an Institutional Grant (NA10OAR4170099) to the Texas Sea Grant College Program from the National Sea Grant Office, National Oceanic and Atmospheric Administration, US Department of Commerce. JEC was funded in part by US National Science Foundation grants EF-1038337 and DMS-1225529. JEC thanks Priscilla K. Rogerson for assistance.

Supplementary material

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Wildlife and Fisheries SciencesTexas A&M UniversityCollege StationUSA
  2. 2.Laboratory of PopulationsRockefeller and Columbia UniversitiesNew YorkUSA

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