Population dynamics in orchid conservation: a review of analytical methods based on the rare species...

Chapter 10

POPULATION DYNAMICS IN ORCHID CONSERVATION: A REVIEW OF ANALYTICAL METHODS, BASED ON THE RARE SPECIES LEPANTHES ELTOROENSIS

Raymond L. TremblayDepartment of Biology, University of Puerto Rico - Humacao Campus, Humacao, Puerto Rico, 00791, USA.

Michael J. HutchingsSchool of Biological Sciences, University of Sussex, Falmer, Brighton, Sussex, BN1 9QG, United Kingdom.

Prediction of population persistence and prevention of population extinction requires gathering and evaluating data on population growth rate and analysis of the factors that infl uence it. Individual, demographic, and environmental variation, and the interactions between these sources of variation, can all affect population growth rate. Environmental variation can be further broken down into temporal, spatial and catastrophic variation. This chapter describes methods for quantifying variation in population growth rate caused by each of these factors and provides a detailed analysis of the use of the Lefkovich model to describe calculation of population growth rate. The benefi ts of elasticity analysis are demonstrated as a means to determine those components of species’ life cycles where management is most likely to have signifi cant effects on population growth rate. Use of the methods described is strongly recommended in studies aimed at devising conservation strategies for rare and endangered species. Data collected from the epiphytic orchid Lepanthes eltoroensis are used to illustrate the calculations involved.

1. Introduction1.1. The problem of orchid conservationNumbers of plants are declining in populations of many orchid species. Reversing this trend in many cases requires intervention and the adoption of new management regimes. Conservationists can often suggest measures that might at least partially alleviate population losses, but adopting management changes without full knowledge of their potential effects is risky. For small populations of endangered species it may be irresponsible. The potential consequences for a population of a change in management should, whenever possible, be subject to predictive modelling. This requires a demographic approach. Demography is the study of the statistics of births, deaths, diseases etc. in populations. The purpose of this chapter is to advocate this approach as a basis for practical orchid conservation and more informed management decisions. We fully agree with Schemske et al. (1994) in concluding that a demographic approach provides the best way of realistically assessing the current status of a species or its individual populations, and of determining the life history stages that most strongly affect growth of its populations.

K.W. Dixon, S.P. Kell, R.L. Barrett and P.J. Cribb (eds) 2002. Orchid Conservation. pp. 163–183. © Natural History Publications Kota Kinabalu, Sabah.

Suzanne

Sticky Note

Suzanne

Sticky Note

2003 not 2002

In this chapter we describe the most important methods available for analysis of population statistics.Some of the methods may appear daunting at fi rst sight, but they provide unrivalled potential for comparing and understanding the effects of different management regimes on populations. Understanding the effects of different management regimes is a vital part of the conservationist’s job: although increasing the numbers of plants in populations may prevent imminent extinction, deeper understanding of the behaviour of populations of rare species will permit better predictions of their future fates, and in turn lead to a greater probability of avoiding actions that will cause their decline.

1.2. The demographic approachA demographic approach to understanding population behaviour requires repeated censuring of populations at several time intervals. The ultimate aim is to use the data collected on different census dates to predict population size at some future time, Nt+1, from the current population size, Nt. Between two census dates, a population gains individuals through the two processes of birth (B) and immigration (I), and loses individuals through the processes of death (D) and emigration (E). This can be expressed in the equation: Nt+1 = Nt + B + I - D - EThe simplicity of this equation belies the diffi culties in collecting accurate data on any of these aspects of population fl ux. It is often assumed that numbers of immigrants and emigrants are equal, and that counting them can therefore be ignored because their effects cancel out. In practice, the assumption rarely rests on more than guesswork and hope, and is often questionable. Moreover, there are often differences between the genetic properties of immigrating and emigrating organisms (Ellstrand and Elam, 1993). Quantifi cation of immigration and emigration in species with seeds as small as those of the orchids is currently virtually impossible. Assessing recruitment (births) via seeds, and the fate of these recruits is equally problematic in orchids. Little is known about germination of orchid seeds under fi eld conditions (Rasmussen and Whigham, 1993; Ackerman, Sabat and Zimmerman, 1996; Batty, Dixon and Sivasithamparam, 2000) and, at least in terrestrial orchids, it is often impossible to count new recruits or even record the year of their germination with certainty (Hutchings 1987a,b). This is because new individuals of many orchid species pass through a subterranean phase following germination, from which they may not emerge above ground for several years (e.g. Wells, 1981). Lack of above ground activity can be construed by the unwary as indicative of death in terrestrial orchids, but caution is necessary in making this conclusion, as detailed study of several species has shown that reappearance of plants at a later date is not uncommon. Death of individual plants cannot be assumed to have occurred in many terrestrial species until they have failed to appear for three or more consecutive years (Hutchings 1987a,b; Light and MacConaill, 1991).

As well as indicating whether populations are stable, decreasing or increasing in size, repeated censuses allow analysis of fl ux - i.e. quantifi cation of recruitment and death in the intervals between censuses. Such information is valuable, but somewhat limited. For example, a population could have a complete lack of fl ux, or high but matching rates of recruitment and mortality. Neither situation would change the numbers of plants recorded, but the population’s age structure at later censuses would be very different in each case. If population size is maintained solely by established but senile plants, and there is no recruitment, an unpredicted and rapid fall in numbers might eventually occur, culminating in population extinction. A population with high mortality may remain vigorous if recruitment is buoyant, but be at serious risk if recruitment ceases. Thus, it is desirable to obtain information on the categories of plants that die, and on life spans. As Schemske

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et al. (1994) state, neither censuses nor predictions of future population behaviour from average birth and death rates will tell the observer what is controlling population growth rate. To design effective conservation strategies it is imperative to determine the life history stage(s) that exert(s) the greatest control on population growth rate. Such life history stage(s) should be the focus around which management is designed.

1.3. The life-stage based modelMatrix projection models are the most powerful methods for predictive analysis of population dynamics. For plants, stage-based matrix projection models (Lefkovitch, 1965) are more appropriate for analysing plant population dynamics than age-based models (Caswell, 2001 and references within) because life history stage (for example, seed, seedling, juvenile, adult, senescent) and size are more important than age in determining demographic parameters (Harper, 1977). For example, in Lepanthes spp., survivorship (Tremblay, 2000a; Tremblay and Ackerman, 2001) and reproductive effort are more highly correlated with plant stage than with age (Tremblay and Rodriguez Ortiz, unpublished), while in the bromeliad Tillandsia brachycaulus, size categories are an effective method for categorizing the life cycle (Mondragón et al., 1999). Accessible introductions to stage-based matrix projection models can be found in Caswell (1989, 2001), Ebert (1999), Akçakaya, Burgman and Ginzburg (1999) and Cochran and Ellner (1992). The use of such models is described in this chapter.

Natural variation in demographic parameters complicates predictions about population persistence. These parameters alter signifi cantly with life history stage and plant size, and also vary because of temporal and spatial variation in environmental conditions. For example, recruitment may be low and mortality high over a period when a population experiences drought, but recruitment may rise and mortality fall during a period when rainfall is closer to the long-term average. Such temporal variation or temporal stochasticity in demographic parameters should ideally be overcome by collecting data at numerous time intervals, enabling variation in population behaviour to be analysed over time. Examples of such analyses will be given below. Plants in populations located in different places may also differ in recruitment and mortality rates, and in fecundity, because of differences in the environmental conditions they experience. Consequently, it is desirable to collect data from more than one population, to enable spatial variation or spatial stochasticity to be taken into account when interpreting population behaviour. In practice this is often diffi cult or logistically impossible. An example of such an approach will also be presented below. Finally, and of great importance in dealing with small populations of rare species, the results of matrix modelling may depend strongly on the age- or stage structure of sampled populations. For example, if many of the plants in a small population are in the pre-reproductive stage, the population will have a lower mean plant fecundity than if it contained many reproducing plants. Large populations might be expected to consistently contain similar proportions of plants in each life-history stage, but substantial differences in the proportional representation of different life-history stages can easily occur in small populations. This effect, which is referred to as demographic stochasticity or demographic variance, can have a signifi cant impact upon predictions of population behaviour.

In this chapter, data collected from natural populations of the orchid Lepanthes eltoroensisStimson will be used to show how demographic parameters calculated from stage-based transition matrices can be affected by temporal and spatial variation, and by demographic stochasticity, and how these factors can alter predictions about population survival.

2. Materials and methods

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2.1. Lepanthes eltoroensisThe genus Lepanthes contains more than 600 species (Luer, 1986), approximately 92 of which occur in the Caribbean (Stimson, 1969; Hespenheide, 1968, 1969, 1973; Hespenheide and Dod, 1989, 1990, 1993; Barré and Feldmann, 1991; Tremblay and Ackerman, 1993). L. eltoroensis is an epiphyte endemic to Puerto Rico, where it is restricted to mountain ridges along the El Toro and Tradewinds trails and the Cerro El Cacique in the Caribbean National Forest (CNF). For the purposes of this study, a population is defi ned as the set of individuals growing on a single phorophyte (i.e. host tree). There is a mean of 47.6 adult L. eltoroensis plants per colonized phorophyte and a mean distance of 38.6m between phorophytes. Most populations have fewer individuals than the mean (Tremblay, 1997). Thus, populations are clumped in space and separated from each other by considerable distances.

2.2. Data collectionA total of 228 individuals of L. eltoroensis in fi ve populations were tagged and monitored at monthly intervals for 26 months from September 1994 until November 1996 (Table 1). No data were collected in July 1995 or in June or October 1996. The number of individuals recorded in different populations is shown in Table 1.

Table 1. Number of individuals, and life-history stage structures at the beginning of the sampling period, in fi ve populations of Lepanthes eltoroensis. Populations were visited once per month between September 1994 and November 1996, except for July 1995, and June and October 1996.

Population Seedlings Juveniles Adults Total1 9 8 43 602 25 10 45 803 0 12 40 524 1 5 12 185 10 3 5 18Total 45 38 145 228

2.3. The life stagesAt every recording date each plant of L. eltoroensis was recorded as being in one of four stages, defi ned as follows: (i) seedlings: small plants without well-developed petioles on any leaf (Fig. 1a), (ii) juveniles: individuals with at least one lepanthiform sheath on the petiole and no current or previously-produced infl orescences (Fig. 1b), (iii) non-reproducing adults: individuals that were not currently fl owering, but which carried dried infl orescences from a previous fl owering event, (iv) reproducing adults: individuals with photosynthesising (green) infl orescences (Fig. 1c). Reproduction by all reproducing individuals was also measured at every recording date, as male success (pollinaria removal) or female success (fruit set). Although pollinaria removal is not used in the projection matrix analysis, it is evidence of pollinator activity and thus can be used to determine whether pollinator activity is equal between populations and time periods. Fruits remain on plants for at least 1.5 months. Therefore, it was possible to count all fruits produced during the sampling period.

2.4. Statistical programsPrograms for analysing demographic parameters and population growth rate are readily available. Among them are free programs on the web site: http://www.sci.sdsu.edu/Cornered_Rat/ which illustrates examples given in Ebert (1999), and programs available from Akçakaya, Burgman

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and Ginzburg (1999). Contingency table analysis programs based on Monte Carlo simulations are available from: http://www.wisc.edu/genetics/CATG/engels/. Statview 4.5 was employed for calculation of basic statistics (Abacus Concepts, Inc.).

Figure 1. Lepanthes eltoroensis Stimson. A. Seedling (note the absence of lepanthiform sheaths on petiole. B. Juvenile (petiole with lepanthiform sheaths (LS) indicated by an arrow). C. Adult without infl orescence. D. Adult with infl orescence and fl owers. E. Flower. Drawing by Erick Noel Bermúdez Carambot.

