Abstract

Computer models are used by ecologists for studying a broad range of research questions, from long-term forest dynamics to the functional traits that theoretically give one species an advantage over others. Despite their increasing popularity, these models have been criticized for simulating complex biological phenomena, involving numerous biotic and abiotic variables, using seemingly overly simplistic computational approaches. In this article, we review the usefulness and limitations of spatially explicit individual-based models for forested ecosystems by focusing on the attributes of a recent model, called SERA (for spatially explicit reiterative algorithm), that employs seven allometric formulas and a few physical principles. Despite its simplicity, SERA successfully predicts forest self-assembly and dynamics. It also predicts phenomena that are not part of its mathematical structure. Because of this, SERA simulations can be used to explore the consequences of experimentally manipulating plant communities in ways that cannot be achieved using real communities.

Modeling forested communities to guide conservation and resource usage has a long history. The first yield tables for forests were compiled in the early eighteenth century and served as mathematical models that standardized mensuration methods in the late nineteenth century (Pretzsch et al. 2008, Pretzsch 2009). Although they continue to be useful for making informed conservation and management decisions, yield tables are unable to shed light on the complex interactions among multiple species competing for light and space in an environment that can change over the hundreds of years for which individual trees might live. For this reason, beginning in the early twentieth century, ecologists sought other, more theoretical ways to understand and predict plant population and community dynamics (Clements 1916, Watt 1925, Gleason 1926, Tansley 1935, Whittaker 1953). However, the construction of quantitatively and empirically robust theories about species ensembles has been an arduous undertaking, in large part because of inadequate data for many long-term abiotic ecological processes and the multitude of environmental factors (and associated feedback mechanisms) that influence them. In addition, even in an otherwise physically homogeneous environment, the dynamics of even a single population are extremely complex because of numerous short- as well as long-term processes involving propagule dispersal, the intensity of herbivory, competition for resources, biomass allocation patterns, and density-dependent mortality that collectively result in reiterative changes in the size—frequency distributions and physical locations of individuals (Harper 1982).

The evolution of computer models

One approach to dealing with experimentally intractable problems has been the use of computer modeling. By the middle of the twentieth century, computational technology was sufficient to numerically explore the dynamics of forested communities. By the 1960s and early 1970s, a number of models had been developed to emulate the establishment, growth, and mortality of plants (albeit in comparatively small patches, or world spaces) as a function of competition for limited resources in response to abiotic factors, such as climate and soil type (e.g., Siccama et al. 1969, Botkin et al. 1970, 1972a, 1972b). One feature that repeatedly emerged in these simulations was the appearance of gaps resulting from the deaths of large trees, the subsequent release of competitively suppressed seedlings, and an increase in the rate of recruitment. One particular "gap model," the northeastern forest growth simulator (called JABOWA), developed by Botkin and colleagues (1972b), was pivotal in stimulating the development of a new class of models that has been referred to generically as JABOWA-FORET models (e.g., Gignoux et al. 1995, Humphries et al. 1996, Shugart and Smith 1996; for an extensive review, see Bugmann 2001).

SORTIE is one of the best-known descendants of this class of models. It is a spatially explicit stochastic model of northeastern forests that describes local competition among nine dominant or subdominant tree species growing in mid- and late-successional stands using the empirically derived responses of individual plants (Pacala et al. 1993, Canham etal. 1994, Pacala et al. 1996). Community dynamics emerge in SORTIE as the result of fine-scale competition for light among neighboring plants (Pacala et al. 1993, Canham et al. 1994, Pacala and Deutschman 1995, Pacala et al. 1996, Deutschman et al. 1997). SORTIE simulations that preserve vertical structure while ignoring the effects of horizontal heterogeneity report different community dynamics; for example, successional displacement of shade-intolerant species occurs twice as quickly, and total biomass is reduced by one-half (Pacala and Deutschman 1995, Deutschman et al. 1997). These differences indicate that fine-scale horizontal interactions among trees are ecologically important.

In its original form, SORTIE is an especially complex and computationally expensive model (see Deutschman et al. 1999). These distractions stimulated the creation of the perfect-plasticity approximation (PPA) model (Adams et al. 2007, Purves et al. 2007), which, like SORTIE, is based on height-structured competition for light (which is widely acknowledged as a critical driver of canopy structure and forest dynamics; see Oliver and Larson 1996, Purves and Pacala 2008). However, PPA ignores many of the other, finer details that characterize SORTIE. In PPA simulations, the growth and mortality of trees depend on how much light they intercept, which is calculated with a 3-D light-tracing algorithm that accounts for canopy geometry and crown transmissivity. Propagules are distributed according to spatially explicit dispersal kernels. In contrast to SORTIE (which assumes that canopies are rigid, species-specific 3-D structures), the architecture of canopies in PPA simulations can extend into any unoccupied horizontal parcel, provided that the parcels minimize overlap with other canopies and conserve total canopy area. One potential drawback of PPA is that the assumption of perfect plasticity is violated when trees experience heterogeneous physical environments or dissimilar neighborhoods (Purves et al. 2008). Nevertheless, despite its idealization of canopy plasticity, PPA accurately approximates SORTIE dynamics when SORTIE is modified to include a realistic level of crown plasticity (Strigul et al. 2008), and it accurately mimics the observed species-specific size dependencies among canopy radius, depth, and vertical location (understory versus canopy) in US forests (Purves et al. 2007).

