Takahashi (1989) hypothesized that the basic architecture of a chromosome is tree-like, consisting of a concatenation of 'mini-chromosomes'. A fractal dimension of D = 2.34 was determined from an analysis of first and second order branching patterns in a human metaphase chromosome. Xu et al. (1994) hypothesized that the twistings of DNA binding proteins have fractal properties.
Lewis and Rees (1985) determined the fractal dimension of protein surfaces (2 <= D <= 3) using microprobes. A mean surface dimension of D = 2.4 was determined using microprobe radii ranging from 1-3.5 angstroms. More highly irregular surfaces (D > 2.4) were found to be sites of inter-protein interaction. Wagner et al. (1985) estimated the fractal dimension of heme and iron-sulfur proteins using crystallographic coordinates of the carbon backbone. They found that the structural fractal dimension correlated positively with the temperature dependence of protein relaxation rates.
Smith et al. (1989) used fractal dimension as a measure of contour complexity in two-dimensional images of neural cells. They recommend D as a quantitative morphological measure of cellular complexity.
Self-similarity has recently been found in DNA sequences (summarized in Stanley 1992; see also papers in Nonnenmacher et al. 1994). Glazier et al. (1995) used the multifractal spectrum approach to reconstruct the evolutionary history of organisms from m-DNA sequences. The multifractal spectra for invertebrates and vertebrates were quite different, allowing for the recognition of broad groups of organisms. They concluded that DNA sequences display fractal properties, and that these can be used to resolve evolutionary relationships in animals. Xiao et al. (1995) found that nucleotide sequences in animals, plants and humans display fractal properties. They also showed that exon and intron sequences differ in their fractal properties.
The kinetics of protein ion channels in the phospholipid bilayer were examined by Liebovitch et al. (1987). The timing of openings and closings of ion channels had fractal properties, implying that processes operating at different time scales are related, not independent (Liebovitch and Koniarek 1992). López-Quintela and Casado (1989) developed a fractal model of enzyme kinetics, based on the observation that kinetics is a function of substrate concentration. They found that some enzyme systems displayed classical Michaelis-Menten kinetics (D = 1), while others showed fractal kinetics (D < 1).
Fractal dichotomous branching is seen in the lung, small intestine, blood vessels of the heart, and some neurons (West and Goldberger 1987; Goldberger et al. 1990; Glenny et al. 1991; Deering and West 1992). Fractal branching greatly amplifies the surface area of tissue, be it for absorption (e.g. lung, intestine, leaf mesophyll), distribution and collection (blood vessels, bile ducts, bronchial tubes, vascular tissue in leaves) or information processing (nerves). Fractal structures are thought to be robust and resistant to injury by virtue of their redundancy and irregularity. Nelson et al. (1990) examined power-law relationships between branch order and length in human, dog, rat and hamster lung tissue. Differences between the human lung and those of other species were hypothesized to be related to postural orientation. Long (1994) relates Leonardo da Vinci's ratio of branch diameters in trees (= 0.707) to observed dichotomous fractal bifurcations.
Tyler and Wheatcraft (1990) offer a useful overview of the application of fractal scaling to soil physics. Tyler and Wheatcraft (1989) used particle-size distributions to determine the fractal dimension of various soils, and to relate D to such soil properties as percolation and surface water retention. Perfect and Kay (1991) used a similar method to examine soil fragmentation, while Bartoli et al. (1991) used various methods to estimate the mass, pore and surface fractal dimensions of silty and sandy soils. Eghball et al. (1993) used Rosin's Law to demonstrate that different tillage methods and crop sequences affected soil fragmentation (fractal dimension). Perfect et al. (1993) modelled the relationship between soil aggregate size and tensile strength using a multifractal approach. Frontier (1987: 340) suggests that it would be interesting to examine the relationship between soil microflora-fauna diversity and soil fractal geometry.
