Taylor's law and abrupt biotic change in a smoothly changing environment
 Joel E. Cohen
 … show all 1 hide
Abstract
Taylor's law (TL), widely verified in empirical ecology, states that the variance of population density approximates a power function of the mean population density, with exponent denoted b. A model of multiplicative increments in population density, where the increments are determined by a Markovian environment, predicts TL with an explicit formula for b. We give a simple theoretical example where, unexpectedly, smooth changes in environmental autocorrelation lead to an abrupt, infinite discontinuity in b. As the daily probability of change in environmental state increases from 0 to 1, b rises from 2 slowly at first, then explodes to +∞ when the population becomes critical, drops to ∞, and rises again to 2. In this model, an exponent b of large magnitude (positive or negative) signals the proximity of a population's criticality and of a singularity in b. A comparable realworld singularity in b could adversely affect fisheries, forestry, agriculture, conservation, and public health.
 Altenberg L (2013) A sharpened condition for strict convexity of the spectral radius via the bipartite graph. Linear Algebra Appl. http://dx.doi.org/10.1016/j.laa.2013.01.008. Accessed 14 Feb. 2013
 Anderson, RM, Gordon, DM, Crawley, MJ, Hassell, MP (1982) Variability in the abundance of animal and plant species. Nature 296: pp. 245248 CrossRef
 Azevedo, RBR, Leroi, AM (2001) A power law for cells. Proc Natl Acad Sci U S A 98: pp. 56995704 CrossRef
 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: pp. 12131220
 Ballantyne, F, Kerkhoff, AJ (2007) The observed range for temporal meanvariance scaling exponents can be explained by reproductive correlation. Oikos 116: pp. 174180 CrossRef
 Barnosky, AD, Hadly, EA, Bascompte, J, Berlow, EL, Brown, JH, Fortelius, M (2012) Approaching a state shift in Earth's biosphere. Nature 486: pp. 5258 CrossRef
 Broecker, WS (2003) Does the trigger for abrupt climate change reside in the ocean or in the atmosphere?. Science 300: pp. 15191522 CrossRef
 Caswell, H (2001) Matrix population models: construction, analysis and interpretation. Sinauer Associates, Inc., Sunderland, MA
 Caswell, H, Cohen, JE (1995) Red, white and blue: environmental variance spectra and coexistence in metapopulations. J Theor Biol 176: pp. 301316 CrossRef
 Cohen, JE (1976) Ergodicity of age structure in populations with Markovian vital rates, I: countable states. J Am Stat Assoc 71: pp. 335339 CrossRef
 Cohen JE (2013a) Taylor's power law of fluctuation scaling and the growthrate theorem. Theor Popul Biol. http://www.sciencedirect.com/science/article/pii/S004058091300035X
 Cohen JE (2013b) Cauchy inequalities for the spectral radius of products of diagonal and nonnegative matrices. Proc Am Math Soc, in press (accepted)
 Cohen JE (2013c) Multiplicative dynamics in a Markovian environment implies Taylor's power law of fluctuation scaling. Submitted
 Cohen, JE, Xu, M, Schuster, WSF (2012) Allometric scaling of population variance with mean body size is predicted from Taylor's law and densitymass allometry. Proc Natl Acad Sci U S A 109: pp. 1582915834 CrossRef
 Cohen, JE, Plank, MJ, Law, R (2012) Taylor's law and body size in exploited marine ecosystems. Ecol Evol.
 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: pp. 20122955 CrossRef
 D'Odorico, P, Ridolfi, L, Laio, F (2013) Precursors of state transitions in stochastic systems with delay. Theor Ecol.
