Hurricane disturbance and the population dynamics of a tropical understory shrub: megamatrix elasticity analysis

John B. Pascarella


Many organisms live in habitats that are characterized by extreme, but relatively infrequent temporal variants of the environment. Spatial and temporal variation in ecological factors that influence population dynamics is common for many plant species (Clark and Clark 1994, Horvitz and Schemske 1995). Nevertheless, estimates of demographic parameters are often obtained from single populations, single patches, and single points in time (Sarukhan and Gadgil 1974, Hartshorn 1975). In other cases, weighted, mean, or multiplicative matrices have been used to simulate environmental variation in space and time (Bierzychudek 1982, Pinero et al. 1984, Huenneke and Marks 1987, Horvitz and Schemske 1995). Conclusions about the relative importance of life history stages drawn from single-environment models may differ from conclusions drawn from models that specifically incorporate environmental variation (Menges 1986, 1988, 1990, Schemske et al. 1994). The effects of environmental variation on plant population dynamics has been modeled using analytical matrix models that incorporate information on patch transitions (Horvitz and Schemske 1986, Cipollini et al. 1994) or stochastic sequence models (Tuljapurkar 1980, 1982, 1989, Tuljapurkar and Orzack 1980, 1989, Silva et al. 1991, Kalisz and McPeek 1993).

For understory plants in forest environments, canopy disturbance is often strongly positively correlated with growth and reproduction (Levey 1988, 1990, Denslow et al. 1990, Niesenbaum 1992, Pascarella 1995). In general, many understory plants are not density limited, but light limited. Their dynamics are thus not driven by intraspecific density effects but rather by the opening and closing of the forest canopy above them (Horvitz and Schemske 1986, 1995, Horvitz 1991). Many understory plants are long lived; they are not usually killed by the events causing the opening of the canopy. Instead, they may survive and respond to the change in canopy cover In many forests, the predominant disturbance event affecting understory plants is individual treefalls, which create relatively small patches of open canopy. In these environments, models of understory plant demography have been constructed that use information on patch-specific demographic transitions relating to gap-phase successional states and transitions between different patch types that encapsulate gap-phase successional processes (Horvitz and Schemske 1986, Cipollini et al. 1994). One of the main foci in these models is understanding the effect of dispersal rate on population growth rate.

In contrast to spatial variation in canopy cover caused by treefalls, temporal variation in canopy cover is the predominant effect caused by hurricanes. Understory plants in some Caribbean subtropical forests experience hurricane disturbance at frequent intervals (Brokaw and Walker 1991, Tanner et al. 1991). Although hurricanes are usually considered catastrophes for canopy tree species (Horvitz et al. 1995), understory shrubs usually suffer very little direct mortality (Pascarella 1995). For light-limited understory plants, posthurricane light environments may offer critical windows for population growth. The scale of hurricane disturbance is much larger than individual treefalls. Thus, temporal variation in patch-specific demographic transition rates may be more important than small-scale (individual tree-sized) spatial variation. In these environments, dispersal among patches at the landscape level is relatively unimportant as nearby forest patches may experience similar levels of disturbance.

Our goal in this study was to determine the importance of environmental variation caused by hurricanes to the demography of the tropical understory shrub Ardisia escallonioides using a megamatrix approach. The megamatrix approach results from cross-classifying individual plants by life stage and patch type and assuming that stage transitions and patch transitions occur alternately within each year. Our principal questions were: (1) What is the expected proportion of the habitat in different states of canopy openness, according to a hurricane-driven model of forest dynamics? (2) How did the demography of the plant vary as a function of canopy openness? (3) What is the relative importance (elasticity) of the different patches to overall population growth? (4) What is the relative importance of different life history stages to overall population growth rate according to single-environment matrix analysis as compared to dynamic-environment megamatrix analysis?


Study species and sites

Ardisia escallonioides Schlecht. & Cham. (Myrsinaceae) is an understory shrub in the subtropical forests of south Florida (Long and Lakela 1971). It also occurs in the coastal limestone forests of the Bahamas, Cuba, Hispaniola, Eastern Mexico, Belize, Guatemala, and northern Honduras (Lundell 1966, CIQRO 1982). Ardisia escallonioides flowers in the fall following vegetative growth during the summer (Tomlinson 1980). The principal habitat of A. escallonioides in south Florida is subtropical hardwood forest on the Miami rock ridge and in the Florida Keys (Tomlinson 1980, Snyder et al. 1990); the total range of the species in Florida includes small populations north of this area, primarily along the coast (Little 1978). Subtropical hardwood forests in south Florida are found on localized outcrops of limestone, embedded in a dominant landscape of pineland or wet prairie, and contain primarily tropical tree and shrub species of wide distribution (Tomlinson 1980, Snyder et al. 1990). They are similar floristically and structurally to forests in the West Indies and the Yucatan Peninsula (CIQRO 1982, Smith and Vankat 1992). The study region is classified as subtropical moist forest (Dohrenwend and Harris 1975) using the life-zone system of Holdridge (1967). Mean annual temperature in Miami is 23 [degrees] C and ranges from 20 [degrees] C in December to 28 [degrees] C in August. Mean annual rainfall is 1500 mm/yr but is highly seasonal, with 78% total rainfall occurring between May and October. The dry season is marked by extended rainless periods of 1525 d (Chen and Gerber 1990). Soils in subtropical hardwood forests in the Miami rock ridge typically consist of thin sandy soils with extensive areas of surface limestone rock outcroppings; sinkholes are common (Brown et al. 1990).

