Demographic PVA’s

Assessing Population Growth and Viability

Structured populations in a deterministic environment

• Deterministic projection models, when we do not have (or use) estimates of variation

vector: a representation of the population structure

• It is a column of numbers that indicates the densities of individuals in each class in the population at one point in time

n(t) =1456

123

9

Each entry aij(t) in a projection matrix A(t) gives the number of individuals in class i at census (t+1) produced on average by a single individual in class j at census (t)

a11(t) a12(t) a13(t)

a21(t) a22(t) a23(t)

a31(t) a32(t) a33(t)

A(t) =

If we know the densities at census t n(t), we can project the densities at

the next census n(t+1)

a11(t) a12(t) a13(t)

a21(t) a22(t) a23(t)

a31(t) a32(t) a33(t)

n1(t+1)

n2(t+1)

n3(t+1)

=

n1(t)

n2(t)

n3(t)

If we know the densities at census t n(t), we can project the densities at

the next census n(t+1)

n1(t+1)

n2(t+1)

n3(t+1)

=

a11(t) n1(t+1)+ a12(t) n2(t+1)+ a13(t) n3(t+1)

a21(t) n1(t+1)+ a22(t) n2(t+1)+ a23(t) n3(t+1)

a31(t) n1(t+1)+ a32(t) n2(t+1)+ a33(t) n3(t+1)

In a constant environment…• A(t)=A

• Population convergence

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1 2 3 4 5 6 7 8 9 10 11

year

pro

po

rtio

n in

cla

ss..

one year

two year

three year

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5 6 7 8 9 10 11 12

year

bir

ds

one year

two year

three year

total

The stable distribution (w) is

0.1189

0.1047

0.7764

w=

λ1=0.6389

The ultimate or long term growth rate

(λ) is

The unique vector containing the ultimate proportions of the population in each class given the constant projection matrix A

The reproductive values (v) is

1.0000

1.0973

1.1139

v=

The relative contribution to future population growth an individual currently in a particular class is expected to make

Reproductive values take into account the number of offspring an individual might produce in each of the classes it passes through the future, the likelihood of the individual reaching those classes, the time required to do so, and the population growth rate

Eigenvalue sensitivities

• The ultimate rate of population growth in a constant environment, λ1, depends on the magnitudes of all the elements in A, so changing any of them will change λ1.

• Sij Sensitivity: is a useful measure of how much changes in a particular matrix element will change λ1

Sij Sensitivity:

• It is the partial derivative of λ1 with respect to aij

• It measures the change in λ1 that would result from a small change in aij , keeping all other elements of the matrix A fixed at their present values

1k kk

ji

ijij wv

wv

aS

Sensitivities

0.1082 0.0953 0.7067

0.1187 0.1046 0.7755

0.1205 0.1062 0.7872

S=

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

size of matrix element

do

min

ant

eig

enva

lue.

.

Sensitivity

Slope=0.787

How to include stochastic environmental effects on matrix

models?

• Matrix selection

Modeling using matrix selection

Year 1 matrix 1

Year 1 matrix 3

Year 2 matrix 1

Choose 1:

Year 2 matrix 3or

or

Year 0 matrix 1

Year 0 matrix 3

Choose 1:

or

or

or or

Year 2 matrix 2Choose 1:

If environmental conditions are aperiodic and uncorrelated, and moreover the probability of choosing a particular matrix does not change over time then environmental conditions are said to be:

“independently and identically distributed” or “iid”

Year 0 Matrix 2

Yar 1 matrix 2

Mountain golden heather

0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300

350

400

450

Population size at t = 50

Nu

mb

er o

f re

aliz

atio

ns

Year 0 fire matrix

Year 1 matrix 1.1

Year 1 matrix 1.2

Year 1 matrix 1.3

Year 3 matrix 3.1

Modeling samples from matrices by time since fire.In this (simplified) example, the fire return interval is 3 years:Use this:

Choose 1:

Other years

Use this:

reset

or or

Beyond interpolation, input pooled matrices

Year 3 matrix 3.2

or

fire

Estimating the Stochastic log Growth rate λs

• By simulation

Log[N(t+1)/N(t)]

over all pairs of adjacent years

Estimating the Stochastic log Growth rate λs

• Tuljapurkar’s approximation (an analytical solution)

21

2

2

1loglog

s

klijl klijj kiSSaaCov

11 11

2 .,...

yyxxn

yxCav ii

11

,

Calculating Quasi-Extinction probability

• Box 7.5

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Years into the future

Cum

ulat

ive

prob

abili

ty o

f qua

si-e

xtin

ctio

n