Extending Nonlinear Analysis to Short Ecological Time Series

Nonlinearity is important and ubiquitous in ecology. Though detectable in principle, nonlinear behavior is often difficult to characterize, analyze, and incorporate mechanistically into models of ecosystem function. One obvious reason is that quantitative nonlinear analysis tools are data intensive (require long time series), and time series in ecology are generally short. Here we demonstrate a useful method that circumvents data limitation and reduces sampling error by combining ecologically similar multispecies time series into one long time series. With this technique, individual ecological time series containing as few as 20 data points can be mined for such important information as (1) significantly improved forecast ability, (2) the presence and location of nonlinearity, and (3) the effective dimensionality (the number of relevant variables) of an ecological system.

[1]  W. Ditto,et al.  Chaos: From Theory to Applications , 1992 .

[2]  D. Kleinbaum,et al.  Applied Regression Analysis and Multivariable Methods , 1999 .

[3]  A. Hastings,et al.  Stochastic Dynamics and Deterministic Skeletons: Population Behavior of Dungeness Crab , 1997 .

[4]  George Sugihara,et al.  Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series , 1990, Nature.

[5]  C. S. Holling.  Resilience and Stability of Ecological Systems , 1973 .

[6]  J. King,et al.  Nonlinear time series analysis of food intake in the dab and the rainbow trout. , 2007, Journal of theoretical biology.

[7]  Holly T. Dublin,et al.  Elephants and Fire as Causes of Multiple Stable States in the Serengeti-Mara Woodlands , 1990 .

[8]  Stephen P. Ellner,et al.  Chaos in a Noisy World: New Methods and Evidence from Time-Series Analysis , 1995, The American Naturalist.

[9]  Russ E. Davis,et al.  Red noise and regime shifts , 2003 .

[10]  Tang,et al.  Self-Organized Criticality: An Explanation of 1/f Noise , 2011 .

[11]  B T Grenfell,et al.  Noisy Clockwork: Time Series Analysis of Population Fluctuations in Animals , 2001, Science.

[12]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[13]  S. Levin.  THE PROBLEM OF PATTERN AND SCALE IN ECOLOGY , 1992 .

[14]  J. Sutherland,et al.  Multiple Stable Points in Natural Communities , 1974, The American Naturalist.

[15]  William M. Schaffer,et al.  Stretching and Folding in Lynx Fur Returns: Evidence for a Strange Attractor in Nature? , 1984, The American Naturalist.

[16]  S. Carpenter,et al.  Catastrophic regime shifts in ecosystems: linking theory to observation , 2003 .

[17]  D. Ruelle.  Chaotic evolution and strange attractors : the statistical analysis of time series for deterministic nonlinear systems , 1989 .

[18]  George Sugihara,et al.  Nonlinear forecasting for the classification of natural time series , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[19]  D. Kleinbaum,et al.  Applied regression analysis and other multivariable methods, 3rd ed. , 1998 .

[20]  S. P. Ellner,et al.  Measles as a case study in nonlinear forecasting and chaos , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[21]  G Sugihara,et al.  Nonlinear control of heart rate variability in human infants. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[22]  M. Kot,et al.  8. Differential systems in ecology and epidemiology , 1986 .

[23]  George Sugihara,et al.  Distinguishing random environmental fluctuations from ecological catastrophes for the North Pacific Ocean , 2005, Nature.

[24]  B. Bolker,et al.  Chaos and biological complexity in measles dynamics , 1993, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[25]  Nathan J. Mantua,et al.  Methods for detecting regime shifts in large marine ecosystems: a review with approaches applied to North Pacific data , 2004 .

[26]  George Sugihara,et al.  Noise and Nonlinearity in an Ecological System , 2001 .

[27]  S. Levin.  The problem of pattern and scale in ecology , 1992 .

[28]  Nancy Knowlton,et al.  Multiple “stable” states and the conservation of marine ecosystems , 2004 .

[29]  Kikukatsu Ito,et al.  Nonlinear dynamics of homeothermic temperature control in skunk cabbage, Symplocarpus foetidus. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Stephen P. Ellner,et al.  Living on the edge of chaos: population dynamics of fennoscandian voles , 2000 .

[31]  S. Carpenter,et al.  Catastrophic shifts in ecosystems , 2001, Nature.

[32]  S. Levin,et al.  Fundamental Questions in Biology , 2006, PLoS biology.

[33]  Jean-Marc Fromentin,et al.  Recurrent and density-dependent patterns in long-term fluctuations of Atlantic bluefin tuna trap catches , 2006 .

[34]  G Sugihara,et al.  Residual delay maps unveil global patterns of atmospheric nonlinearity and produce improved local forecasts. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[35]  S. Ellner,et al.  Chaos in Ecology: Is Mother Nature a Strange Attractor?* , 1993 .

[36]  Stephen R. Carpenter,et al.  Management of eutrophication for lakes subject to potentially irreversible change , 1999 .

[37]  Steven R. Hare,et al.  Empirical evidence for North Pacific regime shifts in 1977 and 1989 , 2000 .

[38]  George Sugihara,et al.  A comparison of long-term trends and variability in populations of larvae of exploited and unexploited fishes in the Southern California region: A community approach , 2005 .

[39]  Y. Wong,et al.  Differentiable Manifolds , 2009 .

[40]  R. May.  Thresholds and breakpoints in ecosystems with a multiplicity of stable states , 1977, Nature.

[41]  C. S. Holling,et al.  Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest , 1978 .

[42]  L Landini,et al.  Non-linear prediction for oesophageal voice analysis. , 2002, Medical engineering & physics.

[43]  George Sugihara,et al.  Fractals: A User's Guide for the Natural Sciences , 1993 .

[44]  David W. Pierce,et al.  Distinguishing coupled ocean–atmosphere interactions from background noise in the North Pacific , 2001 .

[45]  Dixon,et al.  Episodic fluctuations in larval supply , 1999, Science.

[46]  T. Clutton‐Brock,et al.  Noise and determinism in synchronized sheep dynamics , 1998, Nature.