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: