Scaling Relationships in Biology including Community Ecology Big

Scaling Relationships in Biology (including Community Ecology) Big fleas have little fleas on their back to bite them, and little fleas have lesser fleas and so ad infinitum. Swift 1733 (? ) Photo of fiddler crabs from Gilbert (2000) Developmental Biology, 6 th ed. ; other photos from Wikipedia

Scaling Relationships Organisms range over 21 orders of magnitude in body size! Statistic from West et al. (1997); Fig. from Bonner (1988)

Scaling Relationships Biologically relevant processes operate over an enormous range of spatial & temporal scales Figure from Levin (1992)

Scaling Relationships For example… gas exchange through individual stomata & global warming represent phenomena that occur at vastly different scales of space & time Processes are naturally linked across scales, so how can we extrapolate from one scale to another (e. g. , leaf forest globe)? What are the mechanistic links among patterns and processes across scales? Photos from Wikipedia

Scaling Relationships A good starting point is to identify scaling relationships Scaling often assesses how attributes change with changes in a fundamental dimension (e. g. , length, mass, time) The attributes of the organism, community or ecosystem are generally the dependent variables (Y), whereas the fundamental dimension is the independent variable (X)

Scaling Relationships Many scaling relationships can be expressed as power laws: Y = c Xs X is the independent variable – measured in units of a fundamental dimension; c is a constant of proportionality; and s is the exponent (or “power” of the function) The relationship is a straight line on a log-log plot: Log 10(Y) = Log 10(c) + s Log 10(X) …and by rearranging, this is the form of the familiar equation for a straight line: y = mx + b

Scaling Relationships Consider the scaling of squares & cubes as functions of the length of a side (the fundamental dimension) Area = Length 2 Area Length 2 Surface area = 6 * Length 2 Surface area Length 2 Volume = Length 3 Volume Length 3

Scaling Relationships Area Y = X 2 (accelerating function) Length

Scaling Relationships Area Y = X 2 (accelerating function) Length

Scaling Relationships Area Y = X 2 (accelerating function) Length

Scaling Relationships Y = 2 X (accelerating function) Log 10(Area) Area Y = X 2 Length Log 10(Length) Etc…

Surface area Y = 6 * X 2 (accelerating function) Log 10(Surface Area) Scaling Relationships Length Y = 2 X + 0. 778 Log 10(Length) Etc…

Scaling Relationships Y = 3 X (accelerating function) Log 10(Volume) Volume Y = X 3 Length Log 10(Length) Etc…

Scaling Relationships Consider the ways in which surface area & volume of a sphere scale with its radius Surface area = 4 r 2 Surface area r 2 Volume = 4/3 r 3 Volume r 3

Scaling Relationships Surface-to-volume ratio: Surface area r 2 Volume r 3 Surface area 1/2 Volume 1/3 Surface area 1/2 r Volume 1/3 r Surface area Volume 2/3

Scaling Relationships Y=4. 83 * X 0. 667 (decelerating function) Log 10(Surface area) Surface area Slope = 1 Volume Y=0. 667 * X + 0. 68 Log 10(Volume) Etc… Volume increases proportionately faster than surface area

Scaling Relationships Y=4. 83 * X 0. 667 (decelerating function) Log 10(Surface area) Surface area Slope = 1 Volume Y=0. 667 * X + 0. 68 Log 10(Volume) Etc… This simple fact has myriad important implications for biology (e. g. , Bergmann’s “rule”? )

Scaling Relationships Y=4. 83 * X 0. 667 (decelerating function) Log 10(Surface area) Surface area Slope = 1 Volume Y=0. 667 * X + 0. 68 Log 10(Volume) Etc… For example, endoparasite S should increase more rapidly than ectoparasite S as host body size increases

Surface area / Volume Scaling Relationships Y = 3 * X-1 Radius Etc… As you could infer from the earlier figures, the surface area to volume ratio changes with the radius of the sphere

Y = 3 * X-1 Log 10(Surface area / Volume) Surface area / Volume Scaling Relationships Radius Y= -1 * X + 0. 48 Log 10(Radius) Etc… …and the rate of change of the ratio is constant in log-log plotting space

Scaling Relationships Allometry – Coined by Julian Huxley (1932) for the study of size & its relationship to characteristics within individuals (due to ontogenetic changes) & among organisms (due to size-related differences in shape, metabolism, etc. ) For example, size is related allometrically to basal metabolic rate in birds & mammals: B M 3/4 The red line’s slope = 1

Scaling Relationships in Community Ecology Metabolic Ecology Theory Geoff West, James Brown & Brian Enquist proposed that many allometric relationships in biology are governed by the physical properties of branching distribution networks (e. g. , blood vessels, xylem & phloem) Figure from West et al. (1997)

Scaling Relationships Allometric relationship: Height vs. diameter in trees The critical buckling height for cylinders is: Hcritical = k * (E/ )1/3 * D 2/3 Therefore, if trees maintain “elastic similarity”: H D 2/3 Giant sequoia See Greenhill (1881); Figure from Mc. Mahon (1975) Douglas Ponderosa fir pine

Scaling Relationships Allometric relationship: Height vs. diameter in trees If trees maintain “elastic similarity”: H D 2/3 Both lines have slopes = 2/3; the broken line is 1/4 the magnitude of the complete line Trees avoid buckling under their own weight, with a 4 x safety factor See Greenhill (1881); Figure from Mc. Mahon (1975) Dataset for U. S. record trees.