SERA

Despite their tremendous successes, SORTIE and PPA have been tested using only a limited number of species. These models have also been criticized as being either unnecessarily complex or too simplistic (see Pacala and Hurtt 1993, Bugmann and Martin 1995, Fischlin et al. 1995, Loehle and LeBlanc 1996, Schenk 1996, Deutschman et al. 1999).

Here, we explore some of these limitations and criticisms in the context of a more recent model, called SERA (for spatially explicit reiterative algorithm). As do SORTIE and PPA, SERA has statistical and mechanistic elements in its computational logic. Its statistical element involves the numerical implementation of probability functions that dictate propagule dispersal and that specify stochastic or size-dependent mortality (Hammond and Niklas 2009, 2011). The model's mechanistic element is based on the proposition that complex population and community dynamics emerge as the direct result of competition for light and space that in turn is dictated by how plants allometrically allocate mass to the construction of their reproductive and vegetative organs.

SERA was specifically designed as a tool to examine whether ¼-power rules govern plant allometry (West et al. 1997, Enquist et al. 2009). However, SERA has other pedagogical and research functions because it can be tailored to explore how specific assumptions about plant allometry affect the behavior of real or purely hypothetical plants. It may also advance conservation efforts by predicting when and how particular species survive or are driven to local extinction.

Seven allometric formulas

The algorithm that drives SERA simulations has been presented in detail elsewhere (Hammond and Niklas 2009, 2011). Briefly, each plant is modeled as having a single photosynthetic canopy that conforms to any specified geometry (but that is a uniformly thick hemisphere by default) elevated by a single untapered cylindrical stem (figure 1). Seven allometric formulas are required to simulate growth, one of which is necessary only if a species undergoes a change in its energy or mass allocation when reaching reproductive maturity table 1). The mathematical transparency resulting from the use of these seven formulas allows for the unambiguous resolution of whether a simulation's output is the direct a priori result of using specific numerical values to characterize a species or a consequence of plant—plant interactions that emerge during a simulation. These seven formulas are sufficient for the construction of a simple spreadsheet that accurately models individual tree growth when parameterized using values from a real or hypothetical species. Additional allometric relationships are used to define the properties of propagules.

Figure 1.

Plant schematics and flow chart for SERA (spatially explicit, reiterative algorithm) computational logic. (a) Tree canopy leaf mass (ML) and projected area (AL , with k as a proportionality constant) determine the energy (E) captured and converted into total annual growth (GT) that is subsequently allotted to new leaf (ML) and stem mass (MS). Partial shading reduces the effective ALand therefore GT . (b) An object's status as a propagule or nonpropagule (tree) is assessed at each iteration. If a propagule successfully germinates, its mass is allocated to the construction of seedling MLand MS. If the object is an established tree that has sufficient energy to continue growing, it is evaluated in the next iteration for vegetative and reproductive status (total reproductive mass equals propagule number, n, times the mass of the individual propagule [dispersal unit], MD). Reproductive maturity is initiated on the basis of the inequality of an equation dictating how stem biomass is apportioned between trunk diameter and height growth that in turn depends on how much light a tree received in previous growth cycles.

Figure 1.

Plant schematics and flow chart for SERA (spatially explicit, reiterative algorithm) computational logic. (a) Tree canopy leaf mass (ML) and projected area (AL , with k as a proportionality constant) determine the energy (E) captured and converted into total annual growth (GT) that is subsequently allotted to new leaf (ML) and stem mass (MS). Partial shading reduces the effective ALand therefore GT . (b) An object's status as a propagule or nonpropagule (tree) is assessed at each iteration. If a propagule successfully germinates, its mass is allocated to the construction of seedling MLand MS. If the object is an established tree that has sufficient energy to continue growing, it is evaluated in the next iteration for vegetative and reproductive status (total reproductive mass equals propagule number, n, times the mass of the individual propagule [dispersal unit], MD). Reproductive maturity is initiated on the basis of the inequality of an equation dictating how stem biomass is apportioned between trunk diameter and height growth that in turn depends on how much light a tree received in previous growth cycles.

Table 1.

The seven formulas and the numerical values for the species-specific allometric (scaling) exponents (α) and constants (β) required to initialize SERA simulations forAbies alba and three hypothetical plant species.

Table 1.

The seven formulas and the numerical values for the species-specific allometric (scaling) exponents (α) and constants (β) required to initialize SERA simulations forAbies alba and three hypothetical plant species.

SERA's world space defaults to a 1.0-hectare square, whose edges permit no influx of propagules—that is, the world space can be thought of as an island or very tall mesa. As in SORTIE, PPA, and other forest models, plants compete with neighboring plants for light, space, and—if desired—soil nutrients that stimulate growth. Trees shaded by neighbors receive less light and therefore grow less rapidly than taller, unshaded plants (figure 1). The angle of solar incidence is averaged for each growing season (which equals one simulation cycle) such that light comes, on average, from directly above, although this stipulation can be relaxed to cope with daily changes in the solar angle or differences in latitude; in the default setting, the photosynthetic area available for each plant is the projected area of its canopy (AL) at noon on the summer solstice at the equator. SERA prioritizes the order in which trees receive sunlight on the basis of tree height: The tallest trees receive light first.

