Showing papers by "Roger Bradbury published in 1984"

Fractals in ecology: methods and interpretation

Abstract: We agree that the method of computing fractals described by Mark (1984) is both more appropriate and more informative than the method we used (Bradbury and Reichelt, 1983). In this note, we argue that it is the departure from strict self-similarity in real structures that gives them their ecological interest. We support these arguments with the analysis of new finer-resolution data from the previously studied coral reef which suggests that reefs are extremely smooth structures at the scale of coral colonies, and that this smoothness is reduced at both finer a n d coarser scales. David Mark (1984) has voiced what we ourselves have also realized. The method of computing fractals that h e describes (Mandelbrot, 1977; Goodchild, 1980) is both more appropriate and more informative than the method we used (Bradbury and Reichelt, 1983). However, we feel that there is a risk that Mark's discussion may leave some readers with the impression that lack of self-similarity invalidates the whole fractal approach to ecosystem studies. In this note we wish to emphasize that this is not so; it is precisely the lack of self-similarity that is of greatest ecological relevance and interest. Problems with variograms. The variogram approach to fractal estimation assumes that one is dealing with a continuous function W(t) of some independent variable t. In this context, the only possible interpretation of t is 'straight-line' distance along a transect, while W must represent the corresponding distance measured along the reef surface. Fractal estimates are then based on the variance of increments where T = step length used (Berry and Lewis, 1980). The chief practical difficulty is that in the reef transects, our steps with dividers were made along the reef surface, whereas T in this context represents increments in the straight line distance. A further problem is that V(T) should be computed over all values of t (or at AIMS Publication No. 21 1 O Inter-Research/Printed in F. R. Germany least be estimated from some representative subsample), whereas with our sampling method we effectively used only those values of t that are multiples of T, thus introducing a systematic bias into the results. Ecological interpretation of fractal dimensions. True fractals are abstractions; no real-world phenomenon is self-similar at all scales. Individual biological (and other) processes always operate over a restricted range of scales. However, the geometric properties of the patterns they produce, i f extended over all possible scales, would define true fractal curves. Thus estimates of fractal dimension made over the restricted range of scales do reflect the geometric properties of the processes. Peaks and dips in the value D of this 'fractal index' (our term for the curve showing estimates of fractal dimension against scale) indicate shifts in the sources of biological pattern. The fractal index is thus a statistical tool for examining patterns and processes in coral communities. Estimates of D for real processes (whether obtained by the variogram or ratio method) may stray outside the range 1.0 to 2.0 in certain circumstances. Such transgressions should not be interpreted as evidence that the application of fractal theory is not valid. They might, on the contrary, provide useful information about the phenomenon concerned, for they may indicate limits to the range over which particular processes affect the system (growth of individual corals, for instance). Reanalysis of our data. We have now reanalysed the data presented in Bradbury and Reichelt (1983), together with a more comprehensive suite from the same site (Fig. 1) which w e have since collected. These new data range over 7 scales, with a finer degree of resolution than we were able to report before. Since the method recommended by Mark allows us to make estimates of D at each scale, w e are now able to examine the variation of D both within-scale and between-scale as well as with the finer resolution 296 Mar. Ecol. Prog. Ser. 14: 295-296, 1984 offered by the new data. We consider our new data set to be superior not only in terms of resolution but also in terms of accuracy (as will be explained below). Therefore we will focus on these new data in the remainder of this reply. Within-scale variation of D. We have split our samples in the following ways to obtain additional information on the within-scale variability of D: (1) In order to examine the stationarity (or stability) of D at each scale, the sequence of observations at each scale was split into 2 subsequences that join end-to-end. (2) In order to examine the local variability of D at each scale, 2 interlaced subsequences were formed by selecting alternate observations from the original sequence of data. 1.0 I 0 10 20 50 100 200 50

Determination of the physical parameters of coral distributions using line transect data

Abstract: We present a new method for extracting a comprehensive suite of biologically significant parameters from line transect data of coral communities. In addition to the percentage coral cover (the traditionally extracted parameter), the method extracts the population density of the coral colonies, their mean diameter and associated standard deviation and, for adequate data, their size frequency distribution. The method assumes only that the coral colonies form a system of non-overlapping circles in the plane, that the diameters of the circles are random quantities with an unknown distribution function, and that the transects are placed randomly. We test the method on both theoretical and real data to show that it performs as well as, if not better than, current methods in extracting the traditional parameter as well as being able to extract the additional useful parameters indicated. Because the method makes few restrictive assumptions and seems robust when used with field data, we suggest that it has wide application wherever line transects are used for ecological survey. The method is implemented in a Fortran program available from the senior author.

Spatial patterns in coral reef benthos: multiscale analysis of sites from three oceans

Abstract: Modalites de la repartition spatiale du benthos recifal dans trois sites: recif Grande Barriere, Mer des Caraibes, Mer Rouge