Scaling Relationships in Community Ecology Species-area relationships The Arrhenius equation describes a power-law scaling relationship: S = c. Az log (S) = log (c) + z * log (A) Figure from Rosenzweig (1995)

Scaling Relationships in Community Ecology Mainland vs. island size relationships (Foster’s “rule”) a. Insular races of mammals compared to their nearest mainland relatives; the scaling relationship suggests an “optimum size of 100 g” Figure from Brown’s Macroecology (1995)

Scaling Relationships in Community Ecology Mainland vs. island size relationships (Foster’s “rule”) a. Insular races of mammals compared to their nearest mainland relatives; the scaling relationship suggests an “optimum size of 100 g” b. The largest (solid circles) & smallest (open circles) mammals of a landmass as a function of area; as area – and thus the number of species – decreases, the sizes of the mammals converge on 100 g Figure from Brown’s Macroecology (1995)

Scaling Relationships in Community Ecology Size vs. density in plant communities The relationship between size & number for plants grown in monoculture gave rise to the empirical “self-thinning rule”, i. e. , the mean size of individuals in the stand is proportional to their density raised to the -3/2 power Self-thinning rule: m = k N -3/2 Plantago asiatica Figure from Yoda et al. (1963)

Scaling Relationships in Community Ecology Size vs. density in plant communities The similarity to geometric constraints suggested this possibility: Self-thinning rule: m = k N -3/2 Area Volume 2/3 Area Density-1 Volume Mass Density-1 Mass 2/3 Mass Density-3/2 Plantago asiatica Figure from Yoda et al. (1963)

Scaling Relationships in Community Ecology Size vs. density in plant communities Enquist & colleagues have challenged the traditional -3/2 thinning rule by re-examining sizedensity relationships & by providing a new, mechanistic way to approach the problem Enquist et al. ’s prediction: m = k N -4/3 Figure from Enquist et al. (1998)

Scaling Relationships in Community Ecology Size vs. density in plant communities Enquist & Niklas (2001) suggested Metabolic Ecology Theory as the explanation for the apparent consistency of size-density relationships across forests Their prediction translates into: N = k DBH -2 Figure from Enquist & Niklas (2001)

Scaling Relationships in Community Ecology Empirical curves do not exactly match predictions (slope = -2; shaded region) from Metabolic Ecology Theory (Enquist & Niklas 2001 ) 1000000 10000 1000 N / km 2 100 10 100 Diameter (mm) Figure redrawn from Muller-Landau, Condit, Harms et al. (2006) Ecology Letters 1000

Scaling Relationships in Community Ecology Species-genus & species-family ratios Enquist et al. (2001) suggested three hypotheses for the relationship between species richness and number of higher taxa within a local community a. A positive relationship with a shallow slope; as species are added they come from an increasingly limited subset of higher taxa b. A slope of unity represents the upper constraint boundary; addition of new species occurs only upon addition of higher taxa Figure from Enquist et al. (2002) c. Communities could be scattered within the shaded region below the constraint line, such that the variance in abundance of higher taxa increases with S; higher taxa abundance would be effectively unpredictable from S

Scaling Relationships in Community Ecology Species-genus & species-family ratios Enquist et al. (2001) found surprising similarity among tropical forests worldwide Figure from Enquist et al. (2002)

Scaling Relationships – Fractals Fractal models describe the geometry of a wide variety of natural objects E. g. , the branching distribution networks of organisms… Within an object, as a fundamental dimension changes, fractal properties of the object obey scaling (power function) relationships Figure from West et al. (1997)

Scaling Relationships – Fractals Fractal objects may also exhibit the property of self similarity (self-similar objects maintain characteristic properties over all scales) The Sierpinski Triangle If you are curious, visit: http: //www. arcytech. org/java/fractals/sierpinski. shtml

Scaling Relationships – Fractals “In the natural world, there is no guarantee that… elegant selfsimilar properties will apply” Sugihara & May (1990) Even so, fractal properties (and self-similarity over finite scales) appear throughout the natural world: Barnsley’s fractal fern

Scaling Relationships – Fractals “In the natural world, there is no guarantee that… elegant selfsimilar properties will apply” Sugihara & May (1990) Even so, fractal properties (and self-similarity over finite scales) appear throughout the natural world: Clematis fremontii – Fremont’s leather flower; endemic to KS, NE, MO (Original from Erickson 1945) Figure from Brown’s Macroecology (1995)

Scaling Relationships – Fractals It has become customary to introduce fractals with reference to measuring the coast of Britain (e. g. , Mandelbrot 1983) , a project that first suggested the intriguing fact that as the scale of the ruler decreases, the length of the coast increases: L = K δ 1 -D L = Total length δ = Length of the ruler D = Fractal dimension Figure from Sugihara & May (1990) Self-similarity characterizes the object of interest if D is constant over all scales (δ), i. e. , if the power term of the function is constant

Scaling Relationships – Fractals The fractal dimension (D) can be thought of as the “crookedness, ” “tortuosity” or “complexity” of the object: D=1 D = 1. 26 D = 1. 5 (d) D=2 Figure from Sugihara & May (1990)

Scaling Relationships – Fractals In practice there are many ways to estimate D, and to use D in community ecology (see Sugihara & May 1990). Morse et al. (1985) used the boundary-grid method to show that the areas of leaf surfaces in nature display fractal properties, and that D changes with δ D ≈ 1. 5 for the boundaries of vegetation surfaces Morse et al. (1985) show that this means that for an order of magnitude decrease in body length, there is 3. 16 more area to occupy! Figure from Morse et al. (1985, Nature)

Scaling Relationships – Fractals Morse et al. (1985), used their analysis to suggest an explanation for the observation that as the body sizes of arthropods increase, their numbers (densities) decrease more rapidly than expected if available area remained constant Figure from Morse et al. (1985, Nature)