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(American Journal of Botany. 2003;90:911-914.)
© 2003 Botanical Society of America, Inc.

Ecology

Saguaro (Carnegiea gigantea, Cactaceae) age–height relationships and growth: the development of a general growth curve1

Taly Dawn Drezner2

Department of Geography, Bolton Hall 410, P.O. Box 413, University of Wisconsin, Milwaukee, Wisconsin 53201-0413 USA

Received for publication October 22, 2002. Accepted for publication January 10, 2003.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 LITERATURE CITED
 
Because the growth rate of saguaros varies across different environments, past studies on saguaro population structure required extensive data collection (often over many decades) followed by site-specific analysis to estimate age at the sampled locale. However, when height–growth data from different populations are compared, the overall shape of the growth curves is similar. In this study, a formula was developed to establish saguaro age–height relationships (using stepwise regression) that can be applied to any saguaro population and only requires a site-specific factor to adjust the curve to the local growth rate. This adjustment factor can be established more efficiently and requires less data than the full analyses required for previous studies. Saguaro National Park East (SNP-E) was used as the baseline factor, set to 1.0. Results yielded a factor of 0.743 for SNP West. When the formula was applied to 10-yr interval data from Organ Pipe Cactus National Monument (OPCNM) in Arizona, USA, this location had a factor of 0.617 (relative to SNP-E). With this formula and relatively little field sampling, the age of any individual saguaro (whether the individual was sampled or not) in any population can be estimated.

Key Words: age determination • age–height relationship • Arizona • Cactaceae • Carnegiea gigantea • growth–height relationship • saguaro cacti • Sonoran Desert


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 LITERATURE CITED
 
Determining the age of the large columnar saguaro cactus [Carnegiea gigantea (Engelm.) Britt. and Rose] has long been problematic, despite many attempts to do so and many studies on population age structure. Saguaros, which may live upwards of 200 yr, do not put on annual rings. Thus, researchers rely on height and growth to estimate age (e.g., Hastings, 1961 ; Hastings and Alcorn, 1961 ; Parker, 1993 ; Pierson and Turner, 1998 ). Extensive data collection, often over many decades, is needed to estimate age and reconstruct population structure (Steenbergh and Lowe, 1977 ; Turner, 1990 ; Pierson and Turner, 1998 ). However, results are applicable only to the population sampled because growth rates may vary by 100% or more from one locale to the next due to variation in environmental conditions (Steenbergh and Lowe, 1977 , 1983 ).

Steenbergh and Lowe (1983) collected annual growth data for saguaros at Saguaro National Park, East (SNP-E) and SNP, West (SNP-W). Subsequent studies have applied the values of Steenbergh and Lowe's (1977 , 1983 ) and others' (Hastings and Alcorn, 1961 ) for SNP to other locales (Brum, 1973 ; Jordan and Nobel, 1982 ; Parker, 1993 ). Saguaro National Park is near the species' environmental optimum (Niering et al., 1963 ), thereby yielding age underestimations for most locations. For example, Brum (1973) specifically states (p. 198), "[reported findings]... could suggest error in applying growth rate data from Saguaro National Monument [=SNP] to those individuals near Parker Dam." Age–height data from Organ Pipe Cactus National Monument (OPCNM) are also presented in Steenbergh and Lowe (1983) and are poor, relative to SNP data for application to other locales (e.g., Parker, 1993 ). In addition to a relatively small sample size (N = 30), data were collected once in 1967 and once in 1977, not annually (Steenbergh and Lowe, 1983 ). Also, the relationship between growth and height is poorly established in some age classes.

A saguaro age–height relationship applicable to multiple populations has never been established. Steenbergh and Lowe (1983) established two curves, one for SNP-E and one for SNP-W using stepwise regression. Also, Turner (1990) provided his observed growth–height curve for MacDougal Crater in Mexico. I observed that the growth curves for saguaros in SNP-E, SNP-W, and MacDougal Crater were similar. Growth is slow when saguaros are seedlings (see also Jordan and Nobel, 1982 ), and growth increases with age and size. Maximum growth occurs in saguaros between about 3 and 4 m in height (Steenbergh and Lowe, 1983 ; Turner, 1990 ), and then declines (Steenbergh and Lowe, 1983 ; Turner, 1990 ; Niklas and Buchman, 1994 ). In this paper I suggest that although growth rate varies from locale to locale, the growth pattern (i.e., the shape of the curve) is relatively consistent for the species, and the general growth pattern may be approximated. Each population (or locale) then requires only a local multiplicative adjustment factor.