Vlcek and Cheung (1986) measured the fractal dimension of leaf edges in a number of species. Although D was found to be highly variable in some species (e.g. oaks), they felt that D might have potential as a taxonomic character. The fractal dimension of root systems was examined by Tatsumi et al. (1989) using the box-counting method. They found fractal dimensions in the range 1.46 - 1.6 for mature crop plants. Fitter and Strickland (1992) demonstrated that the fractal dimension of root systems increases over time (to a maximum D of approximately 1.35), and varies between species. Corbit and Garbary (1994) found no differences in the fractal dimension of three algal species, though D increased with both developmental stage and frond structural complexity.
Zeide and Gresham (1991) estimated the fractal dimension of the crown surface of loblolly pine trees in North Carolina, and found evidence that D varies with site quality and thinning intensity. Osawa (1995) determined that trees with higher crown fractal dimensions have less negative self-thinning exponents. It was hypothesized that species-specific changes in foliage packing over time account for this relationship. Chen et al. (1994) developed a fractal-based canopy structure model to calculate light interception in poplar stands.
The fractal geometry of fungal foraging is described by Ritz and Crawford (1990). Fractal dimension varies between fungal species, and tends to be greater when nutrient availability is higher (Bolton and Boddy 1993).
Nonlinear dynamics is the study of systems that respond disproportionately to stimuli. A simple deterministic nonlinear system may behave erratically (though not randomly), a state which has been termed chaos. Chaotic systems are characterized by complex dynamics, determinism, and sensitivity to initial conditions, making long-term forecasting impossible. Chaos, which is closely related to fractal geometry, refers to a kind of constrained randomness (Stone and Ezrati 1996). Wherever a chaotic process has shaped an environment, a fractal structure is left behind.
Goldberger et al. (1990) state that physiology may prove to be one of the richest laboratories for the study of fractals and chaos as well as other types of nonlinear dynamics. A good example is the study of heart rate time series (Goldberger 1992). Conventional wisdom states that the heart displays 'normal' periodic rhythms that become more erratic in response to stress or age. However, recent evidence suggests just the opposite: physiological processes behave more erratically (chaotically) when they are healthy and young. Normal variation in heart rate is 'ragged' and irregular, suggesting that mechanisms controlling heart rate are intrinsically chaotic. Such a mechanism might offer greater flexibility in coping with emergencies and changing environments. Lipsitz and Goldberger (1992) found a loss of complexity in heart rate variation with age. Based on this result, they defined aging as a progressive loss of complexity in the dynamics of all physiological systems. Sugihara (1994), using a different analytical approach, found that prediction-decay and nonlinearity models are good predictors of human health. Healthy patients have a steeper heart rate decay curve, and have greater nonlinearity in their heart rhythms. Teich and Lowen (1994) found that human auditory neuron transmissions are best modelled as fractal point processes, and that such transmissions display long-term persistence (H > 0.5). Hahn et al. (1992) examined thermoregulation responses to heat stress in cattle. Fractal dimensions of thermoregulation profiles were found to decrease with increasing stress. They also found that the interval between temperature reading was critical to the detection of changes in thermoregulatory profiles. Similar results were obtained by Escós et al. (1995) in a study of stress in wild goats (spanish ibex). These authors also found that plants under stress show greater variability in allometric relationships, and reduced branch structure complexity.
Basic ideas of chaotic dynamics in population biology are summarized by Schaffer and Kot (1986). The question of whether natural population cycles are deterministic or purely stochastic was examined by Sugihara et al. (1990; also, Sugihara and May 1990). They state that populations are embedded in a dynamic web of other species and environmental forces, implying that irregularities in population cycles (which have traditionally been 'smoothed' prior to modelling) may provide important information regarding their dynamics. Sugihara et al. (1990) found that for pure additive noise, the correlation of adjacent values was independent of the prediction interval, but for chaotic trends correlations decline as the prediction interval increased. They found that measles epidemics display chaotic properties, but that chickenpox epidemic patterns are best modelled as noise superimposed on a strong annual cycle. Ellner and Turchin (1995) have argued that it is potentially misleading to make a strict distinction between chaotic and stochastic dynamics. Using an approach of non-linear time-series modelling and estimation of Lyapunov exponents (see Godfray and Grenfell 1993), they demonstrated that ecological populations vary from noise-dominated, stable dynamics to weakly chaotic ones. However, Sugihara (1994) claims that their approach is fundamentally flawed, and offers an alternative method based on locally-weighted maps. Hastings et al. (1993) summarize the various methods available for detecting deterministic chaos in biological time series.
