matrix population models for wildlife conservation and management 27 february - 5 march 2016...
DESCRIPTION
When the model matrix varies from year to year…. Time-Varying Models: … in a known fashion over finite time window Recorded sequence of bad and poor years Relationship between demographic parameter and env. covariate MAIN AIM: model a known trajectory (retrospective) Random Environment: … in a random fashion over a finite or infinite time window Projection of relationship between parameter and env. covariate Unexplained year-to-year (environmental) variation MAIN AIM: projection, asymptotic behavior (prospective)TRANSCRIPT
Matrix Population Models for Wildlife Conservation
and Management27 February - 5 March 2016
Jean-Dominique LEBRETON Jim NICHOLS
Madan OLI Jim HINES
Lecture 6Time-Varying and Random environment
Matrix models
Shripad TULJAPURKAR ("Tulja") who extensively developed the theory of population models in random environment
When the model matrix varies from year to year….
Time-Varying Models: … in a known fashion over finite time window• Recorded sequence of bad and poor years• Relationship between demographic parameter and env. covariate• MAIN AIM: model a known trajectory (retrospective)
Random Environment: … in a random fashion over a finite or infinite time window• Projection of relationship between parameter and env. covariate • Unexplained year-to-year (environmental) variation• MAIN AIM: projection, asymptotic behavior (prospective)
Survival of storks in Baden-Württemberg estimated with rainfall in Sahel as a covariate
1957 1959 1961 1963 1965 1967 1969 19710.3
0.4
0.5
0.6
0.7
0.8
0.9
1Es
timat
ed s
urvi
val
Phi(rain)
Model (St,p) (Srain,p) (S,p)AIC 1349.50 1339.15 1356.10
log (S / (1- S)) = a + b rain
Storks in Baden-Württemberg: Modelling numbers with survival driven by rainfall in Sahel
Year 57 58 … i i+1 … 74
Rain x57 x58 … xi xi+1 ... x75
Survival 57 58 ... i i+1 ... 75
Matrix M57 M58 … Mi Mi+1 ... M75
Numbers obtained by a « time-varying matrix model » (using N56 based on average stable age structure): Ni+1=Mi*Ni
0
20
80
4060
100120
140
160
180
57 59 61 63 65 67 69 71 73 75
An "ad hoc"comparison(o = model)(x =census)based ona3Ni(3)+Ni(4)(Nr of breeders)
Storks in Baden-Württemberg: Modelling numbers with survival driven by rainfall in Sahel
Likelihood based approach to time-varying models:
State-space modelGreater snow goose
Observed census
smoothed pop size(dotted lines: 95 % CI)
Random Environmentthe scalar exponential model
(no stage/age classes)
n(t)=At n(t-1)
At random scalar (i.i.d.) , with E(At)=
E(n(t) / n(t-1) ) = n(t-1)
E(n(t) / n(0) ) = t n(0)
Random environmentIncreasing variability
At = 1.15 with probability 1 = 1.15
At = 1.2 and 1.1 with prob. 0.5 = 1.15
At = 1.4 and 0.9 with prob. 0.5 = 1.15
At = 2.0 and 0.3 with prob. 0.5 = 1.15
At = 1.15 with probability 1 = 1.15
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10
12
14
Time
log Pop. Size
At = 1.2 and 1.1 with prob. 0.5 = 1.15
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10
12
14
log Pop. Size
Time
At = 1.4 and 0.9 with prob. 0.5 = 1.15
0 10 20 30 40 50 60 70 80 90 100-2
0
2
4
6
8
10
12
14
log Pop. Size
Time
At = 2.0 and 0.3 with prob. 0.5 = 1.15
0 10 20 30 40 50 60 70 80 90 100-25
-20
-15
-10
-5
0
5
log Pop. Size
Time
Where is the paradox ?
n(t) = At n(t-1) n(t) = n(0) Ai
log n(t) - log n(0) = log Ai
Central Limit Theorem: log Ai Normal distribution
log n(t) - log n(0) Normal ( t , ²t)
a = E(log At ) , v = var(log At ) t = a t ²t = v t
The paradox is still there
log n(t) - log n(0) Normal ( t , ²t)
t = a t , ²t = v t
(log n(t) - log n(0) )/t Normal (a , v/t)
1/t log n(t) Normal (a , v/t)
v/t 0 when t
1/t log n(t) log s = E (log At)
However 1/t log E(n(t)) log = log E(At)
Is the paradox still there ?
