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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