Spatio-temporal patterns in plant populations
in small landscape elements
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
outlineoutline
1. research questions1. research questions
2. data recording2. data recording
3. data management3. data management
4. analysis of temporal changes4. analysis of temporal changes
5. analysis of spatial patterns5. analysis of spatial patterns
6. further analyses6. further analyses
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Impact of different ‘management’ regimes on demographic Impact of different ‘management’ regimes on demographic parameters of populations in small landscape elements?parameters of populations in small landscape elements?
barochory ((Primula verisPrimula veris) vs. ) vs. myrmecochory ((Primula Primula vulgarisvulgaris))-> population dynamics & max. dispersal distance?-> population dynamics & max. dispersal distance?
Impact of ‘management’ regimes on shape & tail of the Impact of ‘management’ regimes on shape & tail of the dipsersal curve?dipsersal curve?
Relationship between dispersal distance & subsequent Relationship between dispersal distance & subsequent seedling/juvenile survivalseedling/juvenile survival
general research questionsgeneral research questions11
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data recordingdata recording11
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66Primula vulgaris Primula veris
data recordingdata recording11
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Primula vulgarisPrimula vulgaris
•13 populations13 populations
• forest edges (n=3) forest edges (n=3) vs. vs. grazed parcel margins (n=4) grazed parcel margins (n=4) vs. vs. [1999-2002][1999-2002] clearing of ditches & mowing (n=2)clearing of ditches & mowing (n=2)
• restoration of mowing regime (n=1)restoration of mowing regime (n=1) [1999-[1999-2003]2003]
• changes in adjacent land use (n=1)changes in adjacent land use (n=1) [1999-[1999-2002]2002]
data recordingdata recording11
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Primula verisPrimula veris
•16 populations16 populations
• graft (Voeren) (n=4) graft (Voeren) (n=4) vs. vs. grazed parcel margins (n=4) grazed parcel margins (n=4) vs. vs. [2000-2002] [2000-2002] clearing of ditches & mowing (n=4)clearing of ditches & mowing (n=4)
• restoration of calcareous grassland (n=2)restoration of calcareous grassland (n=2) [1999-[1999-2001]2001] (Voeren)(Voeren)
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0 10 20 30 40 50 60 70 80 Centimeters
2000
1999
2001
data managementdata management11
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Life Life stage stage 19991999
rosetterosettes 1999s 1999
Life Life stage stage 20002000
rosettes rosettes 20002000
……
Plant Plant 11
fl. adultfl. adult 55 n-fl. n-fl. adultadult
1010
Plant Plant 22
seedlinseedlingg
11 juvenilejuvenile 11
Plant Plant 33
juvenilejuvenile 11
Plant Plant 44
n-fl. n-fl. adultadult
11 fl. adultfl. adult 22
Plant Plant 55
seedlingseedling 11
……
analysis of temporal changesanalysis of temporal changes11
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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.
analysis of temporal changesanalysis of temporal changes11
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S J NFA
RA1 RA2
RA3
-> Moloney (1986) vs. empirical approach
YEAR YEAR t+1t+1
LL6666GG6262GG6464GG63630000ReprReprAdult 3Adult 3
LL5656LL5555GG5454GG5353GG5252GG5151ReprReprAdult 2Adult 2
LL4646LL4545LL4444GG4343GG4242GG4141ReprReprAdult 1Adult 1
LL3636LL3535LL3434LL3333GG3232GG3131NFNFAdult Adult
FF66FF55FF4400LL2222GG2121juvenilejuvenile
FF33FF22FF11000000seedlingseedling
ReprReprAdult 3Adult 3(>3 roz.)(>3 roz.)
ReprReprAdult 2Adult 2(2-3 roz.)(2-3 roz.)
ReprReprAdult 1Adult 1(1 roz.)(1 roz.)
NFNFAdult Adult
juvenilejuvenileseedlingseedling
YEAR tYEAR t
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analysis of temporal changesanalysis of temporal changesn(t + 1) = n(t + 1) = AA n(t)n(t)
analysis of temporal changesanalysis of temporal changes11
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SymboSymboll
DefinitionDefinition Demographic Demographic interpretationinterpretation
AA a square matrix containing the coefficients that a square matrix containing the coefficients that represent proportions of – mostly year-to-year – represent proportions of – mostly year-to-year – transitions between life stages.transitions between life stages.
Projection Projection matrixmatrix
aaijijthe element in row i, collum j of the projection matrix A the element in row i, collum j of the projection matrix A matrix matrix
elementelement
the dominant eigenvalue of Athe dominant eigenvalue of A population population growth rate growth rate
A A = = (the right eigenvector of A associated with (the right eigenvector of A associated with ))
stable stagestable stagedistributiondistribution
A = A = ((the left eigenvector of A associated with the left eigenvector of A associated with ))
reproductive reproductive valuesvalues
ssijij / / aaij ij
(the sensitivity of (the sensitivity of to changes in matrix element a to changes in matrix element aijij))sensitivitysensitivity
eeijijaaijij / / aaijij
(the proportional sensitivity of (the proportional sensitivity of to proportional changes to proportional changes in matrix element ain matrix element aijij))
elasticityelasticity
11 / / 22 (ratio between the dominant & subdominant (ratio between the dominant & subdominant eigenvalue)eigenvalue)
damping damping ratioratio
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sensitivity analysissensitivity analysis
(Silvertown (Silvertown et alet al. 1992, . 1992, 1996)1996)
Triangular plot of composite elasticity values Triangular plot of composite elasticity values for for growthgrowth (G), (G), statisstatis (L) and (L) and fecundityfecundity (F) (F)
species populations
C ≈≈ S
R ≈
C ≈≈ S
R ≈
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sites [sites [P. vulgarisP. vulgaris]]
forest edgesforest edgesgrazed parcel marginsgrazed parcel margins
ditch clearing (+mowing)ditch clearing (+mowing)
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population growth rates [population growth rates [P. vulgarisP. vulgaris]]
repeated measures ANOVA:
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1999-2000 2000-2001 2001-2002
po
pu
lati
on
gro
wth
rat
e +
/- S
D
forest edge
grazed
cleared/mown
tests of within-subjects contrastsF P
YEAR 11.019 < 0.05YEAR * MANAGEMENT 6.533 < 0.05
tests of between-subjects effectsF P
MANAGEMENT 6.5650 < 0.05
L
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
G
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
F
0.00.10.20.30.40.50.60.70.80.91.0
L
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
G
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
F
0.00.10.20.30.40.50.60.70.80.91.0
L
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
G
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
F
0.00.10.20.30.40.50.60.70.80.91.0
sensitivity analysis [sensitivity analysis [P. P. vulgarisvulgaris]]
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mown/clearedforest edgegrazed
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population growth rates [population growth rates [P. vulgarisP. vulgaris]]
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1999-2000 2000-2001 2001-2002 2002-2003
not mownmown 1mown 2
different mowing different mowing regimesregimes
?
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sensitivity analysis [sensitivity analysis [P. P. vulgarisvulgaris]]
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A RA
1 RA
2 RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
mown 2
not mown
mown 1
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A RA
1 RA
2 RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A RA
1 RA
2 RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
mown 2
not mown
mown 1
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not mown mown 1 mown 2
sensitivity analysis [sensitivity analysis [P. P. vulgarisvulgaris]]
L
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
G
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
F
0.00.10.20.30.40.50.60.70.80.91.0
L
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
G
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
F
0.00.10.20.30.40.50.60.70.80.91.0
L
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
G
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
F
0.00.10.20.30.40.50.60.70.80.91.0
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population growth rates [population growth rates [P. vulgarisP. vulgaris]]
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1999-2000 2000-2001 2001-2002
scraping of wholeditch bank
scraping of lowerside of ditch bank
clearing (+mowing) of clearing (+mowing) of ditchesditches
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sensitivity analysis [sensitivity analysis [P. P. vulgarisvulgaris]]
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A
RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A
RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A
RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2001 2002
Scrapingof whole
ditch bank
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A
RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A
RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A
RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2001 2002
Scrapingof whole
ditch bank
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Scraping of whole ditchbank
Scraping of lower side of ditch bank
sensitivity analysis [sensitivity analysis [P. vulgarisP. vulgaris]]
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population growth rates [population growth rates [P. vulgarisP. vulgaris]]
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1999-2000 2000-2001 2001-2002
arable field
grassland
changes in adjacent land usechanges in adjacent land use
!
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sensitivity analysis [sensitivity analysis [P. P. vulgarisvulgaris]]
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A
RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A
RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
arable field
grassland
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A
RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A
RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A
RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J
NF
A
RA
1
RA
2R
A3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1999 2000
S J
NF
A
RA
1
RA
2
RA
3 S
J NF
A RA
1
RA
2
RA
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2000 2001
arable field
grassland
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sensitivity analysis [sensitivity analysis [P. P. vulgarisvulgaris]]
arable field
grassland
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clearing / mowingclearing / mowing
grassland marginsgrassland margins
sites [sites [P. verisP. veris]]
graft (Voeren)graft (Voeren)
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tests of within-subjects contrastsF Sig.
