Population Dynamics of Mammillaria Magnimamma Haworth

Population dynamics of Mammillaria magnimamma Haworth. (Cactaceae) in a lava-ﬁeld in central Mexico

Teresa Valverde*, Sandra Quijas, Manuela López-Villavicencio and Silvia Castillo Departamento de Ecología y Recursos Naturales, Facultad de Ciencias, Universidad Nacional Autónoma de México. Ciudad Universitaria, México, 04510, D.F., México; *Author for correspondence (e-mail: [email protected])

Received 23 October 2001; accepted in revised form 23 August 2002

Key words: Cacti, Demography, Elasticity analysis, Long-lived species, Population projection matrices, Stochas- tic simulations

Abstract

One of the habitats occupied by Mammillaria magnimamma is a 2000-year old lava-field, in Mexico City. The great ecological interest on this lava-field and the little knowledge there is regarding cacti population ecology have compelled us to analyse the demography of this species to evaluate its present conservation status at this site. We studied two populations of this species within the lava-field: one in a disturbed site (i.e., recently burned) and another one in a well preserved site. For each population we built two size-based population projection matrices (1996/97 and 1997/98). Demographic data were gathered directly from observations of plant fates from one year to the next. Additionally, seed germination and seedling establishment experiments were carried out in the field to estimate fecundity values and seedling survival probabilities. The four matrices built were used to perform numerical analyses simulating yearly stochastic demographic variation to project the overall population’s long-term behaviour under these changing conditions. Three of the four matrices showed ␭ values slightly below unity. In these cases elasticity values were highest for matrix entries corresponding to plants remaining in their same category. The matrix that showed a ␭ value above unity (well preserved site, 1997/98) had higher elasticity values for entries referring to seedling survival and growth. The numerical simulations of demographic stochasticity showed that the population appears to be growing at a slow rate. According to the simulation results, the variation in overall population size over time may be accounted for by yearly variation in seed germination and seedling survival. Population persistence probability might decrease significantly if fire frequency increases.

Introduction rather than the exception (Bierzychudek 1982; Cipol- lini et al. 1994; Horvitz and Schemske 1995; Valverde The analysis of demographic patterns in plants dur- and Silvertown 1998). Yet, the way in which this de- ing the last three decades has produced a large mographic variation occurs, the life cycle phases it amount of information about important aspects of the affects, and its long-term population consequences are biology and life history of a number of species (Sil- poorly understood, particularly among long-lived vertown et al. 1993). This information has allowed us plant species. The understanding of the factors that to deepen our understanding of the relevance of dif- determine long-term population dynamics still re- ferent population processes, particularly among an- mains a central issue in ecology (Horvitz and Schem- nual or generally short-lived herbs and shrubs from ske 1995). temperate habitats. One aspect that is now generally Cacti are a plant group that has received little at- accepted is that spatio-temporal variation in the de- tention from a demographic point of view. Recent demographic behaviour of plant populations is the rule mographic information on cacti species is adding sig- 168 nificantly to our understanding of plant population dez and Godínez 1994), the demographic analysis of dynamics in nature, since they include long-lived spe- M. magnimamma is expected to offer empirical tools cies that inhabit dry tropical areas, two particularities that could aid in the evaluation of the conservation that are poorly represented within the demographic status of the reserve where the study was performed. literature. The available demographic information on The high demographic vulnerability of cacti is pre- this plant group, as well as on other long-lived plant sumed to be a result of their slow individual growth species, suggest that spatio-temporal fluctuations in rates and frequently high mortality rates during the the success of early life-cycle stages are responsible juvenile phases; apparently these features contribute for long-term population fluctuations (Bowers 1997; to limit their capacity for population growth (Hernán- Pierson and Turner 1998). However, projection ma- dez and Godínez 1994; Contreras and Valverde 2002; trix results and their corresponding elasticity analyses Esparza-Olguín et al. 2002). Additionally, many cacti have shown that, in general, matrix entries corre- species are highly restricted in their distribution and sponding to seed and seedling transitions seem to occupy very specific habitats, which makes them contribute little to population growth rate (Godínez- prone to extinction by habitat destruction. In fact, the Alvarez et al. 1999; Esparza-Olguín et al. 2002). Al- Cactaceae is a plant family with a particularly high though long-term numerical fluctuations appear to be number of threatened species (Hunt 1992; Anderson common within these species, no information is avail- et al. 1994). However, our knowledge of the popula- able as to what kind of ecological processes may ac- tion behaviour of most of these species is very lim- count for them. In this study we analyse the popula- ited. tion dynamics of the cactus Mammillaria In this paper we address the demographic analysis magnimamma Haworth. in a lava-field in Mexico of two populations of Mammillaria magnimamma, City. We incorporated information on the demo- one in a well-preserved and one in a disturbed (i.e., graphic behaviour of this species in different sites and recently burnt) site within the ЉEl XitleЉ lava-field, years into a long-term stochastic numerical simula- through the construction of population projection ma- tion which allowed us to project the potential fate of trices for two growth periods (1996/97 and 1997/98). the population. The aim of this exercise was to con- With the information obtained from the demographic tribute to the understanding of the way in which spa- analysis we carried out numerical simulations, using tio-temporal changes in the demography of plant spe- the stochastic approach developed by Bierzychudek cies give rise to fluctuating population numbers. (1982), incorporating the effect of the spatio-tempo- An additional interest to address the population ral variation in the demography of the species in or- dynamics of Mammillaria magnimamma is related to der to understand the way in which this variation may the ecosystem in which it was studied: a 2000 year result in population fluctuations, as well as to evalu- old lava-field with a particularly high plant diversity ate the long-term persistence probability of this spe- (Carrillo 1995). This lava-field was created after the cies in the area. The stochastic analysis was used to eruption of ЉEl XitleЉ volcano and comprises an im- simulate different disturbance regimes and evaluate portant part of what is now Mexico City, one of the population persistence probability under different most densely populated areas in the world. Unfortu- probability of occurrence of fires, which are the main nately, the lava-field is slowly being turned into ur- cause of disturbance at the site. ban areas and much of its biological value is being lost. In fact, some of its endemic species (e.g., Mam- millaria sanangelensis, Bletia urbana) are virtually Methods extinct. In recent years a small fraction of this lava- field was protected in the form of a reserve, in which The study site this study was carried out. It has been suggested that the analysis of the persistence probability of species This study was carried out at the ЉEl XitleЉ (meaning that are particularly vulnerable to disturbances within ЉnavelЉ in Nahuatl) lava-field, located within Mexico an ecosystem may throw light on the conservation City (19°40Ј N, 99°00Ј W) (Figure 1). This lava-field status of the ecosystem itself by providing informa- is the result of the eruption of the ЉEl XitleЉ volcano, tion on the probability of native species turnover which took place over two thousand years ago (Car- (Menges 1990). Since it has been suggested that cacti rillo 1995). The original area occupied by this lava- are particularly vulnerable to disturbances (Hernán- field was 80 km2, covering an altitudinal gradient 169

Figure 1. Location of the study site (ЉEl XitleЉ lava-field) within Mexico City. from 3,100 to 2,250 m above sea level. The lower part nated by the shrubs Senecio praecox, Verbesina vir- of the lava-field (2,500–2,250 m a.s.l.) was covered gata, Dodonea viscosa and Wigandia urens, the tree by a plant community (described in detail by Rze- Buddleia cordata, and the herbs Muhlenbergia ro- dowski (1954)) dominated by the shrub Senecio prae- busta, Dahlia coccinea and Echeverria gibbiflora, cox. This shrubland, in which 350 plant species could among others (Rzedowski 1954). be found (Rzedowski 1954), comprised 40% of the Within the ЉPedregal de San AngelЉ reserve we original area covered by the lava-field. Today the area chose two sites: a well-preserved (P) and a relatively still occupied by this type of shrubland has been re- more disturbed (D) site. The main difference between duced to only 2.9 km2, most of which is now pro- these two sites is the apparent degree of conservation. tected by the ЉPedregal de San AngelЉ Reserve, be- As noted above, the most prevalent disturbance fac- longing to the National Autonomous University of tor in this area is fire, generally of human origin, oc- Mexico (UNAM). Within this reserve, only 226 of the curring mainly in spring, towards the end of the pro- original plant species were found in 1990, plus 77 longed dry season. Between 1996 and 1998 Site D new records, most of which correspond to ruderal was burnt twice, while Site P was damaged only species that have recently entered the ecosystem (Va- slightly by the 1998 fire. Additionally, Site D is lo- liente-Banuet and de Luna 1990). cated closer to the edge of the reserve, which makes The M. magnimamma populations studied were lo- it more prone to colonisation by ruderals. As a con- cated within the Pedregal Reserve referred to above. sequence, the general features of these two sites are This area has a temperate sub-humid climate with a quite different, with P site showing a relatively more seasonal rainfall pattern. Annual rainfall ranges be- developed tree cover, deeper soils, higher soil water tween 700 and 950 mm and most of it falls between content, and lower solar radiation at ground level June and September, while the rest of the year is compared to site D (Valverde et al. 1999). These dif- rather dry. The total yearly precipitation for 1997 and fering conditions in turn determine a relatively differ- 1998 was 749.4 mm and 907.2 mm, respectively (data ent species composition between the two sites: site P from the Observatorio Meteorológico, UNAM). shows a sparse tree cover dominated by Buddleia cor- Mean annual temperature is 16.3 °C, with an annual data, while site D is colonised mainly by grasses and minimum of −6 °C and an annual maximum of 34 °C. herbs. Despite the relatively high annual precipitation, the reserve shows xerophytic characteristics as a result of The species the basaltic substrate, which has a low water retention capacity. The soils are shallow, produced by the Mammillaria magnimamma is a small globular cac- erosion of the basaltic rock and the decomposition of tus with a green-greyish coloration and with one to organic matter. The vegetation of this area is domi- several cylindric stems. The flowers (20–25 mm long) 170 are pinkish to purpleish. Fruits are clavate, 20–35 mm boxes, 2 cm deep, which were slightly buried in the long, and with a deep red colour. Each fruit contains ground in particular microsites within each study site. between 20 and 100 round, brown seeds, 0.7 mm in In each box we scattered 50 newly collected seeds. diameter. M. magnimamma is endemic from Mexico; Boxes were designed to prevent seeds from being it may be found in the central highlands at altitudes transported by wind or water. Within each site we set from 1,700 to 2,600 m above sea level, in grasslands a total of eight boxes in the following type of micro- on flat and rocky soils (Bravo-Hollis and Sánchez- sites: two in fully illuminated microsites with soil, Mejorada 1991). two in shaded microsites with soil, two in fully illuminated microsites with no soil and two in shaded Field methods microsites with no soil (see below). These experiment started at the beginning of August 1998 and seed ger- Within each of the study sites (P and D) we marked mination was recorded daily for the first four weeks, circular plots of6mindiameter (28.3 m2) within and then every week for two months. Total germina- which M. magnimamma plants were located through tion percentage per box was averaged for each site; coordinates. In June–July 1996 each plant was tagged statistical differences in mean germination percentage with plastic flagging and given an identification num- between sites were evaluated through a student-t test. ber. The sample included 203 plants within 22 plots The effect of sowing treatments within sites was eval- in site D, and 206 within 25 plots in site P. We re- uated only superficially, since the number of repli- corded plant size by measuring the diameter of all the cates per treatment did not allow further statistical stems of each plant in summer 1996; we re-measured treatment. all sampled plants in summer 1997 and later in sum- Seedling survival experiments were carried out in mer 1998. Plant size was evaluated as ’cumulative the field in order to estimate the probability of seed- diameter’, which was the sum of the diameters of all ling survival and growth from one year to the next. stems per plant. Additionally, we visited the sampled These estimates were later used in the construction of plants every two weeks during the flowering and projection matrices, as detailed below. In June 1997 fruiting periods (February to August 1997 and 1998) seeds were germinated in greenhouse conditions in and took records of flower and fruit production per order to obtain seedlings. These seedlings were kept plant. A number of fruits was collected and the aver- in the greenhouse for two months and were later age number of seeds per fruit was calculated. Using planted in the study sites. In August 1997, three 40 × the data on the demographic fate of each individual 40 cm quadrats were marked on the ground with plant from one year to the next we built population wooden sticks in each study site and 40 seedlings projection matrices for the periods 1996/97 and were planted evenly spaced (to allow relocation) 1997/98 for each of the study sites. within them. The size and degree of development of As no germinating seeds or seedlings were ob- these seedlings was similar to that of two- or three- served in the field, the matrix entries concerning these week old seedlings observed in the field as part of the life-cycle stages were calculated from the results of seed germination experiments described above. The field experiments. Seed germination experiments survival of the introduced seedlings was recorded ev- were performed in order to estimate the probability ery two weeks for two months. In 1998 a slightly dif- of seeds germinating immediately after dispersal. ferent experimental design was followed also for this These probabilities were later incorporated in the ma- part of the study. In June 1998 seeds were germinated trix analysis as part of the fecundity values, as de- in greenhouse conditions in order to obtain seedlings. tailed below. In June 1997 (just before the start of the When seedlings were two months old, they were rainy season) 100 newly collected seeds were scat- planted in the field in 20 × 20 cm quadrats marked on tered within each of four 15 × 15 cm plots marked on the ground with wooden sticks; 30 seedlings were the ground with wooden sticks in each study site (D planted evenly spaced within each of these quadrats. and P). These seeds were monitored every two weeks A total of eight seedling quadrats were set in each until August in order to evaluate seed germination. study site (D and P). The type of microsites chosen Average seed germination percentage (arcsine trans- to plant these seedling was the same as in the 1998 formed) was compared between sites using a student-t germination experiment described above (i.e., a fac- test. In 1998 a more detailed experimental design was torial design of fully illuminated and shaded micro- followed. We built 15 × 15 cm plastic mesh open sites, with and without soil; two replicates per treat- 171 ment). Seedling survival was followed daily for a Table 1. Size categories chosen to model the size-based demogra- month, and every two weeks for another two months. phy of M. magnimamma. For the 1998 data, survival curves were obtained for Category Cumulative diameter (cm) each treatment and each site; the effect of treatments within each site was evaluated using the Peto and 1 < 0.3 (seedlings) Peto survival curve pair-wise comparisons (Pyke and 2 0.3–4.5 3 4.6–7.5 Thompson 1986). The four treatments used in the 4 7.6–10.5 seed germination and seedling establishment experi- 5 10.6–13.5 ments of 1998 all represent common microsites 6 13.6–20.5 within the study sites. Spatial heterogeneity in envi- 7 > 20.6 ronmental conditions is tremendous within lava- fields. Thus, the idea of performing these experiments in a variety of microsites was to obtain a more real- by the number of seeds produced in a year by an av- istic picture of actual overall germination and estab- erage individual of each size class, multiplied by the lishment probabilities within the reserve. probability of seed germination. Thus, fecundity was given in terms of the average number of seedlings Numerical methods produced per plant in each size class. This implies that seeds do not remain viable in the soil for long The field data on plant survival and growth were used periods of time; seed viability is high in newly col- to construct population projection matrices for each lected seeds (85–90%, Ruedas et al. (2000)), there- of the study sites (P and D) and for the two periods fore it is reasonable to expect that seeds either germi- studied (1996/97 and 1997/98). These four matrices nate or die within three or four months after seed will be called P-1996/97, P-1997/98, D-1996/97 and shed. With regard to germination probability, we cal- D-1997/98. Lefkovitch matrices were used, character- culated it from the data of the germination experi- ised by the classification of individuals according to ments performed in the field in both the disturbed and size classes. The general matrix model is given by the well-preserved sites. As the 1998 germination ex- nt+1 =Ant, in which A represents a square matrix, periments were carried out in four different microsites and n is a column vector whose elements represent (illuminated/shaded, and with/without soil) in both the number of individuals in each class at time t and the preserved and the disturbed sites, to obtain the t+1. Matrices are composed by aij entries, which re- germination probability for each site we averaged the present the probability of an individual in the j-th results of the four microsites, assuming that their fre- class contributing or becoming an individual in the quency in the environment is approximately the same. i-th class from time t to time t+1 (Caswell 1989). In The transition from class 1 to class 2 (i.e., seed- this case the time step was one year. ling survival and growth) was calculated according to The population was subdivided into seven size cat- the results of the seedling survival experiments per- egories, as defined in Table 1. This number of cate- formed in the field each year. In 1997 all planted gories allowed a detailed dissection of the life cycle; seedlings died within the first five weeks in both the at the same time, we made sure that sample size in preserved and the disturbed sites; thus, to estimate each category was large enough to allow robust esti- their survival probability after a year, we built the mates of transition probabilities (see n in Table 4). We survival curves for these seedlings with the mortality preferred this option rather than using the algorithms data obtained over the first five weeks and extrapo- that have been developed for choosing the number of lated them to 365 days. In 1998 seedlings were fol- categories (Moloney 1986), since we aimed to build lowed for three months, until survival curves ap- matrices that were comparable (i.e., with the same peared to reach a plateau. In this case we also used number of categories, defined by the same size inter- the survivorship data to fit a survivorship curve from vals). The probability that a plant in each size-class which the probability of survival to day 365 was cal- would contribute to a different or the same size-class culated. This probability was incorporated in the ma- from one year to the next was calculated directly from trix model as the probability of seedling survival and field observations, i.e. from the proportion of plants growth to category 2. in each size-class that followed different fates between t and t+1. The fecundity elements were given 172

The probability of plants in the largest size cate- simulations of the behaviour of the M. magnimamma gory (7) remaining in the same category from one population as a whole under the spatio-temporally year to the next was calculated directly from field heterogeneous environment characteristic of the study observation in the cases of matrices P-1996/97 and site. The four resulting matrices represent demo- P-1997/98. However, in the disturbed site no mortal- graphic variation in space and time. Yet, in this case ity was observed in this category; yet, some mortality these four matrices were used to project numerical of large adults must be incorporated in the matrix, changes in time, because the variation that occurs in otherwise it would imply that individuals are immor- space (i.e., the differences between the demographic tal. Thus, the mortality of large adults for matrices behaviour between D and P sites) may also occur from site D was calculated as follows: 1) first we ran through time when disturbances take place in differ- the matrices of the well preserved site (P) in the pro- ent patches of the ecosystem. gram STAGECOACH (Cochran and Ellner 1992), In this section we used the method originally de- from which a maximum longevity estimate was ob- veloped by Bierzychudek (1982), which incorporates tained; 2) then we adjusted the value of the matrix stochasticity in the demographic behaviour of the entry referring to the survival of class-7 individuals studied species by using the information of more than in the matrices of the disturbed site (D) in order to one matrix. An initial population vector was built by equal the longevity estimate obtained for matrices in averaging the observed population structures for each site P. site and year, so that the initial size structure in the The dominant eigen-value (␭) and the correspon- simulations was similar to the actual size structure dent right (w) and left (v) eigen-vectors of the matrix observed in the field. This vector was used to start the were obtained by the power method. The confidence matrix iterations. For each matrix iteration a ran- intervals for ␭ were obtained through the analytical domly selected matrix was used; the selection criteria method described in Caswell (1989, p. 185). Addi- varied in each of the five simulation exercises (from tionally, elasticity matrices were calculated from the now on called ЉrunsЉ) performed. a) In the first run right and left eigen-vectors as follows: each of the four matrix was sampled with the same probability (0.25). b) In the second run we incorpo- ϭ ͑ ͗ ͒͑͘ ␭͒ eij viwj/ v,w aij/ . (1) rated the relative frequency of ЉgoodЉ and ЉbadЉ years according to rainfall data; we analysed the rainfall The elasticity (eij) of each matrix entry, aij, repre- pattern in the study site for the last 40 years (data sents a measure of proportional sensitivity of ␭ to from the Observatorio Meteorológico, UNAM) and proportional changes in aij. Also, since the sum of all calculated the probability of occurrence of ЉgoodЉ the eij in an elasticity matrix equals unity, each eij years (those with a total yearly precipitation above may be interpreted as the contribution of its respec- 900 mm, which represents the 1997–1998 period), ␭ tive aij to the value of (de Kroon et al. 1986). and ЉbadЉ years (with a yearly precipitation below 900 The deterministic time-invariant matrix model de- mm, represented by the 1996–97 period). These prob- scribed above is based on the assumption that demo- abilities were 0.308 and 0.692, respectively. Addition- graphic conditions do not change over time and that ally, we incorporated the probability of disturbances the modelled population grows exponentially at a rate (fires) taking place in two out of three years, on av- given by ␭. This matrix model is therefore used to erage. The resulting probabilities for each matrix are project the future consequences of the present demo- reported in Table 6. c) In the third run we used the graphic scenario, should everything remain the same. same ЉgoodЉ and ЉbadЉ year frequencies as above, but However, the demography of most plant species vary fire occurrence probability was one every other year, over time and space (Horvitz and Schemske 1995; which is close to the actual fire frequency observed Quintana-Ascencio and Menges 1996; Valverde and in the last 10 years at the site. d) The fourth run con- Silvertown 1998; Mandujano et al. 2001). Therefore, sidered ЉgoodЉ and ЉbadЉ years as above, but fire fre- to evaluate the potential long-term demographic be- quency was, on average, of one every three years. d) haviour of a species it is more realistic to incorporate Finally, in the fifth run we used the same conditions demographic information gathered in different sites as above, but with an average fire frequency of one and different years. In this study we used the four every five years (Table 6). projection matrices built (P-1996/97, P-1997/98, In each run, a total of 100 matrix iterations were D-1996/97 and D-1997/98) to perform numerical carried out, after which the numerical behaviour of 173 the population (and of each of the size categories) Table 2. Mean germination percentages (± standard deviation) ob- could be plotted against time. To calculate the popu- tained in the field experiments performed in the Preserved and the lation growth rate under these stochastic conditions Disturbed sites for the two study periods. For the 1998 results ini- ␭ tials refer to sowing treatments: IWS = illuminated, with soil; INS ( s), we carried out 30 series of 100 matrix iterations. = illuminated, no soil; SWS = shaded, with soil; SNS = shaded, no For each series of 100 iterations, we plotted the natu- soil. In each of the 1998 sowing treatments (1998) the averages ral logarithm of average population size against time were obtained from only two replicates per site. In 1997 averages (iterations) to represent the numerical long-term pop- are from four replicates per site. ulation behaviour expected under stochastic demo- % Germination graphic variation and we fitted a linear regression to Well-Preserved site Disturbed site the data. The slope of the fitted line was considered a measure of the intrinsic rate of population increase 1997 1.00 ± 2.00 0.25 ± 0.50 Germination Probability 0.01 0.0025 (r), from which ␭ could be calculated (␭ =er) s 1998 (Bierzychudek 1982). Since we calculated 30 ␭ val- IWS 22.00 ± 14.14 0 ues in each case, we were able to report an average ␭ INS 0 0 value for each run (plus standard deviation) as well SWS 5.00 ± 4.24 6.00 ± 5.65 as to estimate the ‘persistence probability’ of the over- SNS 36.00 ± 25.45 17.00 ± 9.89 all population according to its projected long term Germination probability 0.157 0.058 numerical behaviour over a period of 100 years. The latter were defined as the relative frequency with which ␭ values below unity were obtained. Seedling survival in 1997 is described in Figure 2a. All seedlings died within the first 45 days after planting. Seedlings in site P showed a higher initial Results survival compared to those in site D, but the final result was the same for both sites. According to the ex- Early stages of plant development trapolation of the 1997 survival curves to day 365, the probability of seedling survival was 0.001 in site D, In 1997 the number of seeds that were observed ger- and 0.003 in site P. In 1998 higher seedling survival minating in the field experiment carried out was very rates were observed than in the previous year, and low (Table 2). Four out of 400 seeds germinated in seedling survival was in general higher in site P than site P, while only one out of 400 germinated in site in site D (Figure 2, 2b and 2c). In site P seedlings D. This difference was not statistically significant (t = planted in shaded microsites with soil had a signifi- 1.0, d.f. = 3, p = 0.39). In 1998 mean germination cantly higher survival rate than the other three treat- percentage across the four sowing treatments was ments (Figure 2b), while in site D the most successful 15.8% in site P and 5.8% in site D (Table 2); the dif- treatment were the fully illuminated microsites with ference between these values was statistically signif- soil, which showed a significantly higher survival rate icant (t = 2.64, d.f. = 7, p = 0.03). The higher germi- than the other treatments. In this site, the fully illu- nation percentage obtained this year compared to the minated microsites with no soil resulted in total seed- one observed in 1997 determined much higher fecun- ling mortality within the first nine days after planting. dity values for the 1997–98 period (see results be- Combining the results of all treatments within each low). The more detailed experimental design fol- site, the survival probability to day 365 was 0.0015 lowed in this second experiment also allowed us to for site D, and 0.134 for site P. These data were in- evaluate, at least in a general way, the microenviron- terpreted as the transition probability from class 1 to mental conditions that are more appropriate for the class 2 in the Lefkovitch matrices (Table 4). germination of M. magnimamma seeds. In both sites the shaded microsites with no soil resulted in higher Reproduction germination percentages, while the fully illuminated site with no soil did not show any germination at all M. magnimamma showed a relatively long yearly re- (Table 2). The low number of replicates used for each productive season (Figure 3). Flower buds started in treatment did not allow further statistical analysis of February and ended in June–July in both years and these results. sites. Flowers were found between March and June. 174

