Introduction

The study of rarity as a biological phenomenon is at the core of ecological theory since it addresses the causes that limit the abundance and/or distribution of certain species. Although quite a number of papers have been published on the subject of rarity in the last decades, few common traits have been found among rare species (Esparza-Olguín 2004), and this apparent lack of general trends has left us rather discouraged regarding the possibility of understanding the underlying causes of rarity (Rabinowitz 1981; Gaston 1994; Kunin and Gaston 1997; Bevill and Louda 1999).

An important contribution to the study of rarity was Rabinowitz’s (1981) recognition of three main aspects involved in the identification of rare species: (1) habitat specificity, (2) geographic distribution range, and/or (3) population density. These ideas have been pivotal in our understanding of rarity as a multidimensional phenomenon that must be addressed from different perspectives. Here we propose that a different set of conceptual tools should be used in addressing each of the three aspects of rarity recognized by Rabinowitz (1981): (1) a species’ habitat specificity may be studied by analysing the ecophysiological traits which enable it to occupy and successfully exploit certain habitats, while being at a disadvantage in others; (2) on the other hand, metapopulation theory may be useful in offering some understanding of a species’ geographic distribution range by looking at the factors that affect the foundation and extinction of local populations at a regional scale (Carter and Prince 1981; Hanski 1989); (3) finally, the use of demographic analyses appears appropriate for addressing questions regarding population numbers; in fact, demographic studies have often been acknowledged as potentially important tools for increasing our insight into the causes of rarity (Lande 1988; Gaston 1994; Schemske et al. 1994; Bevill and Louda 1999). In this paper we offer an example of this latter approach by analysing the population dynamics of three cactus species along a rarity gradient. We aim to identify general demographic traits accounting for the differences in the abundance patterns between the rare and the more common species. An additional significant aspect of our study system is that the three species studied belong to the same genus. The advantage of comparative studies on congeneric or closely related species is that they maintain the phylogenetic effect to a minimum when contrasting particular life history traits (Harvey and Pagel 1991; Silvertown and Dodd 1996; Esparza-Olguín 2004). This comparative approach to the analysis of plant rarity has seldom been used in the context of plant demography (but see Fielder 1986, 1987; García et al. 2002 for exceptions).

Most of the recent studies analysing the demography of plant species have used transition matrix models (e.g. van Groenendael and Slim 1988; Menges 1990; Horvitz and Schemske 1995; Oostermeijer et al. 1996; Valverde and Silvertown 1998; Mandujano et al. 2001). These deterministic matrix models assume that demographic behaviour does not change in time and thus project the dynamics of an exponentially growing population. Although this is seldom the case, the results of such deterministic models are valuable in the sense that they project the current demographic behaviour into the future and thus represent the potentiality of the present conditions (Caswell 2001). Since natural populations generally show variation in their demographic behaviour, the comparison of a series of annual matrices for a given population is an effective way of addressing such demographic variation. The differences in abundance which determine the varying degree of rarity of plant species must be related to some degree to the patterns described by this temporal demographic variation (de Kroon et al. 2000; Higgins et al. 2000).

In this study we analyse the demographic behaviour of three columnar cactus species that differ in their degree of rarity. Neobuxbaumia macrocephala (F. A. C. Weber) Dawson is the rarest of the three, followed by Neobuxbaumia tetetzo (Coulter) Backeberg and finally by Neobuxbaumia mezcalaensis (Bravo) Backeberg, which shows a broader distribution range and higher overall population densities compared to the other two (Table 1). We studied one population of each species over a 3-year period and analysed the demographic variation between yearly transition matrices. We aimed to answer the following questions.

  1. 1.

    Are any particular demographic traits associated with the different levels of rarity displayed by these species?

  2. 2.

    How does demographic behaviour vary through time for each species?

We expect that the answers to these questions will contribute to our understanding of the mechanisms that effectively limit population numbers, thus deepening our insight into the demographic causes of rarity.

