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Spatial applications of gap models
Forest Ecology and Management, 42 ( 1991 ) 95-110
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Elsevier Science Publishers B.V., Amsterdam
Spatial applications of gap models D.L. Urban, G.B. Bonan ~, T.M. Smith and H.H. Shugart Environmental Sciences Department, The University of Virginia, Charlottesville VA 22903, USA
ABSTRACT Urban, D.L., Bonan, G.B., Smith T.M. and Shugart H.H., 1991. Spatial Applications of GAP models. For. Ecol. Manage., 42:95-110. Recent developments in individual-based forest simulators have made it possible to extend the basic approach to a wider range of forest ecosystems. One recent trend is toward more general representations of abiotic processes, and more attention to the role of tree life-history traits in generating forest response to environmental gradients. Gap models that explicitly attend spatial aspects of the light regime can be extended to simulate forest pattern at scales larger than the forest gap; examples at landscape and geographic scales are presented.
INTRODUCTION
Forests are difficult systems to study for several reasons. First, the demographic mechanisms underlying forest dynamics - tree establishment, growth, and mortality - operate on very different space and time-scales that may be difficult to reconcile empirically (Shugart and Urban, 1989). Each of these demographic processes has an intrinsic stochastic element, which may be confounded further by chance environmental fluctuations. Rare events in forests (such as episodic mortality and recruitment following an unusual weather event) may persist for the life-spans of trees, so that historical legacies are preserved in forests. Finally, the dynamical couplings among patterns of growth, mortality, and regeneration can lead to emergent behaviors in forest systems, which may not be readily explainable in terms of single demographic mechanisms (Shugart 1987). Individual-based simulation models (gal models sensu Shugart and West, 1980; see also Shugart, 1984) have been valuable tools in exploring the consequences of couplings among the demographic processes of tree establishment, growth, and mortality, as well as the interplay of environmental factors constraining these processes (Shugart, 1984; Shugart and Urban, 1989 ). Three aspects of this type of model contribute to its continued success: (1) The model considers trees individually, so that the natural consequences of size differences and asymmetric competition can be expressed in the model (Huston et al., 1988). ~Present address: National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, U.S.A. 0378-1127/91/$03.50
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(2) The basic formulation of the model is quite general, so that it can be applied to a wide variety of forests characterized by a great diversity of tree species (witness a score or more FORET-derived models). (3) The model is structurally dominated in its behavior, to the extent that model results often depend largely on the rank pattern of relationships in the model, as compared to precise quantitative details of the relationships. This relative insensitivity to fine details is a boon to model implementation, because it allows one to use simple approximations to simulate complicated processes (Shugart, 1984). In this paper we illustrate current applications of a FORET-derived gap model, focusing on examples that emphasize these three aspects of gap models. Each of the examples addresses concerns that are implicitly spatial, yet, as we will emphasize, spatial concerns do not necessarily require a model that is explicitly spatial. Before introducing the examples, however, it will be useful to review the model that served as the basic framework in each of the applications. THE ZELIG MODEL
ZELIG is a basic model framework developed for versatility in application (Smith and Urban, 1988). The model was derived from FORET (Shugart and West, 1977; Shugart, 1984 ), and retains most of the salient features of its parent model (see also Botkin et al., 1972). ZELIG differs from its sibling versions in that it is implemented on a grid or transect of model plots (Fig. 1 ). Grid cells can be made interactive to the extent that trees on a given cell can shade or be shaded by trees on adjacent cells. Alternatively, either the grid can be collapsed to a linear transect, or a large number of model plots can be simulated independently (in which case ZELIG is similar to a conventional FORET-style model). These three implementation modes provide for three degrees of spatial resolution in the model, and in turn, correspond to sampiing designs commonly used in empirical studies: the interactive grid mode provides the greatest spatial resolution in the model, and corresponds to contiguous sample quadrats or stem maps; the transect mode is equivalent to a line-transect sampling design, and is especially appropriate for direct gradient analyses; a grid of independent plots would be equivalent to sampling a large forest tract with a stratified layout of quadrats, and is an efficient way to model larger spatial patterns. The first application of ZELIG was to consider the spatial scaling of forest structural pattern (Smith and Urban, 1988 ). In this, the model was run as an interactive grid of 900 100-m 2 cells (30 × 30 cells). The simulated forest, a 9-ha stand, was resampled with aggregates of 4, 9, 16, ..., 100 grid cells, so that the effect of sampling scale on forest structure could be examined. The analysis emphasized the qualitative difference in forest structure and dynamics as witnessed at fine (100-m 2) versus coarse (1-ha) levels of resolution: at
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Fig. 1. Schematic of roving-window aggregate plot as defined in the ZELIG model. Leaf area shading the central grid cell is aggregated from adjacent cells according to the proportions indicated: one-half for each cardinal neighbor cell; one-fourth for each diagonal neighbor; plus the local leaf area.