3. Basic methods for analysis of population growth rates3.1 Life-cycle graph A life-cycle diagram was prepared to illustrate all types of transitions observed between the four developmental stages in each of the intervals between consecutive census dates (Fig. 2). Straight arrows indicate all observed transitions from one developmental stage (i) to any other stage (j),


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and the production of new seedlings (F41) via sexual reproduction between consecutive census dates. Values labelled Gij represent the probability that individuals advance or regress from one developmental stage to another between consecutive census dates (i.e. from time t to time t + 1). In the case of L. eltoroensis, adult plants can move in either direction between the non-reproductive and reproductive stages. Values labelled Pij and accompanied by circular arrows represent the probability that individuals in a given developmental stage will remain in that stage until the next census date. Developmental stages are given the subscripts 1 (seedlings), 2 (juveniles), 3 (non-reproductive adults) and 4 (reproductive adults). A seed stage is not used in the life-cycle diagram because seed production and the progression from seeds to seedlings are exceptionally diffi cult to quantify accurately in the Orchidaceae. All the transition probabilities and fecundities for L. eltoroensis are monthly estimates because the time intervals between consecutive census dates were one month apart. In many studies, however, the interval between censuses is one year, so that transition probabilities are frequently calculated as a per annum rate.

Figure 2. Schematic diagram of the life-cycle of Lepanthes eltoroensis and probabilities of transitions between life-history stages between successive census dates. G = probability of passing from one stage to another, P = probability of surviving the time period and remaining in the same stage. Arrows labelled with Gij‘s indicate movement of an individual from one stage to another. Pij‘s indicates individuals remaining in the same stage at the next census date. The arrow labelled Fij indicates the production of new individuals by sexual reproduction by a specifi c stage. A0 is a non-reproducing adult stage, representing individuals that have reproduced at some point in their life but which are presently not reproducing. Reproducing adults (A+) have one or more shoots (leaves) with one or more active infl orescences. (Symbols follow Caswell, 2001, chapter 4).

3.2. Basic calculations of demographic parametersThe values calculated for all possible transition probabilities between stages from time t to time t + 1 can be arranged into a projection matrix (A). An example, based on population 2 of L. eltoroensis, is shown in Table 2a. To illustrate, 190 seedlings were recorded at all sampling dates. Throughout the recording period, a total of 180 of these remained as seedlings from one census date to the next (P11), whereas four advanced into the juvenile stage (G12). The probabilities of these two types of transition were therefore 180/190 (P11 = 0.9474) and 4/190 (G12 = 0.0211) respectively (Table 2b).

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The difference between 1 and the sum of these two values is the probability of a seedling dying in a sampled period (1.0000 – 0.9474 – 0.0211 = 0.0315). (Conventionally, probabilities of senescence are not usually included in life-cycle diagrams.) The remaining probabilities are calculated in a similar way (Table 2b). Values in the main diagonal leading from the top left to the bottom right of the matrix represent the probabilities that individuals will survive and remain in the same stage from one census date to the next. The values along the top edge of the matrix except that at the extreme left represent the rate of reproduction for individuals in each stage (in the case of L. eltoroensis only reproducing adults produce offspring). Values below the main diagonal are the probabilities of individuals in particular stage classes advancing to later developmental stages, whereas values above the main diagonal are probabilities that individuals in particular developmental stages will regress to earlier stages. The transition matrices for each of the fi ve populations of Lepanthes eltoroensis, and the transition matrix calculated using the data from all populations, are shown in Table 3.

Table 2 A and B. Number of individuals observed in each of the life stages of Lepanthes eltoroensis, population 2, summed across all recording dates. S = seedlings, J = juveniles, A0 = non-reproducing adults, A+ = reproducing adults. The fi rst row of data indicates the developmental stage occupied by any plant at time t, and the fi rst column indicates the stage occupied at time t + 1. For example, a total of 180 out of 190 plants t + 1. For example, a total of 180 out of 190 plants trecorded in the seedling stage at time t remained in this stage at time t remained in this stage at time t t + 1. The penultimate row of data shows t + 1. The penultimate row of data shows tthat a total of six plants that were seedlings at time t, had senesced by time t + 1. The bottom line shows that a t + 1. The bottom line shows that a ttotal of 190 seedlings were recorded at time t. The value F41 represents an estimate of fecundity. Calculation of this value is described in section [7.4.5.] (2B) The transition probabilities corresponding to the data in (2A). The probability of a seedling recorded at time t remaining in the seedling stage at time t remaining in the seedling stage at time t t + 1 (i.e. one month t + 1 (i.e. one month tlater) is 180/190 = 0.9474. Other probabilities are calculated in similar ways.

Table 2A.Time t S (t + 1)t + 1)t J (t + 1)t + 1)t A0 (t + 1)t + 1)t A+ (t + 1)t + 1)tS 180 F41

J 4 197A0 354 142A+ 11 181 633Dead 6 3 1Total 190 208 538 776

Table 2B.Time t S (t + 1)t + 1)t J (t + 1)t + 1)t A0 (t + 1)t + 1)t A+ (t + 1)t + 1)tS 0.9474 F41

J 0.0211 0.9471A0 0.6580 0.1830A+ 0.0529 0.3364 0.8157Dead 0.0315 0.0056 0.0013


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Table 3. Stage-based monthly transition matrices and fecundity for fi ve populations of L. eltoroensis, and a stage-based monthly transition matrix based on data from all fi ve populations. “?” indicates that transitions were not observed during recording, or that recruitment did not occur in the population. In further analyses, mean values obtained for the whole data set were used as appropriate entries for unknown values in the matrices.

Population 10.94318 0 0 F41F41F0.01136 0.93396 0 00 0 0.72465 0.308390 0.04717 0.26392 0.68481Population 20.94737 0 0 F41F41F0.02105 0.94712 0 00 0 0.65800 0.183000 0.05288 0.33643 0.81572Population 30.96805 0 0 F41F41F0.00319 0.92241 0 00 0 0.79592 0.387200 0.07758 0.18707 0.60061Population 40.72727 0 0 ?0.27273 0.98788 0 00 0 0.74775 0.155170 0.00606 0.24324 0.84483Population 50.93243 0 0 F41F41F? 0.95833 0 00 0 0.93103 0.153850 0.02083 0.05172 0.76926All populations0.95278 0 0 F41F41F0.01473 0.93301 0 00 0 0.73724 0.268120 0.05901 0.25058 0.72694

3.3. Calculating expected population sizeIn addition to the projection matrix A, a population can be described by a vector that summarises the distribution at time t of its constituent individuals between the (four) stages into which it has been t of its constituent individuals between the (four) stages into which it has been tclassifi ed. The vector is written in the following form:

n1

Nt =n2

n3

n4

where n1 - n4 refer to the numbers of individuals in each of the four developmental stages (seedlings, juveniles, non-reproducing adults, reproducing adults) at time t, and the sum∑ (n1 - n4) is equal to the population size. Multiplication of the transition matrix A by the column vector n produced at time t generates the expected numbers of individuals in the population after the next time step has t generates the expected numbers of individuals in the population after the next time step has t

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been reached (i.e. at time t + 1). In the case of the probability matrix presented in Table 2b, the t + 1). In the case of the probability matrix presented in Table 2b, the tcalculation would follow the example:

Nt+1 = Nt * A =

P11 0 0 F41F41F n1G12 P22 0 0 n20 0 P33 G43 n3

0 G24 G34 P44 n4

Nt+1

= Nt * A =

t * A =

t

0.9474 0 0 F41F

41F n

10.0211 0.9471 0 0 n

20 0 0.6580 0.1830 n

3

0 0.0529 0.3364 0.8157 n4

Nt+1

=

0.9474 * n1 + 0 * n

2 + 0 * n

3 + F

41F

41F * n

40.0211 * n

1 + 0.9471 * n

2 + 0 * n

3 + 0 * n

40 * n

1 + 0 * n

2 + 0.6580 * n

3 + 0.1830 * n

4

0 * n1 + 0.0529 * n

2 + 0.3364 * n

3 + 0.8157 * n

4

This calculation generates the population vector that summarises the number of individuals in each of the developmental stages at time t + 1. Similar calculations can be made to predict t + 1. Similar calculations can be made to predict tcorresponding values at further time intervals. Again, summation of the values n1 - n4 gives the projected total size of the population, assuming that the probabilities of transitions between stages and the reproductive outputs do not change either over time or in response to changing population size (i.e. the values are not density-dependent). The validity of these assumptions is discussed below.

3.4. Stable stage distribution and population growth rate, λThe proportions of the population in each developmental stage initially fl uctuate somewhat from one time step to the next when the multiplication process described above is carried out. However, repetition of this process over several time steps results in stabilization of these proportions - i.e. a stable stage distribution is generated, and the structure of the population, as judged by representation of its different developmental stages, has reached equilibrium (Ebert, 1999, Caswell, 2001). Moreover, when this is achieved, the population is changing in size at a constant rate, and therefore either increasing or decreasing exponentially. Thus, when the transition matrix and the population vector are multiplied together, the outcome refl ects the rate at which the population is increasing or decreasing. This rate is denoted by λ (lambda). In mathematical terms, Ant = λnt. λ is referred to as the fi nite rate of increase of the population. When the value of λ is 1.0, population size remains the same from one time step to the next. Values above or below unity indicate that the population size is increasing or decreasing respectively. Clearly, accurate determination of λ should be of prime concern to those involved in species conservation, particularly of rare and potentially threatened species. Caswell (1989, 2001), Ebert (1999), Akçakaya, Burgman and Ginzburg (1999) all provide accessible introductions to stage-based transition matrix models for estimating population growth rates.

At its stable stage distribution, population 2 of L. eltoroensis will contain 19.8% seedlings, 7.6% juveniles, 25.5% non-reproductive adults and 47.4% reproductive adults. The expected stable stage distributions for the fi ve populations differ substantially (Table 4). In some populations, the


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*

*

expected number of adults is larger than juveniles and seedlings. The differences in expected stable stage distributions refl ect the vital rates of each population and the local environmental conditions. These expected frequencies can be compared with observed frequencies to determine whether populations have attained stable stage distributions (Taylor, 1979). The variation in the stable stage distributions between populations suggests that local differences in vital rates and in effects of environmental conditions are important in determining population performance.

Table 4. The expected stable stage distributions in fi ve populations of Lepanthes eltoroensis and the stable stage distribution based on data from all populations. All calculations use F41 = mean number of recruits per populations or the mean value when absent.

PopulationLife Stage

1 2 3 4 5 All

Seedling 0.111 0.198 0.606 0.0037 0.582 0.293Juvenile 0.022 0.076 0.027 0.564 0.240 0.067Non-reproducing adults 0.463 0.252 0.244 0.150 0.127 0.325Reproducing adultsReproducing adults 0.401 0.474 0.123 0.249 0.051 0.315

3.5. Fecundity analysisAccurate estimations of fi tness are diffi cult to make in the Orchidaceae because of the small size of seeds and the large number that can be produced. There are also several diffi culties in determining the number of seedlings produced during any period by the adult plants in a population. Firstly, although most dispersed seeds are deposited close to the parent plant (Ackerman, Sabat and Zimmerman, 1996), some may travel distances at least as great as that between populations. This makes it diffi cult to be certain of the population of origin of newly recruited seedlings. Secondly, there is little information about how long orchid seeds survive in the soil or on phorophytes before germination (Batty, Dixon and Sivasithamparam, 2000). Thus, it is not known whether establishing seedlings come from seed produced during a specifi c window of time. Thirdly, because seedlings of many terrestrial orchid species may remain in a subterranean state for several years following their germination (Ziegenspeck, 1936; Summerhayes, 1951; Wells, 1981; Willems, 1982), they may not be observed until long after recruitment has occurred. Thus, surrogate or indirect measures are often the only practical ways of analysing fi tness in orchids. The number of recruits in a population is often presumed to be correlated with the number of fruits produced by plants in the population. Much effort is needed to increase knowledge of this important step in the life history of orchids.

In the case of L. eltoroensis, sexual reproduction can occur throughout the year rather than being confi ned to a particular season. The population of L. eltoroensis shown in Table 2a produced only three fruits throughout the survey period, during which a total of 776 observations were made on reproducing adult plants. It is not known how many seeds were produced on average per fruit, but the reliability of a mean value would probably be low given the small sample size. However, fruit production per reproducing individual observed can be used as a measure of fi tness. The probability of any of the 776 recorded fl owering orchids bearing a fruit is (3/776) = 0.0039. This value could be used as the entry F41 in Table 2a. Values for the other four populations of L. eltoroensis have been calculated in a similar way (Table 5).

Alternatively, and perhaps with greater validity, fecundity could be calculated from the number of new recruits to the population. A total of 18 newly establishing seedlings were observed in population 2 during the survey period. Consequently, the probability of a reproducing adult producing a recruit is 18/776 = 0.0232 (Table 5). This calculation is an “anonymous” method of

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analysing reproduction, because it does not ascribe seedling production to specifi c parent plants (Caswell, 2001).

Fruit production only occurred in two populations of L. eltoroensis. Recruits (seedlings) were observed in two populations in which no fruit set was observed. Recruitment in populations with no apparent fruiting may have occurred because fruits were produced before the survey started but their seedlings emerged during the monitoring period, or because seeds dispersed from other populations gave rise to the seedlings. Genetic analysis using DNA-fi ngerprinting techniques might distinguish between these possibilities (Hamrick, Godt and Sherman-Broyles, 1995; Hamrick, 1987). Paternity analysis would help clarify the importance of migration between orchid populations (Stacey, Johnson and Taper, 1997; Qamaruz-Zaman, et al., 1998).

Table 5. Mean and standard deviation of fruit set per reproducing adult, number of recruits, mean number of recruits produced per reproducing adult, and number of reproducing adults recorded over the whole survey period for fi ve populations of Lepanthes eltoroensis.

Population Number of adults recorded

Number of fruits produced

Mean fruit set per recorded adult

S.D. of fruit set

Number of recruits

Mean number of recruits per adult recorded

1 441 0 - - 6 0.01362 776 3 0.0039 0.062 18 0.02323 328 5 0.0152 0.116 39 0.11894 174 0 - - 0 -5 13 0 - - 9 0.6923All 1732 8 0.00462 72 0.0416

3.6. Reproductive value of the life stagesIn some species it is possible to distinguish the contribution of plants in different developmental stages, or of plants of different ages to recruitment in future generations. In such cases, a reproductive value can be calculated for each stage or age class. This is a measure of the relative contribution of each class to future generations. Its value depends on the probability of plants surviving to the beginning of the stage or age class in question, and the average fecundity of plants within that stage or age class (Table 6). If the goal of a management program is to increase the size of a population, close consideration should be given to ways of increasing survival up to and within the developmental stage or age class that makes the largest contribution to the reproductive value of the population (Ebert, 1999; Caswell, 2001).

Table 6. The mean reproductive value of the life-stages of L. eltoroensis. For method of calculation, see Caswell (2001).

Life Stage Pop. 1 Pop. 2 Pop. 3 Pop. 4 Pop. 5 All populationsSeedling 1.000 1.000 1.000 1.000 1.000 1.000Juvenile 4.285 2.632 6.977 1.018 4.175 3.047Non-reproducing adults 5.197 2.699 5.927 2.668 5.870 3.206Reproducing adults 5.263 2.766 6.182 2.832 7.109 3.330

3.7. Calculating confi dence intervals on λMost values of λ are calculated from transition probabilities based on samples of study populations, rather than on complete censuses. Because of this, there is uncertainty associated with all calculated transitional values, and consequently the value of λ is also subject to uncertainty. This uncertainty should be expressed as a 95% confi dence interval (CI). The CI is both added to, and subtracted


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from the value of λ to give two values that defi ne the range within which the investigator can be 95% certain that the true value of λ lies. Normally, we would not accept the conclusion that a population is stable in size unless these two values fall on different sides of 1.0, or the conclusion that a population was declining (or increasing) in size unless both values were below (or above) 1.0. Caswell (2001) stated that very few studies qualify estimates of λ with CI or with any other values indicating uncertainty in its estimation. Reliable estimates of CI’s are diffi cult to make because sampling variances in life history parameters may be high, especially if sample sizes are small. Furthermore, there may be covariance between different life history parameters and the distribution of λ itself may not be normal. Caswell (2001) and Ebert (1999) describe appropriate methods for estimating CI’s. The easiest and most commonly used methods assume a normal distribution of the values affecting λ, and calculate CI from the standard error of λ (^λ±1.96∗SE(^λ)).

An alternative method for calculating CI involves bootstrapping. This involves repeated random sampling from the complete set of observations available on different plants. Suppose that data are available on m individuals. A “bootstrap sample” is obtained from this data set by selecting m individuals at random, allowing for all individuals to be re-sampled during this selection (i.e. so that the same individual might be drawn more than once or omitted from the bootstrap sample altogether). This sample is used to draw up a new transition matrix, from which the value of λ is calculated. This whole process is repeated many times (characteristically 5000 times or more), to generate a distribution of values of λ. The 95% CI for λ is then estimated from this new range of values (Caswell, 2001; Meyer et al., 1986). Clearly, the quantity of computation involved in such methods is very large, so that they require the aid of powerful computers. In the present examples we assume for simplicity that the variance in λ is normally distributed and do not elaborate further on the bootstrap analysis, except to calculate the 95% CI of spatial variation.

Assuming that λ can be estimated accurately over a particular time interval, its reliability for predicting how a population will change in size as time passes will depend on whether the demographic parameters on which it is based (i.e. the transition probabilities and fecundities used in the projection matrix) remain constant. In most cases these values are affected both by conditions external to the population, such as climate variables and interactions with other species, and by responses of the population to its own density (density-dependent responses). A further factor that may affect population growth rate prediction is variation in parameter estimates due to fi nite (small) population size. Section 3.6 describes methods for circumventing the assumptions of the stage-based projection matrix model that transition probabilities, survivorship and reproductive output remain constant (Caswell, 1989, 2001; Cochran and Ellner, 1992).

4. Identifying critical life history stages4.1. Perturbation analysesPerturbation analyses are tools that enable ecologists to determine whether changes in demographic parameters affect the value of λ, and by how much. The practical value of these methods is in allowing the conservation biologist to identify the stages in a species’ life-history that have the greatest effect on population growth (Caswell, 2001; Schemske et al., 1984). There are three main approaches. The fi rst involves exploratory matrix projections, to examine the effect on population growth rate of changing transition or fecundity values. Fiedler (1987) used this approach to show that increasing seed production had little effect on λ in rare species of Calochortus, but a great effect on a common species in the same genus. Applied to our species, increasing the mean probability

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of reproducing adults of Lepanthes eltoroensis giving rise to new recruits from the observed value of 0.02 to 0.08 changed the projected value of λ for population 2 from just below one to just above one (Figure 3). Consequently, if we could quadruple recruitment in this species, population size might be maintained over time. However, to produce a population increase of 1% would require an increase in this transitional value to over 0.20.

The second approach, sensitivity analysis, measures the effect of small changes in demographic parameters on λ. This method allows comparison of the effects of changes in the values of different demographic parameters on λ (Meyer, Ingersoll and McDonald, 1987). The third approach is elasticity analysis. This is an extension of sensitivity analysis, and provides a measure of the proportional sensitivity of λ to specifi ed levels of change in different demographic parameters. This method re-scales values in the transition matrix in such a way that the effects of changes in both transition probabilities and fecundities can be directly compared.

Figure 3. Relationship between the probability of reproducing adults of Lepanthes eltoroensis in population 2 giving rise to new recruits and its effect on population growth rate, λ.

The values of all of the elasticities associated with the elements of a transition matrix sum to unity. Thus, the elasticity value associated with each matrix element indicates the proportional contribution of that element to λ. Elasticities can therefore be used to identify the life-cycle stages (i.e. the transitions and fecundities) that have the greatest impact upon population growth rate, by examining the effect of a fi xed proportional change in each of the matrix elements on population growth rate. Direct comparisons can be made of the effects of changing the values of transition probabilities and fecundities, which by defi nition can only lie between 0 and 1 (Caswell, 1984; de Kroon et al., 1986).

Enlightened conservationists would normally aim management at infl uencing the vital rates that have the greatest effect upon λ. The elasticities calculated for population 2 (Table 7) suggest that management that increases the likelihood of reproducing plants continuing to reproduce will have the greatest effect on population growth rate (see elasticity value in the bottom right corner of the elasticity matrix). In contrast, the growth rates of populations 1 and 3 would be most sensitive


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to changes in the probability of non-reproductive adults remaining in this state, and changing the probability of juveniles remaining as juveniles would have the greatest effect on the growth rate of population 5. Note that the effect on population growth rate can be negative or positive depending on how the parameter is changed.

Table 7. Elasticity and sensitivity matrices for fi ve populations of L. eltoroensis, and elasticity and sensitivity matrices based on all data for all fi ve populations.

Population 1Elasticity matrix Sensitivity matrix0.0226 0 0 0.0012 0.0238 0 0 0.08480.0012 0.0199 0 0 0.1019 0.0211 0 00 0 0.3716 0.1373 0 0 0.5086 0.44090 0.0012 0.1370 0.3086 0 0.0259 0.5150 0.4465Population 2Elasticity matrix Sensitivity matrix0.0759 0 0 0.0045 0.0804 0 0 0.19450.0045 0.0761 0 0 0.2141 0.0806 0 00 0 0.1890 0.0992 0 0 0.2882 0.53650 0.0045 0.0992 0.4472 0 0.0854 0.2958 0.5508Population 3Elasticity matrix Sensitivity matrix0.1972 0 0 0.0049 0.2019 0 0 0.04080.0045 0.0595 0 0 1.4091 0.0637 0 00 0 0.3869 0.0946 0 0 0.4817 0.24220 0.0044 0.0949 0.1530 0 0.0566 0.5025 0.2525Population 4Elasticity matrix Sensitivity matrix0.0157 0 0 0.0060 0.0217 0 0 0.14510.0060 0.3283 0 0 0.0221 0.2341 0 00 0 0.1735 0.0598 0 0 0.2333 0.38710 0.0056 0.0599 0.3451 0 0.9298 0.2476 0.4108Population 5Elasticity matrix Sensitivity matrix0.2028 0 0 0.0133 0.2162 0 0 001910.0133 0.3590 0 0 0.0221 0.3724 0 00 0 0.2586 0.0174 0 0 0.2760 0.11200 0.0133 0.0174 0.1050 0 0.6338 0.2442 0.1357All populationsElasticity matrix Sensitivity matrix0.1082 0 0 0.0051 0.1133 0 0 0.12190.0051 0.0733 0 0 0.3452 0.0784 0 00 0 0.2974 0.1050 0 0 0.4024 0.39080 0.0051 0.1050 0.2958 0 0.0857 0.4180 0.4059

It should be pointed out that the different transitions between developmental stages and fecundity are not all equally easy to manipulate by changes in management. Thus, the lessons learnt from the exploratory analyses referred to above cannot always be translated into effective practical action. Despite this, many recent studies have demonstrated the value of elasticity analysis and other perturbation analyses as tools for directing conservationists towards appropriate management

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regimes (de Kroon, van Groenendael and Ehrlen, 2000; Caswell, 2000; Wisdom, Mills and Doak, 2000; Saether and Bakke, 2000; Heppell, Caswell and Crowder, 2000).

5. Sources of error in predicting population growthError and uncertainty in predicting population growth rates from demographic data arises from many sources. Predictions about rare and threatened species are particularly likely to be subject to error, not least because their populations are often small, and therefore estimates of demographic parameters are based on small samples, leading to low reliability. This section examines sources of variation that may contribute to error and uncertainty in obtaining reliable estimations of demographic parameters. The sources of error and uncertainty include individual variation in performance between plants and demographic stochasticity, environmental variation, including temporal and spatial variation, and catastrophic events.

