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Analyzing Matrix Models

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Semipalmated Sandpiper in Manitoba. Three age classes. Initial age distribution (#/ha) ... When population reaches stable age (or size or stage) distribution then all ... – PowerPoint PPT presentation

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Title: 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

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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
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