Analyzing Matrix Models

1
Analyzing Matrix Models

• ESM 211
• Nov. 8, 2006

2
Projecting the model

• Semipalmated Sandpiper in Manitoba
• Three age classes
• Initial age distribution (/ha)

• Projection matrix
• Iterate the model

3
(No Transcript)
4
(No Transcript)
5
(No Transcript)
6
Asymptotic growth rate

• When population reaches stable age (or size or
  stage) distribution then all classes grow (or
  decline) at the same rate

lambda_1 0.6389479 Class Stable
distribution Reproductive value 1
0.1188640 1 2
0.1047353 1.097332 3
0.7764007 1.113922
7
Matrices and growth rate

• The asymptotic growth rate (l1) is the dominant
  eigenvalue of the projection matrix
• The stable age distribution (w1) is the
  associated right eigenvector
• The reproductive value distribution (v1) is the
  associated left eigenvector

8
What good is a deterministic matrix model?

• Assumes that the environment is constant
  unrealistic! But
• If lambda lt 1, population is really in trouble!
• We might not have info on temporal variability
• Insights from sensitivity analysis (next) carry
  over to stochastic case

9
Sensitivity Elasticity

• Sensitivity absolute rate of change of l1 with
  respect to absolute change in a matrix element
• Elasticity relative rate of change of l1 with
  respect to relative change in a matrix element

10
Sensitivity Elasticity of vital rates
11
Sensitivity in management

• For each potential mgmt action
• Define its effect on demography (e.g. reduce
  mortality of adults)
• Estimate cost per unit effort
• Use sensitivity or elasticity to find improvement
  in lambda per unit cost

12
Sea turtle conservation

• All 5 US species are endangered
• Current densities may be only 1 of pre-European
  values
• Populations declining at up to 5 per year
• How can we reverse sea turtle population decline?

13
Sea turtle life history

• Eggs laid on beaches, clutches buried in sand
• Hatchlings emerge and enter sea
• 30 years to reach reproductive maturity

14
Anthropogenic threats

• Eggs preyed on and accidentally destroyed by
  humans
• Hatchlings disoriented by bright lights
• Hatchlings get stuck in vehicle tracks
• Juveniles and adults get caught in fishing gear,
  especially shrimp trawls

15
Protection strategies

• Protect eggs and hatchlings 20-30 years of
  aggressive efforts had little effect
• Install turtle excluder devices (TEDs) on shrimp
  trawls unpopular with fishermen -- politically
  costly

16
Stage-structured sea turtle model
17
Sea turtle elasticities
18
Predicted growth rates
19
Effects of vital rate uncertainty
20
Environmental stochasticity 1

• Construct a projection matrix for each year
• Simulate the population by drawing one of the
  matrices for each simulation year
• Run lots of replicate simulations etc.

21
Environmental stochasticity 2

• Estimate the means, variances, and covariances of
  vital rates
• Calculate the stochastic log growth rate

22
Environmental stochasticity 3

• Estimate the means, variances, and covariances of
  vital rates
• For each simulation year, draw vital rates at
  random construct matrix for that year
• 0-1 parameters use Beta distribution
• Fertility lacking other info, use log-normal
• Run replicate simulations etc.

23
Stochastic simulations with demographic
stochasticity

• For each individual, draw random numbers to
  determine fate, with parameter given by the
  corresponding mean vital rate
• Survival binomial
• Growth sequence of binomials
• Fecundity Poisson?

24
Growth rates

• Sampling distribution of growth transitions is
  multinomial
• Treat as a sequence of binomial processes

25
Adding environmental stochasticity

• Estimate among-year variance covariance in
  vital rates from data
• Each simulation year, the mean vital rate is
  drawn at random
• Binomial variables Use beta distribution
• Fertility lacking other info, use log-normal
• Simulate demographic stochasticity in that year
  using that rate

26
Estimating variance in demographic rates

• Some of the among-year variance in observed
  demography will be due to sampling error
• Need to correct for this
• First cut similar approach as for count-based
  PVA

27
Simple variance correction
28
More sophisticated approach

• Solve the following for Vc
• Find CI by replacing 1 in eqn above with

29
Sensitivity again What we really want to know

• Sensitivities of
• Stochastic growth rate
• Extinction time under demographic stochasticity
• Extinction time under environmental stochasticity
• Equilibrium density under density dependence
• Invasion speed

30
What does sensitivity of l tell us about all this?

• Nothing, directly
• But it might tell us something about the
  sensitivity of those quantities to various stages
  in the life history
• Caswell compare sensitivity of l to sensitivity
  of quantities of interest

31
Sensitivity of stochastic growth rate

• Build a model with demographic stochasticity
• Run model for many iterations, and calculate
  long-term growth rate
• Change one stage specific parameter, and re-run
  the model
• Change in long-term growth rate is its
  sensitivity to that parameter

32
Sensitivity of stochastic growth rate (2)

• Build a matrix model with same stage transitions
  and parameter values
• Calculate sensitivities of l to each parameter
• Compare with sensitivities from previous slide

33
Sensitivity of stochastic growth rate (3)

• Sensitivity of l strongly correlated with
  sensitivity of stochastic growth rate
• element with large effect on l also has large
  effect on stochastic growth rate
• Similarly, sensitivity of l strongly correlated
  with sensitivity of
• extinction time
• equilibrium density
• invasion speed

34
Caveats of Caswells work

• Based on perfect knowledge
• e.g., for environmental stochasticity, analysis
  of l based on average matrix, where average is
  taken over all the generations of the stochastic
  simulation
• Important question how many years of data
  required for this approach to work?

35
Some analytics