 deMenocal, PB, Tierney, JE (2012) Green Sahara: African humid periods paced by Earth's orbital changes. Nat Educ Knowl 3: pp. 12
 deMenocal, PB, Ortiz, J, Guilderson, T, Adkins, J, Sarnthein, M, Baker, L (2000) Abrupt onset and termination of the African Humid Period: rapid climate responses to gradual insolation forcing. Quat Sci Rev 19: pp. 347361 CrossRef
 Dennis, B, Desharnais, RA, Cushing, JM, Costantino, RF (1997) Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments. Ecol Monogr 65: pp. 261281 CrossRef
 Dennis, B, Desharnais, RA, Cushing, JM, Henson, SM, Costantino, RF (2001) Estimating chaos and complex dynamics in an insect population. Ecol Monogr 71: pp. 277303 CrossRef
 Doney, SC, Sailley, SF (2013) When an ecological regime shift is really just stochastic noise. Proc Natl Acad Sci U S A 110: pp. 24382439 CrossRef
 Eisler, Z, Bartos, I, Kertész, J (2008) Fluctuation scaling in complex systems: Taylor's law and beyond. Adv Physiol 57: pp. 89142 CrossRef
 Engen, S, Lande, R, Saether, BE (2008) A general model for analyzing Taylor's spatial scaling laws. Ecology 89: pp. 26122622 CrossRef
 Evans, SN, Ralph, PL, Schreiber, SJ, Sen, A (2013) Stochastic population growth in spatially heterogeneous environments. J Math Biol 66: pp. 423476 CrossRef
 Fracker SB, Brischle HA (1944) Measuring the local distribution of Ribes. Ecology 25(3):283–303, July. Stable URL: http://www.jstor.org/stable/1931277 Accessed 25 Aug 2013
 Fronczak A, Fronczak P (2010) Origins of Taylor's power law for fluctuation scaling in complex systems. Phys Rev E 81(6)
 GarcíaCarreras, B, Reuman, DC (2011) An empirical link between the spectral colour of climate and the spectral colour of field populations in the context of climate change. J Anim Ecol 80: pp. 10421048 CrossRef
 GarcíaCarreras, B, Reuman, DC (2013) Are changes in the mean or variability of climate signals more important for longterm stochastic growth rate?. PLoS One 8: pp. e63974 CrossRef
 Gillis, DM, Kramer, DL, Bell, G (1986) Taylor's power law as a consequence of Fretwell's ideal free distribution. J Theor Biol 123: pp. 281287 CrossRef
 Hastings, A, Wysham, DB (2010) Regime shifts in ecological systems can occur with no warning. Ecol Lett 13: pp. 464472 CrossRef
 Jørgensen, B (1987) Exponential dispersion models. J Roy Stat Soc Ser B 49: pp. 127162
 Jørgensen, B (1997) The theory of dispersion models. Chapman & Hall, London
 Kaltz, O, EscobarPáramo, P, Hochberg, ME, Cohen, JE (2012) Bacterial microcosms obey Taylor's law: effects of abiotic and biotic stress and genetics on mean and variance of population density. Ecol Process 1: pp. 5 CrossRef
 Keeling, MJ (2000) Simple stochastic models and their powerlaw type behavior. Theor Popul Biol 58: pp. 2131 CrossRef
 Kemp, AW (1987) Families of discrete distributions satisfying Taylor's power law. Biometrics 43: pp. 693699 CrossRef
 Kendal, WS (2004) Taylor's ecological power law as a consequence of scale invariant exponential dispersion models. Ecol Complex 1: pp. 193209 CrossRef
 Kilpatrick, AM, Ives, AR (2003) Species interactions can explain Taylor's power law for ecological time series. Nature 422: pp. 6568 CrossRef
 Klein, AM, Simons, BD (2011) Universal patterns of stem cell fate in cycling adult tissues. Development 138: pp. 31033111 CrossRef
 Kleinen TC (2005) Stochastic Information in the Assessment of Climate Change. Doctoral Dissertation. University of Potsdam, February 2005. 123 pp. http://opus.kobv.de/ubp/volltexte/2005/538/ Accessed 25 Aug 2013
 Lawton, JH (1988) More time means more variation. Nature 334: pp. 563 CrossRef
 Lewontin, RC, Cohen, D (1969) On population growth in a randomly varying environment. Proc Natl Acad Sci U S A 62: pp. 10561060 CrossRef
