Matrix Model Approach Life Tables from the Czech Republic and Germany

The populations and their dynamics based on data from Germany and the Czech Republic were compared by analysing the outputs from matrix models. The matrix models are based on transition probabilities between categories that were defined in terms of size categories rather than age. Despite slight differences in the sampling procedure used in these countries, plants were assigned to one of four size categories: seedlings (in the Czech Republic)/small vegetative plants (in Germany); juveniles/medium-sized vegetative plants; adult non-flowering plants/large vegetative plants; flowering plants (in both countries). The matrix model with data obtained at the end of each growth season

Fig. 6.5. Annual early wood rings in the secondary root xylem of H. mantegazzianum make it possible to determine the age of both vegetative and flowering plants. Age of plants was estimated by herb-chronology (Dietz and Ullmann, 1997, 1998; von Arx and Dietz, 2006). The figure shows annual rings in a cross-section of the root of a 7-year-old individual. Lines indicate the transitions between late wood of the previous year and early wood of the following year.

Caucasus/unmanaged

Caucasus/managed

eg 4

CZ/un managed

CZ/managed

10 15 20 25 15 10

CZ/managed

10 15 20 25

Fig. 6.6. Age structure of H. mantegazzianum populations at managed (pastures) and unmanaged sites in the native (Caucasus) and invaded (CZ - Czech Republic) distribution ranges. White bars represent mean number of non-flowering plants (lines = sd) per sample plot. Black bars represent flowering plants. For details of the sampling and method used to determine age see Pergl et al. (2006).

uses the year-to-year transitions for the years 2002-2005 in the Czech Republic and 2002-2004 in Germany.

As the data are for a limited number of transitions and sites, and the matrix models do not incorporate the spatial component of spread, the prediction of a decrease on population development in the future needs to be interpreted with caution (for a discussion of the differences between outputs from matrix and individual based models based on the same data, see Nehrbass et al., 2006). The estimated population growth rates for individual sites, and the number of individuals used for each transition matrix within each plot, are summarized in Table 6.2. For seven out of the eight populations studied in the Czech Republic, it was possible to construct a pooled matrix across 2002-2005. For the majority of sites, however, the number of individuals within a plot was insufficient for constructing a robust transition matrix every year (Table 6.2). The values for the finite rate of population increase (X) at particular sites pooled across years varies from 0.550 to 1.099. Those for particular sites and years are within the range 0.684-1.286 (Pergl et al., unpublished). Pooled across years, values of X are 1.15 and 1.16 for open and dense stands in Germany (Huls, 2005). Populations in open stands have X values of 0.76 and 1.24 for transitions 2002-2003 and 2003-2004 and in dense stands values of 0.75 and 1.38, respectively (Table 6.3). These population growth rates indicate stable or slightly decreasing local populations, which is to be expected as these populations invaded these sites a long time ago. Large-scale invasion dynamics depend on regional-scale processes such as seed dispersal, including long-distance dispersal and successful establishment of new populations. The populations that reach and remain at the carrying capacity act as sources for further invasions. It is clear that once a population

Table 6.2. Summary of finite rates of population increase (X) based on matrix models for populations of H. mantegazzianum in the Czech Republic. For each locality and year, number of analysed plants (Plant no.) is shown as number of living/total number of individuals. Values marked with * are based on insufficient numbers of plants (missing data in diagonal or subdiagonal matrix elements); NA (not available). Numbers of localities correspond to those in the overview in Perglova et al., Chapter 4, this volume.

Table 6.2. Summary of finite rates of population increase (X) based on matrix models for populations of H. mantegazzianum in the Czech Republic. For each locality and year, number of analysed plants (Plant no.) is shown as number of living/total number of individuals. Values marked with * are based on insufficient numbers of plants (missing data in diagonal or subdiagonal matrix elements); NA (not available). Numbers of localities correspond to those in the overview in Perglova et al., Chapter 4, this volume.

Locality

Pooled

2002-

■2003

2003-

2004

2004-2005

X

Plant no.

X

Plant no.

X

Plant no.

X

Plant no.