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larger sampling scales, forest structure converged on the mean diameter distribution and the apparent rate of structural change slowed. The continuous nature of the gridded forest allowed Smith and Urban (1988 ) to depict the relationship between these scales (Fig. 2). RECENT DEVELOPMENTS IN GAP MODELS
The basic framework of ZELIG has recently served as a point of departure fore three new initiatives in forest modeling. The first of these examples emphasizes the individual-based nature of forest dynamics, especially addressing the implications of plant-level adaptations to forest pattern across environmental gradients. The second example extends and generalizes the treatment of abiotic environmental constraints, so that the same model could be used for broadly regional applications. The final examples take advantage of the structure of the model to simulate spatial aspects of forest pattern in a simple but explicit way. Collectively, the examples illustrate a modeling philosophy that takes advantage of the strengths of the model while minimizing the interpretative difficulties of inadequacies that are unavoidable in any model. Again, we emphasize that while the applications are all implicitly spatial, not all spatial concerns require an explicitly spatial model to provide useful insights into the phenomena.
Life-history attributes and succession Huston and Smith ( 1987 ) used a ZELIG-derived gap model, in independent-plot mode, to integrate individual differences in life-history traits to their stand-level consequences. In this, they defined a number of artificial tree species in terms of sets of life-history parameters (growth rate, longevity, shade tolerance, and maximum size). They then simulated succession under a factorial design of scenarios in which different combinations of species were represented. This resulted in a limited set of successional pathways which Huston and Smith categorized as replacement, divergence, convergence, suppression, and pseudocyclic replacement. Which pattern was expressed depended on which species were included in the succession. Smith and Huston (1989) have since extended this approach to consider two resources, above and below ground. They began with the premise that an individual cannot at the same time be well adapted to resource utilization above as well as below ground. Trade-offs that confer competitive advantage in using light (shade tolerance) have the associated cost of reducing wateruse efficiency (hence, reduce drought tolerance), while species that are drought-tolerant cannot also be shade-tolerant. There are several lines of ecophysiological evidence that support this as a reasonable working hypothesis. Smith and Huston ( 1989 ) implemented this premise as hypothetical tree spe-
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Fig. 4. Distribution of permanent, discontinuous, and no permafrost for sites within the domain of boreal forest, as simulated by Bonan (1989).
Explicit spatial effects in gap dynamics Some spatial effects in forests d e m a n d a more explicit consideration of space. T w o examples o f such effects are presented below. The first of these considers the effects o f disturbance patch size in forests: how do gap dynamics vary in response to light gaps of different areal extent? This question was addressed with a gridded version o f ZELIG. The second example is concerned with much broader-scale patterns, and attends latitudinal variation in effective gap size as effected by sun angle. This application used a transect version of ZELIG.
Disturbance patch sizes and gap dynamics The original Z E L I G model was used to illustrate changes in forest composition resulting from light gaps o f different sizes. Such patterns would arise
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naturally as a result of single- versus multiple-treefall gaps (e.g. Pickett and White, 1985 ). Smith artificially created square gaps of 400, 900, 1600, and 2500 m 2 in a mature 9-ha stand, and subsequently tracked the performance of shade-tolerant versus intolerant species in each case. With small gaps ( 4 0 0 - 9 0 0 m 2), ZELIG behavior was equivalent to that of a conventional FORET-style gap model, because disturbance patch size roughly corresponded to the size o f a FORET model plot (gap). As patch size
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Fig. 3. Successions along a soil-moisture gradient, as simulated by Smith and Huston (1989). Species 1 is shade-tolerant and drought-intolerant; species 15 is the converse; and other species are interpolated between these two extremes. Middle panel: relative positions of selected species along a soil moisture gradient at year 500 of a simulation. Top panel: seral trajectories of species 1, 10, and 15 across this gradient (note species 10 is seral on mesic sites, but dominant at maturity on drier sites). Lower panel: species replacement patterns along this gradient (note more species participate on more mesic sites, because of behaviors as illustrated in top panel). cies w i t h i n v e r s e l y r e l a t e d t o l e r a n c e s for s h a d e a n d m o i s t u r e stress, a n d s i m u l a t e d s u c c e s s i o n a c r o s s a soil m o i s t u r e g r a d i e n t . T h e i r s i m u l a t i o n s illustrate the i n t u i t i v e result t h a t a site is u l t i m a t e l y d o m i n a t e d b y t h e m o s t s h a d e - t o l e r a n t species t h a t c a n s u r v i v e t h e m o i s t u r e s t a t u s o f t h a t site. T h e results also
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show that a species can play a different seral role (early versus late successional ) depending on soil moisture availability: the behavior of the species in time depends on its spatial context (Fig. 3 ). Importantly, there is no single global strategy that confers competitive advantage under all environmental regimes; the tradeoffs in life history strategies dictate that forests must generate these patterns in space and time. Both of these examples take advantage of the individual-based nature of the model. This structure allows the model to begin with mechanisms that are expre.ssed at the individual level (and vary with tree size as well as among species), and extends these mechanisms to higher-level implications. An important conclusion from the work of Smith and Huston (1989) is that the same simple mechanism - resource competition among individual trees - can result in the myriad successional patterns observed in real forests. Neither example is explicitly spatial, but each implies general rules about forest distribution along environmental gradients that are implicitly spatial in nature.