 Lorenzo, ED, Ohman, MD (2013) A doubleintegration hypothesis to explain ocean ecosystem response to climate forcing. Proc Natl Acad Sci U S A 110: pp. 24962499 CrossRef
 Perry, JN (1988) Some models for spatial variability of animal species. Oikos 51: pp. 124130 CrossRef
 Perry, JN, Taylor, LR (1985) Ades: New ecological families of speciesspecific frequency distributions that describe repeated spatial samples with an intrinsic powerlaw variancemean property. J Anim Ecol 54: pp. 931953 CrossRef
 Pimm, SL, Redfearn, A (1988) The variability of population densities. Nature 334: pp. 613614 CrossRef
 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: pp. 316319 CrossRef
 Reuman, DC, Desharnais, RA, Costantino, RF, Ahmad, OS, Cohen, JE (2006) Power spectra reveal the influence of stochasticity on nonlinear population dynamics. Proc Natl Acad Sci U S A 103: pp. 1886018865 CrossRef
 Reuman, DC, Costantino, R, Desharnais, R, Cohen, JE (2008) Color of environmental noise affects the nonlinear dynamics of cycling, stagestructured populations. Ecol Lett 11: pp. 820830 CrossRef
 Russell, BD, Harley, CDG, Wernberg, T, Mieszkowska, N, Widdicombe, S, HallSpencer, JM (2012) Predicting ecosystem shifts requires new approaches that integrate the effects of climate change across entire systems. Biol Lett 8: pp. 164166 CrossRef
 Scheffer, M, Carpenter, SR (2003) Catastrophic regime shifts in ecosystems: linking theory to observation. Trends Ecol Evol 18: pp. 648656 CrossRef
 Scheffer, M, Carpenter, S, Foley, JA, Folke, C, Walker, B (2001) Catastrophic shifts in ecosystems. Nature 413: pp. 591596 CrossRef
 Scheffer, M, Bascompte, J, Brock, WA, Brovkin, V, Carpenter, SR, Dakos, V (2009) Earlywarning signals for critical transitions. Nature 461: pp. 5359 CrossRef
 Schreiber, SJ, Benaïm, M, Atchadé, KAS (2011) Persistence in fluctuating environments. J Math Biol 62: pp. 655683 CrossRef
 Smith, HF (1938) An empirical law describing heterogeneity in the yields of agricultural crops. J Agric Sci 28: pp. 123 CrossRef
 Taylor, LR (1961) Aggregation, variance and the mean. Nature 189: pp. 732735 CrossRef
 Taylor, LR (1984) Assessing and interpreting the spatial distributions of insect populations. Annu Rev Entomol 29: pp. 321357 CrossRef
 Taylor, LR, Woiwod, IP, Perry, JN (1980) Variance and the large scale spatial stability of aphids, moths and birds. J Anim Ecol 49: pp. 831854 CrossRef
 Tuljapurkar, SD (1990) Population dynamics in variable environments. Springer, Berlin, New York CrossRef
 Tweedie, MCK (1946) The regression of the sample variance on the sample mean. J Lond Math Soc 21: pp. 2228 CrossRef
 Tweedie, MCK (1947) Functions of a statistical variate with given means, with special reference to Laplacian distributions. Proc Camb Philos Soc 43: pp. 4149 CrossRef
 Tweedie MCK (1984) An index which distinguishes between some important exponential families. In Ghosh JK, Roy J (eds.) Statistics: applications and new directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. Indian Statistical Institute, Calcutta. pp 579–604
 Wilson, LT, Sterling, WL, Rummel, DR, DeVay, JE Quantitive sampling principles in cotton. In: Frisbie, RE, Elzik, KM, Wilson, LT eds. (1989) Integrated pest management systems and cotton production. Wiley, Somerset, pp. 85120
 Title
 Taylor's law and abrupt biotic change in a smoothly changing environment
 Journal

Theoretical Ecology
Volume 7, Issue 1 , pp 7786
 Cover Date
 20140201
 DOI
 10.1007/s120800130199z
 Print ISSN
 18741738
 Online ISSN
 18741746
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Markovian environment
 Regime change
 Power spectra
 Population density
 Fluctuation scaling
 Authors

 Joel E. Cohen ^{(1)}
 Author Affiliations

 1. Laboratory of Populations, The Rockefeller University and Columbia University, 1230 York Avenue, Box 20, New York, NY, 10065, USA