Most probable and expected trajectories differ
EXPECTED MOST PROBABLE
At values log log s
1.15 1.15 0.1398 0.1398
1.2 1.1 1.15 0.1398 0.1388
1.4 0.9 1.15 0.1398 0.1156
2.0 0.3 1.15 0.1398 -0.2554
There is no real paradox n(t) log-normal distribution Median < Expectation
At = 1.2 and 1.1 with prob. 0.5
Distribution for a large t
0 0.5 1 1.5 2 2.5 3 3.5
x 106
0
5
10
15
20
25
30
Nt
12.5 13 13.5 14 14.5 15 15.50
5
10
15
20
25
30
Log Nt
-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 00
5
10
15
20
25
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
10
20
30
40
50
60
70
80
90
100
There is no real paradox n(t) log-normal distribution Median < Expectation
At = 2.0 and 0.3 with prob. 0.5
Distribution for a large t
A few trajectories with large growth rate keep
the expected growth rate equal to log = log E(At)
Most probable trajectories are concentrated
around log s = E ( log At ) (more and more when t )
There is no real paradox
log s is a relevant measure of growth rate
Environmental variability influences population growth
Environmental variability depresses Population growth
A deterministically growing population may decrease
0 0.5 1 1.5 2 2.5-1.5
-1
-0.5
0
0.5
1
log
log s
Jensen’s inequality: log s=E(log At ) log =log E(At)
0 0.5 1 1.5 2 2.5-1.5
-1
-0.5
0
0.5
1
The effect of Environmental variability on Population growth
How do these results extend to
age or (stage-) classified populations ?
The Barn Swallow example
1st y n1(t-1) n1(t)
After 1st y n2(t-1) n2(t)
s0
s1
s2
f2
f1
f1s0 f2s0
A = s1 s2
Average valuess0= 0.2 f1, f2 =3/2, 6/2
s1=0.5 s2= 0.65
+ random Survival with variance ²
E(log N(t)), Constant Environment
0 10 20 30 40 50 60 70 80 90 1002
3
4
5
6
7
8
log s= 0.0488
log =0.0488
E(log N(t)), Random Environment =0.02
0 10 20 30 40 50 60 70 80 90 1002
3
4
5
6
7
8
log s= 0.0581
log = 0.0488
0 10 20 30 40 50 60 70 80 90 1002
3
4
5
6
7
8
log s= 0.0745
log = 0.0488
E(log N(t)), Random Environment =0.05
0 10 20 30 40 50 60 70 80 90 1002
3
4
5
6
7
8
log s= 0.0628
log = 0.0488
E(log N(t)), Random Environment =0.10
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
log s=-0.2862
log = 0.0488
E(log N(t)), Random Environment =0.15
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
log s=-0.2862
log = 0.0488
E(log N(t)), Random Environment =0.15
Same depression of growth as in the scalar case? (most likely trajectories tend to be below average trajectory)
Phase plane, Constant Environment
0 100 200 300 400 500 600 7000
100
200
300
400
500
600
700
800
900
n2(t)
n1(t)
Phase plane =0.02
0 50 100 150 200 250 300 350 400 4500
50
100
150
200
250
300
350
400
450
500
n2(t)
n1(t)
0 100 200 300 400 500 600 700 800 900 10000
200
400
600
800
1000
1200
Phase plane =0.05
n2(t)
n1(t)
Same depression as above …plus a constant mismatch in age structure
log s log(E(At)) = log
log s E(log(At)), At being a matrix
log s log E ((At)), nt eigenvector of A
Convergence of 1/t log c’ n(t) to log s
even under correlated environments(“Ergodic theorems on products of random matrices”)
How to calculate the asymptotic growth rate ?
Simulation or Approximation
“The computation is considerably more involved
than in the scalar case” S.Tuljapurkar (1990)
Estimation of log s Simulation and approximation
Simulation: Matlab, ULM.
Large number of repetitions needed
Approximation : valid from small variability
For uncorrelated parameters: 1 2 log s log - ____ ___ var() 2 ²
APPROXIMATE log log s
0.00 1.05 0.0488 0.0488
0.01 1.05 0.0488 0.0486
0.05 1.05 0. 0488 0.0442
0.10 1.05 0.0488 0.0304
0.15 1.05 0.0488 0.0075
0.20 1.05 0.0488 -0.0246
Estimation of log s Approximation
Random Environment simulation in ULM
{ juvenile survival rate defvar s0 = 0.2
{ subadult survival ratedefvar s = 0.35
{ adult survival ratedefvar v = 0.5
{ juvenile survival rate defvar s0 = 0.1+rand(0.2)
{ subadult survival ratedefvar s = 0.35
{ adult survival ratedefvar v = 0.5
Fixed environment Random Environment
In batch file (or using interpreted command ²changevar²),Just define the parameter of interest as random.Several continuous distributions are available. The parameter value will be evaluated at each time step.² 0.1+rand(0.2)² 0.1 +Unif[0, 0.2] Unif[0.1, 0.3]
