Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.
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Transcript of Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.
![Page 1: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/1.jpg)
Analysis of the Life-Cycle Graph:
The Transition Matrix
Modeling Approach
![Page 2: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/2.jpg)
Parameterized Model
Matrix Analysis: Population GrowthPopulation
Growth Rate
= 0.998
= 0.997
= 1.12
Asymptotic Size Class Distribution
![Page 3: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/3.jpg)
Parameterized Model
Matrix Analysis: Population Projection
Projection of Population into Future
![Page 4: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/4.jpg)
Sensitivity Analysis
How does (population growth rate) change in response to a small change in transition rate?
= 1.12
+ .04
= 1.12
= 1.14
![Page 5: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/5.jpg)
Sensitivity Analysis
![Page 6: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/6.jpg)
Sensitivity Analysis: A Couple of Problems
High sensitivities may be associated with transitions that don’t occur in nature.
There is a basic difference in values associated with survivorship and fecundity.
![Page 7: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/7.jpg)
Elasticity Analysis: a potential solution
How does (population growth rate) change in response to a proportional change in transition rate?
= 1.12
+ 10%
= 1.12
= 1.13
![Page 8: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/8.jpg)
Parameterized Model Elasticity Analysis
![Page 9: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/9.jpg)
Model Predictions
• Life table• Matrix
= 1
< 1
> 1
Key assumptions?
![Page 10: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/10.jpg)
Density Effects Population change over time Birth and Death Rates
![Page 11: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/11.jpg)
Density Effects Birth and Death Rates Impact of increasing density
Decrease in
• Light• Nutrients• H20• Space
Impact of increasing density on the population
• Increase in death rate
• Decrease in reproduction
Increase in
• disease• herbivory
![Page 12: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/12.jpg)
Density Effects Population change over time Birth and Death Rates
![Page 13: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/13.jpg)
Density Effects in
Plant Populations
![Page 14: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/14.jpg)
An Experimental Approach
Increasing density
Basic design
Replicate treatments as many times as possible
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Measures of Density Effects
• Total biomass
• Above ground biomass
• Root biomass
• Seed production
• Population size
General response is often referred to as “Yield”
![Page 16: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/16.jpg)
Density Experiment: Example #1Total yield of the population
• Yield increases with increasing density (to a point)
• Similar pattern in different components of yield
• At higher densities yield tends to stay constant
![Page 17: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/17.jpg)
Density Experiments: Example #2
Total yield may differ among environ-ments, but the same general pattern is observed
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Density Experiments: Example #3
?
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Density Experiments: Example #4
?
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Empirical Data on Yield Density Relationships
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Yield-Density Equations
A General Model of
Intraspecific Density Effects
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Yield-Density Equations
baN
WNNwY
1
max
Y = Total yield of the population per unit area
![Page 23: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/23.jpg)
Yield-Density Equations
baN
WNNwY
1
max
Y = Total yield of the population per unit area
w = average yield of an individual
![Page 24: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/24.jpg)
Yield-Density Equations
baN
WNNwY
1
max
Y = Total yield of the population per unit area
w = average yield of an individual
N = population density
![Page 25: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/25.jpg)
Yield-Density Equations
baN
WNNwY
1
max
Y = Total yield of the population per unit area
w = average yield of an individual
N = population density
Wmax = maximum individual yield under conditions
of no competition
![Page 26: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/26.jpg)
Yield-Density Equations
baN
WNNwY
1
max
Y = Total yield of the population per unit area
w = average yield of an individual
N = population density
Wmax = maximum individual yield under conditions
of no competition
1/a = density at which competitive effects begin to become important
![Page 27: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/27.jpg)
Yield-Density Equations
baN
WNNwY
1
max
Y = Total yield of the population per unit area
