Analysis of the Life-Cycle Graph: The Transition Matrix Modeling Approach

Parameterized Model Matrix Analysis: Population Growth Population Growth Rate  =  =  = 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 = 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 = 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 H 2 0 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 #1 Total 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 = Total yield of the population per unit area

Yield-Density Equations = Total yield of the population per unit area = average yield of an individual

Yield-Density Equations = Total yield of the population per unit area = average yield of an individual N = population density

Yield-Density Equations = Total yield of the population per unit area = average yield of an individual N = population density W max = maximum individual yield under conditions of no competition

Yield-Density Equations = Total yield of the population per unit area = average yield of an individual N = population density W max = maximum individual yield under conditions of no competition 1/a = density at which competitive effects begin to become important

Yield-Density Equations = Total yield of the population per unit area = average yield of an individual N = population density W max = 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)

Total YieldIndividual Yield X X The Two Faces of Yield-Density

Total YieldIndividual Yield

Three General Categories of Yield- Density Relationships b < 1 : under compensation b = 1 : exact compensation (“Law of constant yield”) b > 1 : over compensation

Three General Categories of Yield- Density Relationships b < 1 : under compensation b = 1 : exact compensation (“Law of constant yield”) b > 1 : over compensation

Exact Compensation (b=1) for aN>>>1 x x x C

Exact Compensation (b=1) for aN>>>1 x x x C

Exact Compensation (b=1) log transform

Exact Compensation (b=1) log transform 1/a  density above which competitive effects become important

Exact Compensation (b=1) log transform slope ≈ b

Exact Compensation (b=1) for aN>>>1 xxx x

Exact Compensation (b=1) for aN>>>1

Exact Compensation (b=1) for aN>>>1

Under Compensation (b<1) b = 1 b = 0.8 b = 0.5 b = 0.25 b = 0

Under Compensation (b<1) b = 1 b = 0.8 b = 0.5 b = 0.25 b = 0 b = 1 b = 0.8 b = 0.5 b = 0.25 b = 0

No Density Effects (b=0) b = 0

Over Compensation (b>1) b = 1 b = 1.2 b = 2.0 b = 1 b = 1.2 b = 2.0