Fractal index captures the role of vegetation clumping in ... - CiteSeerX

Functional Ecology 2003 17, 315– 322

Fractal index captures the role of vegetation clumping in locust swarming

Blackwell Publishing Ltd.

E. DESPLAND† Department of Zoology, University of Oxford, Oxford OX1 3PS, UK

Summary 1. This paper evaluates box-counting dimension as a quantitative clumping index for discontinuous plant cover, and applies it to studies of both small- and large-scale ecological processes in desert locust (Schistocerca gregaria) swarming. 2. The quadrat under study is tiled with squares of increasing size, and the relationship between the number of occupied squares and square size gives the box-counting dimension. This index is high for random distributions and decreases when the vegetation is clumped; it is also positively linked to the abundance of vegetation. 3. At the scale of individual locusts, this index captures the space-filling properties of vegetation that promote change to the gregarious phase and the onset of swarming. At the very large scale, box-counting dimension measured from satellite imagery reflects topography, which influences the concentration and migration of locust swarms. 4. Fractal geometry considers scale of measurement in quantifying pattern and provides a framework to interpret ecological phenomena across spatial scales. Demonstration of a statistically significant relationship between the box-counting exponent and locust behaviour shows how this approach can be applied to ecological studies. Key-words: Aggregation, box-counting method, discontinuous plant cover, dispersion, spatial statistics Functional Ecology (2003) 17, 315–322

Introduction The spatial structure of vegetation plays an important role in ecosystem functioning, and can influence the individual behaviour and population dynamics of resident animals (Tilman & Kareiva 1997). However, measuring spatial effects can be difficult because distribution pattern cannot be reduced to a single number. A variety of vegetation distribution indices exist to describe different aspects of pattern: one must identify the aspect of pattern that is biologically relevant to the question under study in order to select the appropriate index. One aspect of pattern that is not described by the traditional spatial statistics used in plant ecology (McAuliffe 1990; Fortin 1999) is the degree of clumping of discontinuous plant cover. While clumping is a relatively intuitive concept to comprehend, it includes such characteristics as patch number, size, shape and spacing, and is therefore difficult to quantify. Fractal geometry provides an alternative measuring system for irregular shapes that has often proved more useful for natural forms than has Euclidean geometry. Many natural objects exhibit fractal structure, that is, nested complexity of pattern-within-pattern appearing © 2003 British Ecological Society

†Author to whom correspondence should be addressed. E-mail: [email protected]

at different scales of measurement. One of the key contributions of fractal geometry is the recognition of scale-dependence in the description of complex forms, the scale of measurement used influences the value obtained. However, the concept of fractal dimension of a natural form is not clearly defined, causing some confusion. For a given object, a variety of characteristics (e.g. diameter, circumference, mass) can exhibit fractal properties and be statistically distributed according to a power law across scales of measurement – dimension will be measured differently in each case. In the study of vegetation, diverse fractal methods have been developed for measuring various aspects of complexity: for instance, fractal dimension can represent the ratio between the perimeter and the area of regions of different vegetation types, as an index of the complexity of their shape (DeCola 1989; van Hees 1994). A different method of calculating fractal dimension evaluates spatial autocorrelation in species composition indices of different plant communities (Palmer 1988), or in reflectance values of satellite images (deJong & Burrough 1996). The appropriateness of the various indices depends on the aspect of pattern one seeks to quantify, and hence on the ecological question under study. Although multiple fractal indices have been proposed, few have been applied to testing biological 315

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© 2003 British Ecological Society, Functional Ecology, 17, 315–322

