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Analysis of the Life-Cycle Graph:
The Transition Matrix
Modeling Approach
Parameterized Model
Matrix Analysis: Population GrowthPopulation
Growth Rate
= 0.998
= 0.997
= 1.12
Asymptotic Size Class Distribution
Parameterized Model
Matrix Analysis: Population Projection
Projection of Population into Future
Sensitivity Analysis
How does (population growth rate) change in response to a small change in transition rate?
= 1.12
+ .04
= 1.12
= 1.14
Sensitivity Analysis
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.
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
Parameterized Model Elasticity Analysis
Model Predictions
• Life table• Matrix
= 1
< 1
> 1
Key assumptions?
Density Effects Population change over time Birth and Death Rates
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
Density Effects Population change over time Birth and Death Rates
Density Effects in
Plant Populations
An Experimental Approach
Increasing density
Basic design
Replicate treatments as many times as possible
Measures of Density Effects
• Total biomass
• Above ground biomass
• Root biomass
• Seed production
• Population size
General response is often referred to as “Yield”
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
Density Experiments: Example #2
Total yield may differ among environ-ments, but the same general pattern is observed
Density Experiments: Example #3
?
Density Experiments: Example #4
?
Empirical Data on Yield Density Relationships
Yield-Density Equations
A General Model of
Intraspecific Density Effects
Yield-Density Equations
baN
WNNwY
1
max
Y = Total yield of the population per unit area
Yield-Density Equations
baN
WNNwY
1
max
Y = Total yield of the population per unit area
w = average yield of an individual
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
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
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
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)
baN
WNY
1
max
baN
WNNwY
1
max
Total Yield
baN
WNNw
1
max
Individual Yield
X X
The Two Faces of Yield-Density
The Two Faces of Yield-Density
baN
WNY
1
max
baN
WNNwY
1
max
Total Yield
baN
Ww
1
max
Individual Yield
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
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
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
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
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
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
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
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
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
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
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
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
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
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