copyright 2006 hal caswell applications of markov chains in demography and population ecology hal...
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Copyright 2006 Hal Caswell
Applications of Markov chains in demography and population
ecology
Hal Caswell
Biology Department
Woods Hole Oceanographic Institution
June 2006 Hal Caswell -- Markov Anniversary Meeting
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Markov chains in demography
• particle = individual organism• states = stages in the life cycle• chains absorbing• questions relate to absorbtion
– when– where– timing– perturbation analysis
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• Patterns and rates of transition reflect hypotheses about biological processes– growth– development– maturation– reproduction– mortality
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Discrete demography
• life cycle
• stages
• projection interval
• transitions
• reproduction
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Age-classified life cycle
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Size-classified life cycles
1. Dormant seeds 12. Dormant seeds 23. Small rosettes4. Medium rosettes5. Large rosettes6. Flowering plants
Teasel (Dipsacus sylvestris)
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Wandering albatross (Diomedea exulans)
extracted from Croxall et al. 1990
Stage-classified life cycles
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North Atlantic right whale
calf post-mommommatureimmature
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Analysis within the life cycle
• partition entries into transitions and reproduction
• recognize that death is present as an absorbing state
• life cycle describes dynamics on transient states of an absorbing Markov chain
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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transition probabilities (s x s)
absorbtion probabilities (a x s)
mortality probabilities (1 x s)
reproduction rates (s x s)
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1=calf
2=immature
3=mature
4=mom
5=post-mom
6=dead
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Importance?
• population dynamics
• conservation (growth and persistence, invasion and recolonization)
• evolution (success of life cycles, aging, environmental response)
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June 2006 Hal Caswell -- Markov Anniversary Meeting
17Magnus and Neudecker 1988, Nel 1980
June 2006 Hal Caswell -- Markov Anniversary Meeting
18hours of fun for everybody!
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The North Atlantic right whale (Eubalaena glacialis)
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calving
feeding
Endangered, by any standard
N < 300 individuals
Minimal recovery since 1935
Ship strikes
Entanglement with fishing gear
June 2006 Hal Caswell -- Markov Anniversary Meeting
212030: died October 1999entanglement
1014 “Staccato” died April 1999 ship strike
Mortality and serious injury due to entanglement and ship strikes
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1=calf
2=immature
3=mature
4=mom
5=post-mom
6=dead
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Parameter estimation
• photographic ID catalog
• multistage mark-recapture analysis
• maximum likelihood estimates
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Results
• data 1980-1998• series of statistical models • evidence for declining trend in survival of
moms, and of birth rate• population growth rate declined to negative
levels
Fujiwara and Caswell 2001 Caswell and Fujiwara 2004
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Time-invariant model
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E(number of reproductive events)
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28calf imm mature mom post-mom
0
2
4
6
8
10
12
14
16
18E
xpec
ted
occ
up
ancy
(yr
s)
Right whale calf
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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June 2006 Hal Caswell -- Markov Anniversary Meeting
32calf imm mature mom post-mom
0
0.5
1
1.5C
V o
f st
age
occ
up
ancy
tim
e
Right whale calf
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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510
1520
25
0
10
20
30
0
200
400
600
vec Uvec N
Se
nsi
tiv
ity
d
ij/du
kl
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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June 2006 Hal Caswell -- Markov Anniversary Meeting
40calf imm mature mom post-mom
0
5
10
15
20
25
30
35
40L
ife
exp
ecta
ncy
(yr
s)
Right whale
June 2006 Hal Caswell -- Markov Anniversary Meeting
41calf imm mature mom post-mom
0
0.2
0.4
0.6
0.8
1
1.2
1.4C
V o
f li
fe e
xpec
tan
cy
Right whale
Stage
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The birth interval
• interval between births• determines birth rate• varies over time• responds to environment• especially important in species producing few
offspring• a special case of the “timing problem”• calculation uses machinery for absorbtion times,
but conditional times
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Modified life cycle graph
died before breeding
reproducebefore dying
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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June 2006 Hal Caswell -- Markov Anniversary Meeting
55calf imm mature mom post-mom
0
2
4
6
8
10
12
Starting stage
Mea
n t
ime
to r
epro
du
ctio
n (
yrs)
Right whale
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Time variation
• the vital rates of the right whale changed between 1980 and 1998
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1980 1984 1988 1992 19960.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96C
alf s
urv
ival
time trendbest model
Year
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1980 1984 1988 1992 19960.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1M
oth
er
surv
ival
time trendbest model
Year
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1980 1984 1988 1992 19960.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5B
irth
pro
bab
ility
time trendbest model
Year
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June 2006 Hal Caswell -- Markov Anniversary Meeting
611980 1982 1984 1986 1988 1990 1992 1994 1996 199810
20
30
40
50
60
70
Year
Lif
e ex
pec
tan
cy
period
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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June 2006 Hal Caswell -- Markov Anniversary Meeting
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June 2006 Hal Caswell -- Markov Anniversary Meeting
661980 1982 1984 1986 1988 1990 1992 1994 1996 199810
20
30
40
50
60
70
Year
Lif
e ex
pec
tan
cy
periodcohort
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Summary
• stage-classified demography from Markov chains– expected stage occupancy times– variance in stage occupancy times– expected longevity– variance in longevity– net reproductive rate– generation time– birth and other intervals– period and cohort effects in variable envionments
• perturbation analyses• connection to data• things don’t look good for the right whale