Table 1 Ecological features of the three Neobuxbaumia species studied
Full size table

Materials and methods

The study site

This study was carried out in the Tehuacán Valley, near Zapotitlán Salinas, in the Mexican State of Puebla (18°20′N, 97°28′W). This area is part of the Tehuacán–Cuicatlán Biosphere Reserve, in Central Mexico, which is well known for its high diversity of cacti and other succulents (Zavala-Hurtado 1982; Dávila-Aranda et al. 1995). The Zapotitlán Salinas area has a sub-arid climate with an annual mean temperature of 18–22°C, and annual total rainfall of ca. 400 mm, 70% of which falls during the rainy season, between June and September (Zavala-Hurtado 1982; Zavala-Hurtado and Díaz-Solís 1995). The soils in this area are calcareous, rocky and shallow, with low organic matter content (8.87%) and extremely low fertility [P, 7 ppm; K, 0.76 ppm; Ca, 39.28 ppm; N, 0.27% (Ruedas et al. 2005)]. The vegetation in the area is a xerophytic shrubland dominated by columnar cacti such as Myrtillocactus geometrizans (C. Mart.) Console, Cephalocereus columna-trajani (Karw.) Schuman and the three Neobuxbaumia species included in this study. Also globose and barrel-like cacti such as Mammillaria sp., Echinocactus sp., Coryphanta sp. and Ferocactus sp., as well as other succulents such as Agave sp. and Yucca sp. are abundant. Other important physiognomic elements are shrubs and small trees such as Cercidium praecox (Ruíz and Pav.) Harms, Prosopis laevigata (Humb. and Bonpl. Ex. Willd) M.C. Johnst., Acacia sp. and Mimosa sp., which colonize open spaces and constitute nurse plants for many cacti and other succulents (Valiente-Banuet and Ezcurra 1991; Godínez-Álvarez et al. 1999; Ruedas et al. 2005). The three Neobuxbaumia species studied occupy slightly different environments within the study area (Ruedas et al. 2005). Thus, a different location was chosen to study the population dynamics of each species, although the three sites were relatively close to each other (3–14 km apart).

The species

N. mezcalaensis is a non-branching columnar cactus that may reach over 10 m in height. It flowers between April and June. Its flowers are borne all along the stem and are white in colour. Its fruits are greenish to reddish (Arias-Montes et al. 1997; Valiente-Banuet et al. 1997). According to its high population densities and relatively broad distribution range, this species may be considered as the most common of the three included in this study (Table 1).

N. tetetzo is a branching columnar cactus that reaches about 15 m in height. It flowers between May and July. Its flowers are produced on branch tips and are white in colour; fruits are green and fleshy (Arias-Montes et al. 1997; Godinez-Álvarez et al. 1999). This species is usually found forming dense populations; yet, its distribution range is narrower than that of N. mezcalaensis, thus we consider it as having an intermediate level of rarity (Table 1).

N. macrocephala is a branching columnar cactus that may also reach ca. 15 m in height. Branch tips bare a reddish pseudo-cephalium from which purple flowers emerge in late spring. Fruits are also purple in colour and ripen early in the summer (Arias-Montes et al. 1997; Valiente-Banuet et al. 1997). This species is found only in the Tehuacán Valley where it generally forms sparse populations. Due to its low population numbers and narrow distribution range, this species is considered the rarest of the three included in this study (Table 1).

The seeds of these Neobuxbaumia species are dispersed by bats and birds (Valiente-Banuet et al. 1997; Godinez-Álvarez et al. 1999); this type of biotic seed dispersal may favour germination and seedling establishment by allowing the seeds to reach microenvironments under the shade of nurse plants, where the success of these life cycle stages is enhanced (Valiente-Banuet and Ezcurra 1991; Godínez-Álvarez et al. 2002). The long flight distances of these dispersal vectors suggest that dispersal limitation is not responsible for their distribution limits.

