Multi-scale trajectory analysis: powerful conceptual tool for understanding ecological change

doi:10.1007/s11515-009-0012-y

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Multi-scale trajectory analysis: powerful conceptual tool for understanding ecological change

László ORLÓCI(

)

Ecologia Quantitativa, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, 91540-000, Brazil

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Abstract

The model at the basis of trajectory analysis is conceptually simple. When applied to time series vegetation data, the projectile becomes a surrogate for vegetation state, the trajectory for the evolving vegetation process, and the properties of the trajectory for the true process characteristics. Notwithstanding its simplicity, the model is well-defined under natural circumstances and easily adapted to serial vegetation data, irrespective of source. As a major advantage, compared to other models that isolate the elementary processes and probe vegetation dynamics for informative regularities on the elementary level, the trajectory model allows us to probe for regularities on the level of the highest process integrity. Theories and a data analytical methodology developed around the trajectory model are outlined, including many numerical examples. A rich list of key references and volumes of supplementary information supplied in the Web Only Appendices rounds out the presentation.

Keywords attractor migration determinism fractal dimension parallelism periodicity phase structure

Corresponding Author(s): László ORLÓCI

Issue Date: 05 June 2009

Cite this article:

László ORLÓCI. Multi-scale trajectory analysis: powerful conceptual tool for understanding ecological change[J]. Front. Biol., 0, (): 158-179.

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https://academic.hep.com.cn/fib/EN/10.1007/s11515-009-0012-y
https://academic.hep.com.cn/fib/EN/Y0/V/I/158

Tab.1 De Smidt’s Heathland data set after Lippe et al. (1985)

Fig.1 The phase space diagram after Orlóci et al. (2002a, 2006). Top: idealized trajectory, BoldItalic_BoldItalic to P (the time dimension). Symbols: BoldItalic_BoldItalic: the state, defined by the triplet (X_O1, X_O2 and X_O3) at which the observer’s time begins; BoldItalic: the state of the momentary target called the attractor. Bottom: stereo mapping of the Atlantic Heathland trajectory (Table 1). See details in the text.

Tab.2 Data sources by location and contact person after Orlóci et al. (2006)

Fig.2 Cwynar’s (1982)palynological spectrum from Hanging Lake, Yukon Territory. Sample and location information are given in Table 2. Graph and accompanied data were downloaded from the Global Pollen Database, 2007 URL address: http://www.ncdc.noaa.gov/paleo/pollen.html and search. Short listed taxa (each taxon represents a different palynomorph named according to their lowest identifiable systematic status) after L. C. Cwynar: 1, Alnus; 2, Betula; 3, Ericaceae; 4, Ericales; 5, Picea; 6, Salix; 7, Artemisia; 8, Asteraceae-Asteroideae; 9, Brassicaceae; 10, Chenopodiacea-Amaranthaceae; 11, Cyperaceae; 12, Fabaceae; 13, Plantago canescens; 14, Poaceae; 15, Rosaceae; 16, Other trees and shrubs; 17, other herbs; bottom scale, pollen counts (%); dark shading, original scale; light shading, 5×exaggerated scale. Black markings on depth scale, dated horizons. Red line demarks a record set called paleorelevé. This is the virtual equivalent of the regional vegetation community 8200 years BP. Interesting to observe the great age of the sediments. From this we can infer that the Hanging Lake site remained essentially undisturbed by glacial ice for at least forty millennia.

Fig.3 Long-term performance of taxa in deviation terms after Orlóci et al. (2006). Taxa are taken from Cwynar’s (1982) Hanging Lake spectrum (Fig. 2, Table 2). Horizontal axis: years before the present; vertical axis: Vostok temperature differences (T) following Petit et al. (2001); SSD: sums of squared deviations; Δ: deviations from random expectation. Random expectation is the imaginary state that we would have if the compositional transitions were ruled completely by chance. Deviations above or below random expectation (the zero line) indicate over or under-performance of the taxon. See Feoli and Orlóci (1985) and Orlóci and Orlóci (1988) for use Δ and SSD graphs in a study of environmental effects in transect based edge detection.

Tab.3 Late Quaternary vegetation history of Hanging Lake reconstructed from the sums of squared deviations (SSD) graph (Fig. 3)

Fig.4 Long-term oscillations of distance (D), angle (AN), velocity (V) and acceleration (A) at the Hanging Lake site (Table 2). The Vostok temperature graph (T) is superimposed. The functions are discussed in the text. Data set used covers 89 taxa. See Petit et al. (2001) for temperature details. Diagram adapted from Orlóci et al. (2006).

Tab.4 Error dampening by moving averages

Tab.5 Regression estimates ρ(V,T), f⁺and f^- in 14 sites

Fig.5 Regression method of estimation using block size (BS) as the scale variable. The site is Hanging Lake (Table 2, Fig. 4). See numerical results in Table 5. Abbreviations: F⁺ or F^–, percent frequency of positive or negative correlation values determined in randomized sitting of windows with randomly chosen BS values; y: regression estimate; R²,coefficient of determination; BS: block size in time step units; r(V,T), product moment correlation coefficient. The ‘best estimate’ used is the point on the regression line corresponding to BS=1. Graph adapted from Orlóci et al. (2002a, 2006).

