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