3

0.749

59/93

0.93*

27/35

0.25*

14/34

1.052*

18/24

6

0.551

51/103

0.376

23/59

NA

12/24

1.233*

16/20

8

1.099

112/151

0.924

25/45

NA

35/44

NA

52/62

9

0.994

111/157

1.08

39/51

0.684

33/56

1.286

39/50

12

NA

24/39

NA

6/14

NA

5/8

NA

13/17

13

0.878

433/600

0.945

167/234

0.867

140/195

0.849

126/171

15

0.83

113/170

0.721

39/66

0.842

31/49

1.012

43/55

16

0.796

140/200

NA

57/80

NA

48/65

NA

35/55

Pooled

0.84

1043/1513

0.935

383/584

0.813

318/475

0.953

342/454

Table 6.3. Finite rate of population increase (X), bootstrap estimate of X (Xb) with lower and upper 95% confidence intervals in brackets (Dixon, 2001; Manly, 1993), expected numbers of replacements (R0), stage distribution (ssd, stable stage distribution; osd, stage distribution observed during the second year of the transition interval), and Keyfitz's A (distance between observed and stable stage distribution) of dense and open stands of H. mantegazzianum in 2002-2003 and 2003-2004. Abbreviations of life cycle stages: sv - small vegetative; mv - medium vegetative; lv - large vegetative; fl - flowering. Matrix analyses were based on paired permanent plots of 1 x 2.5 m established in dense and open stands at five study sites in Hesse, Germany (Hüls, 2005).

Table 6.3. Finite rate of population increase (X), bootstrap estimate of X (Xb) with lower and upper 95% confidence intervals in brackets (Dixon, 2001; Manly, 1993), expected numbers of replacements (R0), stage distribution (ssd, stable stage distribution; osd, stage distribution observed during the second year of the transition interval), and Keyfitz's A (distance between observed and stable stage distribution) of dense and open stands of H. mantegazzianum in 2002-2003 and 2003-2004. Abbreviations of life cycle stages: sv - small vegetative; mv - medium vegetative; lv - large vegetative; fl - flowering. Matrix analyses were based on paired permanent plots of 1 x 2.5 m established in dense and open stands at five study sites in Hesse, Germany (Hüls, 2005).

Stand

Stage distribution

Interval

type

X

Xb

R0

sv/mv/lv/fl

Keyfitz's A

2002-2003

Dense

0.75

0.74

0.24

ssd: 0.29/0.32/0.24/0.15

0.07

(0.56-0.87)

osd: 0.24/0.29/0.27/0.20

Open

0.76

0.73

0.45

ssd: 0.09/0.17/0.36/0.37

0.12

(0.39-0.99)

osd: 0.04/0.10/0.40/0.46

2003-2004

Dense

1.38

1.31

6.45

ssd: 0.38/0.34/0.24/0.04

0.25

(0.70-1.63)

osd: 0.64/0.23/0.10/0.02

Open

1.24

1.07

3.94

ssd: 0.23/0.21/0.48/0.07

0.46

(0.72-1.40)

osd: 0.70/0.13/0.15/0.03

of H. mantegazzianum reaches carrying capacity there is no potential for further growth. The variability in population growth rate at sites in Germany (Huls, 2005) seems to be closely related to annual climatic variation and sto-chasticity. Extremely hot and dry conditions in summer 2003 strongly reduced primary productivity across Europe (Ciais et al., 2005), inducing dramatic changes in the population structure of H. mantegazzianum due to increased mortality and low seedling establishment. This resulted in an increase in recruitment of new plants in gaps and high population growth rates between 2003 and 2004 (Table 6.3).

Matrix models make it possible to estimate a theoretical stable population stage structure, which can be compared with the observed stage of distribution. When the data are pooled across years, G-tests indicate no significant differences between the predicted and observed stage structures for each locality in the Czech Republic (Pergl et al., unpublished). However, when the test was performed using data for individual years, most differences were significant. This suggests that, although there is a significant year-to-year variation in stage structure, the populations show stable long-term dynamics. This is supported by the results from Germany. The perturbation caused by the extreme climatic conditions in 2003 resulted in large deviations between observed and expected stage structures in 2004 (measured as Keyfitz's delta; Table 6.3). Over this period, both open and dense stands showed similar stage structures and population dynamics. However, the differences between the observed and predicted stable stage distributions were smaller in 2002-2003, when populations experienced average weather. This was particularly true for dense stands, which were very close to equilibrium conditions, whereas open stands showed larger deviations from the expected stage structure.