On 24 August 1992, Hurricane Andrew passed over the mainland of south Florida (Stone et al. 1993). Sustained wind speeds in the northern eyewall exceeded 241 km/h with gusts up to 285 km/h, making Andrew a category-4 hurricane on the Simpson-Saffir scale (Pielke 1990, Mayfield et al. 1994). We had four study sites between 80 [degrees] and 81 [degrees] W and 25 [degrees] and 26 [degrees] N in which there were populations of Ardisia escallonioides. Study plots at two populations were established prior to Hurricane Andrew in June 1992 (Castellow Hammock [CAS] and Deering Estate [DEE]), at one after the hurricane in September 1992 (Matheson Hammock [MAT]), and at one in December 1993 (John Pennekamp State Park [JPS]). All study plots are within protected natural areas. DEE and CAS probably experienced the highest sustained winds and gusts while MAT experienced both lower sustained winds and gusts (examination of maps and descriptions from Mayfield et al. 1994, Wakimoto and Black 1994). Due to its location well to the south of the hurricane eye, JPS was not directly affected by Hurricane Andrew (R. Skinner, Florida State Parks, personal communications).

Megamatrix model: introduction

We generated a large matrix whose dimension is given by the product of the number of life history stages of the plant with the number of temporal variants of the environment. Such a “megamatrix” includes two dynamic processes: the dynamics of the environment and the dynamics of the organism within each environmental state. A formal mathematical presentation of the megamatrix can be found in Bharucha (1961), Cohen (1977), and Tuljapurkar (1982), while an intuitive biological presentation is given in Horvitz and Schemske (1986) and Cipollini et al. (1994).

The megamatrix view of the landscape assumes that the landscape consists of many patches, so that in the long run the population experiences a landscape near the stable patch distribution. We think of each patch corresponding to a set of plants inhabiting one hammock (tropical hardwood tree island, ranging in size from 0.1 to 40 ha) and the overall population landscape corresponding to the whole region of southern Florida that includes the Miami Rockridge region, the Tamiami limestone region, and the Florida Keys. Our model assumes that the patchy history of hurricanes over this landscape has caused these patches of forest to be in different states of canopy closure and that the evolutionary ecology of the understory species that inhabit this landscape is best understood in the context of the temporal variability caused by hurricanes.

Matrix model construction: patch dynamics

Patch dynamics were modeled as a linear Markovian process, in which a vector, f, of patch types at time t, [TABULAR DATA FOR TABLE 1 OMITTED] is acted on by a set of patch transition probabilities, C, to become a vector of patch types at time t + 1 [ILLUSTRATION FOR FIGURE 1 OMITTED],

f(t + 1) = Cf(t). (1)

A matrix of patch-type transition probabilities, C, was generated from data for south Florida on historical hurricane frequency and severity. The matrix C predicts a particular stable patch distribution for the dynamic environment. To determine the predicted distribution of patch types in the environment, we found the column eigenvector, [f.sup.*], associated with the dominant eigenvalue of the C matrix.

The main parameters used to calculate the entries of the C matrix are the probabilities of low, medium, and severe canopy disturbance, p1, pm, and ps, respectively, and of no hurricane, nh, occurring in a year (Table 1). For example, a hurricane that generates severe canopy disturbance may cause a closed-canopy forest to become 65% open, with 50% probability (top right-hand corner of Table 1), while the same kind of hurricane would cause patches that were 15% open before the hurricane to be 65% open after it (top row, sixth element in Table 1), with 100% probability. Similarly, any hurricane at all will keep a 65%-open patch in its 65%open state. Even in the absence of hurricanes, many patches will remain quite open, with 75% probability (top row, first element in Table 1). In the absence of hurricanes, forest canopies tend to close over time: they [TABULAR DATA FOR TABLE 2 OMITTED] have a higher probability of getting more closed than they do of staying in the same state. Eventually, they reach the “closed-canopy” state, in which they remain unless there is a hurricane (bottom right-hand corner of Table 1). The values of p1, pm, ps, and nh depended upon: the probability that a hurricane crosses southern Florida in a given year, the probability that a given forest patch is in the path of the hurricane, and the amount of canopy disturbance, which itself depends upon the severity of the hurricane overall as well as the heterogeneity of wind speeds within the hurricane (See Appendix A for details). We note that Hurricane Andrew, a severe storm (category 4), caused severe canopy disturbance at one study site, which was at the northern edge of the inner eye wall of the hurricane, and moderate to low canopy disturbance at other study sites located at differing distances from this area (Horvitz et al. 1995, Pascarella 1995).

Matrix model construction: population dynamics within a patch type

Stage classes. – To characterize population dynamics, we used Lefkovitch (1965) stage-classified transition matrices instead of Leslie age-classified matrices (1945). Same-aged individuals of our study species may vary in size due to different light environments, and in such cases, stage class may be more predictive of fate than age class (Caswell 1989). We distinguished stage classes by biological criteria (Kirkpatrick 1984, Horvitz and Schemske 1995), rather than numerical criteria that are sometimes used (Vandermeer 1975, 1978, Moloney 1986). We used a combination of size (height and dbh [diameter at breast height]) and reproductive criteria (flowering, fecundity) to determine stage classification (Pascarella 1995) (Table 2). Adults were distinguished from juveniles by height (1 m) based on field and shadehouse observations. Seedlings were easily distinguished from small juveniles in the field by their small, serrate leaves, which contrast with large, entire leaves of juveniles (J. B. Pascarella, unpublished data).