YEAR 1.936 n.s.YEAR * MANAGEMENT 0.652 n.s.
between subjects effectsF Sig.
MANAGEMENT 4.369 < 0.05
population growth rates [population growth rates [P. verisP. veris]]
repeated measures ANOVA:
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
2000-2001 2001-2002
po
pu
lati
on
gro
wth
ra
te +
/- S
D
cleared/mown
grazed
graft
sensitivity analysis [sensitivity analysis [P. verisP. veris]]
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F
L
F
G L
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conclusions temporal analysis [conclusions temporal analysis [P. vulgaris & P. verisP. vulgaris & P. veris] - 1] - 1
P. vulgarisP. vulgaris P. verisP. veris
mown / mown / clearedcleared
grazedgrazed forest forest edgeedge
mown / mown / clearedcleared
grazedgrazed graftgraft
λλ:: management management effect?effect?
aa bb abab aa bb abab
λλ:: year effect? year effect? yesyes nono
Sensitivity Sensitivity analysisanalysis
λλ
rel. dep. rel. dep. on on
fecundityfecundity
λλ
higly higly dep. on dep. on survival survival adultsadults
intermedintermed..
λλ
rel. dep. rel. dep. on on
fecundityfecundity
λλ
higly higly dep. on dep. on survival survival adultsadults
intermed.intermed.
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conclusions temporal analysis [conclusions temporal analysis [P. vulgarisP. vulgaris] - 2] - 2
(a) mowing regime
- first year : effect of mowing once or twice a year ≅
- In the “long run”: only mowing twice a year (July & October) efficient to counter population senescence to lift λ above 1
(b) ditch clearing regime (& mowing of ditch bank)
- large impact on population growth rate (depending on intensity)
- temporarily higher mortality levels
- compensated by higher seedling recruitment and survival
c) changes in adjacent land use
- grassland -> arable field: (1) effect on population growth rate and (2) populations more responsive to survival of veg. adults
- population quickly recovers when the original land use is restored
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conclusions temporal analysis [conclusions temporal analysis [P. vulgaris & P. verisP. vulgaris & P. veris] - 3] - 3
in general:
- disturbances of any kind force populations into earlier stages
of
the successional G-L-F trajectory (Silvertown et al. 1996)
- however, these more dynamic populations (i.e. more
dependent on / responsive to fecundity and growth) are not
necessarily associated with higher population growth
rates (’s)
analysis of spatial patternsanalysis of spatial patterns11
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calculation of dispersal distances
dispersal curves for the two species for different management regimes-> tail (max. dispersal distance)?!
dispersal distance and survival -> Janzen (1970)-Connell (1971) vs. Hubbell (1979)
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calculation of dispersal distances
# reproductive adults year x# seedlings year x+1
calculation of dispersal distancescalculation of dispersal distances
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dispersal distance & management regime [dispersal distance & management regime [P. vulgarisP. vulgaris]]
dispersal distance
2.852.65
2.452.25
2.051.85
1.651.45
1.251.05
.85.65
.45.25
.05
forest 200012
10
8
6
4
2
0
Std. Dev = .22
Mean = .29
N = 33.00
dispersal distance
2.85
2.65
2.45
2.25
2.05
1.85
1.65
1.45
1.25
1.05
.85
.65
.45
.25
.05
grazed 200050
40
30
20
10
0
Std. Dev = .26
Mean = .26
N = 164.00
dispersal distance
mown/cleared 2000300
200
100
0
Std. Dev = .44
Mean = .42
N = 724.00
dispersal distance
3.152.95
2.752.55
2.352.15
1.951.75
1.551.35
1.15.95
.75.55
.35.15
forest 200130
20
10
0
Std. Dev = .14
Mean = .25
N = 61.00
dispersal distance
2.852.65
2.452.25
2.051.85
1.651.45
1.251.05
.85.65
.45.25
.05
grazed 200150
40
30
20
10
0
Std. Dev = .22
Mean = .17
N = 99.00
dispersal distance
mown/cleared 2001300
200
100
0
Std. Dev = .51
Mean = .38
N = 621.00
dispersal distance
2.852.65