Extending and generalizing abiotic constraints Bonan (1989 ) undertook to develop a general model that was capable of simulating boreal forest throughout its circumpolar range. Because these forest are largely constrained by abiotic factors such as temperature, soil moisture, and permafrost (Bonan and Shugart, 1989 ), the general model required extensions of these processes as implemented in previous gap models. Among these extensions was a generalized insolation submodel that corrects solar radiation - hence also temperature and potential evapotranspiration - for latitude, elevation, cloudiness, slope, and aspect (Bonan, 1989). A second extension in the model was to link insolation, soil temperature, the insulating properties of ground cover, soil moisture content, and the dynamics of permafrost; with these submodels, Bonan (1989) was able to accurately predict the spatial distribution of permafrost over much of the circumboreal region (Fig. 4 ). Bonan's ( 1989 ) results confer a transportability to his model that is consistent with the FORET tradition: given local climate, soils data, and species parameters, the basic model should work anywhere within the range of boreal forest. Some aspects of the model are general to temperate regions as well. This generality allows users to apply the same model over large regions, which is analogous to a broadly stratified sampling design in an empirical study. The general model serves the same purpose as the sampling design: it provides experimental control via a standardized method. This is especially important in applications that span broad environmental domains wherein many other factors may co-vary to confound ecological interpretation.
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increased, ZELIG behavior diverged from FORET results in that shade-intolerant species were increasingly favored by the large light gaps simulated with ZELIG (Fig. 5 ). This divergence occurs because FORET (and similar versions that use a fixed plot size) cannot account for environmental patterns (here, light availability) at scales larger than the model plot, while the interactive grid in ZELIG does account these effects. Thus, the spatially interactive ZELIG could reproduce differential species response to forest gap size as found in tropical forests (Fig. 6, and Brokaw, 1985 ).
Sun angle and gap dynamics Following Bonan's ( 1989 ) reformulation of the insolation submodel, two of the present authors (Urban and Bonan ) further modified the ZELIG model in order to explore the effects of solar angle on gap dynamics in forests at different latitudes. The goal was to generalize the treatment of the light regime in forests so that the same model could simulate forests of any stature, on any topographic position, at any latitude. This entailed partitioning directbeam and diffuse sky components of radiation, and modeling interception of these components by a mixed-age, mixed-species canopy. This version of the model depends explicitly on the structure of the ZELIG model in its transect mode. The model transect allowed a number of simplifying assumptions and streamlined computations such that the potentially overwhelming details of Diagonal 30 - ~
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the geometry of insolation in a mixed-species, multi-stratal forest could be approximated in a reasonable and straightforward way. This approach also entailed modifying the structure of tree crowns to in-
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clude vertical distribution of foliage, as developed by Leemans and Prentice (1987). In this it was assumed that, given full sun, a tree would hang leaves along the full length of its bole. Shaded trees self-prune from the bottom up, to a height where available light reaches the light-compensation point for that particular species (i.e., shade-tolerance class). Thus, as a modeled forest matures, its canopy rises and thins from below; when the canopy breaks up, trees regenerate in the light gaps created by tree deaths. This crude representation provided the geometry necessary to reproduce the effects of side-lighting and variations in effective gap size, as influenced by tree height, crown ratio, and sun angle. The model determines direct-beam and diffuse radiation as computed by Bonan's model. Direct-beam insolation (Sb) is modeled as impinging on the transect at angle 0, which is the solar altitude angle integrated between sunrise and sunset over the course of the growing season. Interception of this beam is computed by constructing a diagonal leaf-area profile at angle 0, in a stepwise fashion through the vertical leaf-area profile on each cell of the transect (Fig. 7 ). Direct-beam radiation impinging on any height position in the transect is that proportion of Sb transmitted by foliage between that position and the sun (light extinction is modeled according to Beer's Law, as in other gap models). Diffuse radiation (Sd) is similarly modeled, except it is averaged over seven arcs of the isotropic sky (i.e. seven diagonal leaf-area profiles as in Fig. 7; the number of arcs was arbitrary but the average converged rapidly and seven was a sufficient sample). Light available at any position in the canopy is the sum of direct and diffuse radiation reaching that position. This algorithm allows trees on a given transect cell to shade trees on nearby cells, and the degree to which this shading occurs depends strictly on sun angle and tree height (i.e. shadow length). These aspects of the model can be seen in Fig. 8, in which the sun is to the left at roughly at 45 ° angle. Dense foliage at one position in the canopy results in a suppressed understory on cells to the right (i.e., in the shadow); similarly, canopy gaps release a dense understory, not on the same cell, but further to the right. This version of ZELIG has been used in trial applications to gauge the magnitude of the effect of sun angle on gap dynamics, controlling for latitudinal variation in climate, edaphic site conditions, and tree species. The model generates forests with a less dense understory at flat (boreal) sun angles compared with steep (tropical) angles. Compositionally, the model predicts a persistent importance of shade-intolerant species in tropical forests, while undisturbed boreal forests support only shade-tolerant species at maturity (Fig. 9). Preliminary results also imply that different patterns of foliage distribution within tree canopies can drastically alter model results, which sets a high priority on these features for further refinement of the model. Some of these patterns have been suggested previously (Shugart, 1984, 1987 ), but earlier versions of the model were not capable of simulating lati-