hypotheses. The present study defines the box-counting dimension of the mass of a discontinuous set as a measure of clumping for desert vegetation, and draws statistical relationships between this index and animal behaviour. In deserts, the spatial distribution of vegetation depends mainly on the presence and flow of water. Monod (1954) has distinguished between ‘contracted’ and ‘diffuse’ vegetation in Saharan environments. Vegetation that forms a continuous cover is termed diffuse. In more arid habitats the vegetation is contracted: plant cover is discontinuous and limited to wadis and depressions where rainfall is concentrated by the topography. This pattern can be repeated from the centimetre to the kilometre scale depending on the structure of the landscape. Thus desert vegetation occurrence has fractal structure, revealing ever-greater complexity as one zooms in closer. This structure is not self-similar, and the degree of clumping of plant cover depends on the scale of measurement, reflecting the biotic and abiotic factors that control plant growth. For instance, in the area around Akjoujt, Mauritania, vegetation distribution is essentially homogeneous at very large scales (10 × 10 km grid squares). However, at a smaller scale (2·5 × 2·5 km grid), vegetation is concentrated in strips and patches of higher humidity where rainwater is channelled by topography (El-Hadi 1996). At much smaller scales, different patterns appear: clumps of perennial shrubs and grasses form patches of long-lasting vegetation, surrounded by smaller patches of short-lived annuals. The fractal structure of desert vegetation influences ecological processes occurring at multiple spatial scales (Monod 1954), and the lack of a quantitative index to measure clumping has limited the analysis of these relationships. The desert locust, Schistocerca gregaria L., changes phase and forms swarms in response to local increases in population density. At low density, locusts in the solitarious phase are green and cryptic; under crowded conditions they change to the brightly coloured gregarious phase, and aggregate (Uvarov 1977; Simpson et al. 1999). Locusts occupy a wide range of arid habitats across North Africa and the Middle East; however, outbreaks tend to start within the same high risk areas. The topography and vegetation distribution of these areas appears to make them conducive to outbreak formation (Pedgley 1981; Popov et al. 1991; Popov 1997). A locust outbreak begins with the change of individual locusts to the gregarious phase; for the outbreak to be sustained, local gregarious populations must converge over large geographical areas and fuse into swarms that migrate together. These processes depend on vegetation distribution at small and large spatial scales, respectively. The present study examines the link between a fractal index of vegetation clumping and desert locust swarm formation, at two very different spatial scales.

Methods  -  The box-counting method can be used to measure the complexity of a discontinuous set of points in the plane (Broomhead 1985; Lovejoy et al. 1986; Hastings et al. 1992; Hastings & Sugihara 1993; Kenkel & Walker 1996). The box-counting dimension, D, relates the expected number of points, or mass of the set, observed at one scale to observations made at another scale. A set with a small D is one for which the set is concentrated on a decreasing fraction of the total space as the scale of measurement increases (Lovejoy et al. 1986). The fractal dimension of a random subset is the same as that of the original set. Therefore a random distribution of points is of dimension 2. Distributions with lower dimensions fill less than the complete plane at all scales, and are aggregated more than random patterns of comparable density (Milne 1992; Kenkel & Walker 1996). In this method, the vegetation contained in a square quadrat is mapped onto a 64 × 64-square grid. The spatial scale at which D is measured depends on the size of the grid square. The grid is tiled with boxes of varying side length, l = 2–32 grid squares; the inverse slope of the relationship between F(l ) (the number of boxes which contain some vegetation) and l represents D. Values of l equal to the size of a single grid square or of the entire quadrat are not used because they bias the measurement. This process is repeated 10 times, changing the position where the first box is placed on the grid to reduce the influence of stochastic effects. This application of the box-counting dimension characterizes the space-filling properties of a mass fractal, and provides a good estimate of its fractal dimension (Hall & Wood 1993). However, this index is sensitive to variation in methodology and depends on image resolution, thresholding and size (Baveye et al. 1998). Nonetheless, whether the object under consideration is a true mass fractal, and whether the index accurately represents its dimensionality, are of little importance – it is the index’s ability to capture differences between vegetation distribution patterns and to express statistically significant ecological relationships that is of interest.

-       The mass of the set, C, is given by the number of occupied boxes F(l ) when l = 1. This represents the amount of matter in the set which, in the case of vegetation, corresponds to the percentage plant cover. It is a fractal property because it depends on the scale of measurement. As D reflects the space-filling properties of a set, it is tightly linked to the mass of the set. Patterns with large mass will tend to have higher values of D than patterns with lower mass. This effect was investigated using a series of artificial, random and completely clumped

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Fig. 1. Plots of box-counting dimension D vs mass C for patterns mentioned in the text. Hatched lines represent the series of artificial random and clumped distributions used to determine the available parameter space. The values for the small-scale (cages 1– 4) and large-scale (erg, intermediate and massif ) distribution patterns are also shown.

(all resource in a single centrally located patch) distributions, with the number of full grid squares, C = F (1), varying from 100 to 4000. The box-counting index, D, of these artificial patterns was plotted against their mass, C (Fig. 1). For the random distributions, D remained at 2 independent of C. This was expected because the fractal dimension of a random subset of a set (in this case, the plane) has the same dimension as the set itself. The curve for the clumped distributions also increased toward D = 2 as the plane became increasingly filled. When nearly all the grid squares were full, both distribution patterns resembled a simple Euclidean filled plane of D = 2. Thus if C is very large, D no longer contains much information; in this situation a different fractal index appropriate for measuring the complexity of shapes should be used (Kenkel & Walker 1996). The area of parameter space that D can occupy (bounded by the hatched lines in Fig. 1) changes as C increases and the curves for the extreme distributions converge. Thus comparisons of D between patterns must consider the mass of each set.

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© 2003 British Ecological Society, Functional Ecology, 17, 315–322

The relationship between small-scale vegetation distribution and locust behaviour and phase state was evaluated in laboratory and field experiments. The grid size was chosen to measure D at the scale of individual locusts: each grid square side was the size of the locusts used in the experiment (1 cm long nymphs in the laboratory; 3 cm long adults in the field). For the laboratory experiments, artificial fractal patterns of identical mass were generated using an algorithm for threedimensional fractal landscapes. The fractal dimension of the landscape is determined by a parameter, H, which represents the power to which the function

defining the landscape must be raised when changing scales in order to maintain scaling relationships (Saupe 1988). Different random number seeds for the algorithm were used to generate two series of qualitatively different patterns with the same values of H (Fig. 2). Vials of seedling wheat were placed in a 64 × 64 cm arena according to these patterns. Ten solitarious phase locust nymphs were added to the arena; their behaviour was observed for a period of 8 h and their phase state was measured at the end of the trial using a standard behavioural assay – this method returned a value of 0 for insects that had remained in the solitarious phase, and a value of 1 for animals that had switched to the gregarious phase [see Despland et al. (2000) for details]. In the field experiments, D was evaluated for four quadrats of natural vegetation in a habitat occupied by solitarious locusts (Akjoujt, Mauritania 19°45·981′ N; 14°25·387′ W). The study site was located in a depression with a simple plant community consisting of discontinuous cover (c. 25%) composed mainly of a single species, Hyoscyamus muticus (Solanaceae). This is a perennial, rhizomatous plant, and individuals vary considerably in size from about 10 cm to 1 m in diameter (Ozenda 1977). Cages measuring 2 × 2 m were constructed over plants distributed according to very different patterns (Fig. 3). Groups of 10 adult solitarious locusts were placed in each cage, and an egg-laying substrate was provided. At the end of 10 days the eggs were collected and the phase state of the progeny was assayed at hatching [see Despland & Simpson (2000b) for details].

-  Satellite imagery was used to measure D of desert vegetation occurrence at the very large scale, and to relate it to topography. The area of the Sahara desert

318 E. Despland

Fig. 2. Artificially generated fractal patterns used in the small-scale laboratory experiment: each dot represents one vegetation patch. A different random number seed was used in each of the two columns. The three patterns in each column were generated using the same number seed, but decreasing values of H. D and H are shown for each pattern.

© 2003 British Ecological Society, Functional Ecology, 17, 315–322

between 16 –24° N and 16° W−28° E (which corresponds to the Western region of the desert locust’s distribution range) was divided into quadrats of 4 × 4°. Quadrats along the coasts that included significant amounts of sea were discarded. NOAA/AVHRR satellite images for each of these quadrats were obtained from the Food and Agriculture Organization’s African Real Time Environmental Monitoring Information System (FAO-ARTEMIS) programme at 7·6 × 7·6 km spatial resolution (Hielkema 1990; Anonymous 1994). Summary images covering the period from 1988 to 1991 (all months combined) were used to represent the overall structure of the landscape, rather than seasonal patterns of vegetation growth and desiccation. The soil-adjusted vegetation index (SAVI) was calculated from the satellite reflectance data. Remotely sensed vegetation indices measure photosynthetic activity, and therefore large values represent the presence of relatively lush, green vegetation. The SAVI minimizes interference from the soil background, and is particularly useful for comparing images from places with different soil types (Huete 1988). The low biomass and photosynthetic activity of desert vegetation make it difficult to determine thresholds to distinguish vegetation from barren pixels in a

remotely sensed image. For this reason, a threshold was established for the present study at SAVI = 0·05. The pixels above this threshold were considered to contain ‘resource’. This provides an index of the relative amounts of vegetation in the different quadrats. Hielkema et al. (1986) have shown that this approach can accurately document significant vegetation biomass changes in the Sahara desert as they relate to habitat suitability for the desert locust. The box-counting dimension of pixels above the SAVI = 0·05 threshold was calculated. Topographical maps were used to classify the quadrats into three categories of landscape types: sand deserts of low relief (ergs); fragmented landscapes associated with mountain massifs; and intermediate landscapes. The data for each quadrat were plotted in D × C space to test whether this approach distinguished the different landscape types.