2.452.25
2.051.85
1.651.45
1.251.05
.85.65
.45.25
.05
forest 2002140
120
100
80
60
40
20
0
Std. Dev = .11
Mean = .19
N = 254.00
dispersal distance
2.8502.650
2.4502.250
2.0501.850
1.6501.450
1.2501.050
.850.650
.450.250
.050
grazed 200260
50
40
30
20
10
0
Std. Dev = .10
Mean = .158
N = 137.00
dispersal distance
mown/cleared 200250
40
30
20
10
0
Std. Dev = .31
Mean = .36
N = 176.00
max. = 1.04 m
max. = 1.26 m
max. = 0.61 m
mean = 0.29 m
mean = 0.25 m
mean = 0.19 m
max. = 2.12 m
mean = 0.17 m
max. = 0.54 m
mean = 0.16 m
max. = 2.45 m
mean = 0.26 m
max. = 2.62 m
mean = 0.42 m
max. = 3.89 m
mean = 0.38 m
max. = 1.89 m
mean = 0.36 m
a
a
a
ab
b
b
b
c
b
dispersal distance
grazed 2001100
80
60
40
20
0
Std. Dev = .08
Mean = .108
N = 376.00
11
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66dispersal distance
grazed 200240
30
20
10
0
Std. Dev = .08
Mean = .15
N = 187.00
dispersal distance & management regime [dispersal distance & management regime [P. verisP. veris]]
dispersal distance
mown/cleared 2001300
200
100
0
Std. Dev = .19
Mean = .23
N = 2466.00
dispersal distance
mown/cleared 2002300
200
100
0
Std. Dev = .16
Mean = .25
N = 2621.00
max. = 0.57 m
max. = 2.60 m max. = 0.47 m
max. = 5.49 m
mean = 0.10 m
mean = 0.15 m
mean = 0.23 m
mean = 0.25 m
<P < 0.001
≈ ?n.s
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P. vulgarisP. vulgaris
dispersal distance & survivaldispersal distance & survival
Wald χ² = 44.285n = 1720P < 0.001
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P. vulgarisP. vulgaris
dispersal distance & survivaldispersal distance & survival
dispersal distance
Wald χ² = 4.425n = 91; P = 0.035
Wald χ² = 0.065n = 271; P = 0.798
Wald χ² = 53.200n = 1344; P < 0.001
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dispersal distance & survivaldispersal distance & survival
P. verisP. veris
dispersal distance
2.01.81.61.41.21.0.8.6.4.20.0
1.2
1.0
.8
.6
.4
.2
0.0
-.2
survival prob
fate(1=surv, 0=mort)
Wald χ² = 15.223n = 2815P < 0.001
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dispersal distance & survivaldispersal distance & survival
P. verisP. veris
dispersal distance
Wald χ² = 7.141n = 344; P = 0.008
Wald χ² = 16.137n = 2438; P < 0.001
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spatial patterns: conclusionsspatial patterns: conclusions
P. vulgarisP. vulgaris P. verisP. veris
managemenmanagementt
forest forest edgeedge
grazedgrazed mown / mown / clearedcleared
grazedgrazed mown / mown / clearedcleared
dispersal dispersal distancedistance **** **** ****** ** ******
seedling / seedling / juvenile juvenile survivalsurvival
********(63 %) (63 %)
********(62 %)(62 %)
******(47 %)(47 %)
****(19 %)(19 %)
**(8 %)(8 %)
dispersal dispersal (D) ~ (D) ~ survival survival (S)(S)
S ▲ if D ▲S ▲ if D ▲
herbivory / herbivory / pathogens / pathogens / intraspecific intraspecific comp. ?comp. ?
n.s.n.s. S ▲ if D ▲S ▲ if D ▲
herbivory / herbivory / pathogens / pathogens / intraspecific intraspecific comp. comp.
S ▲ if D ▲S ▲ if D ▲
herbivory ?herbivory ?
S ▼ if D ▲S ▼ if D ▲
high high disturbance disturbance regime ?regime ?
J.-C. J.-C. J.-C. H.
further analyses (1)further analyses (1)11
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temporal changes
- relationship with reproductive output
- comparison of population growth rate by randomization tests
- LTRE (mowing regime P. vulgaris, management regime P. veris)
- stochastic population modelling (Caswell 2001, Tuljapulkar & Caswell 1997)
spatial patterns - relationship between aggregation & dispersal distance (calculation of Ripley’s K & comparison between
management regimes)
- aggregation and flower morph types ?
- ? ? ?
incorporating plant performance
- link with recruitment / population dynamics ?
further analyses (2)further analyses (2)11
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0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
0.0 0.5 1.0 1.5 2.0 2.5
distance
rip
ley
's K
L_T_
L_T_2_5
L_T_97_5