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tudinal variation in the forest light environment, nor could previous models partition different components of the light regime. Of course, the geometry of the actual radiation regime of forest canopies is much more complex than is simulated in this version of ZELIG, yet the structure of the transect model permitted an approximate representation that is simple yet useful as a first approximation. Both of the previous two examples are notable in that they explicitly model spatial effect on gap dynamics, and generate patterns that cannot be reproduced with nonspatial versions of this or similar models. For example, Shugart and West (1979) used a spectrum of fixed plot sizes to illustrate the effects of gap size on forest dynamics as simulated with the FORET model. While their results overlap with the above examples in some cases, it remains that a fixed plot size in the model cannot have the flexibility to simulate the variable effect of deaths of trees of different sizes, or disturbance patches larger than a single plot. Both the gridded and transect versions of ZELIG have the flexibility to simulate a broad range of gap sizes as environmental circumstances dictate. Of course, under typical conditions both versions of the model simplify and reduce to a conventional gap model, which retains the ability to reference these spatial applications to previous and current studies with a larger family of gap models (e.g., Davis and Botkin, 1985; Pastor and Post 1986, 1988). CONCLUSIONS AND PROSPECTUS
These examples were selected to illustrate c o m m o n trends in recent applications of gap models. Each of the examples is concerned with phenomena that are implicity spatial in nature, yet is has been shown that space need not always be modeled explicitly to examine these phenomena. For example, in the case of topographic and geographic applications, site parameters that are associated with particular spatial locations can be used in a nonspatial version of a gap model, and model results can then be mapped back onto the spatial locations (e.g. Figs. 3 and 4; see also Solomon, 1986). This approach is especially amenable to use with geographic information system (GIS) mapping technology. In some applications, spatial effects must be modeled explicitly (e.g., Figs. 5-9); in these cases the structure of the ZELIG model provides a useful and versatile framework for model extensions and refinements. Further extensions to this framework, currently under development and testing, include the incorporation of contagiously propagated disturbances such as fire, and the spatial effects of seed dispersal and localized seed pools. Again, the structure of the ZELIG grid or transect simplifies implementation of such spatial extensions. Another c o m m o n theme in recent applications has been an effort to gener-
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attention to the physical processes that control abiotic environmental patterns (e.g. the physics of heat transfer). In contrast to this, efforts to generalize biotic processes and silvics in the model have often been at the expense of fine-scale details. Thus, the recent trend suggests the seeming paradox of incorporating more detail in biophysics and abiotic processes, while including fewer case-specific ecological or biological details. Again, the goal is to confer on the model a degree of generality and ease of implementation so that the same model can be applied across broad regions or diverse forests. Gap models could conceivably be made more universal while retaining the fine-scale details of interest to many forest ecologists. This would require the reformulation of tree growth routines as currently implemented, into a routine based on first principles of eco-physiology in which resource acquisition and allocation under various environmental conditions would dictate annual diameter and height growth, as well as foliage production and other features of interest. At present, quantitative submodels of these processes are not generally available (but see Farquhar and von Caemmer, 1982; Makela, 1986; Reynolds et al., 1986; Running and Coughlan, 1988). Laboratory and field experiments and tree-growth models based on physiology may soon provide such mechanistic submodels. As these are developed, gap models can be used to integrate the tree-level processes to their stand-level implications (Fig. 10 ). In this way, two seemingly competitive approaches to forest modeling can be reconciled and used in tandem as complementary tools. ACKNOWLEDGEMENT
Research supported by grant Nos. BSR-8807882 from the NAtional Science Foundation and NAG 5-1028 from the National Aeronautics and Space Administration to H.H. Shugart and the University of Virginia.
REFERENCES Bonan, G.B., 1989. A computer model of the solar radiation, soil moisture, and soil thermal regimes in boreal forests. Ecol. Modelling, 45: 275-306. Bonan, G.B. and Shugart, H.H., 1989. Environmental factors and ecological processes in boreal forests. Annu. Rev. Ecol.. Syst., 20: 1-28. Botkin, D,B, Janak, J.F. and Wallis, J.R., 1972. Some ecological consequences of a computer model of forest growth. J. Ecol., 60: 849-873. Brokaw, N.V.L., 1985. Gap-phase regeneration in a tropical forest. Ecology, 66: 682-687. Davis, M.B. and Botkin, D.B., 1985. Sensitivity of cool-temperature forests and their fossil pollen record to rapid temperature change. Quat. Res., 23: 327-340. Farquhar, G.D. and von Caemmer, S., 1982. Modeling of photosynthetic response to environmental conditions. In: O.L. Lange, P.S. Noble, C.B. Osmond and H. Ziegler (Editors), Physiological Plant Ecology. Springer, Berlin, pp. 549-587. Huston, M.A. and Smith, T.M., 1987. Plant succession: life history and competition Am. Nat., 130: 168-198.
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Huston, M.A., DeAngelis, D.L. and Post, W.M., 1988. New computer models unify ecological theory. BioScience, 38: 682-691. Leemans, R. and Prentice, I.C., 1987. Description and simulation of tree-layer composition and size distributions in a primaeval Picea-Pinus forest. Vegetatio, 69:147-156. Makela, A., 1986. Implications of the pipe model theory on dry matter partitioning and height growth in trees. J. Theor. Biol., 123: t03-120. Pastor, J. and Post, W.M., 1986. Influence of climate, soil moisture, and succession on forest carbon and nitrogen cycles. Biogeochemistry, 2: 3-27. Pastor, J. and Post, W.M., 1988. Response of northern forests to CO2-induced climate change. Nature, 334: 55-58. Pickett, S.T.A. and White, P.S., 1985. The Ecology of Natural Disturbance and Patch Dynamics. Academic Press, New York. Reynolds, J.F., Bachelet, D., Leadley, P. and Moorhead, D., 1986. Response of vegetation to carbon dioxide. Assessing the effects of elevated carbon dioxide on plants: Toward the development of a generic plant growth model. U.S. Dept. of Energy, Washington, DC, Rep. 023. Running, S.W. and Coughlan, J.C., 1988. A general model of forest ecosystem processes for regional applications. I. Hydrological balance, canopy gas exchange, and primary production processes. Ecol. Modelling, 42:125-154. Smith, T.M. and Huston, M.A., 1989. A theory of the spatial and temporal dynamics of plant communities. Vegetatio, 83: 49-69. Smith, T.M. and Urban, D.L., 1988. Scale and resolution of forest structural pattern. Vegetatio, 74: 143-150. Solomon, A.M., 1986. Transient response of forests to CO2-induced climate change: Simulation modeling experiments in eastern North America. Oecologia, 68: 567-579. Shugart, H.H., 1984. A Theory of Forest dynamics. Springer, New York. Shugart, H.H., 1987. Dynamic ecosystem consequences of tree birth and death patterns. BioScience, 37: 596-602. Shugart, H.H. and Urban, D.L., 1989. Factors affecting the relative abundances of forest tree species. In: P.J. Grubb and J.B. Whittaker (Editors), Toward a More Exact Ecology. Blackwell, Oxford, pp. 249-273. Shugart, H.H. and West, D.C., 1977. Development of an Appalachian deciduous forest succession model and its application to assessment of the impact of the chestnut blight. J. Environ. Manage., 5: 161-179. Shugart, H.H. and West, D.C., 1979. Size and pattern of simulated forest stands. For. Sci., 25: 120-122. Shugart, H.H. and West, D.C., 1980. Forest succession models. BioScience, 30:308-313.
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Elsevier Science Publishers B.V., Amsterdam
Spatial applications of gap models D.L. Urban, G.B. Bonan ~, T.M. Smith and H.H. Shugart Environmental Sciences Department, The University of Virginia, Charlottesville VA 22903, USA
ABSTRACT Urban, D.L., Bonan, G.B., Smith T.M. and Shugart H.H., 1991. Spatial Applications of GAP models. For. Ecol. Manage., 42:95-110. Recent developments in individual-based forest simulators have made it possible to extend the basic approach to a wider range of forest ecosystems. One recent trend is toward more general representations of abiotic processes, and more attention to the role of tree life-history traits in generating forest response to environmental gradients. Gap models that explicitly attend spatial aspects of the light regime can be extended to simulate forest pattern at scales larger than the forest gap; examples at landscape and geographic scales are presented.
INTRODUCTION
Forests are difficult systems to study for several reasons. First, the demographic mechanisms underlying forest dynamics - tree establishment, growth, and mortality - operate on very different space and time-scales that may be difficult to reconcile empirically (Shugart and Urban, 1989). Each of these demographic processes has an intrinsic stochastic element, which may be confounded further by chance environmental fluctuations. Rare events in forests (such as episodic mortality and recruitment following an unusual weather event) may persist for the life-spans of trees, so that historical legacies are preserved in forests. Finally, the dynamical couplings among patterns of growth, mortality, and regeneration can lead to emergent behaviors in forest systems, which may not be readily explainable in terms of single demographic mechanisms (Shugart 1987). Individual-based simulation models (gal models sensu Shugart and West, 1980; see also Shugart, 1984) have been valuable tools in exploring the consequences of couplings among the demographic processes of tree establishment, growth, and mortality, as well as the interplay of environmental factors constraining these processes (Shugart, 1984; Shugart and Urban, 1989 ). Three aspects of this type of model contribute to its continued success: (1) The model considers trees individually, so that the natural consequences of size differences and asymmetric competition can be expressed in the model (Huston et al., 1988). ~Present address: National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, U.S.A. 0378-1127/91/$03.50
© 1991 - - Elsevier Science Publishers B.V.
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(2) The basic formulation of the model is quite general, so that it can be applied to a wide variety of forests characterized by a great diversity of tree species (witness a score or more FORET-derived models). (3) The model is structurally dominated in its behavior, to the extent that model results often depend largely on the rank pattern of relationships in the model, as compared to precise quantitative details of the relationships. This relative insensitivity to fine details is a boon to model implementation, because it allows one to use simple approximations to simulate complicated processes (Shugart, 1984). In this paper we illustrate current applications of a FORET-derived gap model, focusing on examples that emphasize these three aspects of gap models. Each of the examples addresses concerns that are implicitly spatial, yet, as we will emphasize, spatial concerns do not necessarily require a model that is explicitly spatial. Before introducing the examples, however, it will be useful to review the model that served as the basic framework in each of the applications. THE ZELIG MODEL
ZELIG is a basic model framework developed for versatility in application (Smith and Urban, 1988). The model was derived from FORET (Shugart and West, 1977; Shugart, 1984 ), and retains most of the salient features of its parent model (see also Botkin et al., 1972). ZELIG differs from its sibling versions in that it is implemented on a grid or transect of model plots (Fig. 1 ). Grid cells can be made interactive to the extent that trees on a given cell can shade or be shaded by trees on adjacent cells. Alternatively, either the grid can be collapsed to a linear transect, or a large number of model plots can be simulated independently (in which case ZELIG is similar to a conventional FORET-style model). These three implementation modes provide for three degrees of spatial resolution in the model, and in turn, correspond to sampiing designs commonly used in empirical studies: the interactive grid mode provides the greatest spatial resolution in the model, and corresponds to contiguous sample quadrats or stem maps; the transect mode is equivalent to a line-transect sampling design, and is especially appropriate for direct gradient analyses; a grid of independent plots would be equivalent to sampling a large forest tract with a stratified layout of quadrats, and is an efficient way to model larger spatial patterns. The first application of ZELIG was to consider the spatial scaling of forest structural pattern (Smith and Urban, 1988 ). In this, the model was run as an interactive grid of 900 100-m 2 cells (30 × 30 cells). The simulated forest, a 9-ha stand, was resampled with aggregates of 4, 9, 16, ..., 100 grid cells, so that the effect of sampling scale on forest structure could be examined. The analysis emphasized the qualitative difference in forest structure and dynamics as witnessed at fine (100-m 2) versus coarse (1-ha) levels of resolution: at
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Fig. 1. Schematic of roving-window aggregate plot as defined in the ZELIG model. Leaf area shading the central grid cell is aggregated from adjacent cells according to the proportions indicated: one-half for each cardinal neighbor cell; one-fourth for each diagonal neighbor; plus the local leaf area.
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larger sampling scales, forest structure converged on the mean diameter distribution and the apparent rate of structural change slowed. The continuous nature of the gridded forest allowed Smith and Urban (1988 ) to depict the relationship between these scales (Fig. 2). RECENT DEVELOPMENTS IN GAP MODELS
The basic framework of ZELIG has recently served as a point of departure fore three new initiatives in forest modeling. The first of these examples emphasizes the individual-based nature of forest dynamics, especially addressing the implications of plant-level adaptations to forest pattern across environmental gradients. The second example extends and generalizes the treatment of abiotic environmental constraints, so that the same model could be used for broadly regional applications. The final examples take advantage of the structure of the model to simulate spatial aspects of forest pattern in a simple but explicit way. Collectively, the examples illustrate a modeling philosophy that takes advantage of the strengths of the model while minimizing the interpretative difficulties of inadequacies that are unavoidable in any model. Again, we emphasize that while the applications are all implicitly spatial, not all spatial concerns require an explicitly spatial model to provide useful insights into the phenomena.
Life-history attributes and succession Huston and Smith ( 1987 ) used a ZELIG-derived gap model, in independent-plot mode, to integrate individual differences in life-history traits to their stand-level consequences. In this, they defined a number of artificial tree species in terms of sets of life-history parameters (growth rate, longevity, shade tolerance, and maximum size). They then simulated succession under a factorial design of scenarios in which different combinations of species were represented. This resulted in a limited set of successional pathways which Huston and Smith categorized as replacement, divergence, convergence, suppression, and pseudocyclic replacement. Which pattern was expressed depended on which species were included in the succession. Smith and Huston (1989) have since extended this approach to consider two resources, above and below ground. They began with the premise that an individual cannot at the same time be well adapted to resource utilization above as well as below ground. Trade-offs that confer competitive advantage in using light (shade tolerance) have the associated cost of reducing wateruse efficiency (hence, reduce drought tolerance), while species that are drought-tolerant cannot also be shade-tolerant. There are several lines of ecophysiological evidence that support this as a reasonable working hypothesis. Smith and Huston ( 1989 ) implemented this premise as hypothetical tree spe-
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Fig. 4. Distribution of permanent, discontinuous, and no permafrost for sites within the domain of boreal forest, as simulated by Bonan (1989).
Explicit spatial effects in gap dynamics Some spatial effects in forests d e m a n d a more explicit consideration of space. T w o examples o f such effects are presented below. The first of these considers the effects o f disturbance patch size in forests: how do gap dynamics vary in response to light gaps of different areal extent? This question was addressed with a gridded version o f ZELIG. The second example is concerned with much broader-scale patterns, and attends latitudinal variation in effective gap size as effected by sun angle. This application used a transect version of ZELIG.