Figure 2. Seedling survivorship curves for a) the 1996/97 growth period (both study sites), b) the 1997/98 period in the well preserved site and c) the 1997/98 period in the disturbed site. The different lines in b) and c) correspond to planting treatments; different letters above survivorship curves imply signiﬁcant differences between treatments according to Peto and Peto pair-wise comparisons.

Once ﬂowers are pollinated, fruits take a long time to The total number of fruits produced per plant were develop. In the two years of study the fruiting period counted each year and multiplied by the average num- lasted for around ﬁve months, from April to August, ber of seeds per fruit (for 1996/97: 93.5 ± 36.8 for with maximum fruit production occurring between both sites, n = 25; and for 1997/98: 93.5 ± 42.0 for June and July (Figure 3). In 1997 fruit production was site P, n = 69; and 75.4 ± 45.0 for site D, n = 49). slightly higher than in 1998. Yet, the reproductive This gave the estimated number of seeds produced phenology of the studied populations was remarkably per plant, which was in turn multiplied by the seed similar both between sites and between years. germination probability (see Table 2) to estimate fe- 175

Figure 3. Reproductive phenology of M. magnimamma plants for two years at the ЉEl XitleЉ lava-field. a) Well preserved site and b) disturbed site. cundity. Thus, here fecundity is given in terms of the Size-based demography number of seedlings produced per plant in a reproductive season. Average fecundity for each size cat- We built four population projection matrices, two for egory is reported in Table 3; these results show that, each site (1996/97 and 1997/98) (Table 4). Most cat- in general, fecundity was larger as plant size in- egories showed positive transitions towards several creased. In 1996/97 fecundities were lower than in potential fates in all matrices, i.e. growing one, two, 1997/98 (mainly due to the higher germination per- or even three categories, decreasing in size one to centages obtained in 1998), and plants in site P several categories, or staying in the same category. showed consistently higher fecundity values than Most matrices showed the highest values in those en- plants in site D (Table 3). These fecundities were in- tries referring to the persistence of plants in the same corporated in the transition matrices as the contribu- category. The survival of plants in the 7th category tion of each size category to the seedling category (which was estimated for matrices from site D – see from one year to the next (i.e., first row of the ma- Methods) was similar in the four matrices. Plant mor- trix). tality was highest in the first category and decreased with plant size; in general, both fecundity and mortality were higher in the second compared to the first growth period for both sites. Both ␭ values obtained 176

Table 3. Fecundity values (mean number of seedlings produced per plant) for each size category in a) the preserved and b) the disturbed sites for each of the study periods.

Size Category 1996–97 No. fruits/plant No. Seeds/plant Fecundity 1997–98 No. fruits/plant No. Seeds/plant Fecundity a) Preserved Site 10 000 0 2 0 0 0 0.150 14.025 2.22 3 1.057 98.301 0.98 0.941 87.983 13.90 4 3.380 314.34 3.14 1.313 122.766 19.40 5 4.300 399.9 4.00 1.471 137.539 21.73 6 1.632 151.776 1.51 1.731 161.849 25.57 7 6.263 582.459 5.82 2.808 262.548 41.48 b) Disturbed Site 10 000 0 0 2 0.306 28.458 0.07 0 0 0 3 1.627 151.311 0.38 0.396 29.858 1.73 4 2.364 219.852 0.55 1.018 76.757 4.45 5 1.462 135.966 0.34 0.810 61.074 3.54 6 2.692 250.356 0.63 1.817 137.001 8.18 7 4.857 451.701 1.13 1.545 116.493 6.76 in site D were below unity (although not significantly elasticities for fecundity and growth entries are very so); the same occurred for P-1996/97. However, for low. On the other hand, for the matrix with a positive P-1997/98 we obtained a population growth rate sig- population growth rate (P-1997/98) the highest elas- nificantly higher than unity. This was mainly a result ticity value refers to seedling survival and growth, of the high fecundity values (resulting from high ger- also showing relatively high values for the entries mination percentages) incorporated in this matrix, as corresponding to fecundity and growth (Table 5). well as a high seedling survival probability for this year in this site. Numerical simulations of demographic stochasticity For the three matrices with ␭ values below unity the expected population structure at equilibrium (w) When matrix iterations were performed choosing a has a high proportion of individuals in the first cat- different (randomly allocated) matrix for each itera- egory (seedlings), followed by those in the last cat- tion event, each series of 100 iterations resulted in a egory (Table 4). However, for the matrix with a pos- different projection of numerical population behav- itive ␭ value (P-1997/98) the vector of the stable size- iour, depending on the particular order and frequency category distribution describes a population with which different matrices were utilised. In Fig- represented mainly by seedlings, juveniles and young ure 4 only three of such series are represented, plot- adults. Expected size-specific reproductive values at ting the numerical behaviour of a growing and a de- equilibrium (v) show a very small contribution of clining population (Figure 4a and 4b, respectively), as seedlings, particularly in those matrices in which well as the numerical behaviour of each size category seedling mortality is higher; the reproductive value of independently for a growing population (Figure 4c). other categories is relatively uniform, with the excep- In these plots we used a logarithmic scale on the y tion of matrix P-1997/98 in which there is a clear axis to allow visual appreciation of the overall popu- tendency towards an increase in reproductive value lation trend and to fit a linear regression from which with increasing size. the slope (r = intrinsic rate of population increase) The elasticity matrices (Table 5) show two clearly could be calculated. Sincer=ln␭, then overall pop- ␭ r distinct patterns: for the three matrices referring to ulation growth rate was calculated as s =e.Inall populations with ␭ values below unity (P-1996/97, series performed the numerical fluctuations in overall D-1996/97 and D-1997/98) the highest elasticity val- population size were given mainly by fluctuations in ues are those corresponding to the persistence of large seedling numbers (see Figure 4c). High peaks in the adults (7th category) in their same category, while seedling category were generally followed by peaks 177

Table 4. Population transition matrices for sites P and D and for periods 1996/97 and 1997/98. The ␭ values (± 95% confidence intervals) are given above each matrix. nx = number of individuals per size-category from which transitions were calculated; qx = size-specific mortality; w = stable size-category distribution; v = size-specific reproductive values. Entries in the main diagonal are underlined to facilitate reading.