Field methods

At each location we set permanent transects in which all Neobuxbaumia individuals were located; a metal tag with an ID number was attached to each plant. The number and size of permanent transects varied between sites, depending on population density. These functioned as a grid in which individual plants could be allocated x and y coordinates to allow relocation during the study period. In summer 1999 ca. 350 individuals were tagged in each population. The size of each individual plant was recorded at the beginning of the study by measuring stem length with a measuring rod. In the case of branching plants, we marked and measured each branch individually. We added up the stem lengths of all the stems of a plant to obtain total plant size, and used this variable to categorize plants in size classes (Table 2).

Table 2 Size categories (in relation to total plant length; cm) used to describe the demography of N. macrocephala, N. tetetzo and N. mezcalaensis
Full size table

In summer 2000, 2001 and 2002 populations were visited and plants were relocated and measured again. Size-specific survival and growth probabilities were calculated from the proportion of individuals in each size class that died, remained in the same size class or grew to the following size class from one year to the next. Fecundity was given in “seedling” units, since our observations suggest that the seeds produced during each summer (i.e. the rainy season) do not accumulate for long periods in the soil but rather germinate within a few weeks or die. Thus, size-specific fecundities were estimated through the combination of two parameters: (1) observations on fruit and seed production, and (2) seed germination experiments (since naturally germinating seeds and emerging seedlings were very difficult to observe due to their small size and low frequency in natural conditions), as detailed in the following paragraphs.

The number of fruits produced per plant was evaluated each year during the reproductive season (i.e. April–August) through monthly counts. The mean number of seeds per fruit was determined from a collection of ca. 20 fruits per species each reproductive season.

Seed germination experiments were carried out in August 2000 and 2001 in each of the study sites. In August 2000, twenty batches of 25 seeds per species were planted in the field. Each batch of 25 seeds was sown within a mesh cylinder (20 cm in diameter, 5 cm tall) partially sunk in the ground to allow relocation; the effect of the cylinder on seed survival was considered negligible. Ten of these cylinders were set in open areas and ten under the shade of nurse plants. Seed germination was followed daily for the first 10 days and then monthly for 3 months. Given a high seed availability in 2001, seventy-five seeds were used in each of the 20 seed batches sown in the field for each species in August 2001. Again in this second year ten batches were sown in open areas while the remaining ten were sown under the shade of shrubs, and seed germination was followed daily for the first 10 days and then monthly for the following 3 months. Seed germination probability for each species was obtained from these experiments, averaging the results of the two treatments (i.e. in the open and under a nurse plant). By planting half of the experimental seeds in open areas and half under the shade of nurse trees we assumed that the chance of a seed falling in either of these microenvironments is equivalent, which is a reasonable assumption considering that indeed, ca. 50% of the area is covered by vegetation.

As seedlings are extremely rare in the three populations studied, we estimated seedling survival probabilities from one year to the next from seedling survival experiments carried out in 2000 and 2001. We germinated seeds of each Neobuxbaumia species in a greenhouse in May of each year, and introduced the resultant seedlings to the field in August, when they were 2 months old. In August 2000 we introduced 100 seedlings of each species to their corresponding site; we planted ten groups of ten seedlings each, five of these under the shade of shrubs and five in open areas. To allow relocation, seedlings were planted within the same type of mesh cylinders used for the germination experiments. Seedling survival was followed monthly for a year. In August 2001 we carried out a similar experiment, but in this case we used 12 groups of 36 seedlings each for N. macrocephala and N. mezcalaensis, and 12 groups of 30 seedlings each for N. tetetzo. For each year and species, seedling survival curves were compared pairwise (exposed vs. shaded microsites) according to the methods used by Pyke and Thompson (1986). This statistical test involves the calculation of a log-likelihood ratio (LR) which is compared with a χ2 distribution; LR summarizes the difference between two survival curves based on the observed mortality rate of each cohort.

To calculate the matrix entries representing seedling survival, the results of the two treatments (i.e. exposed and shaded microsites) were averaged. We are aware that by planting 2-month-old seedlings in these experiments, we were unable to evaluate the extent of the mortality risk during the very early stages of seedling growth; thus, the seedling survival probabilities reported here must be somewhat overestimated.

Population projection matrices

We built three population projection matrices (periods 1999–2000, 2000–2001 and 2001–2002) for each of the studied species. As mentioned above, transition probabilities among size classes were estimated as the relative frequencies of each observed transition from 1 year to the next. In some cases (N. macrocephala 2000–2001; N. tetetzo 1999–2000, 2000–2001 and N. mezcalaensis 1999–2000, 2000–2001 and 2001–2002) no deaths were observed in the largest size classes, thus, their observed survival probability equalled unity. However, using a value of 1 for the persistence of the largest adults would imply that plants are immortal, which is certainly not the case. Thus, mortality rate for these size category was calculated as the inverse of category length (in years), estimated according to the observed annual growth rate, as detailed by Enright and Ogden (1979). Mortality rates for individuals in smaller size categories were calculated from observed deaths; when no dead plants were found in these categories, individuals were assumed to leave the category by growing and reaching the following size category.

Fecundity entries were calculated by multiplying the mean number of seeds produced per individual by the probability of seed germination obtained from the seed germination experiments described above. The probabilities of seedling survival for the 1999–2000 and 2000–2001 matrices were obtained from the seedling survival experiments set during the summer of 2000, while the results of the experiment that started in the summer of the 2001 were used to calculate the 2001–2002 matrices.

Population growth rate (λ), stable size-class distributions and size-specific reproductive values were calculated through the power method. The 95% confidence intervals for λ were calculated using the analytical method proposed by Álvarez-Buylla and Slatkin (1994). According to this method, the variance (V) of λ (from which confidence intervals were derived) may be calculated as:

$$V(\lambda ) = \sum\nolimits_{ij} {(\delta \lambda /\delta a_{ij} )^2 V(a_{ij} ) + \sum\nolimits_{ij,kj} {(\delta \lambda /\delta a_{ij} )(\delta \lambda /\delta a_{kj} ){\text{COV(}}a_{ij} ,a_{kj} {\text{)}}} }.$$
(1)

Since transition probabilities show a binomial distribution, their V was calculated as:

$${\text{V}}(a_{ij} ) = a_{ij} ((1 - a_{ij} )/n).$$
(2)

The covariance (COV) between a pair of transitions from the same initial category was calculated according to a multinomial distribution as:

$${\text{COV}}(a_{ij} ,a_{kj} ) = - a_{ij} a_{kj} /n.$$
(3)

The V of fecundity entries was calculated according to the variation observed in relation to fruit production between individuals in each size category (Álvarez-Buylla and Slatkin 1994).

Elasticity matrices were calculated as described by de Kroon et al. (1986). Finally, from the three annual matrices obtained per species, we calculated average matrices, as well as the coefficient of variation (CV) of the different matrix entries; the latter allowed us to identify the entries that varied the most between years.

Life table response experiments

We used life table response experiments (LTRE) to identify which matrix entries contributed more importantly to the observed variation in λ for each species. In contrast to elasticity analyses, which address the relative contribution of matrix entries to λ in a prospective way (i.e. by providing information on how much λ would change if matrix entries were modified), LTRE explore the contribution of matrix entries to the observed variation in λ in a retrospective way [i.e. they provide information on how much λ did change as particular matrix entries varied (Horvitz et al. 1997; Wisdom and Mills 1997; Caswell 2000; Sather and Bakke 2000)]. LTRE are based on the construction of contribution matrices derived from the observed variation in matrix entries combined with their sensitivities (Caswell 2000, 2001). These contribution matrices indicate the extent to which each matrix entry was responsible for increasing or decreasing the λ value of its respective transition matrix compared to the λ value of the average matrix. As the entries in a contribution matrix are additive, they can be summed by demographic process (i.e. fecundity, growth, stasis and retrogression), as is traditionally done in the case of elasticities.

In this study we performed one LTRE per species to analyse how inter-annual variation in matrix entries contributed to variation in λ. Additionally, we carried out a LTRE to analyse variation between species and identify which matrix entries were responsible for increasing or decreasing the λA value of each species in relation to the λ value of a grand mean matrix obtained by averaging the three average matrices (one per species). Thus, we used one-way fixed design LTRE, following the methods described in Caswell (2001).

Results

Seed germination and seedling establishment

During the summer of 2000 the number of seeds that were observed germinating in the field experiments was very low for the three Neobuxbaumia species studied (Table 3). In all three cases the difference between germination percentage in exposed and shaded microsites was not statistically significant (N. macrocephala, t=1.865, df=9, P=0.095; N. tetetzo, t=0.430, df=9, P=0.667; N. mezcalaensis, t=0.799, df=9, P=0.445). In 2001 germination percentages were higher than in the previous year for all species (Table 3). For N. macrocephala, 28 out of 250 seeds germinated in shaded microsites, while only seven out of 250 germinated in exposed microsites; this difference was marginally significant (t=1.982, df=9, P=0.079). In the cases of N. mezcalaensis and N. tetetzo, the difference between germination percentages in exposed and shaded microsites was statistically significant this year (t=3.363, df=9, P=0.008 and t=2.667, df=9, P=0.026, respectively).

Table 3 Mean germination percentages (±SD) obtained in the field experiments during the summers of 2000 and 2001 for the three species studied
Full size table

The results of an ANOVA designed to explore the effect of species, year and microsite on seed germination percentage (arcsin transformed for linearity) indicated that the effect of the three factors was significant, as well as the interaction between species and year, which indicates that the effect of year was not the same for all species (Table 4).

Table 4 Results of the ANOVA performed to test the effect of species, year and microsite (shaded vs. exposed) on the germination percentages obtained in the field experiments
Full size table

In 2000 we observed a significantly higher seedling mortality in the exposed than in the shaded microsites in all three species (Fig. 1a). All N. macrocephala seedlings planted in the exposed microsite died within the first month; 100% mortality in these microsites was reached after 2 months in N. tetetzo and after 6 months in N. mezcalaensis. In contrast to these results, seedling survival did not differ significantly between shaded and exposed microsites in 2001 (Fig. 1b). Yet, 100% mortality in the exposed microsites was reached after 7 months in N. tetetzo, while for the other two species some seedlings were still alive in these microsites after 9 months.

Fig. 1
figure 1

Survivorship curves (log l x ) for the seedling cohorts introduced to the field in exposed (filled circle) and shaded (filled diamond) microsites in a summer 2000 and b summer 2001. The values of the log-likelihood ratio (LR) in each graph correspond to the LR obtained from the comparison of the pair of curves in each graph. Note that the scales of the x- and y-axes differ for a and b

Full size image

Population dynamics

The 3-yearly transition matrices built for N. macrocephala, N. tetetzo and N. mezcalaensis are reported in Appendix 1 and the main results are summarized in Table 5. These show several common features; for instance, in all cases fecundity increased and mortality decreased with increasing plant size, thus, seedlings showed the highest mortality rate, whereas all adult categories showed very low mortality. In fact, in the case of N. mezcalaensis the largest size category showed no mortality at all during the whole study period (from 1999 to 2002), while for N. tetetzo the death of the largest adults was observed only in 2002; and for N. macrocephala in 2000 and 2002.

Table 5 λ values with lower and upper confidence interval limits obtained for each species in each of the study periods. Data in bold indicate the cases in which the calculated λ value was significantly above unity
Full size table

Another common feature for all species and years was the slow individual growth rate, which was noticeable in the higher values of the entries corresponding to stasis or persistence in the same size category compared to values corresponding to growth (Appendix 1). All these features are also apparent in the average matrices reported in Table 6; entries in these matrices are mean values of the 3-yearly transition matrices reported in Appendix 1 and thus incorporate the effect of temporal demographic variation. According to the λA values obtained from these average matrices, the populations of N. macrocephala and N. tetetzo are close to the numerical equilibrium; however, the λA value of the N. mezcalaensis matrix suggest a growing population, i.e. λA is significantly above unity (Table 6).

Table 6 Average population projection matrices for each species [N. macrocephala (n=5 matrices), N. tetetzo (n=3) and N. mezcalaensis (n=3)], with their corresponding elasticity matrices and coefficients of variation (CV) for matrix entries. Data given in bold in the elasticity matrix are the highest values
Full size table

λ values obtained from average matrices show an interesting correspondence with observed population structures (Fig. 2). The N. macrocephala and N. tetetzo populations (which show λA values close to unity; Table 6) show population structures in which the smallest size classes have lower relative abundance compared to the intermediate ones, whereas N. mezcalaensis (with a higher λA value) shows a population structure that reflects a growing trend, i.e. with high relative abundance of the smallest size categories (Fig. 2). Note, however, that seedlings are not represented in the population structure histograms, since no seedlings were observed in the field. This confirms that the seedling stage is an important population bottleneck for all three species.

Fig. 2
figure 2

Mean relative abundance of individuals (±SD) observed in each size category between 1999 and 2002 for a N. macrocephala, b N. tetetzo and c N. mezcalaensis. Category 1 is not shown in these graphs because no seedlings were observed in the field

Full size image

The temporal variability observed in matrix entries may be analysed through their CV (Table 6). These CV values were highest in matrix entries corresponding to retrogression, which was related to the fact that, in general, these particular transitions were positive in only one of the three study periods. Regarding variation in other matrix entries, the lowest CV values were observed in matrix entries corresponding to stasis, followed by those denoting growth. The transition representing seedling establishment (categories 1–2) was one of the growth entries that showed high CVs in N. tetetzo and N. mezcalaensis, while in N. macrocephala it showed no variation at all. Finally, fecundity entries showed high CV values in all three species, with N. mezcalaensis and N. tetetzo showing higher CV values in fecundity entries compared to N. macrocephala (Table 6).

Elasticity analysis

The general pattern of the elasticity matrices obtained from the average matrices was similar between species (Table 6). A common feature of elasticity matrices was that retrogressions and fecundity elements contributed very little to λ. In addition, growth transitions made only a modest contribution to the observed λA. In general, matrix entries referring to stasis showed the largest elasticity values. In all cases the highest elasticity values were found in the persistence of individuals in intermediate size categories (4–7). These trends were also apparent in the elasticity of yearly transition matrices (Appendix 2). In all cases, the stasis elements of elasticity matrices contributed 80–90% of total elasticity (Appendix 2).

Although fecundities had low elasticity values, these entries had large CVs (Table 6). Thus, variation in fecundity clearly has the potential to significantly affect λ, as corroborated by the results of the following section. Yet, fecundity entries were a combination of two different vital rates: germination success and seed production. To evaluate which of these vital rates made a larger contribution to λ, we performed numerical simulations using the average matrices (Table 6) modifying the value of each of these two components independently, and recording the resultant changes in λA. The results show that changes in germination success had a lower impact on λA than equivalent proportional changes in fruit production in the three species studied (Fig. 3).

