Demographic PVAs
Simulating Demographic
Stochasticity and Density
Dependence
Demographic stochasticity
• Simulated by performing so-called Monte
Carlo simulations: the fate of each
individual in a certain class and a certain
year is decided by a set of independent
random choices, all of which are based on
the same set of mean vital rates
• However this can greatly slow a program
Variability caused by demographic stocahsticity in
binomial vital rates
0.14
0.12
Variance in outcome
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
Number of individuals
25
30
35
Techniques to how to perform
Monte Carlo simulations
• For vital rates that are probabilities:
Pick a uniform random number and compare
its value to the probabilities of different
fates an individual might experience
How to use a uniform random number to decide
between fates in a Monte Carlo simulation
0
Survive and shrink to class 3
a34
Survive and stay to class 4
a34+a44
Survive and grow to class 5
a34+a44+a54
Survive and grow to class 6
a34+a44+a54+a64
Die
1
Adding demographic stochasticity to
reproduction
• Determine if the individual lives
• Use a Poisson or another discrete
distributions to obtain the individual fertility
40
Observed
35
Predicted
Frequency
30
Number of flowers
25
20
Avg. # Flowers
15
10
5
0
0
1
2
3
5
Number of flowers
Population 1; 2000
40
Observed
35
Predicted
Frequency
30
25
20
15
10
5
0
0
1
2
3
5
pop1
pop2
pop3
pop4
pop5
pop6
2000
1.45
1.39
3.41
3.22
5.71
2.69
2001
0.75
0
1.89
3.62
1.73
0.69
2002
1.26
1.63
4.10
5.00
4.86
1.65
Number of flowers
Population 2; 2000
P (y=r) = (e –μ
μ r) / r!
Sampling from Poisson
distributions to estimate
flower production
Avg. # Seeds
Number of seeds
per flower
Population 1; 2000
8
6
4
2
Std. Dev = 28.97
Mean = 63.2
N = 30.00
0
12.0
40.0
26.0
68.0
54.0
96.0
82.0
124.0
110.0
number of seeds per flower
138.0
2000 2001 2002
pop1
63
63
53
pop2
56
62
42
pop3
54
43
43
pop4
55
49
38
pop5
63
54
41
pop6
68
50
33
Sampling from normal
distributions to estimate
seed production
Including density dependence
•
Two factors make it more difficult to
account for density dependence in
demographic PVAs
1. We will rarely have as many years of
vital rate estimates from a demographic
study
2. There are many more variable that are
potentially density dependent
Three questions we must consider:
• Which vital rates are density-dependent?
• How do those rates change with density?
• Which classes contribute to the density
that each vital rate “feels”?
Two more limited approaches to
building density-dependent
projection matrix models
1. Assume that there is a maximum number
or density of individuals in one or more
classes, or of the population as a whole ,
that the available resources can support,
and construct a simulation model that
prevents the population vector from
exceeding this limit.
Two more limited approaches to
building density-dependent
projection matrix models
2. Choose one or at most a few vital rates
that, on the basis some evidence are
suspected to be strongly density
dependent
Placing a limit on the size of one or
more classes
Caps or ceilings on population density are
most often used to introduce density
dependence when the focal species is
territorial
The Iberian Lynx
Gaona et al. (1998)
lynx population in Doñana National Park post
breeding census birth-pulse population
Cubs
Juveniles
Floaters
Breeders
Cubs
0
0
0
b x c x p x s4
Juveniles
s1
0
0
0
Floaters
0
s2 (1-g)
s3 (1-g)
0
Breeders
0
s2 (g)
s3 (g)
s4
b = probability that a territory-holding female will breed in a given
year
c = number of cubs produced by females that do breed
p = proportion of cubs that are female
Gaona et al. (1998)
lynx population in Doñana National Park post
breeding census birth-pulse population
• Density dependence acts on g, which
represent the probability that a surviving
juvenile or floater will acquire a territory
next year
• Just before the birth pulse that precedes
census t+1, there will be s4n4(t) breeding
females still in possession of a territory
and K- s4n4(t) vacant territories available.
Gaona et al. (1998)
lynx population in Doñana National Park post
breeding census birth-pulse population
• g= [K-s4n4(t)]/[s2n2(t)+ s3n3(t)]
Density dependent functions
s0 E t   s0 0exp  E t 
The Ricker function
Density dependent functions
s0 0
s0 E t  
1  E t 
Beverton-Holt model
Density dependent functions
Et    bi fi ni (t )
I
II
III
IV
V
I
0
0
b3f3s0E(t)
b4f4s0E(t)
b5f5s0E(t)
II
s1
0
0
0
0
III
0
s2
0
0
0
IV
0
0
s2 (1-b3)
0
0
V
0
0
0
s3 (1-b4)
0