The size of the canopy and the extent to which it is shaded dictate the ability of an individual to harvest light energy (E), which in turn defines the net mass accumulation (total annual growth, GT) that is allocated to new leaf and stem mass (ML and MS, respectively; figure 1). The conversion of E into GT follows the formula GT5 b1(AL ML)α1, where β1 and α1 denote a species-specific normalization (allometric) constant and scaling exponent, respectively. The allocation of new stem mass to stem diameter and height (D and H, respectively) is governed by relationships that permit (but do not demand) a transition from geometric self-similarity to geometric dissimilarity (see Nikla 1994, 2004). Depending on growth in previous years, input variables provide the option of using some portion of GT to construct propagules (seeds or fruits that can differ in number, n, and in the individual mass of the dispersal units, MD, among species), or of reserving some GT for reproduction in one or more future growth cycles (figure 1). The allocation of total plant mass (MT) to ML and MS immediately upon a propagule's "germination" is species specific. Death results from light deprivation (which is a function of the attenuation of light through overtopping canopies), stochastic and age-dependent processes, or the violation of physical laws (figure 1). Simulations are initiated with one or more propagules that can be randomly or nonrandomly scattered, depending on the objectives of a particular simulation.

In the simulations explored here, each simulation begins with the same number of randomly dispersed propagules. At maturity, the trees of each species produce the same number of propagules, all of which have the same initial body mass. Propagules are randomly dispersed around each parent plant at a maximum dispersal distance that is proportional to the height of the parent, by means of a ballistic dispersal equation. All propagules germinate and compete for light and space immediately with neighboring plants, if there are any.

Initializing simulations

All of the 15 variables in the seven SERA formulas can be determined empirically from measurements taken over a few years, or they can be numerically estimated from large published data sets. For example, table 1 lists the values for a typical conifer species (Abies alba Miller), for a typical angiosperm shrub or tree species, and for a variant of an average angiosperm. The data for A. alba come from direct measurements of trees reported by Cantiana (1974) and Hellrigl (1974). The values for the two “average” species are based on regression analyses of worldwide data tabulated by Cannell (1982) for 343 conifer species and 332 angiosperm species.

It is also possible to simulate hypothetical species on the basis of theoretical expectations about how energy and mass are allocated as a plant grows. For example, one prominent theory predicts that basal metabolic rates should, on average, scale as the ¾ power to body mass across all organisms and that most allometric relationships will follow ¼- (or multiples-of-¼-) power rules (West et al. 1997, Enquist et al. 2009). To explore the consequences of ¼-power rules, we modified the allometric equations for an average angiosperm species to create a hypothetical species (denoted as “angiosperm variant”). Importantly, the numerical values listed for this variant species differ little from those of the average angiosperm species, which allows us to see whether a few small allometric differences affect coexistence. For hypothetical species, questions such as these can be answered only with mathematical models.

Simulated dynamics

As in SORTIE, PPA, and many other forest computer models, SERA shows that competition for light and space dictate the ability of a plant to survive and coexist. SERA simulations also show that (a) very small niche differences affect the ability of trees to coexist, (b) the probability of species coexistence increases in spatially heterogeneous worlds, (c) disturbance further increases the probability of coexistence, (d) average plant size decreases and relative growth rate increases as the frequency of disturbance increases (i.e., as in r selection), (e) rare species survive along the borders of patches differing in resource availability, (f) the death of large plants creates light gaps that can be occupied by rare species, and (g) populations in communities can coexist over decades but can fluctuate significantly over hundreds of years (i.e., assessments of the success of a particular species require long-term observation). The appearance of these phenomena as a result of the operation of very simple physical principles suggests that the world of real plants may involve fewer and less complex rules than was previously thought.

Consider, for example, the simulations of A. alba, which were initialized on the basis of the data collected from a single population planted at an initial density of 25,000 plants per hectare that was observed every 5 years over a 95-year period, starting 10 years after the initial planting (Cantiana 1974, Hellrigl 1974). A statistical comparison between the scaling relationships predicted by SERA and those actually observed in the field reveals no significant differences table 2). In addition, SERA accurately predicts age-related trends (figure 2). And, as might be expected from the biology of real plants, simulations of trees with a greater reproductive effort (i.e., a larger reproductive biomass commitment) predict smaller values of leaf and stem growth, as well as smaller canopy and stem mass—a phenomenology that closely mimics the behavior of the real A. alba population (figure 2).

Table 2.

Comparisons between observed and predicted scaling exponents (α) and their 95% confidence intervals (CIs) for allometric relationships within anAbies alba population and the populations of an average conifer and angiosperm species (see table 1).

Table 2.

Comparisons between observed and predicted scaling exponents (α) and their 95% confidence intervals (CIs) for allometric relationships within anAbies alba population and the populations of an average conifer and angiosperm species (see table 1).

Figure 2.

Bivariate comparisons among observed age-dependent trends (in five-year intervals) in a real Abies alba population (denoted by the circles) and those predicted by SERA emulating normal reproductive effort and three times normal effort (denoted by the Xs and plus signs, respectively). (a) Annual growth in leaf mass (GL, in kilograms [kg] per year). (b) Annual growth in stem mass (GS, in kg per year). (c) Standing leaf mass (ML, in meters [m]). (d) Standing stem mass (MS, in m). (e) Stem diameter (D, in m). (f) Stem height (H, in m). The arrow in panel (f) shows the age at which trees are predicted to reach reproductive maturity. This conifer species was parameterized as is shown in table 1. Age was measured in years. Abbreviation: added repro., additional reproductive effort.

Figure 2.

Bivariate comparisons among observed age-dependent trends (in five-year intervals) in a real Abies alba population (denoted by the circles) and those predicted by SERA emulating normal reproductive effort and three times normal effort (denoted by the Xs and plus signs, respectively). (a) Annual growth in leaf mass (GL, in kilograms [kg] per year). (b) Annual growth in stem mass (GS, in kg per year). (c) Standing leaf mass (ML, in meters [m]). (d) Standing stem mass (MS, in m). (e) Stem diameter (D, in m). (f) Stem height (H, in m). The arrow in panel (f) shows the age at which trees are predicted to reach reproductive maturity. This conifer species was parameterized as is shown in table 1. Age was measured in years. Abbreviation: added repro., additional reproductive effort.

Predicting the age of reproductive maturity

Perhaps more interesting is the ability of SERA to predict the age at which plants typically reach reproductive maturity. For example, A. albais reported to reach reproductive maturity after 25 years of growth under ideal growing conditions (Wolf 2003), whereas SERA predicted that it would reach maturity at 24 years of age. However, nothing in SERA's computational logic directly bears on when plants become reproductive. Rather, aspects of life history are correlated with empirically observed juvenile-to-mature shifts in the relationship between plant height and stem diameter (formulas 6 and 7 in table 1). Specifically, when β8 + β8InD > β6Dα6 - β7, plants are identified as mature, and their subsequent growth in height conforms with formula 7 table 1, figure 1). This change in the scaling of height with respect to stem diameter explicitly triggers a change in the allometry of leaf growth with respect to stem growth that alters the subsequent ability of a plant to harvest light. In turn, the change in the ability to harvest light permits trees to gain extra energy, which can be diverted in order to produce propagules. Despite changes in the allometry of height with respect to stem diameter, total vegetative growth with respect to total aboveground body mass shows no simultaneous perturbation either for individual plants or for the entire A. alba population (a regression of the predicted versus observed values gives r2 ≥ .971). Therefore, the shift in the allometry of height with respect to stem diameter appears to be an example of a global set of allometric shifts during the growth of both real and simulated plants.

Other emergent properties

Sensitivity analyses involving three or five separate computer simulations for each species, each using different numerical values for parameterization, show that some population-level phenomena are inextricably dependent on the numerical values used to initialize a simulation. However, some key ecological phenomena emerge repeatedly, regardless of how species are parameterized. Consider that regardless of the scaling exponents and allometry constants required to parameterize species, the scaling exponent for the relationship between standing leaf mass and stem diameter is invariably statistically indistinguishable from 2.0, a numerical value that has been reported for ecologically diverse plant populations and communities (Enquist and Niklas 2001, Niklas 2004). Likewise, the scaling exponents governing the relationship between the mass of an average plant and standing plant density (i.e., the exponent for self-thinning), size—frequency distributions, and life span versus body mass are successfully predicted by SERA, despite the absence of input variables that affect these biological relationships in SERA's computational logic. For example, the predicted exponent for A. alba self-thinning in three separate computer runs was −1.79, whereas the observed value for the real population was −1.77 (figure 3). When computer simulations achieve plant density equilibrium, the exponents for log-transformed data for plant density versus stem diameter average −2.0, which is numerically indistinguishable from the exponent reported for real plant populations (Niklas et al. 2003). Finally, all computer simulations thus far predict that plant life spans will scale as the ¼ power of total body mass, a scaling relationship that has been reported for diverse plant species (Marbá et al. 2007). However, there is nothing in SERA's computational logic that dictates these relationships, other than the effects of competition for light on plant survival and size.

Figure 3.

The scaling of self-thinning in a simulated population of Abies alba, parameterized as is shown in table 1. (a) Polar view of a population at the end of 250 years of simulated growth. The dots indicate the locations of the stems of each tree; the circles indicate the projected areas of the canopies of trees in the largest of four different size categories. (b) Bivariate plot of total plant mass (MT) as a function of tree density (N) for 20 different years during the growth of the population shown in panel (a). The curved line is the regression curve for the log transformed data for MT as a function of N, which has a slope of −1.79. The slope for MT as a function of N of the real population used to simulate A. alba is -1.77.

Figure 3.

The scaling of self-thinning in a simulated population of Abies alba, parameterized as is shown in table 1. (a) Polar view of a population at the end of 250 years of simulated growth. The dots indicate the locations of the stems of each tree; the circles indicate the projected areas of the canopies of trees in the largest of four different size categories. (b) Bivariate plot of total plant mass (MT) as a function of tree density (N) for 20 different years during the growth of the population shown in panel (a). The curved line is the regression curve for the log transformed data for MT as a function of N, which has a slope of −1.79. The slope for MT as a function of N of the real population used to simulate A. alba is -1.77.

Species persistence

As a result of heightened extinction rates and the grim effects of invasive species, there is greater interest in understanding factors that contribute to (or that detract from) the ability of different species to coexist. Like many other models, SERA can be used to examine what happens in a computer-generated world when simulated plants differing in their niche requirements for light and space compete under spatially homogeneous or heterogeneous conditions or when their environment is physically disturbed.

Consider one classic observation in field ecology: Angiosperm tree species tend to displace the majority of conifer tree species at low elevations or low latitudes. In contrast, conifers tend to dominate forested communities at high elevations and latitudes, presumably because their xylem is less prone to damage caused by subfreezing temperatures (Harper 1982, Carlquist 2001). Given this context, an interesting question is whether SERA predicts the competitive displacement of conifers by angiosperms when a population composed of an average conifer species is exposed to a population of an average angiosperm species. Another question is whether conifers gain ascendency if the mortality of angiosperms is increased to mimic vascular tissue damage caused by freezing.

The data drawn from conifer- and angiosperm-dominated communities shown in table 1 can be used to parameterize an average angiosperm species and an average conifer species to conduct both of these experiments in silico, although it is important to acknowledge that each of these two generalized species reflects an amalgam of species, some of which are pioneer evergreen or deciduous shrub or tree species, whereas others are shade-tolerant species that are either rare or commonplace.

With this caveat in mind, computer simulations initialized with equal numbers of angiosperm and conifer propagules quickly predicted that conifers would be relegated to progressively smaller nested groupings that are eventually extinguished as propagules germinate and that the plants would increasingly interact as they increase in size (figure 4). The simulations also showed that the reduction in conifer plant density is not entirely attributable to the angiosperms' reaching reproductive maturity, on average, five years earlier than the conifers, because the conifers were competitively excluded, even when both of the species were parameterized to have equivalent reproductive effort and age at maturity. Despite this level playing field, the conifers were displaced by the angiosperms because of light deprivation and reduced growth rates that resulted from how annual growth is partitioned to produce new leaves and stems. Although stochastic death played a small role in influencing their fate, the conifers maintained a presence in the simulated communities (and even gained dominance) when angiosperm mortality was increased by as little as 10% or when the fraction of light required for conifer seedling survival was reduced by 10% in order to mimic increased shade tolerance.

Figure 4.

Four stages in the assembly and maturation of a community initialized with equal numbers of seeds of an average angiosperm species and an average conifer species (parameterized as is shown in table 1) showing the gradual reduction in the number of conifer plants that are slowly aggregated into small clustered populations. Conifers are indicated by filled circles. (a) Community at 15 years. (b) Community at 50 years. (c) Community at 75 years. (d) Community at 100 years. The world space was 100 meters × 100 meters.

Figure 4.

Four stages in the assembly and maturation of a community initialized with equal numbers of seeds of an average angiosperm species and an average conifer species (parameterized as is shown in table 1) showing the gradual reduction in the number of conifer plants that are slowly aggregated into small clustered populations. Conifers are indicated by filled circles. (a) Community at 15 years. (b) Community at 50 years. (c) Community at 75 years. (d) Community at 100 years. The world space was 100 meters × 100 meters.

Can a ¼-power tree species coexist?

The ability of the hypothetical (average) angiosperm species to rapidly displace an average conifer species raises the question of whether the average angiosperm can coexist with a slightly modified species governed by one or more ¼-power rules, which is very similar to the average angiosperm (the angiosperm variant). The only notable exception is the 2/3-scaling exponent dictating the relationship between tree height and stem diameter table 1), which emerges from the first principles governing the mechanical stability of vertical columns such as tree trunks (Niklas 1992, 2004).

Clearly, this speculation cannot be tested empirically. No known plant species abides entirely with all ¼-power rules. Likewise, no perfectly average angiosperm species exists. However, computer models are not limited to simulating real species. Indeed, this is one of the many advantages of employing models: They can be used to perform thought experiments. In this context, we can ask what happens when the angiosperm variant and the average angiosperm species compete for the same resources in the same world space. Duplicate simulations in which equal numbers of propagules of the two species were randomly distributed repeatedly predicted that the two species would coexist, despite significant fluctuations in abundance in an otherwise homogeneous world space (figure 5a). These fluctuations result from the two species' reaching reproductive maturity at approximately the same time. As a consequence, the two populations fluctuate synchronously as a result of the nearly synchronous cycles of cadre birth and death (figure 5a).

Figure 5.

Time courses (in years) of changes in the number of trees representing an average angiosperm species and of trees representing a species complying with ¼-power rules (denoted as angiosperm variant). The species were parameterized as is shown in table 1. Equal numbers of seeds for each of the two species were randomly distributed in a world space measuring 100 meters × 100 meters. (a) Time course for a simulation in a spatially homogeneous world space in which a hypothetical soil nutrient was set at 100% availability (diagramed as a white square; see the insert). (b) Time course of a simulation in a heterogeneous world space partitioned into four equal-sized patches differing in nutrient availability (diagramed as a 2 × 2 square in which the shading denotes the four nutrient availabilities; white = 100%, black = 25%; light gray = 75%; darker gray =50%). (c) Time course of a simulation in a heterogeneous world space (partitioned in the same way as those in panel b) experiencing disturbance every 20 years (simulated as changes in soil nutrient levels in a shifting pattern indicated by the arrows).

Figure 5.

Time courses (in years) of changes in the number of trees representing an average angiosperm species and of trees representing a species complying with ¼-power rules (denoted as angiosperm variant). The species were parameterized as is shown in table 1. Equal numbers of seeds for each of the two species were randomly distributed in a world space measuring 100 meters × 100 meters. (a) Time course for a simulation in a spatially homogeneous world space in which a hypothetical soil nutrient was set at 100% availability (diagramed as a white square; see the insert). (b) Time course of a simulation in a heterogeneous world space partitioned into four equal-sized patches differing in nutrient availability (diagramed as a 2 × 2 square in which the shading denotes the four nutrient availabilities; white = 100%, black = 25%; light gray = 75%; darker gray =50%). (c) Time course of a simulation in a heterogeneous world space (partitioned in the same way as those in panel b) experiencing disturbance every 20 years (simulated as changes in soil nutrient levels in a shifting pattern indicated by the arrows).

Spatiotemporal heterogeneity and a two-species coexistence model

However, if equal numbers of propagules from the two species are distributed in a stable 2 × 2 heterogeneous world space—for example, one in which a hypothetical soil nutrient, such as nitrogen or phosphorus, is partitioned into four equal-sized patches differing in nutrient availability (at 25%, 50%, 75%, and 100% levels)—the simulations predicted very different community dynamics. In this world space, the concentration of the soil nutrient affected total plant growth in direct proportion to its concentration. Under these conditions, the angiosperm variant gradually declined in number relative to the number of average angiosperm trees (figure 5b). Although this result is difficult to depict graphically, these simulations also indicated that the two species have reproductive—mortality cycles that are nearly 90° out of phase. The explanation for the different dynamics observed between a homogeneous versus a stable 2 × 2 heterogeneous world space rests in the prediction that the angiosperm variant can grow more rapidly than the average angiosperm under conditions of very low nutrient availability, but only at the expense of delaying reproduction relative to the advent of reproductive maturity among average angiosperm plants. The trade-off between being successful at gaining space and light and producing propagules that can colonize other locations in the 2 × 2 world space is not equitable. The angiosperm species' tactic, which is to produce propagules earlier in its life history at the expense of growing to a large size, works to its advantage.

This situation changes, however, when the two species compete in a 2 × 2 world space that experiences 20-year cycles of disturbance, which is mimicked by shifting the locations of the different concentrations of the soil nutrient (figure 5c). Under these conditions, both species experience dramatic and nearly synchronous fluctuations in species numbers. Yet, unlike their behavior in a stable 2 × 2 world space, both species persist (even in simulations that have been run for the equivalent of 500 years; data not shown). The explanation for this contrast in behavior lies in the extirpation of both species in parcels distinguished as nutrient poor; the colonization of these parcels by nearly equal numbers of propagules of the two species; and the general absence of competition among the resulting juveniles, owing to their small sizes (due to their slow growth).

Disturbance and species persistence

The foregoing simulations indicated that hypothetical species sharing very similar numerical (allometric) parameterizations can behave very differently when placed in spatially stable world spaces compared with when they are placed in unstable heterogeneous world spaces. This behavior raises the question of whether this is a generalizable phenomenology. To explore this issue, we parameterized four additional hypothetical species to mimic comparatively small niche differences table 3), and, as in all previous simulations, introduced equal numbers of propagules from each species into spatially stable or unstable homogeneous and heterogeneous environments. The objective was to evaluate the effects of habitat and niche differences on species coexistence and survival.

Table 3.

The seven formulas and the numerical values for the species-specific allometric (scaling) exponents (α) and constants (α) required to initialize SERA simulations of communities composed of four hypothetical plant species (A—D).

Table 3.

The seven formulas and the numerical values for the species-specific allometric (scaling) exponents (α) and constants (α) required to initialize SERA simulations of communities composed of four hypothetical plant species (A—D).

More than 118 simulations were performed to explore these effects; we were able to collectively make five predictions: (1) Small niche differences can result in strong competitor advantages or disadvantages, (2) spatially homogeneous environments do not favor species coexistence, (3) spatially heterogeneous and predictable environments enhance the probability of species coexistence, (4) spatially heterogeneous and disturbed environments further increase the probability of species coexistence but in ways that make it extremely difficult to predict which species will persist (i.e., although some species repeatedly emerge as the dominant species in spatially predictable and unpredictable environments, the identities of subdominant and rare species often differ as a result of small stochastic events during the early stages of community assembly), and (5) old plants (and the boundaries among regions differing in nutrient availability) provide opportunities for rare species to survive and even prosper.

Most if not all of these phenomenologies have been mimicked by other computer models, just as they have been repeatedly observed in many real forested communities. In this respect, their appearance in SERA simulations can be construed as trivial. However, this perception misses the point; that is, the appearance of distinctly biological features in computer simulations stripped of most biological details other than competition for light and space (mediated by biomass partitioning patterns dictated by simple allometric relationships) suggests that real forest dynamics may be driven by a set of comparatively simple rules.

Space constraints preclude presenting the details required to support this contention. However, to illustrate just one example, we focus on the prediction that rare species tend to survive by capitalizing on the attrition of old plants or by surviving at the edges of areas differing in resource availability, a prediction that has some interesting implications for conservation efforts, and one that is easily visualized. To select one among many simulations, consider the results obtained from the four hypothetical species parameterized as in table 3 competing among themselves in a spatially stable world space divided into four equal-size patches differing in the availability of a hypothetical nutrient (figure 6). Over the 250-year course of this simulation, a total of 7740 plants grew and died. At the end of the simulation, one species numerically dominated (n > 1000; species C), two species survived only as small populations (n < 500; species A and D), and another was driven to extinction (species B). At the end of the simulation, most of the plants from species A and D survived in the world-space patches with the two highest nutrient concentrations (75% and 100%). However, reasonably substantial numbers of plants (50 ≤ n ≤ 150) persisted in the other two world-space patches, but clustered primarily at or near the boundaries of the adjoining two patches with the higher nutrient concentrations (figure 6). This distribution of plants was the norm throughout the entire simulation, as can be seen from the distribution of the locations of all plants over the entire course of the simulation.

Figure 6.

Polar views of the edge effects created in a heterogeneous world space partitioned into four equal-sized patches differing in nutrient availability (in the same way as those in figure 5b). Equal numbers of seeds for each of four hypothetical species (species A-D, parameterized as is shown in table 3) were randomly distributed and allowed to grow and to compete for light and space. Each dot or symbol indicates the location of the stem of a plant. The dots denote species C (the most abundant species); the circles and triangles denote species A and D (which persisted to the end of the simulation); the location of species B trees are not shown, because this species was competitively excluded midway through the simulation. (a) The spatial distribution of all trees of species A, C, and D throughout the 250-year simulation. The total number of trees was 7740. (b) The spatial distribution of all of the trees of species A and D that survived to the end of the simulation. Many of these plants survived in the two patches with the lowest nutrient availabilities but were aggregated near the boundaries of the world space or near the edges of neighboring patches that differed significantly in nutrient availability.

Figure 6.

Polar views of the edge effects created in a heterogeneous world space partitioned into four equal-sized patches differing in nutrient availability (in the same way as those in figure 5b). Equal numbers of seeds for each of four hypothetical species (species A-D, parameterized as is shown in table 3) were randomly distributed and allowed to grow and to compete for light and space. Each dot or symbol indicates the location of the stem of a plant. The dots denote species C (the most abundant species); the circles and triangles denote species A and D (which persisted to the end of the simulation); the location of species B trees are not shown, because this species was competitively excluded midway through the simulation. (a) The spatial distribution of all trees of species A, C, and D throughout the 250-year simulation. The total number of trees was 7740. (b) The spatial distribution of all of the trees of species A and D that survived to the end of the simulation. Many of these plants survived in the two patches with the lowest nutrient availabilities but were aggregated near the boundaries of the world space or near the edges of neighboring patches that differed significantly in nutrient availability.

Edge effects

The fact that the majority of plants of all four species grow and survive reasonably well in regions with high to medium nutrient availability is not surprising. That substantial numbers of species A and D also survive along the boundaries created by adjoining patches with high and low nutrient concentrations is perhaps more interesting. Recall that propagules that disperse beyond the external boundaries of the flat world space die (i.e., the world space is not curved). Detailed examination of the outputs of simulations such as the one shown in figure 6 indicate that the canopies of these plants straddle the boundaries between patches. Parts of their canopies experience considerable shading by the canopies of the more successful species C. However, other parts of the same canopies are exposed to high or full ambient sunlight, owing to the general absence of taller competing plants. Another contributing factor to the survival of species A and D is the deaths of old plants, the oldest of which generally occupy the two patches with the lower nutrient availabilities. These plants become established early in the assembly of communities, grow slowly because of poor resource availability, but experience little to moderate competition from other plants because of the low density of plants. Because the distribution of propagules at the beginning of a simulation is random, each of the four species has roughly the same probability of landing in one of the two patches with the low nutrient availabilities. Therefore, a nearly equal number of old plants of each of the four species grew in these patches.

Examination of these simulations also shows that the deaths of old plants and the subsequent formation of light gaps give the propagules of all four species nearly equal opportunities to germinate and survive. Accordingly, even though the growth of all plants in two patches is diminished compared with plants growing in the other two patches, all four species have equal opportunities to survive and reach reproductive maturity. The superimposition of edge effects and equal opportunities, on average, gives the more rare species a slight advantage, albeit in very limited locations.

Theoretical advances

A number of ecological concepts are fundamental to any model of population or community self-assembly and dynamics. Among these are competition for resources, the effects of disturbance (stochasticity), the nonequilibrious nature of vegetation (succession), and the concept that ecological dynamics operate simultaneously at different spatial and temporal scales (Levin and Paine 1974, Grubb 1977, Pickett and White 1985). Some of the first mathematical models to cope with these concepts were predictive, nonspatial models developed using a Markovian approach that utilized a matrix of empirically determined transition probabilities to predict species replacement and composition (e.g., Feller 1968). Gap models subsequently took center stage with the increased access to progressively more-powerful computers (e.g., JABOWA; Botkin et al. 1972b) and have been adapted worldwide (for reviews, see Shugart 1984, Urban and Shugart 1992, Bugmann 2001). More recently, advances in computational speed have lead to the generation of a number of spatially explicit stand-level models that are capable of engaging numerous sources of empirical data to predict the behavior of large-scale ecological processes (e.g., SORTIE and PPA).

In parallel with the development of these and other computer models, theoretical advances have contributed greatly to our understanding of population and community dynamics. For example, the formulation of neutral models has provided predictions for the dynamics of plant communities lacking species-specific interactions or niche structures (Hubbell 2001, 2006; see, however, Hammond and Niklas 2011). The predictions emerging from these models provide null hypotheses against which to compare the empirical effects of various ecological processes, many of which were assumed to result from species-specific interactions and differences in niche structure (e.g., Chave 2004). In this way, analytical (mathematical) models and brut-force (computer) models have helped to identify factors that mechanistically drive ecological processes and patterns (Alonso et al. 2006, Zilio and Condit 2007, Clark 2008). For this reason, modeling has a long and distinguished history in ecology and will undoubtedly continue to be used with greater frequency as researchers focus on the behavior of landscapes, ecosystems, and entire biomes.