The purpose of this study was to (1) find the curve of best fit for saguaro growth using Steenbergh and Lowe's (1983) data for SNP-E and SNP-W; (2) establish the population-specific adjustment factor for SNP-W relative to SNP-E; and (3) determine the factor for OPCNM using Steenbergh and Lowe's (1983) 10-yr interval data, as an example of applying the established formula to a population. Ultimately, this technique will enable future studies to estimate age not only more efficiently but also at any given locale.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 LITERATURE CITED
 
Steenbergh and Lowe (1983) provide raw annual growth data for Saguaro National Park East (N = 216) and Saguaro National Park West (N = 220) (their Appendices II and III). Using their data, I created a database with three columns and 436 rows (one for each measurement). The three columns are saguaro height, its measured growth over 1 yr, and a dummy variable (SNP-E = 0, SNP-W = 1) in the third column. Saguaro National Park East was arbitrarily selected to be the baseline.

Following Steenbergh and Lowe (1983) , base-10 logs were calculated for growth and height, as well as log(height) squared, cubed, etc. With log(growth) as the dependent variable, I conducted a stepwise regression (SPSS software [SPSS, Chicago, Illinois, USA], standard defaults) using the following 11 independent variables: log(height), [log(height)]2, [log(height)]3....[log(height)]10, and the dummy variable (0 or 1), to establish the saguaro growth curve.

I then applied the newly developed curve to demonstrate the applicability of this technique to another population, OPCNM. Steenbergh and Lowe (1983) took height readings for 30 saguaros in OPCNM in only 2 yr, 1967 and 1977. Although data from annual measurements lend themselves to simpler calculations, Steenbergh and Lowe did not collected data in this manner. These non-annual data provide an opportunity to demonstrate an additional benefit of the technique presented in this paper. This method can readily be applied to data for saguaros measured in non-annual and/or uneven time increments.

To find the OPCNM factor, I applied the new growth–height regression formula to find the factor that yielded the best fit for the data, using a procedure that mimics the principles of regression. An efficient way to solve for the adjustment factor is by using any spreadsheet software package, such as Microsoft Excel. After listing the 1967 heights of the 30 saguaros in a column, I first estimated the 1968 heights from the 1967 heights by using the general growth–height formula (provided in Results) using a yet undetermined factor. In place of the factor, reference to a blank cell is made (any value can be used temporarily) that will eventually contain the computed factor. Second, the 1969 heights are estimated from the 1968 heights, and so forth. These steps are done for each of the 30 saguaros for each year until 1977. Third, the difference between the logs of the observed growth and calculated (i.e., expected) growth for the 10-yr period is calculated for each of the 30 saguaros. The values are squared (e.g., in the next column) and totaled. Finally, with the spreadsheet formulas in place, I determined the OPCNM factor by minimizing the sum of squares, adjusting only the value of the cell of the yet unknown factor. The factor can easily be computed using, for example, Microsoft's Excel Solver.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 LITERATURE CITED
 
Stepwise regression on log(growth) for the combined SNP-E and SNP-W data sets yield the following expressions in centimeters (Table 1):SNP-E (dummy variable = 0, baseline formula):

{abot-90-06-09-e1}

SNP-W (dummy variable = 1):

{abot-90-06-09-e2}


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  		Table 1. Regression coefficients for the saguaro growth curve, with growth as a function of height for Saguaro National Park-West. For Saguaro National Park-East, the dummy variable is excluded. Data from Steenbergh and Lowe (1983)

 
The equation derived from SNP-W and SNP-E data is statistically significant (P = 3.5 x 10–112). The adjustment factor for SNP-W (relative to SNP-E) is 10–0.129, where –0.129 is the dummy variable's coefficient (Eq. 2, Table 1). When converted back from the log, a factor of 0.743 is obtained. For example, a saguaro of a given height that grows 10 cm in 1 yr at SNP-E will grow only 7.43 cm at SNP-W.

Figure 1 shows the calculated curves and the observations for SNP-E and SNP-W in the log domain. The curves depict the previous formulas. The two curves are the same, except for the addition of the factor (dummy variable coefficient) for SNP-W. Because SNP-E is, by definition, the baseline location, the multiplicative adjustment factor is 1.0, which mathematically results in no adjustment. Every locale other than the population sampled by Steenbergh and Lowe (1983) at SNP-E will have an adjustment factor. When the formula is applied to the decadal data from OPCNM, the established factor is 0.617 (of SNP-E).



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  		Fig. 1. Height and annual growth of saguaro cacti using raw data of Steenbergh and Lowe (1983) . The curve established in this study was fitted to Steenbergh and Lowe's observations for Saguaro National Park (SNP) East and Saguaro National Park West

 

   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
LITERATURE CITED
 
Age estimation
The age–height relationship can be estimated for any factor; examples of eight possible adjustment factors (0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, and 1.2) are graphed (Fig. 2). To construct the age–height curve for a particular site, the factor needs to be established, as earlier for OPCNM.