Sugihara and May (1990) examined persistence (probability of extinction) in time series of population sizes. The higher the value of H (lower fractal dimension), the smoother and more persistent the population trend. Higher persistence (H) makes a species more prone to extinction, since population values increase (or decrease) faster over time than in populations having low H. Hastings and Sugihara (1993: 138-160) expand on these ideas, and present examples based on bird and butterfly population time series.
Stone and Ezrati (1996) discuss potential applications of nonlinear dynamics and chaos theory to the study of ecological variability. They argue that chaos theory may be particularly useful in modelling vegetation change, where non-equilibrium dynamics (e.g. disturbance, natural mosaic cycling, and habitat fragmentation) often prevail.
Morse et al. (1985) argued that since habitat has a fractal structure, there will be more 'useable' space for smaller animals than for larger ones. Working with invertebrates, they found that predictions of the number of individuals (by size class) based on body mass and metabolic rate alone consistently underestimated observed field values for the smaller size classes. Predictions were considerably improved when the fractal dimension of the habitat was incorporated into the model: smaller organisms 'perceive' more space and are therefore comparatively more abundant. Shorrocks et al. (1991) confirmed this general result, as did Gunnarsson (1992) and Jeffries (1993) using artificial substrates of differing fractal dimension.
Fractional Brownian motion models (Frontier 1987: 351-353) have been used to characterize the movement of organisms. Dicke and Burrough (1988) used fractal analysis to examine spider mite movements in the presence and absence of a dispersing pheromone. Wiens and Milne (1989) took a different approach, examining beetle movements in natural fractal landscapes. They found that observed beetle movements deviated from the modelled (fractional Brownian) ones. A follow-up study by Johnson et al. (1992a) found that beetle movements reflect a combination of ordinary (random) and anomalous diffusions. The latter may simply reflect intrinsic departures from randomness, or result from barrier avoidance and utilization of corridors in natural landscapes. Johnson et al. (1992b) discuss the interaction between animal movement characteristics and the patch-boundary features in a 'microlandscape'. They argue that such interactions have important spatial consequences on gene flow, population dynamics and other ecological processes in the community (see also Wiens et al. 1995). In a comparison of path tortuosity in three species or grasshopper, With (1994a) found that the path fractal dimension of the largest species was smaller than those of the two smaller ones. She suggested that this reflects the fact that smaller species interact with the habitat at a finer scale of resolution than do larger species. In a second study, With (1994b) found differences in the ways that gomphocerine grasshopper nymphs and adults interacted with the microlandscape.
Frontier (1987: 337-343) discusses the ecological significance of contact zones (ecotonal boundaries) between ecosystems, and outlines how fractal theory can be used to examine boundary phenomena. For example, consider the contact surfaces created by turbulence in aquatic ecosystems (the geometry of which is fractal, Mandelbrot 1982; Milne 1988: 72). Turbulent regions (e.g. interfaces between warm and cold water) have high phytoplankton productivity due to increased contact with resources (nutrients and light), which in turn 'feeds' higher trophic levels. This cascade effect implies that spatial patterns at fine spatial scales determine patterns at broader scales. Pennycuick and Kline (1986) estimated D to determine bald eagle territory sizes along rocky coastlines in Alaska. Forest-grassland ecotones could also be examined in this way to determine habitat available to foraging animals, or to plant species restricted to ecotonal environments. Ecotone concepts can also be applied to the design of public spaces (Arlinghaus and Nystuen 1990).
