Abstract
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.
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References
Anderson RM, Gordon DM, Crawley MJ, Hassell MP (1982) Variability in the abundance of animal and plant-species. Nature 296:245–248. doi:10.1038/296245a0
Anderson CNK et al (2008) Why fishing magnifies fluctuations in fish abundance. Nature 452:835–839. doi:10.1038/nature06851
Ballantyne F (2005) The upper limit for the exponent of Taylor’s power law is a consequence of deterministic population growth. Evol Ecol Res 7:1213–1220
Beissinger SR, McCullough DR (2002) Population viability analysis. The University of Chicago Press, Chicago
Beverton RJH, Holt SJ (1957) On the dynamics of exploited fish populations. Ministry of Agriculture, Fisheries and Food, London, republished by Chapman & Hall in 1993
Caswell H (2001) Matrix population models: construction, analysis, and interpretation. Sinauer Associates, Inc., Sunderland
Cohen JE (2013) Taylor’s power law of fluctuation scaling and the growth-rate theorem. Theor Popul Biol 88:94–100. doi:10.1016/j.tpb.2013.04.002
Cohen JE (2014) Taylor’s law and abrupt biotic change in a smoothly changing environment. Theor Ecol 7:77–86. doi:10.1007/s12080-013-0199-z
Cohen JE, Plank MJ, Law R (2012a) Taylor’s law and body size in exploited marine ecosystems. Ecol Evol 2:3168–3178. doi:10.1002/ece3.418
Cohen JE, Xu M, Schuster WSF (2012b) Allometric scaling of population variance with mean body size is predicted from Taylor’s law and density-mass allometry. Proc Natl Acad Sci U S A 109:15829–15834. doi:10.1073/pnas.1212883109
Cohen JE, Xu M, Schuster WSF (2013) Stochastic multiplicative population growth predicts and interprets Taylor’s power law of fluctuation scaling. Proc R Soc B 280:10. doi:10.1098/rspb.2012.2955
Eisler Z, Bartos I, Kertesz J (2008) Fluctuation scaling in complex systems: Taylor’s law and beyond. Adv Phys 57:89–142. doi:10.1080/00018730801893043
Fromentin JM, Powers JE (2005) Atlantic bluefin tuna: population dynamics, ecology, fisheries and management. Fish Fish 6:281–306. doi:10.1111/j.1467-2979.2005.00197.x
Fujiwara M (2012) Demographic diversity and sustainable fisheries. PLoS One 7:14. doi:10.1371/journal.pone.0034556
Fujiwara M, Mohr MS, Greenberg A (2014) The effects of disease-induced juvenile mortality on the transient and asymptotic population dynamics of Chinook salmon (Oncorhynchus tshawytscha). PLoS One 9:10. doi:10.1371/journal.pone.0085464
Gurney WSC, Nisbet RM, Lawton JH (1980) Nicholson’s blowflies revisited. Nature 287:17–21. doi:10.1038/287017a0
Hilborn R, Walters CJ (1992) Quantitative fisheries stock assessment: choice, dynamics & uncertainty. Kluwer Academic Publishers, Boston
Hsieh CH, Reiss CS, Hunter JR, Beddington JR, May RM, Sugihara G (2006) Fishing elevates variability in the abundance of exploited species. Nature 443:859–862. doi:10.1038/nature05232
Jiang J, DeAngelis DL, Zhang B, Cohen JE (2014) Population age and initial density in a patchy environment affect the occurrence of abrupt transitions in a birth-and-death model of Taylor’s law. Ecol Model. doi:10.1016/j.ecolmodel.2014.06.022
Jury EI (1974) Inners and stability of dynamic systems. Wiley, New York
Keeling MJ (2000) Simple stochastic models and their power-law type behaviour. Theor Popul Biol 58:21–31. doi:10.1006/tpbi.2000.1475
Kilpatrick AM, Ives AR (2003) Species interactions can explain Taylor’s power law for ecological time series. Nature 422:65–68. doi:10.1038/nature01471
McArdle BH, Gaston KJ, Lawton JH (1990) Variation in the size of animal populations—patterns, problems and artifacts. J Anim Ecol 59:439–454. doi:10.2307/4873
Neubert MG, Caswell H (2000) Density-dependent vital rates and their population dynamic consequences. J Math Biol 41:103–121
Perry JN (1994) Chaotic dynamics can generate Taylors power-law. Proc R Soc B 257:221–226. doi:10.1098/rspb.1994.0118
Ralston S, O’Farrell MR (2008) Spatial variation in fishing intensity and its effect on yield. Can J Fish Aquat Sci 65:588–599. doi:10.1139/f07-174
Ramsayer J, Fellous S, Cohen JE, Hochberg ME (2012) Taylor’s Law holds in experimental bacterial populations but competition does not influence the slope. Biol Lett 8:316–319. doi:10.1098/rsbl.2011.0895
Taylor LR (1961) Aggregation, variance and mean. Nature 189:732–735. doi:10.1038/189732a0
Taylor LR, Woiwod IP (1980) Temporal stability as a density-dependent species characteristic. J Anim Ecol 49:209–224. doi:10.2307/4285
Taylor LR, Woiwod IP (1982) Comparative synoptic dynamics. 1. Relationships between interspecific and intraspecific spatial and temporal variance mean population parameters. J Anim Ecol 51:879–906. doi:10.2307/4012
Tsukamoto K (2012) Unagi: Daikaiyuu no Nazo. PHP Institute, Tokyo
Yamamura K (1990) Sampling scale dependence of Taylor’s power law. Oikos 59(1):121–125. doi:10.2307/3545131
Yamamura K (2000) Colony expansion model for describing the spatial distribution of populations. Popul Ecol 42:161–169
Acknowledgments
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.
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Fujiwara, M., Cohen, J.E. Mean and variance of population density and temporal Taylor’s law in stochastic stage-structured density-dependent models of exploited fish populations. Theor Ecol 8, 175–186 (2015). https://doi.org/10.1007/s12080-014-0242-8
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DOI: https://doi.org/10.1007/s12080-014-0242-8