5.1. Individual variation and demographic stochasticityThe source of this variation is natural differences between individual plants in populations. The causes may be genetic or developmental, or due to an interaction between genotype and the environment (phenotypic plasticity), resulting in different plants displaying phenotypic differences that may infl uence their survival and reproductive success.

Demographic stochasticity occurs because real populations, and the samples from which data are collected in demographic studies, do not contain an infi nite number of individuals. Estimates of population parameters are less reliable when based on small numbers of individuals because plants differ in survivorship, fecundity and transition probabilities. Moreover, reliable estimation of the probability of occurrence of rarer types of transition between developmental stages requires collection of more data.

One method of evaluating variation in demographic vital rates involves calculation of demographic variance, V(λ) (Ebert, 1999; Caswell, 2001). The method requires calculation of three components, namely the vital rates of the transition matrix, the sample sizes used for the calculation of each transitional value, and the sensitivities associated with each transition. The survivorship and transition probabilities have a binomial sampling distribution. Consequently we can calculate

the variance associated with each vital rate as , where x = the value of each of the transition and survivorship probabilities and Nx is the sample size on which calculation of each of x is the sample size on which calculation of each of xthe vital rates has been based. The contribution (aij) of each vital rate to the demographic variance V(λ) is calculated as its sensitivity (S) squared, multiplied by its associated variance (Vx). The total demographic variance for the population is then calculated by summing all values of aij (Table 8). V(λ) is then used to calculate the 95% confi dence interval. The square root of V(λ) is the standard error of λ. Thus the 95% CI is equal to λ± 1.96 * SE(λ).

The value of the intrinsic growth rate for population 2 of L. eltoroensis was calculated as 1.0027. Calculation of demographic variance for this population (Table 8) gave a 95 % CI of 0.9470– 1.0584. This leads to the more cautious conclusion that population growth rate, taking into account demographic variation, is 95% certain to lie within this range.

5.2. Environmental variation5.2.1. Spatial variationDifferences in environmental conditions experienced by populations of the same species located in different places can affect demographic parameters. Table 9 includes the 95% CI values to illustrate


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the effects of spatial variation in environmental conditions on predictions of λ. The transition matrix from which λ is calculated, is based on the fi ve monitored populations of L. eltoroensis, each of which is supported by a different phorophyte. Causes of differences in these separate transition matrices could include the identity of the phorophyte species, differences in light intensity, humidity, altitude, moss cover, fungal associates, and orchid and pollinator densities. The effects on L. eltoroensis demography of each of these environmental variables could be investigated separately by appropriate experiments. However, the important question in the current context is the extent of the variation in demographic properties between these spatially separated populations.

Table 8. Calculation of the contribution of different vital rates (aij) to demographic variance V(λ), for population 2 of Lepanthes eltoroensis. The intrinsic population growth rate (calculated as described in section 7.4.3) is 1.0027 ± (1.96 * 0.02844), resulting in a 95% CI of 0.9470 – 1.0584.Transition Value of

transition (x)

Nx Variance Sensitivity = S Contribution of the demographic variance associated with each vital rate (aij) to the population demographic variance, V(λ).

P11 0.94737 180 0.00027 0.0804 0.0000017G12 0.02105 4 0.00412 0.2141 0.0002361P22 0.94712 197 0.00029 0.0806 0.0000017G24 0.05288 11 0.00238 0.0854 0.0000332P33 0.65800 354 0.00071 0.2882 0.0000528G34 0.33643 181 0.00136 0.2958 0.0001079G43 0.18300 142 0.00106 0.5365 0.0003031P44 0.81572 633 0.00029 0.5508 0.0000720

Total V(λ)=0.0008086SE = 0.02844

Heyde and Cohen (1985) present a method for estimating the confi dence limits on λ due to variation in environmental conditions. It uses the variation in the transition matrices calculated for each of a set of spatially separated populations and accesses these values at random to calculate a projected population growth rate. The method does not assume normal distribution of variances or of population growth rate. A simple program for calculating confi dence limits on λ due to spatial variation is available at http://www.sci.sdsu.edu/Cornered_Rat/ (Ebert, 1999).

5.2.2. Temporal variationTemporal variation in environmental conditions, which is often unpredictable in direction and magnitude, can also affect individual survivorship and reproductive success. Although it is often impossible to assign the cause of variation in population behaviour to particular environmental variables, we can estimate the extent of this variation and predicts possible effect on population growth rate. The method referred to above (5.2.1) for analysing the effect of spatial variation in environmental conditions on population behaviour can also be used to determine the effect of temporal variation. However, there is another method that requires less computation. This approach assumes that variance in the matrix elements and in λ is normally distributed (although again there is no a priori reason to expect this to be true). It involves construction of a transition

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matrix for every time period under consideration (Nt to Nt+1), and calculation of the variance for each element, as illustrated in the calculations for the effect of demographic stochasticity (Section 7.6.1). For example, the probability of a non-reproducing adult plant making the transition to the reproductive state is shown for each monthly interval in Figure 4.

Table 9. Estimates of λ, and 95% confi dence intervals, based on differing sources of variation (demographic, spatial and temporal), calculated by combining data collected from all fi ve populations of Lepanthes eltoroensis. The CI values suggest that for this example, demographic variation was smaller in magnitude than spatial variation, and that temporal variation was greater than both of these. A major reason for the large scale of temporal variance in these small populations was that some transitions were not observed in some months of the study, but were common in other months. λ = 0.9975.

Source of variation 95% CI for λ Range of 95% CI for Range of 95% CI for λDemographic variation 0.95.91 – 1.0359 0.0767Spatial variation 0.9725 – 1.1322 0.1597Temporal variation 0.7821 – 1.2128 0.4307

There is a large variation in the probability of plants achieving the transition from non-reproducing to reproducing between consecutive months (range 8% - 67%), indicating that some periods are much less benefi cial for fl ower production than others. The variance in the transition from non-reproducing to reproducing was 0.0238. Variances are calculated in the same way for all other transitional values in the matrix, and summed to provide the total population variance V (λ). This value is then used to calculate the 95% confi dence interval (=λ ± 1.96 * S. E. (λ)).