Results -  The box-counting method was effective in recovering the artificial patterns used in the laboratory experiments,

319 Fractal measure of clumping

Fig. 3. Distribution of plant cover in the four small-scale field cages. Dark patches represent vegetation mapped on a 64 × 64 grid. D is also shown for each pattern.

Fig. 4. Box-counting dimension D and locust crowding (based on number of neighbours less than one body length away), activity (based on number of observations in which locomotion occurred), and phase state (0, solitarious phase; 1, gregarious phase) for the laboratory experiments. Large, medium and small symbols represent patterns with H = 1, 0 and −1, respectively.

© 2003 British Ecological Society, Functional Ecology, 17, 315–322

as similar D values were obtained for the patterns with the same values of H (Fig. 4), whereas significant differences in D were observed for the different values of H (confidence intervals based on linear regres-

sion do not overlap; Sokal & Rohlf 1995). Locust behaviour and phase state varied significantly with differences in dimension (F2,720 = 11·4, 93·9 and 7·4; P = 0·001, 0·001 and 0·002 for crowding, activity and

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Table 1. Dimension statistics and progeny phase state for the small-scale field cage vegetation patterns shown in Fig. 3: the box-counting dimension, D, is shown with its 95% confidence interval and regression coefficient. The phase state on hatching of locust offspring confined to the cages is given with its 95% confidence interval Cage

D

95% CI

r2

Phase state

95% CI

1 2 3 4

1·67 1·49 1·49 1·10

1·61–1·76 1·37–1·60 1·39–1·59 1·00–1·31

0·99 0·98 0·99 0·90

0·83 0·60 0·61 0·33

0·80–0·87 0·42–0·78 0·46–0·75 0·14–0·40

phase state, respectively): when the food was more clumped, the insects experienced more crowding and were more active. When D was high the insects remained solitarious, but they gregarized when the food filled less space in the arena. No significant differences in locust behaviour or phase state were observed between the different patterns of similar box-counting dimension (F1,720 = 2·5, 1·4 and 0·5; P = 0·1, 0·4 and 0·8 for crowding, activity and phase state, respectively). D is thus a significant predictor of locust activity, crowding and behavioural phase (Fig. 4). The vegetation distribution patterns from the four field cages are represented in the D × C space shown in Fig. 1: D decreased as the vegetation became concentrated into large clumps. D values from the four cages were significantly different, except for cages 2 and 3 (Table 1). In cage 3 the vegetation appeared more clumped than in cage 2, but it was also more abundant such that the space-filling properties of the two distributions were essentially the same. The significant differences in resource distribution observed among cages were reflected by significant differences in phase state among the offspring of the locusts kept in the enclosures (Table 1). The progeny of insects from cages where vegetation was more clumped were significantly more gregarious than those kept in the cages with more dispersed vegetation. The phase state of insects from cages 2 and 3 was not statistically different. Although these two distribution patterns had qualitatively different appearances, their spacefilling characteristics were the same; this was both represented by their D values and reflected in the response of the locusts.

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The quadrats are represented in D vs C space in Fig. 1. The range of D values observed depended on C. The quadrats occupied almost the entire parameter space available, except the region of low C and high D, suggesting that when resources were sparse they were not distributed randomly, but concentrated in certain parts of the space. Each landscape type spanned a considerable range of resource amounts, reflecting mainly differences in

rainfall (Fig. 1). Analysis of covariance, testing the effect of landscape type on D with C as a covariate, showed that not only was the relationship with mass significant (F1,34 = 2·7; P < 0·001), but D also differed significantly among landscape types (F2,34 = 6·9; P = 0·03), even when the effect of C was taken into consideration. Regardless of the abundance of resource, D was generally lower for ergs, and higher for mountain massifs (Fig. 1). The ergs’ low boxcounting dimensions suggest that these images contained few, large clumps of resource, presumably representing depressions or wetter areas. In the mountain massifs, rainfall and hence vegetation are concentrated by the rugged topography into channels and riverbeds, and these landscapes therefore present fragmented distribution patterns with high D. Thus box-counting dimension, as measured from satellite images, reflects the degree of fragmentation of the landscape and shows significant differences between landscape types.