Disturbance patch sizes and gap dynamics The original Z E L I G model was used to illustrate changes in forest composition resulting from light gaps o f different sizes. Such patterns would arise
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naturally as a result of single- versus multiple-treefall gaps (e.g. Pickett and White, 1985 ). Smith artificially created square gaps of 400, 900, 1600, and 2500 m 2 in a mature 9-ha stand, and subsequently tracked the performance of shade-tolerant versus intolerant species in each case. With small gaps ( 4 0 0 - 9 0 0 m 2), ZELIG behavior was equivalent to that of a conventional FORET-style gap model, because disturbance patch size roughly corresponded to the size o f a FORET model plot (gap). As patch size
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show that a species can play a different seral role (early versus late successional ) depending on soil moisture availability: the behavior of the species in time depends on its spatial context (Fig. 3 ). Importantly, there is no single global strategy that confers competitive advantage under all environmental regimes; the tradeoffs in life history strategies dictate that forests must generate these patterns in space and time. Both of these examples take advantage of the individual-based nature of the model. This structure allows the model to begin with mechanisms that are expre.ssed at the individual level (and vary with tree size as well as among species), and extends these mechanisms to higher-level implications. An important conclusion from the work of Smith and Huston (1989) is that the same simple mechanism - resource competition among individual trees - can result in the myriad successional patterns observed in real forests. Neither example is explicitly spatial, but each implies general rules about forest distribution along environmental gradients that are implicitly spatial in nature.
Extending and generalizing abiotic constraints Bonan (1989 ) undertook to develop a general model that was capable of simulating boreal forest throughout its circumpolar range. Because these forest are largely constrained by abiotic factors such as temperature, soil moisture, and permafrost (Bonan and Shugart, 1989 ), the general model required extensions of these processes as implemented in previous gap models. Among these extensions was a generalized insolation submodel that corrects solar radiation - hence also temperature and potential evapotranspiration - for latitude, elevation, cloudiness, slope, and aspect (Bonan, 1989). A second extension in the model was to link insolation, soil temperature, the insulating properties of ground cover, soil moisture content, and the dynamics of permafrost; with these submodels, Bonan (1989) was able to accurately predict the spatial distribution of permafrost over much of the circumboreal region (Fig. 4 ). Bonan's ( 1989 ) results confer a transportability to his model that is consistent with the FORET tradition: given local climate, soils data, and species parameters, the basic model should work anywhere within the range of boreal forest. Some aspects of the model are general to temperate regions as well. This generality allows users to apply the same model over large regions, which is analogous to a broadly stratified sampling design in an empirical study. The general model serves the same purpose as the sampling design: it provides experimental control via a standardized method. This is especially important in applications that span broad environmental domains wherein many other factors may co-vary to confound ecological interpretation.
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increased, ZELIG behavior diverged from FORET results in that shade-intolerant species were increasingly favored by the large light gaps simulated with ZELIG (Fig. 5 ). This divergence occurs because FORET (and similar versions that use a fixed plot size) cannot account for environmental patterns (here, light availability) at scales larger than the model plot, while the interactive grid in ZELIG does account these effects. Thus, the spatially interactive ZELIG could reproduce differential species response to forest gap size as found in tropical forests (Fig. 6, and Brokaw, 1985 ).
Sun angle and gap dynamics Following Bonan's ( 1989 ) reformulation of the insolation submodel, two of the present authors (Urban and Bonan ) further modified the ZELIG model in order to explore the effects of solar angle on gap dynamics in forests at different latitudes. The goal was to generalize the treatment of the light regime in forests so that the same model could simulate forests of any stature, on any topographic position, at any latitude. This entailed partitioning directbeam and diffuse sky components of radiation, and modeling interception of these components by a mixed-age, mixed-species canopy. This version of the model depends explicitly on the structure of the ZELIG model in its transect mode. The model transect allowed a number of simplifying assumptions and streamlined computations such that the potentially overwhelming details of Diagonal 30 - ~
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the geometry of insolation in a mixed-species, multi-stratal forest could be approximated in a reasonable and straightforward way. This approach also entailed modifying the structure of tree crowns to in-
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clude vertical distribution of foliage, as developed by Leemans and Prentice (1987). In this it was assumed that, given full sun, a tree would hang leaves along the full length of its bole. Shaded trees self-prune from the bottom up, to a height where available light reaches the light-compensation point for that particular species (i.e., shade-tolerance class). Thus, as a modeled forest matures, its canopy rises and thins from below; when the canopy breaks up, trees regenerate in the light gaps created by tree deaths. This crude representation provided the geometry necessary to reproduce the effects of side-lighting and variations in effective gap size, as influenced by tree height, crown ratio, and sun angle. The model determines direct-beam and diffuse radiation as computed by Bonan's model. Direct-beam insolation (Sb) is modeled as impinging on the transect at angle 0, which is the solar altitude angle integrated between sunrise and sunset over the course of the growing season. Interception of this beam is computed by constructing a diagonal leaf-area profile at angle 0, in a stepwise fashion through the vertical leaf-area profile on each cell of the transect (Fig. 7 ). Direct-beam radiation impinging on any height position in the transect is that proportion of Sb transmitted by foliage between