In the oceanographic literature, the problem of patchiness and spatial heterogeneity in the distribution of planktonic species has been a topic of considerable attention since the middle of this century (Kierstead and Slobodkin, 1953; Steele, 1978). Information is collected routinely from ships navigating spatial transects through regions of interest, and analyzed to provide information regarding the spatial distributions of physical and biological variables. These data are complemented by information derived from remote sensing. Standard statistical methods, such as Fourier analysis, are applied to determine the scales of aggregation and distribution, and the correlations among variables of interest. These are compared with the predictions from a variety of theories explaining patchiness, as the beginnings of a basic theory (Kierstead and Slobodkin, 1953; Platt and Denman, 1975; Levin et al., 1989b; Morin et al., 1989).

   In terrestrial environments such approaches are less common among ecologists; in contrast, in the soils literature, there is a sophisticated and well-developed body of work on spatial statistics (Bras and Rodriquez-Iturbé, 1985; Burrough, 1983a,b). Methods such as Fourier analysis, kriging, semi variograms, and other approaches for dealing with the interrelated problems of scale and measurement are central to the field, and to the related subject of geographical information systems (Ripley, 1981; Burrough, 1986). In vegetation studies, Kershaw (1973), Greig-Smith (1983), Oosting (1956) and others recognized the importance of stratified random sampling and other techniques to deal with scale, and R.H.Whittaker led plant synecology into the development of methods for describing the spatial distributions of species along gradients. The latter methods were primarily one-dimensional, however; and despite Whittaker's fascination with the importance of mosaic phenomena (Whittaker and Levin, 1977), he did not have the same success in dealing with multi-dimensional patterns as with one-dimensional distributions. In general, most of the approaches in the terrestrial environment have been static and descriptive, without adequate relation to mechanistic and dynamic theories. On the other hand, the large theoretical literature dealing with the dynamics of spatial pattern (e.g. Levin, 1976), has not made much of an attempt to compare its predictions with the static descriptions.

   Now, in part because of vastly improved high-speed computers and advances in parallel processing, and in part because of advances in remote sensing, there exists tremendous potential to build bridges between theoretical investigations and empirical analyses. Spatial statistics, which represents an active area of theoretical investigation on its own, can and must be applied to describe the distributions of important physical, chemical, climatic, and edaphic variables. They must be coupled with similar analyses of biological variables to produce correlations that are at least suggestive of mechanistic interrelationships, and that provide tests of mechanistic models. Such mechanistic models are essential to making predictions beyond the range of our experiences, because reliance on correlations and extrapolations based solely on them can mislead badly (Lehman, 1986).

   Our own approach (see Levin, 1989; Levin et al., 1989a,b) has been to proceed simultaneously on multiple levels, developing suites of models of increasing complexity. These provide a bridge between complex data sets, and oversimplified caricatures that capture the essential features of data sets. Such approaches have proved immensely useful in other applications in making clear what the basic reasons for observed patterns are (e.g., Ludwig et al., 1978), and include Lorenz's powerful analysis of the problems of weather prediction, described earlier in this paper.

   As a case in point, Kirk Moloney, Linda Buttel, and I have been interested in the gap phase patterns typical of many forests, grasslands, and intertidal habitats. These systems have the common feature that localized disturbances enhance regional coexistence by providing new opportunities for colonization, thereby creating a spatiotemporal mosaic from a system that at equilibrium would otherwise be low in diversity. In the serpentine grassland, where disturbances are the work of gophers and ants, we have joined forces with Harold Mooney and his research group to model the Jasper Ridge community. For forests, we are using a variety of local growth simulators, and will relate to data from both terrestrial and tropical forests.

   Our approach is threefold. First, from remote sensing and ground surveys in the serpentine, the spatiotemporal dynamics of the plant species will be described, and spatial statistics of a variety of sorts applied. Simultaneously, a computer model has been developed that couples local growth and competition dynamics with the forces of disturbance and dispersal (Levin, 1989; Levin et al., 1989b).Random disturbances of a variety of sizes reset the successional clocks in local environments, and open the way for re-colonization. Local environments are spatially correlated with one another due to the spatial correlation in disturbance patterns, the local nature of propagule dispersal, and (in the more complicated versions of the model) neighbor competition. Such correlations can be represented through spatial correlograms, or equivalently, by semivariograms (e.g., Bras and Rodriquez-Iturbé, 1985). In the semivariogram the mean squared difference between the measurement of a variable at two spatially distinct points is given in relation to the spatial distance or lag between those points. Thus, the curve flattens out at large distances, when it exceeds the correlation length introduced by the already mentioned processes. Such relationships are observed in model results (e.g. Levin et al., 1989a) or typically in data such as those reported by Robertson et al. (1988) in their interesting analyses of old field data (Fig. 12.1).

   Another way to represent the patterns produced by such models, and to compare them with data, is to use a stratified technique in which the spatial or temporal variance of a measure is related to the size of the sampling window. For a system without underlying patterns of spatial heterogeneity, in which the only variability arises due to random local events, one expects variance to fall off with the size of the window. This is indeed the case; what is startling, however, is that for the simplest models we considered, and for scales on the order of the correlation length, there was a remarkable fit to a power law. This can be seen in Figure 12.2, in which log variability is plotted against log of scale (window size). This phenomenon is reminiscent of similar relationships seen in statistical physics in the study of critical phenomena, and this similarity has led us to a collaboration with Richard Durrett at Cornell, with the intent to develop simpler versions of our model that still capture the main features, reproducing the observed scaling relationships. In this way we hope to understand the reasons underlying the pattern seen in Figure 12.2. By simulations and such analyses we endeavor to relate the critical exponents from the power law to statistics of disturbance and dispersal, and to provide a basis for extrapolation.