Fig. 3
figure 3

Changes in the value of λ that resulted from increases or decreases in the different components of the fecundity values, i.e. germination success (continuous lines) and seed production (dashed lines). Circles N. macrocephala, triangles N. tetetzo, squares N. mezcalaensis. The proportional decrease or increase in the relevant value is indicated on the x-axis. Orig λ value of the original (unmodified) average matrices

Full size image

Life table response experiments

These analyses allowed us to identify which demographic processes (i.e. fecundity, growth, stasis or retrogression) contributed more importantly to the increase or decrease in the yearly λ values compared to the corresponding λA for each species. In N. macrocephala the low λ value of the 2000–2001 period was given by a decrease in growth, and an increase in stasis, whereas the higher λ values of 1999–2000 and 2001–2002 were characterized by the inverse pattern (Fig. 4a). In contrast, in the case of N. tetetzo large fecundity values in 2001–2002 and low fecundity values in 1999–2000 and 2000–2001 had an important impact on λ (Fig. 4b). Finally, the impact of variation in fecundity was clearly apparent in N. mezcalaensis, in which high fecundity values contributed to a marked increase in λ in the 2001–2002 period (Fig. 4c). Note that the graphs of Fig. 4 show varying scales in the x-axis, which reflect that the extent of the variation in λ differed in each species; N. mezcalaensis showed by far the largest variation.

Fig. 4a–d
figure 4

Results of the life table response experiments. Differences between years are analysed for a N. macrocephala (N. mac.), b N. tetetzo (N. tet.) and c N. mezcalaensis (N. mez.). a–c λ value of the relevant average matrix (λ A ) is given above, the annual λ values are given on the right-hand side. 1 Year 1 (1999–2000), 2 year 2 (2000–2001), 3 year 3 (2001–2002). d Overall differences between species (each averaged between years) are analysed; the λ value of the grand average matrix (λ GM ) is given above and the average λ values per species are given on the right-hand side. Contributions were grouped by demographic process: fecundity (white bars), growth (grey bars), stasis (hatched bars) and retrogression (dotted bars). Note differences in the scale of the x-axis between figure parts

Full size image

The results of the LTRE in which the three species were compared show that high fecundity values were responsible for the high λA value of N. mezcalaensis (Fig. 4d).

Discussion

In this study we have addressed the differing level of rarity of three Neobuxbaumia species by exploring the yearly variation in their population dynamics. Despite the fact that the studied species are long-lived perennials (apparently plants may live for over 300 years) in which relevant population processes may occur at the scale of decades (Pierson and Turner 1998), several interesting trends that appear to relate to the species’ level of rarity were apparent. First of all, the only annual transition matrix with a λ value significantly above unity corresponded to the most common species, N. mezcalaensis (λ=1.091±0.088, for the 2001–2002 period; Table 5). This fact determined that the λA value was highest for this species. Thus, despite the short time scale of our study compared to the longevity of these plants, the results appear to account to some degree for the higher population densities of N. mezcalaensis compared with the other two species.

Secondly, several demographic features help explain the low population numbers of N. macrocephala: This species showed the lowest germination rates (see also Ramírez-Padilla and Valverde 2005), and the lowest variation in seedling establishment and fecundity values; in addition, the frequency of adult mortality in this species was higher than in the other two.

Thus, a certain relationship was observed between λ and rarity; the integration of the demographic variation observed over 3 years in the average matrices resulted in a higher λA value for the most common species, N. mezcalaensis, compared to the other two.

Most of the population projection matrices obtained in this study yielded a λ value statistically indistinguishable from unity. This was also the case for two annual matrices previously reported for N. macrocephala (Esparza-Olguín et al. 2002). Considering that the three Neobuxbaumia species studied are long-lived, slow-growing plants, λs close to unity are not a surprise (Silvertown et al. 1993). This type of demographic behaviour has also been found in other rare plants and may be accounted for by high survival rates of adult plants and low recruitment due to the high vulnerability of early life-cycle stages (Fielder 1987; Damman and Cain 1998; Contreras and Valverde 2002; Rae and Ebert 2002; Valverde et al. 2004). The latter was apparent in the low seed germination and seedling survival rates observed in all three Neobuxbaumia species.

Germination of Neobuxaumia seeds is limited mainly by the poor soil moisture and high temperatures that characterize semi-arid environments (Ramírez-Padilla and Valverde 2005). When seeds germinate, the resultant seedlings are highly vulnerable to drought, predation and other mortality factors (Valiente-Banuet and Ezcurra 1991; Altesor et al. 1993; Mandujano et al. 1996; Contreras and Valverde 2002; Valverde et al. 2004). Both seed germination and seedling survival showed low values in the three Neobuxbaumia species studied. The highest average values were obtained in N. mezcalaensis in 2001–2002 (germination percentage=17.6%, seedling survival=11%), which were responsible in part for the high λ value obtained for this species in this period. Thus, the differences in λ between species and years were determined to some extent by the dynamics of the early life-cycle stages (Valverde et al. 2004), as confirmed by the results of the LTRE and were related to their level of rarity.

The high yearly variation in germination success observed in N. mezcalaensis was responsible, in part, for the high CV of fecundity entries in this species. The other factor that accounted for these high CV values was the massive fruit production observed in this species during the summer of 2001 [i.e. 10 times more fruits per plant compared to the other two species (Esparza-Olguín and Valverde 2003)]. This kind of mast seeding phenomenon appears to be characteristic of N. mezcalaensis, and apparently also of N. tetetzo. Both local people and biologists that have studied these columnar cacti for several years, report the occurrence of reproductive periods of massive blooming followed by several years in which almost no fruits are produced (H. Godínez-Álvarez, personal communication). During our study period we were able to witness one of these massive fruit production events, which was particularly dramatic in N. mezcalaensis (Esparza-Olguín and Valverde 2003). Thus, the high λ value of the 2001–2002 period observed in this species was accounted for by the substantial increase in fecundity (i.e. fruit production and seed germination), which in turn determined its high λA value.

The discussion above emphasizes that high fecundity values resulted in high deterministic λs. Yet, despite the clear relationship between λ and fecundity, this demographic process had low elasticity values in the three species (Table 5, Appendix 2). In general, many long-lived species and some rare plants with λ values close to unity, show this type of elasticity pattern in which population dynamics appear to depend mainly on juvenile and adult survival rather than on seed production and seedling establishment (Enright and Odgen 1979; Oyama 1993; Silvertown et al. 1993; Álvarez-Buylla et al. 1996; Godínez-Álvarez et al. 1999; Contreras and Valverde 2002; Esparza-Olguín et al. 2002; Rae and Ebert 2002; Valverde et al. 2004). Yet, it is important to bear in mind that large variations in a demographic transition with a low elasticity value may have a greater impact on λ than small variations in a demographic transition that has a larger elasticity (Silvertown et al. 1996; Ehrlen and van Groenendael 1998; de Kroon et al. 2000). The results of our LTRE emphasize the importance of fecundity in this respect and highlight the contrasting results that may be obtained with elasticity (i.e. prospective) and LTRE (i.e. retrospective) analyses.

In addition to average matrices, another way in which yearly demographic variation may be incorporated into matrix population models is through stochastic analyses (Mandujano et al. 2001; Mondragón et al. 2004; Valverde et al. 2004). When we applied such analyses to our study system, the results (i.e. stochastic long-term λ values, data not shown) were very similar to the λA values reported above. Thus, here we do not explore them further.

Although a 3-year demographic analysis is indeed limited when describing the demographic behaviour of long-lived species, we have shown that even within this time period interesting results may emerge that could contribute to our understanding of the contrasting abundance patterns displayed by different species. As long-term demographic studies become more available in the literature, we will be able to confirm the role of demographic variability in explaining long-term increases or decreases in population numbers. In addition to contributing to our knowledge of rarity patterns, these studies would allow us to deepen our understanding regarding the best ways to preserve rare and threatened species.