Matrix model analysis

Patch-specific matrices. – The projection matrix model for each patch-specific matrix was:

n(t + 1) = An(t) (2)

where n(t) is a vector of all the individuals in the population at time t, classified by stage, n(t + 1) is the vector for the population at the next time interval, and A is the matrix that shows how individuals in each stage class at one time may become or contribute to each stage class one time unit later, in which the columns refer to stage at time t and the rows refer to stage at time t + 1. For this study, A is an 8 x 8 matrix, in which each entry, [a.sub.ij], refers to the contribution of individuals in the jth class at time t to the ith class at one time unit later [ILLUSTRATION FOR FIGURE 2 OMITTED].

Four populations, which were observed for two posthurricane years, spanned the gradient from no canopy disturbance (JPS), low canopy disturbance (MAT), moderate canopy disturbance (DEE), and high canopy disturbance (CAS). These populations provided the patch-specific demographic transition rates. Data for the specific matrices came from the following sites and years: [less than]5% canopy openness, JPS 1993-1994; 15% canopy openness, MAT 1993-1994; 25% canopy openness, MAT 1992-1993; 35% canopy openness, DEE 1993-1994; 45% canopy openness, CAS 1993-1994; 55% canopy openness, DEE 1992-1993; and 65% canopy openness, CAS 1992-1993. Analysis of hemispherical canopy photographs at these sites and years confirmed these canopy openness categories (Pascarella 1995).

The empirical data from the study plot at the closed-canopy site (JPS) produced a matrix that was not “full rank” (sensu Caswell 1989), because some life history events in such places occur very rarely. Specifically, none of the plants of reproductive size were observed reproducing and none of the juveniles grew sufficiently to make a transition to prereproductive size. We do not believe that these events never occur in the closed-canopy habitat, but rather that they are so low in frequency that a much larger sample size would be needed to detect them empirically. A matrix that is not full rank and not connected among all life stages does not produce asymptotic behavior comparable to matrices that are full rank. To produce a matrix that would have comparable asymptotic behavior to the others, we estimated these transition values (reproduction and growth of juveniles) at a low but nonzero value. We used data on fecundity for the [less than]5% open patch from prehurricane data from DEE (1991-1992) and included a small transition (0.01) from juveniles to prereproductives, while reducing the observed transitions of juveniles-juveniles from 0.96 to 0.95. To investigate the sensitivity of the model to these nonempirically determined transition parameter values, we ran simulations at 10, 50, 110, and 150% for each estimated adult fecundity and juvenile transition.

To determine the fates of all stages except seeds and seedlings, five 5 x 10 m plots were established in each population to span the natural variation in density of Ardisia escallonioides for a total area of 250 [m.sup.2]/population. In December 1993, the density of all stages excluding seeds and seedlings ranged from 0.39 plants/[m.sup.2] (JPS) to 1.45 plants/[m.sup.2] (CAS). Within each plot, all stems over 1 m tall were marked with numbered plastic bird bands. To sample juveniles, 10 1 x 1 m subplots (50 subplots/population) were randomly established within each plot. For each adult and juvenile, we counted the number of stems [greater than] 1 m in height, measured the height of the tallest stem, and recorded the dbh of all stems [greater than]1 m in height. To determine total seed production, every fruit produced was physically counted; only one seed is produced per fruit (Tomlinson 1980).

No natural seedling cohorts could be found in any population in December 1992 (seed production prior to the hurricane was very low) (Pascarella 1995). To estimate transition rates from seeds to seedlings and first-year seedling survival, we conducted field germination experiments in both 1993 and 1994 (Pascarella 1995). Seed to seedling transitions were determined by multiplying: the percentage of seeds germinating in the census interval x survival of seedlings to the end of census period. First-year seedling survival was estimated using seedlings derived from the seed germination experiments and was calculated by dividing the number seedlings alive at December 1994 by the number of seedlings alive in December 1993. Neither seed germination nor seedling survival varied significantly with light availability (Pascarella 1995); thus, the mean of all populations was used as an estimate of these two transitions for all matrices.

Growth and reproduction were highly seasonal, associated with the rainy season (May-October) [ILLUSTRATION FOR FIGURE 3 OMITTED]. We used a December-December census interval. We have taken care to multiply the probabilities of events that occur on a less than 12-mo time frame to generate the December-December matrix entries. These calculations are dependent upon the phenology of the biological events with respect to a chosen census interval (Caswell 1989). For example, in our study, seeds are counted as the mature seeds found on the plant in early December. How can we predict the number of seeds at t + 1 from the size of a plant at time t? This depends upon the amount of flowering the plant will do during the year from t to t + 1; the amount of flowering depends upon the growth that will occur during the early-to-mid rainy season (flowers are produced at the tips of the new branches) [ILLUSTRATION FOR FIGURE 3 OMITTED]. In this context, the top row of the matrix represents the probability a plant of a given size to contribute to next year’s seeds; one must ask for each plant whether it will be a reproductive of a given size next year and how many seeds it will make if it is a reproductive of a given size (see Horvitz and Schemske 1986, 1995, for a discussion of this kind of parameterization of the top row for plant populations). Mathematically, this process is represented as

[a.sub.ij] = [Sigma] [a.sub.ij] x [r.sub.i] (3)

summed over all reproductive stages (i goes from 3 to 8), where [a.sub.ij] = the probability that a plant of stage j will become a reproductive of stage i and [r.sub.i] = the number of seeds produced per plant of stage i. This means that any stage that includes some plants that may become reproductive by time t + 1 may have a nonzero entry in the top row (Horvitz and Schemske 1986, 1995, Caswell 1989). Another example of how the matrix parameters are calculated with respect to the phenology is the calculation of [a.sub.21], the probability that a seed will become a seedling. Seeds counted on plants in early December have to survive on the plants, be dispersed (sometime during February-May), “wait” through the beginning of the rainy season, germinate in mid-to-late-rainy season (August-November), and survive as seedlings until the census in early December [ILLUSTRATION FOR FIGURE 3 OMITTED].

The dominant eigenvalue of each patch-specific matrix A, [[Lambda].sub.a], is an estimate of the asymptotic population growth: [[Lambda].sub.a] corresponds to the average fitness of individuals in a given environment; In [[Lambda].sub.a] = r, the instantaneous growth rate (Fisher 1930, Charlesworth 1980, Caswell 1989). Values of [[Lambda].sub.a] were calculated for each patch type. The associated right column eigenvector, [w.sub.a], gives the stable stage distribution, while the associated left row eigenvector, [v.sub.a], gives the relative stage-specific reproductive values (Caswell 1989). These vectors can be used to yield a measure of proportional sensitivity, elasticity (Caswell et al. 1984, de Kroon et al. 1986):

[e.sub.ij] = [a.sub.ij]/[[Lambda].sub.1] x [Delta][[Lambda].sub.1]/[Delta][a.sub.ij] = [a.sub.ij]/[[Lambda].sub.1] x [v.sub.i][w.sub.j]/ (4)

for each entry of the matrix, [a.sub.ij]. The concept of demographic elasticity is useful for determining the proportional contribution of matrix elements to long-term population growth (Caswell et al. 1984, de Kroon et al. 1986, Mesterton-Gibbons 1993). The elements of the elasticity matrix [E.sub.a], [e.sub.ij], are the proportional effects on population growth rate of proportional changes in each element of the matrix (Caswell 1989).

Megamatrix analysis

The hurricane season (June-November) is characterized by a peak occurrence of hurricanes in mid-to-late August (Pielke 1990, Doehring et al. 1994). It is necessary to carefully consider the relationship of the plant phenology, hurricane phenology, and the timing of our census in the formulation of the megamatrix model. The essence of the megamatrix model is the cross classification of individuals by life stage and patch type. The megamatrix entries give the probability of a plant in a given life stage-patch type to become or contribute to another life stage-patch type by the next census. The probabilities arise from alternating the life-stage transition process (represented by an 8 x 8 matrix) with the patch-type transition process (represented by a 7 x 7 matrix) within a year. This yields a process that can be represented by a 56 x 56 matrix.

The order of multiplication does matter in matrix processes. How do we predict the fates of plants at t + 1 from their life stages and their patch types at time t? Starting in early December, the patch state is not likely to change until the peak hurricane point, 8.5 mo later [ILLUSTRATION FOR FIGURE 3 OMITTED]. During the first 8.5 mo, the plants are doing all their life-stage transitions according to the “rules” of the patch type they were in at time t. In fact the main life-stage events, including those that determine reproduction, happen before the peak of the hurricane season, since growth occurs mostly before the peak in hurricanes [ILLUSTRATION FOR FIGURE 3 OMITTED] and growth is what determines the amount of flowering. Thus, we model the sequence of dynamic processes as life-stage transitions first, followed by patch-type transitions. To clarify this sequence, consider an alternative scenario. If hurricanes occurred in April and plants did most of their growing in June and we continued to census December-December, then we would multiply the dynamic processes in a different order: patch-type transitions first, life-stage transitions of plants in their new habitats second. The megamatrix model is

n(t + 1) = Mn(t) (5)

where M is the 56 x 56 megamatrix, in which each entry gives the probability that a plant in a given patch type and stage class at time t will contribute to or become another stage class in the same or another patch type by time t + 1. The [m.sub.ij[Alpha][Beta]] entry refers to the transition probability from stage j in patch-type [Beta] to stage i in patch-type [Alpha]. For a mathematical presentation of calculating the entries in M, see Appendix B.

The dominant eigenvalue of the megamatrix, M, [[Lambda].sub.m] is an estimate of asymptotic overall population growth in the dynamic system where hurricane disturbance causes patch types to change. The associated right eigenvector, [w.sub.m], is proportional to the stable-stage by patch-type distribution. There are two approaches to calculating reproductive value for temporally varying environments; one would be to simply report the left [TABULAR DATA FOR TABLE 3 OMITTED] eigenvector of the megamatrix. Here, we followed an alternate protocol and we calculated the stage-specific reproductive values according to Tuljapurkar (1989); [v.sub.m], the associated left eigenvector, was multiplied by [f.sup.*], which weights the reproductive value by the stable patch frequency.

Elasticity analysis of the megamatrix: new uses of this parameter

To evaluate the relative contribution to population growth of proportional perturbations in life-stage transitions summed within patch types as well as the relative contribution of proportional perturbations in lifestage transitions summed by stage both within and across patch types, we present three types of summed elasticities. These were found by summing across certain elements of the elasticity megamatrix, [E.sub.m], where the elements are [e.sub.ij[Alpha][Beta]] (defined above as for the M matrix). In the following, we use the shorthand “importance” to refer to the magnitude of an impact on [Lambda], which would be produced by a proportional perturbation to a subset of matrix elements.

1) Patch elasticity can be defined as follows: sum for each [Beta] over ij and [Alpha]. It is defined as

[summation over ij[Alpha]] [e.sub.ij[Alpha][Beta]] = [E.sub.[Beta]]. (6)

This yields one such sum for each patch type and a vector (1 x 7) of patch elasticities. It gives the importance of life history events within one patch type compared overall to life history events in another patch type.

2) Stage-class elasticity by initial patch can be defined as follows. For a given patch type, [Beta], and a given stage, j, summed over all i and [Alpha]:

[summation over i[Alpha]] [e.sub.ij[Alpha][Beta]] = [E.sub.j[Beta]]. (7)

This yields 56 such sums, a (1 x 56) vector of stage class elasticities by initial patch type. It gives the importance of life history events involving a particular stage within a patch type relative to other stages and/or the same stage in other patches.

3) Summed stage-class elasticity can be defined as follows: for a given stage class, j, summed over [Alpha][Beta] and i,

[summation over i[Alpha][Beta]] [e.sub.ij[Alpha][Beta]] = [E.sub.j]. (8)

This yields one sum for each stage class and a vector (8 x 1). It gives the importance of life history events involving a particular stage compared overall to other life history stages.

The “eig” function of PC-MATLAB Version 3.2 (Mathworks 1987) was used to obtain the eigenvalues and the eigenvectors for both patch-specific and megamatrix analysis. We obtained the right eigenvector from the matrix itself and the left eigenvector from the transpose of the matrix. We then scaled the right and left eigenvectors so that all the elements of the right eigenvector summed to one and the scalar dot product (w, v) equaled one (Caswell 1978). We calculated elasticity values using the scaled vectors.

Elasticity is one of two commonly utilized “prospective” sensitivity analyses (sensu Horvitz et al. 1996). By “prospective,” it is meant that these analyses address the hypothetical question of what would happen to population growth rate if particular parameters were to vary, while others remain unchanged. Elasticity addresses proportional perturbations to life history parameters, while the other commonly utilized “prospective” analysis, called “sensitivity,” addresses perturbations by fixed linear amounts (Caswell 1989, Horvitz et al. 1996). Elasticity parameters are additive across a matrix, summing to one. This feature lends itself to asking questions about the relative importance of “groups” of parameters to overall population growth, addressing the question of the relative impact on population growth of changing the transitions of one group vs. another group. For example: compare the effect of changing all seedling transitions by 10% to the effect of changing all adult transitions by 10%. The present paper is concerned with examining the relative importance of both stages and habitats to overall population growth; elasticity analysis is the key to addressing these questions.


Patch dynamics

Values for the patch-transition matrix C are shown in Table 3. The stable patch type distribution ([f.sup.*]) predicts mostly closed-canopy patches ([less than]5% open light availability), 58% of all forest patches [ILLUSTRATION FOR FIGURE 4 OMITTED]. Very open patches (65% open and 55% open) are the second most common, with 9.4 and 9.1% of forest patches, respectively. Intermediate patch types (15-45% open) were the least frequent, ranging from 4.9 to 6.8% of forest patches.

Population growth rates

Several demographic parameters varied over the different patch types. In general, adult fecundity and growth were enhanced in more open patches, while survival was variable across the range of patch types (Table 4, [ILLUSTRATION FOR FIGURE 5 OMITTED]). Long-term patch-specific population growth rates ([Lambda]) ranged from 1 in the closed-canopy patch ([less than]5% open) to 1.96 in the most open patch (65% open) [ILLUSTRATION FOR FIGURE 4 OMITTED]. In intermediately shady patches (1555% open), population growth rates were intermediate but varied; highest values were found at 25%-open patches (1.80) and 45%-open patches (1.72). The lowest growth rate (1.45) was in the 35%-open patch. Long-term population growth rate ([Lambda]) of the megamatrix was 1.71.

With respect to the simulations of variable adult fecundity and juvenile transition rate in the closed-canopy patch, population growth rate decreased slightly with lower adult fecundity ([less than]0.01%) but was not sensitive to increased fecundity. Population growth rate increased slightly (0.2%) with higher juvenile transition rates and decreased slightly (0.07%) with lower juvenile transition rates.

Summing across all stages by patches, the stable-stage by patch-type distribution was characterized by a higher frequency of plants in the more open patches [ILLUSTRATION FOR FIGURE 6A OMITTED]. Summing across all patches by stages, the stable stage by patch type distribution was characterized by a higher frequency of seeds than of any other stage class [ILLUSTRATION FOR FIGURE 6B OMITTED]. Following seeds, other small stage classes such as seedlings and juveniles were the most common stage classes. Reproductive value was dependent upon both stage and patch type. Summing across stages, reproductive values were concentrated in the open patch (65-45% open) [ILLUSTRATION FOR FIGURE 6C OMITTED]. Summing across patches, reproductive values increased with increasing stage class [ILLUSTRATION FOR FIGURE 6D OMITTED].

Elasticity analyses

The megamatrix had the highest patch elasticity in the most open patches [ILLUSTRATION FOR FIGURE 7 OMITTED]. The most open patch (65% open) contributed 64% total elasticity and the second most open patch (55% open) contributed 22% total elasticity. Less than 1% total elasticity was contributed from the closed-canopy patch ([less than]5% open), the most common patch type. With respect to the simulations of adult fecundity and juvenile transition rate in the closed canopy patch, summed patch elasticity was relatively insensitive to variation in adult fecundity ([less than]0.01% difference). Summed patch elasticity was slightly more sensitive to variation in juvenile transition rates (2.2% difference at 65% open patch). The rank order of elasticity of patches remained the same under all simulations except for the highest simulated juvenile transition rate (150% estimated), in which the closed patch ([less than]5% open) had double the elasticity of the [less than]15%-open patch.

Patch-specific elasticity of stage classes analyzed from individual matrices differed dramatically between the closed-canopy patch type ([less than]5% open) and all other patch types (15-65% open) (Table 5). In the closed-canopy patch, all the elasticity was contained in very large individuals, while the elasticity was more evenly distributed across stage classes in the 15-65%-open patch types. In these patches, small juveniles had the greatest proportional elasticity in all but the 45%-open patch where they were second in importance to prereproductives. In all but the closed-canopy patch ([less than]5% open), other important stage classes were seedlings, seeds, and prereproductives. For most patches, elasticity contributions of adults declined with larger stage class (Table 5).

With respect to the simulations of adult fecundity and juvenile transition rate in the closed canopy patch, patch-specific elasticities of stage classes of the most closed patch ([less than]5%) were sensitive to the choice of estimated juvenile transition rates but not to adult fecundity. Single-matrix elasticities were similar at 10 and 50% estimated juvenile transition rates but differed greatly at 110 and 150% estimated juvenile transition rates. At the higher rates, stage-class elasticity was similar to that observed at the more open patch types (1565%). Juveniles had the greatest elasticity with small adults and prereproductives contributing [less than]6% elasticity.

The summed stage-class elasticity of the megamatrix (Eq. 8) was similar to the patch-specific matrices for environments that ranged from 15 to 65% open and distinct from the patch-specific matrix for the closed-canopy environment ([less than]5% open). Elasticity of the megamatrix [TABULAR DATA FOR TABLE 4 OMITTED] was highest for juveniles, followed by seeds, seedlings, and prereproductives, while adult stage classes contributed the least amount of elasticity. The stage-class elasticity within a given patch type of the megamatrix (Eq. 7) varied across patch types, with stage classes within more open patches contributing the greatest amounts of elasticity [ILLUSTRATION FOR FIGURE 8 OMITTED]. However, the relative contributions of particular stage classes within a patch type was similar to the summed stage-class elasticity of the megamatrix. In all but the 15%open patch type where they were second, juveniles were the most important stage class.

With respect to the simulations of adult fecundity and juvenile transition rate in the closed canopy patch, summed stage-class elasticity of the megamatrix was relatively insensitive to variation in adult fecundities (0.01% change in elasticity values). Summed stage-class elasticities were slightly more sensitive to variation in juvenile transition rates (maximum difference 1% in juvenile elasticity), but the rank order of stage classes was similar in all simulations with juveniles contributing the greatest elasticity.


Patch dynamics

Under the historical hurricane frequency for the study region, Ardisia escallonioides populations in hardwood hammocks occurred under closed-canopy forests most of the time, according to the model. Very open forests were rare, occurring only after severe hurricane disturbance. This result is surprising given that south Florida experiences one of the highest frequencies of hurricane disturbance in the United States and the Caribbean (Simpson and Lawrence 1971, Simpson and Riehl 1981, Pielke 1990, Scatena and Larsen 1991). However, canopy recovery is quite rapid, even following severe hurricane disturbance (C. C. Horvitz et al., unpublished data; Pascarella 1995). Canopy openness, as measured using hemispherical canopy photographs, decreased from 7 to 28% in the three hurricane-damaged sites from 1993 to 1994, with the greatest decrease in the most damaged sites (Pascarella 1995). This was due to regeneration of lateral branches of standing trees, growth of new vertical stems of fallen trees, and also rapid height growth of pioneer species (Horvitz et al. 1995).

The patch-transition matrix is the long-term average of the system. The distribution of patch types results from the stable-stage distribution of the patch-transition matrix. The probabilities of patch transitions were determined from historical data on hurricane frequency, strength, size, and impact. The actual distribution of patch types in nature at any point in time may depend on variation in the parameters that determine the patch-transition matrix. For example, hurricane abundance may be cyclical, with periods of greater hurricane abundance followed by periods of fewer hurricanes (Pielke 1990).

It might be suggested that validation of the predictions of the habitat model in our analysis could be done through examination of current canopy status of hardwood forests in south Florida through remote sensing or field data in relation to the passage of recent hurricanes as in Boose et al. (1994) and Lugo (1995). However, the usefulness of this approach in our study region is complicated by the additional natural disturbances of fires and freezes (Loope and Urban 1980, Snyder et al. 1990). Similar heterogeneity of disturbances was not a factor in the forests studied by Boose et al. (1994). In addition, in the greater south Florida area, anthropogenic habitat destruction through land clearing for urbanization and agriculture, arson fires, and exotic vine invasion have also impacted canopy status of many remnant hardwood forests (Snyder et al. 1990, Horvitz et al. 1995). Because of these logistic complications, we judge the validation of the model to be beyond the scope of the present paper.

Single-matrix population dynamics

Under the closed-canopy condition ([less than]5% open) populations had the lowest growth rates, but they were not decreasing. Adults had minimal growth but high survival. Persistence of A. escallonioides in closed-canopy forest is not surprising as other Ardisia species are believed to be shade tolerant (Kappelle 1993, Shimizu 1994). Reproduction was limited in these low-light conditions and seed production was minimal. Following hurricane disturbance, which removes much of the overstory canopy, flowering increased and fecundity increased (Pascarella 1995). In addition, plants exhibited increased vegetative growth (Pascarella 1995). The highest population growth rate occurred in the most open patch (65% open). However, population growth did not increase monotonically with openness in the intermediate patch types. This variation may be due to effects of biotic interactions with an insect seed predator, which is a specialist on flowers and/or fruits of Ardisia escallonioides (Pascarella 1996). For example, fecundity was highly reduced in the 35% open patch type due to the effects of this insect. In this patch (DEE 1993-1994), moth populations had recovered substantially following a dramatic drop in population size immediately following Hurricane Andrew (Pascarella 1995).

Single-matrix elasticities varied considerably among environments. If only the most common patch types were examined ([less than]5% open, single matrix), only very large adults would appear to be critical to population growth elasticity. In contrast, if some of the rarer patches (55% open, 65% open) were examined, the perturbation of small juveniles would have a more critical impact on population growth than perturbations of other stages. However, this result was influenced by the [TABULAR DATA FOR TABLE 5 OMITTED] parameter values used in the closed canopy patch. With respect to the simulations of adult fecundity and juvenile transition rate in the closed-canopy patch, single-matrix elasticity of the common closed-canopy patch ([less than]5%) was sensitive to juvenile transition rates. At higher juvenile transition rates, no differences were noted in the relative rankings of stage classes among patch types.

Megamatrix dynamics

Population growth and disturbances. – According to the megamatrix model, long-term population growth rate was substantially greater than 1. These high population growth rates suggest Ardisia escallonioides should be abundant in subtropical forest understory communities in south Florida. In several community studies of subtropical forests in the Miami rock ridge area, A. escallonioides is usually the dominant or codominant (with Psychotria nervosa Sw.) understory shrub (Alexander 1958, 1967, Olmsted et al. 1983, Molnar 1990).

Although our analysis suggests that hurricane disturbance is beneficial for populations of this understory shrub, other disturbance types (droughts, fires, freezes) may negatively affect A. escallonioides populations in south Florida. As an evergreen species, A. escallonioides may be susceptible to periodic droughts. During normal years, plants often appear wilted by the end of the dry season (J. B. Pascarella, personal observation). In addition to physiological effects, severe droughts are associated with extensive fires in adjacent pineland patches that infrequently enter and burn through subtropical forest patches (Loope and Urban 1980, Loope and Dunevitz 1981, Snyder et al. 1990). Most hammock adults of A. escallonioides are top-killed from these fires but may resprout from the underground roots (Taylor and Herndon 1981). In nondrought years, fires in adjacent pinelands usually do not enter into the moister, shaded hammocks. However, fire disturbance may not be as common for many of the hammock preserves outside of Everglades National Park that no longer are surrounded by pineland due to urbanization. A. escallonioides populations in pinelands are also susceptible to freezes, which occur irregularly (Olmsted et al. 1993; J. B. Pascarella, personal observation). Understory populations in subtropical forest are not usually harmed by freezes, however (Snyder et al. 1990). For both fires and freezes, mortality is usually low as plants rapidly resprout from underground tubers and rhizomes, but plant stature is reduced with potential impacts on fecundity (Olmsted et al. 1993; J. B. Pascarella, personal observation). All three types of disturbances may limit population growth, either through increased mortality, reduced growth, or reduced fecundity.

In other parts of its range in the West Indies, Mexico, and Central America, fires are probably not common except in pineland patches; most collections of Ardisia escallonioides from Mexico and Central America are from subtropical moist forest, not pinelands (Luridell 1966). Most populations, with the possible exception of populations in extreme northeastern Mexico, do not experience freezes. Thus, we believe that hurricane disturbance may be the primary disturbance type that affects populations of A. escallonioides throughout most of its range.

Stable-stage distributions, reproductive values, and elasticity

Stable-stage distributions had a numerical dominance of seeds, a pattern typical of many plant species classified by age, size, or stage (Werner and Caswell 1977, Bierzychudek 1982, Pinero et al. 1984, Horvitz and Schemske 1995). Reproductive values increased with maturity of the stage reaching a peak in very large adults. This pattern is generally consistent with other stage-classified plants (Caswell and Werner 1978, Caswell 1986, Huenneke and Marks 1987, Horvitz and Schemske 1995) in which senescence is lacking or in which large plants die from stochastic rather than physiological processes (Sarukhan 1980). We found that reproductive values of large adults varied across patch types, a pattern that has not been found in other studies of plant demography that include environmental variation (Silva et al. 1991, Cipollini et al. 1994).

Life history events in different patches contribute differentially to population growth, according to the megamatrix analysis. Even though very open patches (55-65% open) were rare according to the stable-patch distribution, the most important patches for population growth were the most open patches. According to the elasticity analysis, this means that small changes to life history transitions in open patches would have a disproportionate impact on population growth rate compared to equivalent proportional changes in transitions in all the other kinds of patches.

Comparison of patch-specific matrices and megamatrix analyses

The insights from single-environment matrix and dynamic-environment megamatrix analyses differed in several respects. If only closed-canopy patches, the most common patch type, were examined (since these are typical patches for the species under a single-environment matrix approach), the general conclusion would have been a population that was stable in size in which very large adults are most critical to population dynamics. In comparison, the megamatrix model suggested a population that was growing quite rapidly, in which the most critical patches were the high-light posthurricane patches, and in which changes in transitions of many stage classes from seeds to small adults would contribute proportionately quite a lot to changes in population growth. Although single-matrix analysis was sensitive to the choice of parameter values for juvenile transitions, the megamatrix analysis was not as sensitive to the choice of parameter values.

The megamatrix approach emphasizes the average population demography at the stable patch-type distribution. An alternative view is a stochastic analysis in which a time series of environments is generated at each patch, the relative abundances of patch states vary stochastically and the long-run overall demography is examined by looking at the overall environment over a time series (Tuljapurkar 1982, 1996). This kind of analysis is also underway and will be the subject of another paper (S. Tuljapurkar, J. Pascarella, and C. Horvitz, personal communication). Megamatrix and stochastic approaches that sample the range of potential patches and model the transitions among patch types are likely to be more realistic than single-environment analyses in determining both population growth rates and predicting which stages and patches are the most critical for population growth.


This paper is part of a doctoral dissertation submitted to the University of Miami. Funding was provided by a University of Miami Curtis Plant Sciences Fund and a Graduate Research Award. I was supported by a National Science Foundation Predoctoral Research Fellowship and a University of Miami Maytag Fellowship. I wish to thank Metro-Dade County Natural Areas Management for permission to work at the Dade County Parks and Florida State Parks, Division 7, for permission to work at John Pennekamp State Park. T. Fleming, S. Tuljapurkar, S. Ellner, and four anonymous reviewers provided helpful comments on earlier drafts of this paper. This is contribution number 641 from the University of Miami Program in Tropical Biology, Ecology, and Behavior.


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We parameterized the canopy disturbance model based on the historical record of hurricane frequency and severity in extreme southern Florida. We combined data on the historical record of hurricane occurrence along 80.5 km of coastline in southern Florida (Simpson and Lawrence 1971), p(h),

p(h) = 0.135 (A.1)

with data on the width of hurricanes (48.3 km, Pielke 1990) to calculate the probability that a hurricane would cross a given patch of forest, p(hur),

p(hur) = 0.6 x 0.135 = 0.081. (A.2)

Then we considered data on the relative frequency of hurricanes of categories 1, 2, 3, 4, and 5 in the study region to generate a vector of probabilities (Doehring et al. 1994),

p(hurclass) = [0.125 0.25 0.375 0.1888 0.063]. (A.3)

We estimated the probability of low, medium, and severe canopy disturbance for a given class of hurricane. These probabilities were estimated from our comparative data on canopy openness after Hurricane Andrew (a category-4 hurricane) at study sites along a gradient of disturbance (Loope et al. 1994, Armentano et al. 1995, Horvitz et al. 1995). The following matrix was used to generate the probability that a population would experience low (25% canopy removal), medium (45% canopy removal), or severe (65% canopy removal) canopy disturbance (rows) for each class of hurricanes (columns):

[Mathematical Expression Omitted]. (A.4)

For example, in a category-1 hurricane, most (80%) populations would experience low canopy disturbance, 20% of populations would experience medium canopy disturbance, and 0% of populations would experience severe canopy disturbance. In a category-5 hurricane, 10% of the populations would experience low, 40% medium, and 50% severe canopy disturbance.

Then the vector of probabilities that a forest containing an A. escallonioides population in southern Florida will suffer low, medium, or severe canopy disturbance, [p1 pm ps], as a result of hurricanes is given by

p(lms) = [p(disturbance by category) p(hurclass) p(hur)], (A.5)

and the probability that a population will not experience any hurricane, p(nh), is given by

p(nh) = 1 – p(hur). (A.6)


A formal way to represent the process that was intuitively described in the text is by use of a Kronecker product. The megamatrix combines the patch-transition matrix with each patch-specific population dynamics matrix. First, we define two matrices, an 8 x 8 identity matrix ([I.sub.s]), and an 8 x 8 matrix that is all zeros ([Z.sub.s]). Next, we define a 56 x 56 matrix, [A.sub.17], composed of the [A.sub.i]’s and [Z.sub.s]’s, as follows:

[Mathematical Expression Omitted] (B.11)

where [A.sub.1], . . ., [A.sub.7] are the patch-specific plant demography matrices and C is the patch-transition matrix (Table 1). Then the megamatrix, M, is defined as:

M = (C [cross product] [I.sub.8]) x [A.sub.17] (B.2)

where C [cross product] [I.sub.8] is the Kronecker product between C and [I.sub.8]. A Kronecker product between two matrices [(C [cross product] [I.sub.8]) in our example] takes each element of the first of the matrices (C) and multiplies it by the whole other matrix ([I.sub.8]). In our example, (C [cross product] [I.sub.8]) yields a 56 x 56 matrix that is composed of 49 submatrices, each of which is an 8 x 8 matrix that is a diagonal matrix composed of zeros on the off-diagonal and one of the elements of C along the diagonal. [c.sub.11] is the diagonal of the first 8 x 8; [c.sub.12] is the diagonal of the 8 x 8 to the right of it; and [c.sub.13] is the diagonal of the 8 x 8 adjacent to it, as follows:

[Mathematical Expression Omitted]

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