Experimental validation of models

Despite these advances, a number of conceptual and philosophical problems continue to emerge whenever a model is used for any purpose. We have purposely alluded to these problems when presenting our SERA simulations. Here, we return to two of the most important and challenging problems: How do we know if a model is giving the correct answers for the wrong reasons, and when is a model sufficiently robust?

In the context of the first question, like other models, SERA can successfully mimic the behavior of real plant populations and communities. SERA can also predict ecological phenomena that have been observed for real forests as well as phenomena that, at the very least, seem biologically reasonable in light of what we know about plant ecology. However, because real species (e.g., A. alba) must be parameterized using empirically determined relationships, it is not surprising that simulations give “the right answers.” By the same token, in the case of predictions based on hypothetical species, there is no guarantee that simulations give us any insight into ecological mechanisms. On one hand, we expect a model to retrieve the relationships used to characterize species (models that fail to do so are suspect), whereas, on the other hand, we have no proof that a model provides mechanistic insights simply because it makes “reasonable” predictions.

How then do we test models like SERA? We believe that there are two obvious, yet powerful, tests. The first is to experimentally manipulate real and simulated populations or communities. Logically, these experiments must manipulate variables that are considered in the model, which requires that they be designed to uncover trends or patterns that are not used to initialize simulations or that are used to parameterize species used in experiments. For example, the data used to initialize the A. alba simulations were gathered from a real population generated from planting 25,000 seedlings. The simulations resulting from this initial plant density predict that the growing population self-thins according to very specific power rules, which is consistent with empirical observation, even though there is nothing in SERA's computational logic that dictates how a population or a community will self-thin. A simple experimental test, therefore, is to see whether SERA accurately predicts the scaling exponents for self-thinning in populations initiated with different seedling densities or whether different planting arrangements of seedlings give different scaling relationships. Indeed, a large number of experiments can be designed and implemented in this and other ways. The point is not that these experiments can be performed but, rather, that they illustrate an approach that can be used to test models like SERA.

Emergent properties

A second test is whether a model has emergent properties. The concept of emergent properties is contentious, in part because it means different things to different people. For example, in the context of the SERA simulations presented here, we could define emergent properties as phenomenologies that change significantly when plants are grown in isolation compared with when they are grown together. Self-thinning is an obvious example. A less obvious emergent property is that SERA accurately predicts the age at which trees reach reproductive maturity in populations or communities. This feature materializes because SERA identifies a plant as reproductively mature whenever its trunk undergoes a transition from geometric self-similarity to elastic self-similarity on the basis of the relationship β8 + β8 InD > β6Dα6 − β7 (see figure 1). However, the age at which this transition occurs is dictated only by the extent to which the individual has harvested sunlight and has therefore grown in height and trunk girth, and this capacity is dictated by the extent to which it has been shaded by its neighbors, which is driven entirely by density-dependent packing. This is evident from comparisons of the ages at which isolated and crowded plants reach reproductive maturity: Isolated plants invariably reach reproductive maturity at significantly earlier ages than plants growing in SERA populations or communities.

Another issue about modeling is the trade-off between the level of detail that we expect a model to provide and the extent to which a model has general applicability. That is, when is a model sufficiently robust? Clearly, there is no general answer to this question. It all depends on our expectations of a particular model and how it will be used. For example, we designed SERA to address a central question: What happens when trees compete for light and space and obey very simple physical laws and processes? For this reason, our model does not directly consider competition belowground (i.e., competition for water and soil nutrients among roots is not modeled). Philosophically, we justified this tactic in part on the basis of the fact that what we see happening aboveground is an indirect reflection of what is happening belowground, because the organism is an integrated phenotype. Put differently, the allometric trends observed for how real plants partition their annual growth to construct new leaves and stems must a priori reflect how they invest in the production of new roots. Therefore, parameterizing a species on the basis of aboveground allometric trends indirectly embeds the allometry of root biomass growth and allocation. Another justification for neglecting roots is a standard premise in modeling: If it's not required, don't use it. In general, the more complicated a model becomes, the less general its predictions. SERA makes the fewest assumptions required to predict what appears to be credible biological behavior.

Mechanistic explanations

Another more fundamental concern is whether models like SERA give insights into ecological mechanisms. Do these models tell us how things work? This is the most difficult question to answer when dealing with any model, even those that give the right answers, because a “mechanistic explanation” means different things to different people. Consider that SERA simulations suggest that competition for space and light is sufficient to explain numerous aspects of plant population and community dynamics. This raises the question of whether competition serves as a mechanistic explanation or merely a description of a more deeply embedded phenomenon. Clearly, there are different forms of competition, and different organisms compete for different things differently. In this sense, competition is not a mechanism sensu stricto; it is a phenomenon in search of one.

Many, if not all, of the issues raised here will continue to be discussed and debated as long as science relies on modeling. In the case of SERA, we believe it is best to view models as mechanisms to test null hypotheses that make the logical (mathematical) consequences of their assumptions immediately transparent or as simple heuristic tools that can be used in the classroom to explore “what if” questions. Either application provides an opportunity to learn about how we cast our questions and what we put into our models.

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Author notes

Sean T. Hammond and Karl J. Niklas (kjn2@cornell.edu) are affiliated with the Department of Plant Biology, Cornell University, Ithaca, New York.