Burrough (1981) used the semivariogram method to estimate D for various environmental transects (e.g. soil factors, vegetation cover, iron ore content in rocks, rainfall levels, crop yields). He found high fractal dimensions in all cases, from D = 1.4 (iron ore content at 3 m intervals) to D = 2.0 (soil pH at 10 m intervals). Very high fractal dimensions indicate spatial independence of successive values. While some of the series displayed self-similarity over many scales (i.e. a linear log-log plot slope), other trends suggested variation in D with changing scale. Palmer (1988) used the same method to examine spatial dependence of vegetation along transects. Values were generally high but not scale-invariant. Based on a fractal analysis, Phillips (1985) concluded that erosion processes along a portion of the Delaware coast could not be easily predicted.
Dispersal distances of crop plant pathogens display power-law relationships (van der Plank 1960), and similar relationships have been suggested for plant propagules (Harper 1977). Based on these observations, Kenkel and Irwin (1994) hypothesized that the dispersal of diaspores and pathogens have fractal properties. They suggested that Lévy or Cauchy flights (Mandelbrot 1982: § 32) are appropriate models of dispersal. Species producing diaspores adapted for long-distance dispersal (e.g. 'weeds') have a low fractal dimension. These species advance through the landscape in large leaps, continually establishing new colonies or epicenters (a 'guerilla' strategy). As a result, they display highly patchy distributions at all spatial scales. Conversely, species lacking adaptations for long-distance dispersal move through the landscape more conservatively (a 'phalanx' strategy), with only occasional 'forays' to establish new epicenters. These species have a higher fractal dimension, resulting in less patchy, more continuous spatial distributions. If this model is correct, outbreaks of pathogens having a low fractal dimension will be difficult to predict, since new outbreaks will seem to appear from nowhere.
Shaw (1994) expands on these ideas, noting that classical dispersal probability models are exponential (that is, all their moments are defined). Exponential models assume that dispersal has a characteristic scale, implying that long-distance dispersal is completely negligible. Exponential-based simulation models result in a 'wave-expanding' dispersal pattern, where wave velocity is proportional to the intrinsic population growth rate. However, empirical studies typically demonstrate that gene flows are much greater than those predicted by exponential models. More realistic models are obtained by using dispersal probability distributions having infinite first and higher moments. Shaw (1994, 1995) uses the Cauchy distribution (analogous to the 'Cauchy flight' of Mandelbrot 1982) to model dispersal. Cauchy-based models produce patterns in which 'daughter foci' are continuously formed, so that dispersal is best described as a disjoint set of locations (c.f. Kenkel and Irwin 1993). Mayer and Atzeni (1993) used the Cauchy distribution to model dispersal distance in the screwworm fly.
Wallinga (1995) modelled weed dynamics under the assumption that weed populations are maintained at low densities (through tillage practices, application of herbicides, and so forth). Under such a scenario, populations are expected to display 'critical phenomena' (Grassberger 1983), resulting in their dynamics and spatial pattern being scale-invariant. Fractal analysis (correlation dimension) of a mapped point pattern of cleavers, a European weed, confirmed the fractal (scale-invariant) nature of weed populations.
Collins and Glenn (1990) argue that competition and dispersal act together to create fractal patterns in tall-grass prairie plant communities. They found evidence of self-similarity in these grasslands (i.e. small-scale patterns are repeated at larger spatial scales).
The hyperbolic distribution, because it lacks a characteristic scale, describes the sizes of self-similar phenomena (Goodchild and Mark 1987). Meltzer and Hastings (1992) examined the size distribution of grazed areas in Zimbabwe, and related H to the relative stability of vegetation patches. Overall, they found that increases in cattle density decreased patch stability. Using similar methods, Hastings et al. (1982) found lower stability in earlier successional patches. Kent and Wong (1982) used the size-frequency distribution of lakes to estimate the fractal dimension of littoral zone habitat in the Precambrian Shield of Ontario, while Hamilton et al. (1992) estimated terrain fractal dimension based on lake size distributions in the Amazon and Orinoco river floodplains. The hyperbolic distribution has also been fit to taxonomic systems (Burlando 1990, 1993) and the size-distribution of seeds (Hegde et al. 1991). Frontier (1987:359-367) discusses applications of fractal theory to rank-frequency diagrams of the distribution of individuals among species.