1 234567(w)(v)

Site P 1996/97 ␭ = 0.956 ± 0.086 10 0 0.983 3.143 4 1.518 5.825 0.849 6.3E-05 2 0.003 0.613 0.019 0 0 0 0 0.001 0.201 3 0 0.355 0.509 0.064 0 0 0 0.002 0.195 4 0 0 0.415 0.762 0.200 0.053 0 0.009 0.180 5 0 0 0.038 0.095 0.350 0 0 0.002 0.167 6 0 0 0.019 0.032 0.400 0.737 0 0.006 0.149 7 0 0 0 0.016 0.050 0.211 0.947 0.131 0.108 nx 120 32 54 64 20 18 18 qx 0.997 0.032 0 0.0318 0 0.0001 0.0526 Site P 1997/98 ␭ = 1.333 ± 0.174 10 2.216 13.901 19.397 21.731 25.572 41.483 0.819 0.003 2 0.134 0.450 0.073 0 0.059 0 0 0.128 0.034 3 0 0.200 0.667 0.156 0 0.039 0 0.041 0.111 4 0 0 0.177 0.590 0.235 0.039 0 0.010 0.137 5 0 0 0 0.108 0.471 0.115 0 0.001 0.152 6 0 0 0 0.012 0.118 0.692 0.039 0.000 0.199 7000000.039 0.923 4.1E-05 0.362 nx 240 41 51 83 17 26 26 qx 0.866 0.3500 0.0784 0.1326 0.1177 0.0768 0.0385 Site D 1996–97 ␭ = 0.967 ± 0.095 10 0.070 0.380 0.550 0.340 0.630 1.130 0.539 1.7E-05 2 0.001 0.270 0.010 0 0 0 0 7.8E-05 0.164 3 0 0.550 0.290 0.030 0 0 0 6.7E-05 0.177 4 0 0.100 0.560 0.240 0 0 0 6.3E-05 0.169 5 0 0 0.090 0.360 0.150 0 0 3.6E-05 0.172 6 0 0 0.040 0.270 0.850 0.540 0 1.2E-04 0.165 7 0 0 0 0.060 0 0.460 0.967 0.461 0.153 nx 120 50 76 34 14 14 15 qx 0.999 0.0800 0.0100 0.0400 0 0 0.0333 Site D 1997/98 ␭ = 0.945 ± 0.099 10 0 1.732 4.452 3.542 8.182 6.757 0.832 2.0E-04 2 0.002 0.579 0.038 0.036 0 0 0 0.010 0.129 3 0 0.211 0.623 0.250 0.048 0 0 0.035 0.148 4 0 0.105 0.226 0.554 0.143 0.065 0 0.033 0.155 5 0 0 0.019 0.089 0.571 0.162 0 0.018 0.205 6 0 0 0.019 0 0.238 0.548 0.046 0.019 0.195 7000000.194 0.881 0.055 0.168 nx 240 19 53 56 21 31 22 qx 0.9985 0.1053 0.0755 0.0714 0 0.0323 0.0735 in category 2 and so on; larger size categories were The results of the stochastic simulations were a numerically relatively stable. function of the particular probability of occurrence 178

Table 5. Elasticities matrices for sites P and D and for periods 1996/97 and 1997/98. Entries in the main diagonal are underlined to facilitate reading. The three highest elasticity values in each matrix are typed in bold letters.

1234567

Site P 1996/97 10 0 7.8E-06 1.1E-04 2.5E-04 3.1E-05 0.003 2 0.003 0.006 4.8E-04 0 0 0 0 3 0 0.004 0.013 0.007 0 0 0 4 0 0 0.009 0.076 0.004 0.003 0 5 0 0 8.0E-04 0.009 0.006 00 6 0 0 3.6E-04 0.003 0.006 0.036 0 7 0 0 0 9.5E-04 5.4E-04 0.007 0.797 Site P 1997/98 10 0.054 0.108 0.038 0.005 0.002 3.3E-04 2 0.208 0.109 0.006 0 1.5E-04 0 0 30 0.160 0.170 0.010 0 1.1E-04 0 4 0 0 0.056 0.046 0.002 1.3E-04 0 5 0 0 0 0.009 0.005 4.3E-04 0 6 0 0 0 0.001 0.002 0.003 1.8E-05 7000003.4E-04 7.8E-04 Site D 1996/97 10 1.4E-09 6.2E-09 8.5E-09 3.0E-09 1.8E-08 1.3E-04 2 1.3E-04 4.9E-05 2.1E-06 0 0 0 0 3 0 1.1E-04 5.1E-05 4.9E-06 0 0 0 4 0 1.9E-05 9.2E-05 3.7E-05 000 5 0 0 1.6E-05 5.7E-05 1.4E-05 00 6 0 0 6.4E-06 4.1E-05 7.3E-05 1.5E-04 0 7 0 0 0 8.5E-06 0 1.2E-04 0.999 Site D 1997/98 10 0 4.7E-04 0.001 4.8E-04 0.001 0.003 2 0.006 0.029 0.006 0.006 0 0 0 3 0 0.012 0.121 0.045 0.005 0 0 4 0 0.006 0.046 0.105 0.015 0.007 0 5 0 0 0.005 0.022 0.078 0.023 0 6 0 0 0.005 0 0.031 0.075 0.018 7000000.023 0.305 allocated to each of the four matrices used (see Ta- long-term ␭ value below unity. From this we esti- ble 6). The highest ␭ value was obtained in the run in mated the persistence probability of the overall popu- which all four matrices were given the same proba- lation as P = 1 − (1/30) = 0.97 (Table 7). Finally, bility of occurrence (Table 7). When the probability when the probability of fire events was set at 0.66, of ЉgoodЉ and ЉbadЉ years was incorporated, along the ␭ value was just above unity, while the persistence with different disturbance regimes (given by different probability of the population was 0.66 (i.e., 10 out of fire frequencies), interesting patterns emerged. Low 30 series described a crashing population) (Table 7). fire probabilities (i.e., 0.2 and 0.3) resulted in ␭ values well above unity. When the probability of occurrence of a fire event for each particular year was 0.5 Discussion (which is close to the actual disturbance regime at the site), the overall ␭ value was also above unity; how- From the four population matrices obtained in the ever, from the 30 series performed with these condi- present study, three of them showed ␭ values slightly tions, one described a crashing population with a below unity. Yet, in these cases the confidence inter- 179

Figure 4. Examples of the numerical behaviour of the overall M. magnimamma population under stochastic demographic variation. a) A growing population, obtained by allocating the same probability of occurrence to each of the four matrices. b) A declining population, incorporating the probability of occurrence of ЉgoodЉ and ЉbadЉ years and a relative frequency of ﬁre events of 0.66. c) A growing population showing the variation in numbers of each size category, for a ﬁre frequency of 0.5. Line codes for size categories are as follows: 1 = dark dotted; 2 = Dark dashed; 3 = Dark continuous; 4 = Dark dashed-dotted; 5 = Fine continuous; 6 = Fine dotted; 7 = Fine dashed. 180

Table 6. Probabilities used for the occurrence of different matrices in the stochastic simulations. ЉGoodЉ years refer to periods with high rainfall (above 900 mm – probability 0.308) and ЉbadЉ years to those with low rainfall (below 900 mm – probability 0.692). In addition to the probability of occurrence of ЉgoodЉ and ЉbadЉ years, different ﬁre frequencies were simulated. The larger numbers in each column are the resulting probabilities of occurrence of each matrix for each particular run.

Matrix ␭ 1st run Equal prob. 2nd run Good and bad years; 3rd run Good and bad years; 4th run Good and bad years; 5th run Good and bad years; fire 2 out of 3 years fire every other year fire every 3 years fire every 5 years

P-96/97 0.956 0.250 (0.692 × 0.33) = 0.230 (0.692 × 0.5) = 0.346 (0.692 × 0.66) = 0.461 (0.692 × 0.8) = 0.554 P-97/98 1.333 0.250 (0.308 × 0.33) = 0.102 (0.308 × 0.5) = 0.154 (0.308 × 0.66) = 0.205 (0.308 × 0.8) = 0.246 D-96/97 0.967 0.250 (0.692 × 0.66) = 0.457 (0.692 × 0.5) = 0.346 (0.692 × 0.33) = 0.231 (0.692 × 0.2) = 0.138 D-97/98 0.945 0.250 (0.308 × 0.66) = 0.203 (0.308 × 0.5) = 0.154 (0.308 × 0.33) = 0.103 (0.308 × 0.2) = 0.062 181

Table 7. Population growth rate (␭) obtained in the different sto- to be determined by the dynamics of the early stages chastic simulations. Each figure is the average of 30 series of 100 of plant development, during which a tremendous iterations each; standard deviations are a measure of variation mortality occurs in this and other cacti species (Sten- within each of these 30 series in each case. See Table 6 for details on the conditions for each run. Persistence probability refers to the nbergh and Lowe 1977; Jordan and Nobel 1981; Va- relative frequency of ␭ values above unity in the 30 series per- liente-Banuet and Ezcurra 1991). Therefore, it is im- formed. portant to address this particular aspect of the species’ ␭ St. Dev. Persistence probability life-cycle to understand one of the most important determinants of population dynamics (Godínez-Alva- 1st run 1.086 0.020 1.00 rez et al. 1999). It is noticeable, for instance, that the 2nd run 1.008 0.018 0.66 seeds of many cacti species germinate readily under rd 3 run 1.036 0.019 0.97 controlled conditions. M. magnimamma seeds reach th 4 run 1.063 0.019 1.00 85% germination in ca. 20 days (Ruedas et al. 2000) th 5 run 1.075 0.022 1.00 and other cacti (Echinocactus platyacanthus, Fero- cactus robustus, F. recurvus, F. flavovirens, Cephalo- cereus chrysacanthus, Pachyrereus hollianus, vals for ␭ were relatively wide, thus, these matrices Neobuxbaumua tetetzo, N. macrocephala) show the appear to represent populations that are close to the same type of germination behaviour (Rojas-Aréchiga numerical equilibrium. The fourth matrix, P-1997/98, et al. 1997, 1998; Godínez-Alvarez and Valiente-Ban- had a positive ␭ value which was significantly differ- uet 1998; Esparza-Olguín et al. 2002). However, in ent from unity, representing a growing population. In natural conditions the frequently low soil moisture this case the high ␭ may be accounted for by the high content appears to dramatically decrease the probabil- fecundity and seedling survival observed in this site ity of successful germination. for the second study period. In turn, these fecundity After seed germination, very few seedlings survive values were given by high seed germination percent- the harsh conditions imposed by high solar radiation ages in the field. These results could have been re- and low nutrient and water availability (Steenbergh lated to two particular events. First, in 1997/98 the and Lowe 1969; Stennbergh and Lowe 1977; Jordan results of the field experiments on seed germination and Nobel 1981; Valiente-Banuet and Ezcurra 1991). and seedling survival were recorded daily; thus, seed The results of our field experiments on seed germina- germination (and therefore fecundity) might have tion and seedling survival suggest that the probability been underestimated the previous year, in which re- that a germinated seed reaches the stage of a success- sults were recorded every two weeks. However, note fully established seedling is tremendously low. Sev- that the general increase in fecundity observed in eral factors may determine this. First, cacti seedlings 1997/98 did not result in a positive ␭ in the case of show very low relative growth rates (Jordan and No- the population in the disturbed site, which suggests bel 1981; Nolasco et al. 1996), and thus the highly that the increase in population growth rate observed fragile phase of seedling establishment becomes ex- in P-1997/98 was not an artefact of the differences in traordinarily long (relative growth rate of M. magni- the frequency of experimental observations between mamma seedlings is ca. 0.015 g g−1 d−1 – Ruedas et the two years. The second event that could have de- al. (2000)). Secondly, seedlings show a relatively low termined these results was that rainfall was higher in water use efficiency due to C metabolism during the 1998 than in 1997, as reported above. This could have 3 early stages of seedling development (Altesor et al. determined the high seed germination and seedling 1993). Thirdly, the high water content of cacti tissues survival values obtained for 1997/98. Water availabil- determines a high incidence of seedling predation ity is clearly one of the most important limiting fac- (Valiente-Banuet and Ezcurra 1991; Mandujano et al. tors for the growth and establishment of cacti species 1996). As a result, seedling survival probability vary (Valiente-Banuet and Ezcurra 1991; Mandujano et al. greatly from year to year or between different micro- 1996); the reduced water retention capacity of the sites. Our results show that for M. magnimamma, this bare basaltic substrate at ЉEl XitleЉ lava-field deter- variation may be of two orders of magnitude (i.e. mines that water availability is directly related to the from 0.001 to 0.134), which accounts for an impor- amount of rainfall. tant variation in ␭ between years or between sites. The most important differences in population The variation in fecundity values also resulted in growth rate between years and between sites appears important changes in ␭. Fecundity was given by two 182