w = average yield of an individual
N = population density
Wmax = maximum individual yield under conditions
of no competition
1/a = density at which competitive effects begin to become important
b = resource utilization efficience (i.e., strength of competition)
![Page 28: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/28.jpg)
baN
WNY
1
max
baN
WNNwY
1
max
Total Yield
baN
WNNw
1
max
Individual Yield
X X
The Two Faces of Yield-Density
![Page 29: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/29.jpg)
The Two Faces of Yield-Density
baN
WNY
1
max
baN
WNNwY
1
max
Total Yield
baN
Ww
1
max
Individual Yield
![Page 30: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/30.jpg)
Three General Categories of Yield-Density Relationships
baN
WNNwY
1
max
b < 1 : under compensation
b = 1 : exact compensation (“Law of constant yield”)
b > 1 : over compensation
![Page 31: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/31.jpg)
Three General Categories of Yield-Density Relationships
baN
WNNwY
1
max
b < 1 : under compensation
b = 1 : exact compensation (“Law of constant yield”)
b > 1 : over compensation
![Page 32: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/32.jpg)
Exact Compensation(b=1)
Density
0 50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
baN
WNY
1
max
baN
Ww
1
max
NNY
1.01
10
aN
WNY
1
max
for aN>>>1
aN
WNY
1
max
x xx
Ca
WY max
C
![Page 33: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/33.jpg)
Exact Compensation(b=1)
Density
0 50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
baN
WNY
1
max
baN
Ww
1
max
NNY
1.01
10
aN
WNY
1
max
for aN>>>1
aN
WNY
1
max
x xx
Ca
WY max
C
![Page 34: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/34.jpg)
Density
0 50 100 150 200 250
Ave
rag
e In
div
idu
al Y
ield
0
2
4
6
8
10
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
Exact Compensation(b=1) baN
WNY
1
max
baN
Ww
1
max
Nw
1.01
10
baN
Ww
1
max
)1log()log()log( max aNbWw
log transform
![Page 35: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/35.jpg)
Density
0 50 100 150 200 250
Ave
rag
e In
div
idu
al Y
ield
0
2
4
6
8
10
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
Exact Compensation(b=1) baN
WNY
1
max
baN
Ww
1
max
Nw
1.01
10
baN
Ww
1
max
)1log()log()log( max aNbWw
log transform
1/a density above which competitive effects become important
![Page 36: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/36.jpg)
Density
0 50 100 150 200 250
Ave
rag
e In
div
idu
al Y
ield
0
2
4
6
8
10
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
Exact Compensation(b=1) baN
WNY
1
max
baN
Ww
1
max
Nw
1.01
10
baN
Ww
1
max
)1log()log()log( max aNbWw
log transformslope ≈ b
![Page 37: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/37.jpg)
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
Exact Compensation(b=1) baN
WNY
1
max
baN
Ww
1
max
aN
Ww
1
max
Density
0 50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
aN
WNY
1
max for aN>>>1
x xxx
![Page 38: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/38.jpg)
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
Exact Compensation(b=1) baN
WNY
1
max
baN
Ww
1
max
aN
Ww max
Density
0 50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
a
WY max
for aN>>>1
![Page 39: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/39.jpg)
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
Exact Compensation(b=1) baN
WNY
1
max
baN
Ww
1
max
N
C
aN
Ww max
Density
0 50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
Ca
WY max
for aN>>>1
![Page 40: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/40.jpg)
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
Under Compensation(b<1) baN
WNY
1
max
baN
Ww
1
max
Density
0 50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
Density
0 50 100 150 200 250
To
tal Y
ield
0
100
200
300
400
b = 1
b = 0.8
b = 0.5
b = 0.25b = 0
![Page 41: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/41.jpg)
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
Under Compensation(b<1) baN
WNY
1
max
baN
Ww
1
max
Density
0 50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
Density
0 50 100 150 200 250
To
tal Y
ield
0
100
200
300
400
b = 1
b = 0.8
b = 0.5
b = 0.25b = 0
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
b = 1
b = 0.8
b = 0.5
b = 0.25b = 0
![Page 42: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/42.jpg)
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
No Density Effects(b=0) baN
WNY
1
max
baN
Ww
1
max
Density
0 50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
Density
0 50 100 150 200 250
To
tal Y
ield
0
100
200
300
400
b = 0
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
b = 0
![Page 43: Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfe41a28abf838cb56dd/html5/thumbnails/43.jpg)
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.01
0.1
1
10
100
Over Compensation(b>1) baN
WNY
1
max
baN
Ww
1
max
Density
0 50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
Density
50 100 150 200 250
To
tal Y
ield
0
20
40
60
80
100
120
b = 1
b = 1.2
b = 2.0
Density
1 10 100 1000
Ave
rag
e In
div
idu
al Y
ield
0.0001
0.001
0.01
0.1
1
10
100
b = 1
b = 1.2
b = 2.0