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  		Fig. 2. The saguaro age–height relationship for eight possible adjustment factors: 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, and 1.2. A unique adjustment factor can be calculated for every saguaro population, and once substituted into the general growth curve, the age–height relationship for that population can be calculated, with examples depicted here

 
Once the adjustment factor is known, height at year 1 can be determined by multiplying the factor by 0.353. This value is the height of 1-yr-old saguaros reported by Steenbergh and Lowe (1983) for SNP-E in centimeters (N = 9). Height at year 2 can be calculated by substituting the height at year 1 and the factor into the formula:

{abot-90-06-09-e3}

Saguaro height at year two is the sum of the height at year 1 and the growth (in centimeters) derived from Eq. 3. Year 2 height is then substituted and height at year 3 can be computed, etc., until heights for all years (e.g., up to 200) are computed. These steps create a table reflecting the estimated height at each year (age) for that population. It is important to note that the actual calculation of the age–height relationship occurs when the factor is initially established, using all of the data. The final step, that of making the table, is not part of the calculation. Rather, it simply creates a tabular representation of the already derived curve solely for the purpose of making the formula results easier to use and interpret. For a given factor, older/taller saguaro values in the table are as reliable as those for younger/smaller saguaros because there are no propagation errors from application of the general formula.

Contribution
A baseline formula was established with which saguaro age can be estimated more efficiently. A similar approximation in past studies typically required extensive field data collection followed by site-specific statistical analyses.

This method has several advantages. First, only one number, the adjustment factor, is required to establish the complete age curve of any population, and it can be obtained from relatively few individuals. Second, saguaros of all heights do not need to be sampled. The general shape of the curve is assumed to be similar for the species, in all locales. In principle, a single individual measured in any two years will provide enough information to estimate the ages of all individuals in that population, regardless of their height. As with all studies, more data provide better approximations. Finally, this study provides a way to quantify growth in any population, where every population has a unique value. The creation of this index provides a means with which different populations can be compared and may include populations in different parts of the species' range or local or topographical comparisons such as differences in slope or aspect.

This study is the first of its kind for saguaros, one that provides an effective solution to the perpetual problem of estimating the age of individuals, a species whose growth varies with varying environmental conditions across populations through its range. Although growth rate varies, it is reasonable based on the literature and empirical observations (e.g., Steenbergh and Lowe, 1983 ; Turner, 1990 ) that growth rate waxes and wanes proportionally relative to height over the life cycle of any one individual, though actual growth varies. This paper develops a method that can be applied to all saguaro populations to approximate individual age.


   FOOTNOTES
 
1 I thank W. Stein and an anonymous reviewer for helpful suggestions. Back

2 Tel: (414) 229-4866; Fax: (414) 229-3981; drezner{at}uwm.edu Back


   LITERATURE CITED
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 LITERATURE CITED
 
Brum G. D. 1973 Ecology of the saguaro (Carnegiea gigantea): phenology and establishment in marginal populations. Madroño 22: 195-204

Hastings J. R. 1961 Precipitation and saguaro growth. University of Arizona Arid Lands Colloquia 1959: –60/1960–61 30-38

Hastings J. R. S. M. Alcorn 1961 Physical determinations of growth and age in the giant cactus. Journal of the Arizona Academy of Science 2: 32-39

Jordan P. W. P. S. Nobel 1982 Height distributions of two species of cacti in relation to rainfall, seedling establishment, and growth. Botanical Gazette 143: 511-517[CrossRef]

Niering W. A. R. H. Whittaker C. H. Lowe 1963 The saguaro: a population in relation to environment. Science 142: 15-23[Free Full Text]

Niklas K. J. S. L. Buchman 1994 The allometry of saguaro height. American Journal of Botany 81: 1161-1168[CrossRef][ISI]

Parker K. C. 1993 Climatic effects on regeneration trends for two columnar cacti in the northern Sonoran Desert. Annals of the Association of American Geographers 83: 452-474[CrossRef][ISI]

Pierson E. A. R. M. Turner 1998 An 85-year study of saguaro (Carnegiea gigantea) demography. Ecology 79: 2676-2693[CrossRef][ISI]

Steenbergh W. F. C. H. Lowe 1977 Ecology of the saguaro II: reproduction, germination, establishment, growth, and survival of the young plant. National Park Service Scientific Monograph Series No. 8. National Park Service, Washington, D.C., USA

Steenbergh W. F. C. H. Lowe 1983 Ecology of the saguaro III: growth and demography. National Park Service Scientific Monograph Series No. 17. National Park Service, Washington, D.C., USA

Turner R. M. 1990 Long-term vegetation change at a fully protected Sonoran Desert site. Ecology 71: 464-477





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