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Page 1: n(t + 1) = A n(t)

n(t + 1) = A n(t + 1) = A

n(t)n(t)

A basic introduction A basic introduction to the construction, analysis and interpretation to the construction, analysis and interpretation

of matrix projection models of matrix projection models

(for the evaluation of plant population viability and management)(for the evaluation of plant population viability and management)

Patrick EndelsLaboratory for Forest, Nature and Landscape Research, KULeuvenVital Decosterstraat 102B-3000 Leuven, BelgiumTel +32(0)16.32.97.69 Fax +32(0)16.32.97.60

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• what’s a matrix projection model?• what data is needed?• assumptions / restrictions• calculation and interpretation of the projection matrix and

its associated values• applications

– perturbation analysis• Retrospective• Prospective

– Grime’s CRS vs. Silvertown’s GLF: a demographic interpretation of functional plant groups

– Projections– Stochastic modelling

• suggested reading

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Method:

• The linear time-invariant matrix model: n(t + 1) = A n(t)

• Age vs. Stage classification

• Determination of the number of life stages: 2 approaches- numerical approach: algortihms for calculating the number of life stages (Moloney, Vandermeer), based on minimising the error due to within cell variance while minimising the error due to small samples- biological approach: relies on field observations of developmental states (Lefkovitch, Werner) => best solution if multiple populations need to be compared

• Calculating fecundities: - the actual seedling recruitment in year X divided by the number of reprodructive adults in year X-1- proportional distribution over the different reproductive stages according to fitness characteristics

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Assumptions:

1) Discrete model: taking population relevees at pre-determined time intervals results in a realistic impression of (actual) population dynamics?=> continuous processes (flowering, growth, etc.) are reduced to discrete events.

2) Transitions (better: transition prob.) between different stages are the same for all the individuals in one life stage

3) Time-invariance (important restriction in projection analysis)

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Case study: Primula veris & P. vulgaris

Stage classification = a combination of both reproductive and size criteria: • Seedlings: individuals developed directly after the germination of seeds,

with cotyledons still present and often also one ‘normal’ leaf-pair.• Juveniles: immature plants without cotyledons and with only one rosette of

leaves. Juveniles can only be distinguished from vegetative adults with one rosette by means of their size: an individual is considered

as an adult when its leaf size is comparable to flowering plants in the same population; if leaf size is significantly smaller then the individual is assigned to the juvenile category.

• Vegetative adults: non-flowering individuals without cotyledons, with one or more rosettes, often showing signs of overwintering leaves. Leaf size is comparable to generative adults which are

growing under similar conditions.• Reproductive adults: plants baring one ore more flowering stalks, having one

or more rosettes and often showing signs of older, overwintering leaves. These flowering adults were divided into three size categories according to the number of rosettes.

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P. veris P. vulgaris

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0 10 20 30 40 50 60 70 80 Centimeters

2000

1999

2001

Data?

Life stage 1999

# rosettes 1999

Life stage 2000

# rosettes 2000

Plant 1 fl. adult 5 n-fl. adult 10

Plant 2 seedling 1 juvenile 1

Plant 3 juvenile 1

Plant 4 n-fl. adult 1 fl. adult 2

Plant 5 seedling 1

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-> going from life cycle graphs to projection matrices

S J NFA RA1 RA2 RA3

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YEAR t

seedling juvenile NFAdult

ReprAdult 1

ReprAdult 2

ReprAdult 3

seedling 0 0 0 F1 F2 F3

YEAR t+1

juvenile G21 L22 0 F4 F5 F6

NFAdult

G31 G32 L33 L34 L35 L36

ReprAdult 1

G41 G42 G43 L44 L45 L46

ReprAdult 2

G51 G52 G53 G54 L55 L56

ReprAdult 3

0 0 G63 G64 G62 L66

The projection matrix, A

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Symbol Definition Demografic interpretation

A a square matrix containing the coefficients that represent proportions of – mostly year-to-year – transitions between life stages.

projection matrix

aijthe element in row i, collum j of the projection matrix A matrix element

the dominant eigenvalue of A population growth rate

A = (the right eigenvector of A associated with )

stable stagedistribution

A = (the left eigenvector of A associated with )

reproductive values

sij / aij

(the sensitivity of to changes in matrix element aij)sensitivity

eijaij / aij

(the proportional sensitivity of to proportional changes in matrix element aij)

elasticity

1 / 2 (ratio between the dominant & subdominant eigenvalue)

damping ratio

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Projection matrices and population growth rate

