Scales and Scaling in Biology and Ecology

1. Scales and Scalingin Biology and Ecology Mathematical Models of Dynamics and Control of Environmental Change, Scaling from Genomes to Ecosystems, PICB, Shanghai, April 26-May 1, 2009 B. Larry Li (李百炼) Professor and Director International Center for Ecology&Sustainability University of California, Riverside, USA

2. Heterogeneity is a fundamental characteristic of nature, which is present in most variables representing natural phenomena. Heterogeneity appears at any scale of ecological systems. • Ecological systems are organized hierarchically over a broad range of interrelated space-time scales.

3. Definition of Hierarchy • Partially ordered set (part-whole relation) • Partial breakdown of continuous scale invariance into discrete scale invariance • Complex critical exponents or fractal dimensions

4. (Li, 2000)

7. In general, we need to consider the following scales: • Temporal scale: (a) the lifetime/duration; (b) the period/cycle; and (c) the correlation length/integral scale; • Spatial scale: (a) spatial extent; (b) space period; and (c) the correlation length/integral scale; • "Organism" scale: (a) body size/mass; (b) species-specific growth rate; (c) species extinction rate; (d) the life span; (e) the home range; (f) niche, and so on.

8. In practice, we have to identify: • Process scale • Observation scale • Modeling/working scale • Management scale

14. Scale invariance or symmetry= Self-similarity = Criticality= Scale independenceEcological scale invariance  Ecological equivalence of all lengths

15. From scale invariance to scale covariance • The scale dependence (covariance) is a spontaneously broken scale symmetry. • That means that we have to take non-linearity in scale into account.

16. (Hierarchy theory: characteristic scales or rates.)

17. Ecological signals in a hierarchical ecosystem can be decomposed multi-rated or scaled signal components that may be related with processes of lower level in the ecosystem via wavelet analysis.

18. y1(x) = 1.58  1.01 x, r2 = 0.77, P < 10^5 y2(x) = y1(1.43) + 2 (x 1.43), 2 = 0.04 ± 0.02 (Modified after Makarieva, Gorshkov & Li, 2003. J. Theor. Biol., 221: 301-307)

20. In general, boundary conditions, finite size effects, forcing or dissipation spoil this scale invariance, and the solution is not power-law anymore. The concept of scale covariance is then very useful to study the breaking of scale symmetry.

21. Scale (or across-scale) dynamics • To identify and study the scale-force responsible for the scale distortion (i.e., for the deviation to standard scaling, mono- or multi-fractals). • The methodology includes, such as, scale relativity, scale-acceleration, the Lagrange scale-equation, discrete scale invariance, etc.

22. Ecological Space/Time Series • NONSTATIONARY • IRREGULAR • NONLINEAR • MULTISCALE • LOCALIZED

23. HOW TO IDENTIFY AND CHARACTERIZE COMPLEX ECO-SIGNALS? HOW TO EXTRACT INSTAN-TANEOUS FREQUENCIES OF THE SIGNAL VARY IN TIME OR SPACE ACROSS SCALES?

25. (Li, 1986. Ecol. Model. 32: 227-236.)

27. GABOR/STFT

28. WAVELET ANALYSIS --- A Relatively New Mathematical Theory

29. WAVELETS

30. Multiresolution dyadic tree: L and H represent low-pass and high-pass filter, respectively. A signal can be represented by a low-pass or coarse signal at a certain scale (corresponding to the level of the tree), plus a sum of detail signals at different resolutions. The subband dyadic tree structure conceptualizes the wavelet multiresolution decomposition of a signal.

31. Ecological signals in a hierarchical ecosystem can be decomposed multi-rated or scaled signal components that may be related with processes of lower level in the ecosystem via wavelet analysis.

32. Mexican Hat Function

34. Estimating the scale: Li & Loehle, 1995, Geophys. Res. Lett. 22: 3123-3126; Correction, 23: 1059-1061 (1996). Nezlin & Li, 2003, J. Marine Systems, 39: 185-202.

38. We need to understand the full law including the crossover rather than just the power law or fractal regime.

39. Spatial Scaling and Ecotone Transition: Diffusion Entropy Analysis (Larry Li and Andrew Morozov, UCR)

40. The variance based methods • Hurst rescaled range analysis • the detrended fluctuation analysis (DFA) • the standard deviation analysis (SDA) • spectral analysis • wavelet based analysis • etc.

41. Diffusion Entropy Analysis (DEA)(Scafetta, et al. 2001. Fractals 9, 193)

42. DEA (cont.) • Let’s plug the above scaling condition into Shannon entropy, Based on changing the integration variable from x to y=x/tδ, we obtain • where

43. Gaussian Statistics (H = δ = 0.5)

44. Lévy Statistics (H ≠ δ = 0.67)

45. Great Plains – Chihuahuan Desert Transition Zone Topography, geology, soil nutrients, and hydrology interact with major air mass dynamics. They provide a spatial and temporal template that has resulted in the Sevilleta being an important biome transition zone. There are four major biomes in New Mexico, Three of them – Great Plains grassland, Great Basin shrub steppe, and Chihuahuan Desert – intersect at the lower elevations of the Sevilleta. And all of the three have transitions to conifer woodland at higher elevations.

46. Site Description: Deep Well and Five Points Deep Well and Five Points are located in biome transition zone between Chihuahuan Desert and Great Plain that make these two sites inherently advantageous to evaluate temporal and spatial dynamics of vegetation transitions. Deep Well and Five Points are two major research sites on the Sevilleta NWR. They are both located on the east side of the Sevilleta.

48. (Anand and Li, 2001. Community Ecology, 2: 161-169.)

49. Blue grama, spring 1993(HSDA=0.6991, HDFA=0.6892, and δ =0.7627)

50. Diffusion scaling exponents δ of black and blue grama grasses in the Great Plains shortgrass steppe and Chihuahuan desert grassland ecotone Great Plains shortgrass steppe Chihuahuan desert grassland