Tab.6 Characteristic values of the topological similarity coefficient and related statistics

Fig.6 Top row: graphs of the topological similarity index and the 95% statistical confidence limits; bottom row: the similarity coefficient’s deviation from random expectation determined in Monte Carlo experiments; horizontal scale: the radius of the tolerance sphere (scale variable) extending from 0 to 100% in 1% steps. See method description and references in Web Based Appendix F and numerical results in Table 6. Data transformation to equal time steps is applied (see Web Based Appendix H). Abbreviations: a, topological similarity; b, upper limit of 95% (statistical) confidence interval about random expectation; c, expectation of a under assumption of random compositional transitions; d, lower limit of 95% confidence interval. Localities are identified in Table 2: HL, Hanging Lake, Yukon Territory; CA, Cambará, Rio Grande do Sul; RND, series of random numbers emulating the case of chance driven compositional transitions; identicals, a trajectory and its exact copy. We take as an optimal estimate of parallelism the value of the similarity coefficient at maximum deviation from random expectation. To attain this value about 50% of random variations have to be bridged by the tolerance radius. Any similarity value (points on curve ‘a’) outside the confidence limits is deemed significant at better than 2.5%. Graph adapted from Orlóci et al. (2006).

Tab.7 Regression estimates of entropy and information in the Hanging Lake site (Table 2) for BS=1 corresponding to Figs. 7 and 8

Fig.7 Long-term evolution of taxon diversity (H) and information divergence (I) at the Hanging Lake site. Refer to Rényi (1961) and Web Based Appendix G for technique. Vostok temperature graph follows Petit at al. (2001). Site detail is in Table 2.

Fig.8 Regression estimation of the correlation of entropy (H) and information divergence (I) with Vostok temperature (T), and the frequency distribution of the correlation as a function of block size BS at the Hanging Lake site (Table 2). See oscillograms in Fig. 7. F⁺ or F^–: percent frequency of positive or negative correlation values; y: regression estimate; R²: coefficient of determination; BS: block size in time step units; r(H,T) or r(I,T): product moment correlation coefficient. Regression estimates at BS=1 are given in Table 7. Note the negative correlation coefficient for entropy and the positive correlation for information divergence. These support the hypothesis that climate warming destabilizes the vegetation.

Tab.8 Phase structure in the Atlantic Heathland trajectory (Table 1, Fig. 1)

Tab.9 Probing the sample trajectories for the intensities of Markov type determinism

Tab.10 Probing sample trajectories for the strength of directedness based on the rank correlation r(D,BoldItalic)

Fig.9 Evolution of disorder-based compositional diversity H, Anand’s structural complexity C, and composition based process velocity V in the Atlantic Heathland site (Table 1). Graphs are scaled for clear viewing. Reference and explanations are found in the text and in Web Based Appendix G.

Fig.10 Long-term evolution of disorder-based compositional diversity H, Anand’s structural complexity C, and composition based process velocity V in the Lagoa das Patas site (Table 2). Graphs are scaled for clearer viewing. Explanations are found in the text and in Web Based Appendix G. Equal time step (100 yr) transformation applies (Web Based Appendix H).

Tab.11 Product moment correlation coefficient (r) of the Atlantic Heathland graphs H, C, and V in Fig. 9

Tab.12 values of the product moment correlation (r) of the H, C, V graphs in Fig. 10 for Lagoa das Pata (Table 2)

Tab.13 Interpretation of the measuring convention L(r)

Tab.14 Hausdorff (fractal) dimensions of trajectories

Tab.15 Cumulative averaging of the Hausdorff dimension D for Vostok (Fig. 11)

Fig.11 The Vostok temperature graph after Petit et al. (2001). The Vostok temperature graph spans 3311 readings over 423 thousand years. Relevant statistics are given in Table 15.

Fig.12 Vegetation map of Eastern North America (after Espenshade and Morrison, 1990, modified) and graphs showing the historic expansion of the Eastern Cool Temperate Deciduous Forest over latitudes and time. Statistics in graph follow Orlóci (2008). Second graph is adapted from Delcourt and Delcourt (1987, Fig. 1.4) with modifications. The rates shown for temperature are from Vostok transformed to global mean average surface rates and adjusted to show the effect of the local thermal flax rate (Orlóci, 2008). Compare the projected rates to the actual historic rates to see the wide gap, and the enormity of the likely consequences. Formation symbols: T: tundra; B: Boreal Forest; M: Mixed Conifer-Northern Hardwood Forest; D: Cool-Temperate Deciduous Forest; E: Warm-Temperate South-Eastern Evergreen Forest; S: Subtropical. See Table 18 for warming rates and Braun (1950) for detailed description of D.

Tab.16 Frequency distribution of 100 yr warming rates based on 4228 steps each 100 yr wide in the complete Vostok set (Fig. 14)

Tab.17 100 yr warming/cooling rates for Vostok (Fig. 11) within periods as identified

Tab.18 Local thermal flux rates after Orlóci (1994), modified

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