main components: seed production and seed germi- rather erratic variation in population numbers, offer a nation. Although fruit and seed production showed more complete picture of the species’ actual demo- only small variations, seed germination probabilities graphic behaviour. In these cases, the variation in (as estimated from our field experiments) varied 100- population size which resulted from the use of differ- fold, ranging from 0.002 to 0.157. This variation was ent matrices for the iteration process, was mainly a the main reason for the observed changes in fecun- result of changes in seedling establishment success, dity between years and between sites. which was the main difference between matrices. This The elasticity values of fecundity elements were phenomenon has also been documented for other low in all cases, especially in the matrices that had a cacti species (Ferocactus cylindraceus and Carnegia ␭ value below unity. This appears to suggest that gigantea – Bowers (1997) and Pierson and Turner changes in fecundity entries would have only a minor (1998)). The underlying causes of this type of numer- effect on ␭, yet, when fecundity increased we ob- ical fluctuations in populations of wild plants have served a significant increase in ␭. However, note that long been a concern of plant population ecologist. fecundity varied from 0.1 to 41.5 seedlings per plant, Although there is a great lack of empirical evidence which represents a ca. 1000-fold variation, whereas on this issue, our simulation results suggest that tem- the value of other matrix entries changed little be- poral variation in seedling survival (including all the tween years or between sites (de Kroon et al. 2000). life-cycle phases involved in it) may be responsible For instance, the persistence of adults in category 7 for the erratic numerical behaviour observed in many ranged from 0.88 to 0.97, and although this particular plant populations. entry had high elasticity values, the changes in this In the first run of stochastic simulations we gave entry did not account for much variation in ␭. The the same probability of occurrence to each of the four high elasticity values of this entry in the matrices with matrices. However, each matrix represent a particular ␭ < 1 suggest that those populations, given the lack scenario that may occur at different frequencies. In of seedling recruitment, rely mainly on the survival 1998 the rainfall was above average, while in 1997 it of old adults and so population numbers would de- was below average. So an important aspect to incor- cline at the rate these adults disappear. A different porate in the simulations was the relative frequency picture emerges when analysing matrix P-1997/98, with which rainy and dry years occur. Additionally, which showed a ␭ > 1. In this case the elasticity val- there are years with widespread fire events, while in ues for fecundity entries added up to 20% of total other years fires are kept to a minimum. Thus, the for elasticity, and the elasticity of seedling establishment matrices represent rainy years with low fire impact was also particularly high. As other authors have (P-1997/98), rainy years with high fire impact (D- stated, this implies that the dynamics of the early 1997/98), dry years with low fire impact (P-1996/97) stages of plant development are fundamental for a and dry years with high fire impact (D-1996/97). Al- population that is growing, and changes in such though we could have access to the information on stages dramatically affect population growth rate the relative frequency of rainy and dry years at the (Horvitz and Schemske 1995; Silvertown et al. 1996; site, no information is available so far concerning the Valverde and Silvertown 1998; Mandujano et al. frequency and intensity of fire events. Thus we simu- 2001). lated the overall population growth rate under differ- When incorporating the observed spatio-temporal ent theoretical fire probabilities (ranging from 0.20 to ␭ variation in the demography of M. magnimamma into 0.66). In all cases the analyses yielded s values numerical simulations of overall population behav- above unity; however, as fire frequency increased, iour, allocating the same probability of occurrence to population persistence probability decreased. If fires all four matrices, we obtained a population growth were as common as two every three years, the likeli- ␭ rate of s = 1.09. The average population growth rate hood that the M. magnimamma population would per- ␭ of the four matrices ( a = 1.05) did not differ much sist in the long run would be unacceptably low. from the value obtained from this stochastic analysis. Regarding the stochastic simulations, it is impor- Yet, it is important to note that this average popula- tant to take into account that the four matrices used tion growth rate does not offer any insights into the are certainly not the only possible ways in which the temporal variation in population size. On the other demography of M. magnimamma could be character- hand, the results of the different stochastic simula- ised. A more complete picture would emerge if popu- tions performed in this study, which resulted in a lation dynamics was evaluated over longer periods of 183 time and over a wider range of conditions within the Anderson E., Arias S. and Taylor N. 1994. Threatened cacti of ЉPedregal de San AngelЉ reserve. Mexico. Royal Botanic Gardens Kew, UK. Cacti species are generally thought to be highly Bravo-Hollis H. and Sánchez-Mejorada H. 1991. Las cactáceas de México. Vol. 3. U.N.A.M., México, D.F., México. vulnerable to disturbances. The results of our study Bierzychudek P. 1982. The demography of jack-in-the-pulpit, a support this idea, since the only matrix that showed a forest perennial that changes sex. Ecological Monographs 52: ␭ significantly higher than unity was the one that cor- 335–351. responded to a population in a well preserved site and Bowers J.E. 1997. 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Elasticities: a tive vegetation if the reserve at the ЉEl XitleЉ lava- review of methods and model limitations. Ecology 81: 607– field is to maintain its original ecological interest and 618. biodiversity. Esparza-Olguín L., Valverde T. and Vilchis-Anaya E. 2002. Demo- graphic analysis of a rare columnar cactus (Neobuxbaumia macrocephala) in the Tehuacan Valley, Mexico. Biological Conservation 103: 349–359. Godínez-Alvarez H. and Valiente-Banuet A. 1998. Germination Acknowledgements and early seedling growth of Tehuacan Valley cacti species: the role of soil and seed ingestion by disperses on seedling growth. Journal of Arid Environments 39: 21–31. This project was carried out with the support of CON- Godínez-Alvarez H., Valiente-Banuet A. and Valiente-Banuet L. ACyT (5-3181 PN). Two of the authors (MLV and 1999. 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