lambda 2.0827 lambda 1.4104cleared 99-00 cleared 00-01

seedl juv nfl ad r ad 1 r ad 2 r ad 3 seedl juv nfl ad r ad 1 r ad 2 r ad 3seedl 0.000 0.000 0.000 1.593 2.205 14.333 seedl 0.000 0.000 0.000 0.934 1.871 6.928juv 0.217 0.118 0.000 0.065 0.090 0.583 juv 0.343 0.250 0.000 0.000 0.000 0.000nfl ad 0.174 0.235 0.125 0.167 0.038 0.000 nfl ad 0.052 0.250 0.353 0.034 0.024 0.000r ad 1 0.391 0.471 0.175 0.278 0.000 0.000 r ad 1 0.070 0.313 0.176 0.276 0.000 0.000r ad 2 0.043 0.059 0.450 0.444 0.500 0.000 r ad 2 0.000 0.000 0.118 0.483 0.463 0.000r ad 3 0.000 0.000 0.175 0.056 0.423 1.000 r ad 3 0.000 0.000 0.059 0.034 0.512 1.000

lambda 0.8901 lambda 0.5935forested 99-00 forested 00-01

seedl juv nfl ad r ad 1 r ad 2 r ad 3 seedl juv nfl ad r ad 1 r ad 2 r ad 3seedl 0.000 0.000 0.000 1.259 2.051 14.387 seedl 0.000 0.000 0.000 0.604 1.744 9.967juv 0.166 0.192 0.000 0.066 0.108 0.757 juv 0.064 0.157 0.000 0.048 0.138 0.789nfl adults 0.009 0.231 0.400 0.500 0.364 0.000 nfl ad 0.023 0.137 0.523 0.182 0.250 0.167r ad 1 0.000 0.077 0.222 0.400 0.091 0.000 r ad 1 0.006 0.039 0.023 0.500 0.250 0.000r ad 2 0.000 0.000 0.044 0.100 0.500 0.143 r ad 2 0.000 0.000 0.000 0.000 0.438 0.500r ad 3 0.000 0.000 0.000 0.000 0.000 0.857 r ad 3 0.000 0.000 0.000 0.000 0.000 0.333

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Reproductive values

0.000.100.200.300.400.50

0.600.700.800.901.00

seedl juv nfl adults fl adults1

fl adults2

fl adults3

alt 00-01

vro 00-01

0.000.100.200.300.400.500.600.700.800.901.00

seedl juv nfl adults fl adults1

fl adults2

fl adults3

alt 99-00

vro 99-00

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Stable stage distribtution

0.000.100.200.300.400.500.600.700.800.901.00

seedl juv nfl adults fl adults1

fl adults2

fl adults3

alt 00-01

vro 00-01

0.00

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0.60

seedl juv nfl adults fl adults1

fl adults2

fl adults3

alt 99-00

vro 99-00

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Perturbation analysis: determining the relative importance of changes in vital rates to population growth rate

• Which of the stages (better: vital rates) is most important to population growth?

• Prospective (hypothetical changes, sensitivity and elasticity analysis) vs. retrospective (actual changes, LTRE) methods

• Prospective: Sensitivity (additive) vs. Elasticity (proportional)– E: takes the actual life cycle into account– S: appropriate technique for evolutionary questions

• retrospective: aims at quantifying the contribution of each of the vital rates to the variability of in different situations

• Applications:– Life history theory: relationship between changes in vital rates

and fitness– Conservation biology: relationship between changes in vital rates

and population growth rate– Ecotoxicology,…..

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YEAR t

seedling juvenile NFAdult

ReprAdult 1

ReprAdult 2

ReprAdult 3

seedling 0 0 0 F1 F2 F3

YEAR t+1

juvenile G21 L22 0 F4 F5 F6

NFAdult

G31 G32 L33 L34 L35 L36

ReprAdult 1

G41 G42 G43 L44 L45 L46

ReprAdult 2

G51 G52 G53 G54 L55 L56

ReprAdult 3

0 0 G63 G64 G62 L66

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0 10 20 30 40 50 60 70 80 Centimeters

2000

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Life table response experiments (LTRE)

vs.

Sensitivity / elasticity analysis

Perturbation analysis: determining the relative importance of changes in vital rates on population growth rate

YEAR t

seedling juvenile NFAdult

ReprAdult 1

ReprAdult 2

ReprAdult 3

seedling 0 0 0 F1 F2 F3

YEAR t+1

juvenile G21 L22 0 F4 F5 F6

NFAdult

G31 G32 L33 L34 L35 L36

ReprAdult 1

G41 G42 G43 L44 L45 L46

ReprAdult 2

G51 G52 G53 G54 L55 L56

ReprAdult 3

0 0 G63 G64 G62 L66

A1 => S1 , E1

A2 => S2 , E2

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S J

NFA

RA1

RA2

RA3 S

J NFA R

A1 RA

2 RA3

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

1999 2000

S J

NFA

RA1

RA

2

RA3 S

J NFA R

A1 RA

2 RA3

0.000

0.200

0.400

0.600

0.800

1.000

1.200

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2000 2001

S J

NFA

RA

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RA

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RA3 S

J NFA R

A1 RA2 R

A3

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1999 2000

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RA1

RA2

RA3 S

J NF

A RA1 R

A2 RA3

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1.000

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2000 2001

grassland restoration site

forested sitesensitivities

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Elasticities

• Sum to unity => comparison of species or individual populations of the same species

• Stages with highest mortality (‘bottlenecks’) are not neccessarely those with the highest elasticity (sensitivity) values

• G-L-F approach: demographic interpretation of Grime’s C-S-R system

• Differences in G-L-F can be interpreted as trade-offs among life history parameters (<-> Shea et al.)