Discussion At the small spatial scale, the present study shows that locusts confined in a habitat will experience more crowding, be more active and gregarize further if the box-counting dimension of the vegetation is lower. Conversely, a pattern with high D fills more of the plane and thus, for a given amount of vegetation, effectively provides more ‘space’ for solitarious locusts to disperse. Solitarious locusts tend to avoid each other, but when the vegetation is clumped they are forced to come together to feed. Crowding increases, and contact between individuals causes them to change to the gregarious phase. This leads locusts to become more active and to aggregate, starting the positive feedback process that can give rise to swarm formation. The field trials show that this effect is transmitted to the following generation. This mechanism has been demonstrated by computer simulations (Collett et al. 1998) and laboratory (Despland et al. 2000) and field (Despland & Simpson 2000b) experiments, and also depends on the quality of the vegetation (Despland & Simpson 2000a). The two experiments above show how D captures the space-filling properties of vegetation distribution that influence the crowding experienced by locusts, and hence their behaviour and phase state. It thus provides a quantitative index to evaluate this relationship statistically. At the very large spatial scale, landscape fragmentation influences the migration and concentration of locust swarms (Kennedy 1939; Roffey & Popov 1968; Pedgley 1981; Popov 1997). The present study shows that D, as measured from satellite images, reflects the degree of fragmentation of the landscape and shows significant differences among landscape types. This approach will be used in future studies to relate the fractal characterization of different landscapes to the historical record of locust outbreaks in those areas.

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Thus box-counting dimension reflects the structure of desert vegetation at different spatial scales; there is no reason to expect patterns at different scales to be self-similar or to play equivalent ecological roles, and fractal geometry recognizes this scale-dependence in ecological processes. Environmental complexity plays a key role in many ecological processes, and fractal geometry provides a scale-dependent, quantitative index that permits empirical, analytical investigation of these relationships. Various fractal indices of habitat structure have provided relevant information in terms of its use by animals (for examples see Pennycuick & Kline 1986; Williamson & Lawton 1991). For instance, Morse et al. (1985) hypothesized that, if vegetation has fractal structure, it should provide more usable space at smaller scales and therefore house more small animals than larger ones. This principle has been validated in several different systems, both in terms of number of individuals (Morse et al. 1985; Gunnarsson 1992; Gee & Warwick, 1994; Haslett 1994) and number of species (Palmer 1992). Fractal dimension of the environment has also been shown to influence insect movement (Johnson et al. 1992; Wiens et al. 1995) and foraging (Ritchie 1998). D represents the space-filling properties of discontinuous vegetation and, as such, reflects the way in which vegetation distribution influences crowding and hence phase change in resident locusts.

Conclusions Fractal geometry provides a conceptual framework to study phenomena that depend on processes operating at multiple spatial scales. In this application, the boxcounting dimension, D, quantifies clumping of discontinuous plant cover in a way that is robust to variability in pattern, and integrates the constituent parameters (patch size, distance, etc.) into a quantitative index that can be used to test ecological hypotheses in a statistically rigorous fashion. D represents the space-filling properties of a discontinuous set, and depends on the mass of the set, C: this index is thus particularly well suited to analysing the way in which vegetation distribution influences the amount of space available to resident animals. For equal mass, D is high for randomly dispersed patterns and low when matter is concentrated in a few large clumps. Plotting results in D vs C spaces elucidates the relationship between boxcounting dimension and mass, and shows how the available parameter space for D depends on C. This technique was developed for analysing vegetation distribution in desert ecosystems, but can be adapted for measuring clumping of discontinuous elements in other contexts (e.g. occurrence of individual plant species). © 2003 British Ecological Society, Functional Ecology, 17, 315–322

Acknowledgements Thanks go to Matthew Collett, with whom this project began, and to Stephen J. Simpson for his continued

support. The field work was made possible by Hans Wilps, GTZ Mauritania. The satellite imagery was provided by David Rogers. Thanks also to David Rogers, Gilles Houle and Marie-Josée Fortin for comments on previous versions of this manuscript. This study was funded by a Fonds pour la Formation des Chercheurs et Aide à la Recherche (Québec, Canada) studentship.

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