The accuracy of matrix model projections can be verified by comparing expected and observed population age structure or age at reproduction if available (Cochran and Ellner, 1992). This was done by estimating the age at flowering (based on data from Czech Republic) using the program stagecoach (Cochran and Ellner, 1992) and the results of a study on the age structure of H. mantegazzianum (Pergl et al., 2006). Results for a site in the Czech Republic indicated an estimated age at flowering of 4.36 ± 1.41 years (mean ± sd), while the observed median age was 3 years, with the oldest plant being 4 years old (Pergl, J. et al., unpublished). This is corroborated by the results from the German study that indicate a generation time (age at flowering in monocarpic species), estimated according to Caswell (2001), of about 3 years for dense stands for the transition 2002-2003 (Hüls, 2005). The close match between the observed data and the results of independent matrix analyses for two regions indicate that, although there are some limitations in the use of simple, time-invariant, deterministic matrix models (Hoffmann and Poorter, 2002; Nehrbass et al., 2006), they accurately describe the essential properties of the H. mantegazzianum populations studied. Elasticity analysis of these matrix models (Caswell, 2001) was used to identify transitions in the life cycle that have large effects on population growth rate, which might be used for developing management or control measures. Elasticity matrices for pooled populations in the Czech Republic and two stand types in Germany are shown in Table 6.4. The elasticity matrices averaged across years are similar in both regions and the highest elasticities are related to growth (transition to higher stage classes, i.e. sub-diagonal) and stasis (remaining within the same stage class, i.e. matrix diagonal). However, when analysed separately for each year-to-year transition, dense stands exhibit a high elasticity for stasis and open stands high elasticity for growth (Hüls, 2005). These results suggest that despite the enormous seed production, survival is crucial; the role of seed

Table 6.4. Elasticity matrices of H. mantegazzianum for pooled data from the Czech Republic, and for open and dense stands in Germany averaged across years and sites. Abbreviations: seedl - seedlings; juv - juveniles; ros - rosette plants; flow - flowering plants; sv - small vegetative; mv - medium vegetative, lv - large vegetative. Although the definition of stage classes varied slightly between regions, seedlings and small vegetative plants, juveniles and medium vegetative plants, and rosettes and large vegetative plants are considered to be equivalent developmental stages.

Table 6.4. Elasticity matrices of H. mantegazzianum for pooled data from the Czech Republic, and for open and dense stands in Germany averaged across years and sites. Abbreviations: seedl - seedlings; juv - juveniles; ros - rosette plants; flow - flowering plants; sv - small vegetative; mv - medium vegetative, lv - large vegetative. Although the definition of stage classes varied slightly between regions, seedlings and small vegetative plants, juveniles and medium vegetative plants, and rosettes and large vegetative plants are considered to be equivalent developmental stages.

Czech Republic

Germany (open stands)

Germany (dense stands)

seedl

juv

ros

flow

sv mv lv

flow

sv mv lv flow

seedl 0.07

0.03

0

0.11

sv 0.01 0 0

0.17

0.05 0.01 0.00 0.13

juv 0.11

0.13

0.02

0.07

mv 0.11 0.03 0

0.05

0.11 0.10 0 0.06

ros 0.04

0.13

0.11

0

lv 0.06 0.15 0.16

0.02

0.03 0.16 0.15 0.01

flow 0

0.04

0.14

0

flow 0 0 0.23

0

0 0 0.19 0

Fig. 6.7. Position of H. mantegazzianum in a rescaled elasticity space based on vital rates of survival (S), growth (G) and fecundity (F) using species from Franco and Silvertown (2004). Note that values of S, G and F are not simply sums of elasticity matrix elements; H H. mantegazzianum (pooled across localities and years; data from the Czech Republic), ▲ iteroparous (polycarpic) forest herbs, ▲ iteroparous herbs from open habitats, • semelparous (monocarpic) herbs.

Fig. 6.7. Position of H. mantegazzianum in a rescaled elasticity space based on vital rates of survival (S), growth (G) and fecundity (F) using species from Franco and Silvertown (2004). Note that values of S, G and F are not simply sums of elasticity matrix elements; H H. mantegazzianum (pooled across localities and years; data from the Czech Republic), ▲ iteroparous (polycarpic) forest herbs, ▲ iteroparous herbs from open habitats, • semelparous (monocarpic) herbs.

production is diminished by the poor establishment and high mortality of seedlings. This accords with the fact that within established populations there is little recruitment, while in the open the role of colonization is high. Compared to the other species, H. mantegazzianum fits near the group of polycarpic perennials in the fecundity, survival and growth 'elasticity space' (Hüls, 2005). Although the strictly monocarpic behaviour of H. mantegazzianum is confirmed and its average age at reproduction is between 3 and 5 years, this species seems to be rather isolated from the other short-lived monocarpic species analysed by Franco and Silvertown (2004) (Fig. 6.7).

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