The effect upon L. eltoroensis of temporal variation in the values of different transitions differs in magnitude. Moreover, the impact of temporal variation changes with different populations. For example, the standard deviation of the transition from seedling to juvenile was small in populations 1 - 3 (0.0111 – 0.0370), larger in population 4 (0.2085) and zero in population 5. These between-population differences may refl ect both differences in ecological conditions between sites and the small size of each population. In general, as also shown by the summary 95% CI values calculated from all the data for the fi ve populations (Tables 9,10), the vital rates of L. eltoroensis life stages were more strongly affected by temporal variation than by demographic or spatial variation. A large proportion of this variation was associated with the transitions between the reproducing and non-reproducing states. The conditions controlling this transition are not known.

Figure 4. Temporal variation in the probability of a non-reproductive adult (A0) making the transition to the reproductive stage (A+) from one recording date (Nt) to the next (Nt+1).


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5.3. Catastrophic eventsSince many orchid populations are small and fragmented, environmental perturbations can have a catastrophic impact on their size, structure and persistence. For the interested reader we recommend publications by Caswell (2001), Grant and Benton (2000), Akçakaya, Burgman and Ginzburg (1999), and Foley (1997) that address this topic in some detail.

Table 10. Estimates of demographic and temporal intrinsic population growth rate for fi ve populations of Lepanthes eltoroensis. Two values of reproductive success (F41) are applied, fruit production per reproducing individual and number of recruits per reproducing individual (see text for details and calculations of F41). See Table 3 for transition values. If no values were observed in a population, the mean for the species is used. Mean fecundity from fruit set = 0.0046, and mean fecundity from recruits = 0.0416. Demographic (DS) and temporal stochasticity (TS) ranges are calculated from the mean fecundity from fruit set. Demographic stochasticity was measured from the series approximation. Temporal stochasticity is measured as the variance in the matrix elements in time (see section 7.6.2.2).

PopNo.

λ (F41from fruits)

λ (F41from recruits)

S.E. 95% CI of DS 95% CI of TS Range of CI for DS

Range of CI for TS

1 0.9912 0.9919 0.0335 0.9262 - 1.0576 0.6328 - 1.3510 0.1313 0.71812 0.9984 1.0027 0.0284 0.9470 - 1.0584 0.7183 - 1.2871 0.1115 0.56883 0.9857 0.9909 0.0836 0.8271 - 1.1574 0.7869 - 1.1949 0.3276 0.40804 0.9982 1.0054 0.0818 0.8450 - 1.1658 0.6857 - 1.3251 0.3208 0.63935 0.9710 0.9937 0.1066 0.7849 - 1.2026 0.5957 - 1.3917 0.4178 0.7961

6. ConclusionsConservation action for endangered species should involve three stages (Schemske et al., 1994). The fi rst is determination of the biological status of the species. In this stage, demographic information is accumulated to determine whether the number of individuals is increasing, decreasing or stable. The second stage involves determination of the life history stages that have the greatest effects on population growth rate and persistence. Thirdly, the biological and environmental causes of variation in the life-history stages that most affect population growth rate should be determined. These steps need to be allied to a conservation management program that includes ecological considerations but which may also involve economic and political decisions. Regardless of non-ecological constraints, there is no alternative methodology that will reliably ensure species conservation.

Other methods may help in prediction of population persistence. These include extinction probability models and population viability analysis (Possingham, Lindenmayer and Norton, 1993; Schemske and Shaffer, 1987). Another important but as yet underdeveloped area, which may hold some promise in orchid conservation is metapopulation analysis. As noted by Tremblay (1997) and Ackerman (1998), many orchids are distributed in small, hyperdispersed populations. It may be that individual populations are always bound to undergo extinction over a relatively short period of time but that a continual process of migration and colonization of empty sites permits metapopulations to persist for long periods. The question then becomes: how many populations, and of what size, linked within such a metapopulation, and how many sites, occupied and potentially occupiable, need to be conserved to ensure regional persistence of a species? (Hanski, 1999) Knowledge of the genetic structure of species with small population size may also infl uence conservation strategies (Ackerman, 1998; Barrett and Kohn, 1991).

While the mathematics behind the methods described above may appear daunting, calculation

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of λ and its confi dence intervals can now be done by using widely accessible and user-friendly computer programs. There are several good source texts describing the theory behind the methods and showing worked examples to orient the conservation biologist (Caswell, 2001, Schemske et al., 1994). Several high-profi le studies have used the methods described above; for example in evaluating the probability of extinction of the southern hemisphere shrubs Banksia goodii and Protea neriifolia (Higgins, Pickett and Bond, 2000); for determining whether populations of the orchid Lepanthes caritensis (Tremblay 1998, 2000b), the spotted owl Strix occidentalis (Noon and McKelvey, 1996; Forsman et al., 1996) and the sandpiper Calidris pusilla (Hitchcock and Gratto-Trevor, 1996) are declining; for determining whether incidental mortality due to fi shery by-catch constitutes a risk for populations of the harbour porpoise, Phocoena phocoena (Caswell et al., 1998); and for evaluating different management strategies for conservation of the loggerhead sea turtle Caretta caretta (Crouse, Crowder and Caswell, 1987; Crowder et al., 1994). The benefi ts of analysing population behaviour by matrix-based techniques have been well illustrated in these and many other studies. We recommend such methods as the most dependable basis for making decisions about appropriate management for orchid conservation.

AcknowledgmentsWe thank Edwin Camacho, Ingrid Negrón, María Díaz, Iris Rodríguez, Iris Neireida Rodríguez, Ramón Cao, Lee-Marie Tormos, Ismael González, Ernesto Ortiz, Luis Artache, Edwin Camacho, Lisa Silva Sanchez and Ledith Resto for assistance in the fi eld and laboratory. We also thank the Department of Natural Resources of Puerto Rico, and the U. S. Forest Service, and the US Fish and Wildlife service for permission to work at our study sites. This work was supported by institutional funds from UPR – Humacao. We are grateful for careful editing of draft versions of this paper by Shelagh Kell.

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