• G-l-f depends on the number of stages

YEAR t

seedling juvenile NFAdult

ReprAdult 1

ReprAdult 2

ReprAdult 3

seedling 0 0 0 F1 F2 F3

YEAR t+1

juvenile G21 L22 0 F4 F5 F6

NFAdult

G31 G32 L33 L34 L35 L36

ReprAdult 1

G41 G42 G43 L44 L45 L46

ReprAdult 2

G51 G52 G53 G54 L55 L56

ReprAdult 3

0 0 G63 G64 G62 L66

E

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C ≈≈ S

R ≈

Grime’s CSR vs. Silvertown’s GLF:

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L

0.0

0.1

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1.0

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F0.00.10.20.30.40.50.60.70.80.91.0

CL 99-00

CL 00-01

FOR 00-01

FOR 99-00

REF99-00REF00-01

Fitting individual populations into the successional trajectory

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LTRE

• aims at quantifying the contribution of each of the vital rates to the variability of

• Demographic alternative for ANOVA

• Fixed & random designs=> Factorial designs:– Main effects: combination of differences in individual matrix entries and

sensitivity of that entry, calculated from a “mean” matrix– Interactions

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Projections & stochastic modelling

• projections vs. predictions

• Incorporating stochasticity- Mean matrix = deterministic behaviour- Stdev = stochastic component

• Several model runs with different settings, based on the actual projection matrix and a series of stdev (popproj, ramas,…) => PVA, extinction probability

• Applications: guidelines for management (disturbance-recovery cycles,…), extinction risk assessment…

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Conclusions

• Time-consuming & not possible for every (plant) species

• Very easy to compare the demographical behaviour of populations / species

• More reliable in combination with long term (less-detailed) population studies

• Management actions need to be based on a combination of sensitivity analysis and common sense

• Projections vs predictions vs modelling

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Suggested reading:

• Matrix projection models, theory and methods- van Groenendael, J., de Kroon, H., & Caswell, H. (1988) Projection matrices in population biology. TREE. 3(10), 264-269.

- Tuljapurkar, S. & Caswell, H. (1997) Structured population models in Marine, terrestrial, and freshwater systems. Chapman & Hall, New York. 643p.

-Caswell, H. (2001) Matrix population models - construction, analysis, and interpretation. Sinauer, Sunderland, MA. 722p.

• Applications: conservation and management of plant populatons- Valverde, T. & Silvertown, J. (1998) Variation in the demography of a woodland understorey herb (Primula vulgaris) along the forest regeneration cycle projection matrix analysis. Journal of Ecology. 86, 545-562.

- Fiedler, P. L., Knapp, B. E., & Fredericks, N. (1997) Rare plant demography: lessons from the Mariposa lilies (Calochortus Liliaceae). p28-48. in: Conservation biology for the coming decade (eds. Fiedler P. L. and Karieva P. M.) Chapman & Hall, New York.

• Algortihms for calculating the number of life stages- Vandermeer, J. (1978) Choosing category size in a stage population matrix. Oecologia. 32, 79-84.

- Moloney, K. A. (1986) A generalized algorithm for determining category size. Oecologia. 69, 176-180.

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Suggested reading (continued):

•Sensitivity and elasticity analysis- de Kroon, H., Plaisier, A., van Groenendael, J., & Caswell, H.(1986) Elasticity, the relative contribution of demographic parameters to population growth rate. Ecology. 67, 1427-1431.

- Silvertown, J., Franco, M., & McConway, K. (1992) A demographic interpretation of Grime's triangle. Functional Ecology. 6, 130-136.

- Silvertown, J., Franco, M., & Menges, E.( 1996) Interpretation of elasticity matrices as an aid to the management of plant populations for conservation. Conservation Biology. 10, 591-597.

• LTRE- Caswell, H. (1989) Analysis of life table response experiments I. Decomposition of effects on population growth rate. Ecological Modelling. 46, 221-237.

- Ehrlén, J. & van Groenendael, J. (1998) Direct perturbation analysis for better conservation. Conservation Biology. 12, 470-474.

- Caswell, H. (2000) Prospective and retrospective perturbation analyses their roles in conservation biology. Ecology. 81, 619-627.

• More on the demography of P. veris in Voeren- Endels, P., Jacquemyn, H., Brys, R., Hermy, M. (2002) Response of Primula veris populations to ecological restoration: linking fitness-related characteristics with demography (subm. J. App. Ecol.)