that position and the sun (light extinction is modeled according to Beer's Law, as in other gap models). Diffuse radiation (Sd) is similarly modeled, except it is averaged over seven arcs of the isotropic sky (i.e. seven diagonal leaf-area profiles as in Fig. 7; the number of arcs was arbitrary but the average converged rapidly and seven was a sufficient sample). Light available at any position in the canopy is the sum of direct and diffuse radiation reaching that position. This algorithm allows trees on a given transect cell to shade trees on nearby cells, and the degree to which this shading occurs depends strictly on sun angle and tree height (i.e. shadow length). These aspects of the model can be seen in Fig. 8, in which the sun is to the left at roughly at 45 ° angle. Dense foliage at one position in the canopy results in a suppressed understory on cells to the right (i.e., in the shadow); similarly, canopy gaps release a dense understory, not on the same cell, but further to the right. This version of ZELIG has been used in trial applications to gauge the magnitude of the effect of sun angle on gap dynamics, controlling for latitudinal variation in climate, edaphic site conditions, and tree species. The model generates forests with a less dense understory at flat (boreal) sun angles compared with steep (tropical) angles. Compositionally, the model predicts a persistent importance of shade-intolerant species in tropical forests, while undisturbed boreal forests support only shade-tolerant species at maturity (Fig. 9). Preliminary results also imply that different patterns of foliage distribution within tree canopies can drastically alter model results, which sets a high priority on these features for further refinement of the model. Some of these patterns have been suggested previously (Shugart, 1984, 1987 ), but earlier versions of the model were not capable of simulating lati-
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tudinal variation in the forest light environment, nor could previous models partition different components of the light regime. Of course, the geometry of the actual radiation regime of forest canopies is much more complex than is simulated in this version of ZELIG, yet the structure of the transect model permitted an approximate representation that is simple yet useful as a first approximation. Both of the previous two examples are notable in that they explicitly model spatial effect on gap dynamics, and generate patterns that cannot be reproduced with nonspatial versions of this or similar models. For example, Shugart and West (1979) used a spectrum of fixed plot sizes to illustrate the effects of gap size on forest dynamics as simulated with the FORET model. While their results overlap with the above examples in some cases, it remains that a fixed plot size in the model cannot have the flexibility to simulate the variable effect of deaths of trees of different sizes, or disturbance patches larger than a single plot. Both the gridded and transect versions of ZELIG have the flexibility to simulate a broad range of gap sizes as environmental circumstances dictate. Of course, under typical conditions both versions of the model simplify and reduce to a conventional gap model, which retains the ability to reference these spatial applications to previous and current studies with a larger family of gap models (e.g., Davis and Botkin, 1985; Pastor and Post 1986, 1988). CONCLUSIONS AND PROSPECTUS
These examples were selected to illustrate c o m m o n trends in recent applications of gap models. Each of the examples is concerned with phenomena that are implicity spatial in nature, yet is has been shown that space need not always be modeled explicitly to examine these phenomena. For example, in the case of topographic and geographic applications, site parameters that are associated with particular spatial locations can be used in a nonspatial version of a gap model, and model results can then be mapped back onto the spatial locations (e.g. Figs. 3 and 4; see also Solomon, 1986). This approach is especially amenable to use with geographic information system (GIS) mapping technology. In some applications, spatial effects must be modeled explicitly (e.g., Figs. 5-9); in these cases the structure of the ZELIG model provides a useful and versatile framework for model extensions and refinements. Further extensions to this framework, currently under development and testing, include the incorporation of contagiously propagated disturbances such as fire, and the spatial effects of seed dispersal and localized seed pools. Again, the structure of the ZELIG grid or transect simplifies implementation of such spatial extensions. Another c o m m o n theme in recent applications has been an effort to gener-
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attention to the physical processes that control abiotic environmental patterns (e.g. the physics of heat transfer). In contrast to this, efforts to generalize biotic processes and silvics in the model have often been at the expense of fine-scale details. Thus, the recent trend suggests the seeming paradox of incorporating more detail in biophysics and abiotic processes, while including fewer case-specific ecological or biological details. Again, the goal is to confer on the model a degree of generality and ease of implementation so that the same model can be applied across broad regions or diverse forests. Gap models could conceivably be made more universal while retaining the fine-scale details of interest to many forest ecologists. This would require the reformulation of tree growth routines as currently implemented, into a routine based on first principles of eco-physiology in which resource acquisition and allocation under various environmental conditions would dictate annual diameter and height growth, as well as foliage production and other features of interest. At present, quantitative submodels of these processes are not generally available (but see Farquhar and von Caemmer, 1982; Makela, 1986; Reynolds et al., 1986; Running and Coughlan, 1988). Laboratory and field experiments and tree-growth models based on physiology may soon provide such mechanistic submodels. As these are developed, gap models can be used to integrate the tree-level processes to their stand-level implications (Fig. 10 ). In this way, two seemingly competitive approaches to forest modeling can be reconciled and used in tandem as complementary tools. ACKNOWLEDGEMENT
Research supported by grant Nos. BSR-8807882 from the NAtional Science Foundation and NAG 5-1028 from the National Aeronautics and Space Administration to H.H. Shugart and the University of Virginia.
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