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4 x 4 size-structured matrix(also called Lefkovitch matrix)
Pij=probability of growing from one sizeto the next or remaining the samesize
(need subscripts to denote newpossibilities)
F=fecundity of individuals at each size
In this case, there are three pre-reproductive sizes (maturity at agefour).
**additional complexities like shrinkingor moving more than one class backor forward is easy to incorporate
!!!!
"
#
$$$$
%
&
=
4443
3332
2221
411
2
000000
00
PPPP
PPFP
A
What we’ve covered so far:
Translating life histories into stage/age/size -based matrices
Understanding matrix elements (survival and fecundity rates)
Basic matrix multiplication in fixed environments
Deterministic matrix evaluation (λ1 , stable stage/age)
Initial framework for sensitivity analysis
Next:Incorporating demographic & environmental stochasticity
Life cycle models put impacts in context
Simple (deterministic):
650
85%
7%15%
10%
45%
30 years
Adu
lt #’
s
10%
1%Population grows (or shrinks)exponentially as a function of thecombination of fixed vital rates
Life cycle models put impacts in context
More realistic (stochastic simulation):
0 100survival
prob.
0 100survival
prob.
0 1000fecundityprob.
0 100survival
prob.
1000 survival
prob.
1000 survival
prob.
30 years
Adu
lt #’
s
Population varies from year toyear as a function of thecombination of randomly drawnvital rates
Simulation-based stochastic model:
0 100survival
prob.
0 100survival
prob.
0 1000fecundityprob.
0 100survival
prob.
1000 survival
prob.
1000 survival
prob.
30 years
Adu
lt #’
s
Life cycle models put impacts in context
More realistic (stochastic simulation):
Stochastic projections
1. Form of stochasticity in matrix elements/vital rates
-Environmental stochasticity?Series of fixed matrices (annual)
-env. conditions ‘independent’ (no autocorrelation)-preserves within year correlations among vital rates
Vary individual vital rates per timestep -separate from sampling variation
-draw vital rates from specified distributions(Lognormal, beta, etc.)
-mechanistic: vital rates affected by periodic conditions(ENSO/PDO, flood recurrence, etc.) ==> probablisitic
Issues to consider:
-Demographic stochasticity:Small population sizes
-Monte Carlo sims of individual fate given distributions ofvital rates (quasi-extinction is easier…)
-Density-dependence in specific vital rates-vital rate is a function of density (difficult to parameterize)
-Quasi-extinction threshold?-minimum ‘viable’ level
-Correlation structure?-within years (common), across years (cross-correlation, harder)
-OUTPUTS: Stochastic lambda, extinction probability CDF
Stochastic projections
Issues to consider:
Simulation-based stochastic model:
0 100survival
prob.
0 100survival
prob.
0 1000fecundityprob.
0 100survival
prob.
1000 survival
prob.
1000 survival
prob.
30 years
Adu
lt #’
s
Life cycle models put impacts in context
More realistic (stochastic simulation):
Some useful distributions
-Uniform
-Normal (Gaussian)
-Log-Normal
-Beta
Some useful distributions
-Uniform
-Normal (Gaussian)
-Log-Normal
-Beta
where µ and σ are the mean and standard deviation of the variable’s natural logarithm(by definition, the variable’s logarithm is normally distributed).
Some useful distributions
-Uniform
-Normal (Gaussian)
-Log-Normal
-Beta
where α and β are the two parameter governing the shape,and B is a normalization constant to ensure that the totalprobability integrates to unity.
The beta distribution can take on different shapes depending on the values of the two parameters. Hereare some examples:
α = 1,beta = 1 is the uniform [0,1] distribution α < 1,beta < 1 is U-shaped (red plot) α < 1beta ≥ 1 or α = 1,beta > 1 is strictly decreasing (blue plot) α = 1,beta > 2 is strictly convex α = 1,beta = 2 is a straight line α = 1, 1 < beta < 2 is strictly concave α = 1,beta < 1 or α > 1,beta ≤ 1 is strictly increasing (green plot) α > 2,beta = 1 is strictly convex α = 2,beta = 1 is a straight line 1 < α < 2,beta = 1 is strictly concave α > 1,beta > 1 is unimodal (purple & black plots)
Moreover, if α = β then the density function is symmetric about 1/2 (red & purple plots).
How do we estimate α and β fromdemographic data?
!
" = x x 1# x ( )
v$
% &
'
( )
!
