Detecting chaos in a citrus orchard: Reconstruction of nonlinear dynamics from very short ecological time series

Abstract The reconstruction of nonlinear dynamics from short ecological time series has been an attractive subject in ecology for the last two decade since May’s classical work. Nonlinear time series analysis (NTSA) is used to investigate deterministic chaos. However, most ecological time series are too short to perform NTSA, which usually requires a time series whose size is in the thousands. Here we propose a way to reconstruct local dynamics from a very short ecological time series whose data point is smaller than ten. For most tree crops such as citrus, nuts and acorns, the yield alternates between high- and low-bearing years. Isagi et al. [Isagi Y, Sugimura K, Sumida A, Ito H. How does masting happen and synchronize? J Theor Biol 1997;187:231–9] proposed a mechanistic model that describes masting as chaos and can be applied to alternate bearing. Here we have used an ensemble data set consisting of the yields of 48 individual trees over seven years to test our proposed method and have successfully validated this method by a one-year forward prediction three times in 2002, 2003 and 2004. We also show the applicability of NTSA tools such as Lyapunov exponents, correlation dimension and deterministic nonlinear prediction on the reconstructed local dynamics.

[1]  Ricard V. Solé,et al.  Control, synchrony and the persistence of chaotic populations , 2001 .

[2]  J. Silvertown,et al.  Population cycles caused by overcompensating density-dependence in an annual plant , 1986, Oecologia.

[3]  B. Kendall,et al.  WHY DO POPULATIONS CYCLE? A SYNTHESIS OF STATISTICAL AND MECHANISTIC MODELING APPROACHES , 1999 .

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

[5]  Horst Malchow,et al.  Experimental demonstration of chaos in a microbial food web , 2005, Nature.

[6]  Sawada,et al.  Measurement of the Lyapunov spectrum from a chaotic time series. , 1985, Physical review letters.

[7]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

[8]  David Tilman,et al.  Oscillations and chaos in the dynamics of a perennial grass , 1991, Nature.

[9]  Hiroki Ito,et al.  How Does Masting Happen and Synchronize , 1997 .

[10]  A. Sasao,et al.  Estimation of citrus yield from airborne hyperspectral images using a neural network model , 2006 .

[11]  A. Hastings,et al.  Demographic and environmental stochasticity in predator–prey metapopulation dynamics , 2004 .

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

[13]  Farmer,et al.  Predicting chaotic time series. , 1987, Physical review letters.

[14]  Robert M. May,et al.  Patterns of Dynamical Behaviour in Single-Species Populations , 1976 .

[15]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[16]  A. N. Sharkovskiĭ.  Dynamic systems and turbulence , 1989 .

[17]  Mercedes Pascual,et al.  LINKING ECOLOGICAL PATTERNS TO ENVIRONMENTAL FORCING VIA NONLINEAR TIME SERIES MODELS , 2000 .

[18]  Martin Casdagli,et al.  Nonlinear prediction of chaotic time series , 1989 .

[19]  O. Sexton,et al.  Ecology of mast-fruiting in three species of North American deciduous oaks. , 1993 .

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

[21]  L. Tsimring,et al.  The analysis of observed chaotic data in physical systems , 1993 .

[22]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .

[23]  Peter Turchin,et al.  Rarity of density dependence or population regulation with lags? , 1990, Nature.

[24]  Jim M Cushing,et al.  A chaotic attractor in ecology: theory and experimental data , 2001 .

[25]  Peter Turchin,et al.  Complex Dynamics in Ecological Time Series , 1992 .

[26]  R M May,et al.  Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos , 1974, Science.

[27]  A. Hastings.  Transients: the key to long-term ecological understanding? , 2004, Trends in ecology & evolution.

[28]  R. Costantino,et al.  NONLINEAR DEMOGRAPHIC DYNAMICS: MATHEMATICAL MODELS, STATISTICAL METHODS, AND BIOLOGICAL EXPERIMENTS' , 1995 .

[29]  J. Elsner,et al.  Nonlinear prediction as a way of distinguishing chaos from random fractal sequences , 1992, Nature.

[30]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[31]  Yoh Iwasa,et al.  The synchronized and intermittent reproduction of forest trees is mediated by the Moran effect, only in association with pollen coupling , 2002 .