Krummel et al. (1987) examined the fractal dimension of forest patches ('islands') using the perimeter-area method. They found that smaller forest patches had lower mean D than larger patches. The transition zone from low to high fractal dimension occurred at approximately 60-73 ha. They concluded that small forest patches are the result of anthropogenic activities. This decrease in landscape complexity with increasing anthropogenic activity was also reported by O'Neill et al. (1988) and Turner and Ruscher (1988). De Cola (1989) used the perimeter-area method to determine fractal dimensions of eight natural and anthropogenic landscape-level classes in northern Vermont. Bian and Walsh (1993) used two-dimensional semivariance to examine scale dependency in the relationship between topography (elevation, slope angle, and slope aspect) and reflectance/absorbance of vegetation at Glacier National Park, Montana. Studies involving fractal dimension estimation of geomorphological features are summarized in Goodchild and Mark (1987), Lam (1990) and Lam and Quattrochi (1992).
A simplifying assumption of many classical ecological models is that habitats are uniform, and that they vary linearly with distance. Some recent studies have examined these assumptions and/or modified the classical models in light of the recognized fractal nature of habitats. Scheuring (1991) modified the classical species-area relationship model to include the fractal nature of vegetation. Palmer (1992) modified the 'competition gradient' model of Czárán (1989) to include fractal habitat complexity. He found that species coexistence increased as landscape fractal dimension increased. Milne et al. (1992) examined mammalian herbivore foraging in artificial fractal landscapes. They concluded that the fractal nature of landscapes is an important determinant of resource utilization rates. Milne (1992) examined the fractal geometry of landscapes from the viewpoint of habitat fragmentation. He concluded that habitat fragmentation affects ecosystem processes, and that this must be recognized in developing an ecologically meaningful view of landscapes and habitats. Haslett (1994) found that the fractal dimension of mountain meadow landscapes correlated well with the abundance of syrphid flies, suggesting that more spatial heterogeneous habitats may support more complex ecological communities. Additional potential applications of fractal analysis to vegetation complexity are outlined by van Hees (1994).
The spatial dependency of image elements (e.g. pixels) is referred to as texture. A 'textural feature' is a combination of image elements that cannot be individually differentiated (Musick and Grover 1990). A number of image segmentation methods for the extraction of textural features are available (Davis 1981; van Gool et al. 1985; Blacher et al. 1993). Fractal-based texture methods overcome some of the problems inherent in classical resolution-sensitive techniques (van Gool et al. 1985), and are particularly well-suited to complex natural scenes (Keller et al. 1987; Pentland 1984).
Keller et al. (1989) describe a modified box-counting texture analysis technique based on the probability density function. They characterized simulated (Brodatz) textures in terms of fractal dimension and lacunarity. An alternative box-counting method was proposed by Sarkar and Chaudhuri (1992). In their method, each x,y coordinate has an associated third dimension (z-coordinate) representing pixel intensity (e.g. gray shade). The box count is determined as the number of cells in a column intercepted by the surface. Using simulated textures, they found that their method was more computationally efficient than that proposed by Keller et. al. (1989); see also Chaudhuri et al. (1993) and Chaudhuri and Sarkar (1995). De Cola (1993) describes a hierarchical grid method for the analysis of surface texture in remotely sensed images. It was found that fractal dimension varied with scale, implying multifractal behaviour.
Pentland (1984) developed a fractional Brownian motion (fBm) approach to image texture analysis based on a modified Fourier algorithm. An analysis of photographs of natural objects found that texture was more effective than spectral properties in characterizing major image features. Keller et al. (1987) used the fBm approach to examine interface complexity of vegetation/landform types. Using the same approach, Dennis and Dessipris (1989) found that anti-aliasing filtering techniques improved estimates of the fractal dimension of 'natural' images, but had little effect on synthetic images.
Image analysis has also been used in medicine and cellular biology. For example, Fortin et al. (1992) analyzed local and large-scale structures in cardiac magnetic resonance images and bone x-rays. They provide detailed descriptions of fBm image analysis models. Note that the methods for fractal analysis of self-affine signals described by Schepers et al. (1992) can also be used in image analysis.