" = 1# x ( ) x 1# x ( )v
#1$
% &
'
( )
We can convert the familiar mean (xbar) andvariance (v) to the relevant parameters for theBeta distribution:
0 0 0 a14
a21 0 0 0
0 a32 0 0
0 0 a43 a44
FROM CLASS (j’s)
TO CLASS (i’s)
XAt=
Nt
matrix of transition probabilities population vector
n1
n2
n3
n4
Matrix population model with four life stagesStochastic 30yr. simulations (10,000 runs)
Do hydrologic stressors onearly life stages affect population dynamics?
eggs
larvae
juvenile
adult
product of component life stagetransitions
Annual transitionprobability
aij
a21= sembryo1 x sembryo2 x sembryo3 x stadpole xsmetamorph
=
Calculation of transition probabilities
0 0 0 a14
a21 0 0 0
0 a32 0 0
0 0 a43 a44
FROM CLASS (j’s)
TO CLASS (i’s)
XAt=
Nt
matrix of transition probabilities population vector
n1
n2
n3
n4
Matrix population model with four life stagesStochastic 30yr. simulations (10,000 runs)
* Starting population size
* Quasi-extinction threshold
* Distributions of survival rates (transitionprobabilities) of each life stage
* Fecundity of adult females
**varied to evaluate different hydrologic andpopulation scenarios**
Scenarios and Outputs:
Response ‘variables’ = 30 yr probability of extinction
stochastic population growth rate
multivariate sensitivity analysis
0 5 10 15 20 25 30
0.01
0.02
0.03
0.04
0.05
0.06
years
pro
bab
ilit
y o
f exti
nct
ion
Reference Modelcumulative extinction probability
Starting Population Sizes
0.0
0.2
0.4
0.6
0.8
1.0
Starting population size
30
yr.
pro
bab
ilit
y o
f exti
nct
ion
SF Eel. (1050)
Average un-regulated (320)
Average regulated (46)
NF Feather, Cresta Reach (21)
0.0
0.2
0.4
0.6
0.8
1.0
Starting population size
30
yr.
pro
bab
ilit
y o
f exti
nct
ion
SF Eel. (1050)
Average un-regulated (320)
Average regulated (46)
NF Feather, Cresta Reach (21)
SF Eel. (1050)
Average un-regulated (320)
Average regulated (46)
NF Feather, Cresta Reach (21)
Starting Population Sizes
0.0
0.2
0.4
0.6
0.8
1.0
Starting population size
30
yr.
pro
bab
ilit
y o
f exti
nct
ion
SF Eel. (1050)
Average un-regulated (320)
Average regulated (46)
NF Feather, Cresta Reach (21)
0.0
0.2
0.4
0.6
0.8
1.0
Starting population size
30
yr.
pro
bab
ilit
y o
f exti
nct
ion
SF Eel. (1050)
Average un-regulated (320)
Average regulated (46)
NF Feather, Cresta Reach (21)
SF Eel. (1050)
Average un-regulated (320)
Average regulated (46)
NF Feather, Cresta Reach (21)
Average regulatedpopulation size 5xhigher chance ofextinction
Without any specifichydro impacts
0.0
0.2
0.4
0.6
0.8
1.0
Summer pulses (0.61)
30
yr.
pro
bab
ilit
y o
f exti
nct
ion
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Summer pulses (0.61)
Sto
chast
ic lam
bd
a
1 2 3 4 1 2 3 4 1 2 3 4
Summer pulses (1-4) with ~40% tadpolemortality each event
Model λ30 yr Extn
prob.∆ Extn prob.
Reference 1.21 0.05 ----Spring pulse flowsw/lower fecundity
0.87 0.85 17x
1 Summer pulse flow,high larval survival
1.03 0.27 5x
Spring scour +Summer pulses (2)
0.84 0.91 18x
Sm starting pop +Spring scour +Summer pulse (2)
0.87 0.99 20x
Reference and Scenario Summary
Sensitivity analysis: larval stage > egg scouring > juvenile 1 = adult
R = αS*e(-S/k)
The traditional form of density dependence
s0[Et] = s0(0) * e(-βEt)
s0[Et] = s0(0) / (1+βEt)
Where s0[Et] = survival (s0) as a function of density [E] at time t
Converting to forms relevant to individual vital rates (sij)
s00 = survival when density is almost 0
β = density dependent coefficient (larger = bigger penalty for survival)
**Substitute sij for relevant stage/age
s0[Et] = s0(0) * e(-βEt)
s0[Et] = s0(0) / (1+βEt)
**Beware that even for deterministic models, imposingdensity dependence (esp. Ricker) can result in cyclicalpopulation dynamics. See Fig. 8.9 in Morris and Doak.
**And stochastic lambda is now meaningless (populationbounded), but extinction prob. still informative
Converting to forms relevant to individual vital rates (sij)
s0[Et] = s0(0) * e(-βEt)
s0[Et] = s0(0) / (1+βEt)
Usually requires enough datapoints to regress logsurvival against initial stage density.
Where a negative slope roughly equals -β, and theintercept is log[s0(0)] (for Ricker)
Where the inverse of the intercept is s0(0), and the slopedivided by the intercept is β (for B-H)
Estimating density dependence: