Ecology 8310
Population (and Community) Ecology
• HW #1
• Age-structured populations
• Stage-structure populations
• Life cycle diagrams
• Projection matrices
Context: Sea Turtle Conservation
(But first … background)
Population
Structure:
From vianica.com
Life Cycle
Diagram:
Age 0
Age 1
Age 2
Age-based approach. What now?
Age 3
Life Cycle
Diagram:
More transitions?
Age 0
Dead
Age 1
Age 2
Age 3
Are we done?
Life Cycle
Diagram:
Add values?
Pij = Per capita transition from
group j to i
P14
P13
Group 1,
Age 0
1-P21
Dead
P21
Group 2,
Age 1
1-P32
P32
1-P43
Group 3,
Age 2
1-P54
P43
Group 4,
Age 3
Now what?
Projections:
Project from time t to
time t+1….
P13
Group 1,
Age 0
1-P21
Dead
P21
Group 2,
Age 1
1-P32
P32
1-P43
P14
Group 3,
Age 2
1-P54
P43
Group 4,
Age 3
Projections:
• nx,t = abundance (or density) of class x
at time t.
• So, given that we know n1,t, n2,t, ….
• And all of the transitions (Pij's)…
• … What is n1,t+1, n2,t+1, n3,t+1, … ?
Projections:
n2,t+1 = ??
= P21 x n1,t
P14
P13
Group 1,
Age 0
Dead
P21
Group 2,
Age 1
P32
Group 3,
Age 2
P43
Group 4,
Age 3
Projections:
n1,t+1 = ??
= (P14 x n4,t) +
(P13 x n3,t)
P14
P13
Group 1,
Age 0
Dead
P21
Group 2,
Age 1
P32
Group 3,
Age 2
P43
Group 4,
Age 3
Project what?
Is there a way to write this out more formally
(e.g., as in geometric growth model)?
Matrix
algebra:
nt 1  Ant
n is a vector of abundances
for the groups;
A is a matrix of transitions
Note similarity to:
nt+1 = lnt
é
ê
ê
n =ê
ê
ê
ë
é
ê
ê
A =ê
ê
ê
ë
n1
n2
n3
n4
Matrix
algebra:
ù
ú
ú
ú
ú
ú
û
For our age-based
approach
P11
P12
P13
P14
P21
P22
P23
P24
P31
P32
P33
P34
P41
P42
P43
P44
ù é
ú ê
ú ê
ú=ê
ú ê
ú ê
û ë
0
P12
P13
P21
0
0
0
P32
0
0
0
P43
P14 ù
ú
0 ú
ú
0 ú
0 úû
Matrix
algebra:
P11
P
21

nt 1  Ant 
P31

P41
 P11n1  P12 n2
P n  P n
21 1
22 2

nt 1 
 P31n1  P32 n2

P41n1  P42 n2
P12
P13
P22
P23
P32
P33
P42
P43
P14   n1,t 
n 

P24   2,t 
P34  n3,t 
 
P44  n4,t 
 P13 n3  P14 n4   n1,t 1 
n 

 P23 n3  P24 n4   2,t 1 

 P33 n3  P34 n4  n3,t 1 

 
 P43 n3  P44 n4  n4,t 1 
Our age-based
example:
P14
P13
Group 1,
Age 0
P21
Group 2,
Age 1
P32
Group 3,
Age 2
P43
Group 4,
Age 3
 0 0 P13 P14   0 0 F3 F4 
P



0 0 0
P21 0 0 0
21


A
 0 P32 0 0   0 P32 0 0 

 

 0 0 P43 0   0 0 P43 0 
A simpler
example:
P12
Group 1
P21
Group 2
 0 P12

A  P21 0
 0 P32
P13
P32
Group 3
P13   0 1 3



0   .6 0 0
0   0 .5 0
Simple
example:
 0 P12 P13   0 1 3 

 

A  P21 0 0   .6 0 0 
 0 P32 0   0 .5 0 
100 


nt   0 
What is nt+1?
 0 
Simple
example:
 0 1 3 100




nt 1  Ant  .6 0 0  0  
 0 .5 0  0 
 (0 x100)  (1x 0)  (3 x 0)   0 
(.6 x100)  (0 x 0)  (0 x 0)  60

  
(0 x100)  (.5 x 0)  (0 x 0)  0 
Simple
example:
P12
Group 1
P21
Group 2
 0 P12

Ant  P21 0
 0 P32
P13
P32
Group 3
P13   n1   0 1 3 100







0  n2   .6 0 0  0 
0  n3   0 .5 0  0 
Simple
example:
Time:
1
n1,t 100
2
3
4
5
6
7
0
60
90
36
108 103
n2,t
0
60
0
36
54
22
65
n3,t
0
0
30
0
18
27
11
Nt
100
60
90
126 108 157 179
Time:
1
2
3
4
5
6
n1
100
0
60
90
36
108
n2
0
60
0
36
54
22
n3
0
0
30
0
18
27
N
100
60
90
126
108
157
n1/N
n2/N
n3/N
1.0
0
0
.60
0
1.0
0
1.50
0.33
0.50
0.17
1.45
0.69
0.14
0.17
1.14
Annual growth
rate=(Nt+1/Nt)
0.67 0.71
0
0.29
0.33
0
1.40 0.86
Let's plot this…
Dynamics:
What about a longer timescale?
300
Abundance
250
200
N
150
n1
100
n2
50
n3
0
0
1
2
3
Time (years)
4
5
6
Dynamics:
Are the age classes growing at similar rates?
4000
Abundance
3500
N
3000
2500
n1
2000
1500
n2
1000
n3
500
0
0
5
10
15
Time (years)
20
25
Dynamics:
Thus, the composition is constant…
Abundance
10000
N
n1
n2
n3
1000
100
10
1
0
5
10
15
Time (years)
20
25
Age
structure:
1
Stable
Age
Distribution
(SAD)
0.8
Proportion
Constant proportions
through time =
Group 1
0.6
0.4
Group 2
0.2
Group 3
0
0
5
10
15
20
25
Time (years)
If no growth (Nt=Nt+1), then:
1. SAD  Stationary Age Distribution
2. SAD is the same as the “survivorship curve” …
(return later)
Dynamics:
If A constant,
then SAD, and
Geometric growth
Nt+1/Nt = l
4000
Abundance
3500
Nt=N0lt
N
3000
2500
n1
2000
1500
n2
1000
n3
500
0
0
5
10
15
Time (years)
20
25
Here, l=1.17
How do we obtain a survivorship schedule from our
transition matrix, A?
Survivorship
schedule:
p(x) = Probability of surviving from age x to age x+1
(same as the “survival” elements in age-based
transition matrix: e.g. p(0)=P21).
l(x) = Probability of surviving from age 0 to age x
l(x) = Pp(x) ; e.g., l(2)=p(0)p(1)
Survivorship
schedule:
Recall:
“Group”
1
2
3
4
é 0
ê
A = ê P21
ê
êë 0
Age, x
0
1
2
3
P12
0
P32
P13 ù é 0 1 3 ù
ú ê
ú
0 ú = ê .6 0 0 ú
ú ê 0 .5 0 ú
0 úû ë
û
Px+2,x+1=p(x)
0.6
0.5
0.0
0.0
l(x)
1.0
0.6
0.3
0.0
Survivorship
curves:
Age specific survival?
Back to the question:
The age distribution should mirror the
survivorship schedule.
Does it?
Survivorship
curves:
Does the age distribution match the
survivorship curve?
“Group”
Age, x
l(x)
1
2
3
0
1
2
1.0
0.6
0.3
Stable
A.D.
0.58
0.30
0.13
Rescaled
AD
1.0
0.52
0.22
Why not?
Survivorship
The population increases 17% each year
curves:
So what was the original size of each
cohort? And how does that affect SAD?
4000
Abundance
3500
N
3000
2500
n1
2000
1500
n2
1000
n3
500
0
0
5
10
15
Time (years)
20
25
Survivorship
curves:
Population Growth!
How can we adjust for growth?
l(x)
1.0
0.6
0.3
Stable
A.D.
0.58
0.30
0.13
Adjusted Rescaled
by growth
=0.58/1.172
1.0
=0.30/1.17
0.6
0.13
0.3
Survivorship
curves:
1. Static Method: count individuals at
time t in each age class and then
estimate l(x) as n(x,t)/n(0,t)
Caveat: assumes each cohort
started with same n(0)!
2. Cohort Method: follow a cohort
through time and then estimate
l(x) as n(x,t+x)/n(0)
Reproductive
Value:
• Contribution of an individual to
future population growth
• Depends on:
• Future reproduction
• Pr(surviving) to realize it
• Timing (e.g., how soon – so
your kids can start
reproducing)
Reproductive
Value:
• How can we calculate it?
• Directly estimate it from
transition matrix (requires math)
• Simulate it
• Put 1 individual in a stage
• Project
• Compare future N to what you
get when you put the 1
individual in a different stage
Reproductive
Value:
Group (Age
class)
1 (0)
2 (1)
3 (2)
N (t=25)
34
67
88
Reproductive
Value
1.0
1.9
2.6
Reproductive
Value:
• Always increase up to
From vianica.com
maturation (why?)
• May continue to increase after
maturation
• Eventually it declines (why?)
• Why might this be useful for
turtle conservation policy?
Issues we've ignored:
•
•
•
•
Non-age based approaches
Density dependence
Other forms of non-constant A
How you obtain fecundity and survival
data (and use it to get A)
• Issues related to timing of the projection
vs. birth pulses
• Sensitivities and elasticities
• How you obtain the SAD and RV's (right
and right eigenvectors) and l (dominant
eigenvalue)
Generalizing
the approach:
Age-structured:
Age 0
Age 1
Age 2
Age 3
Stage 3
Stage 4
Stage-structured:
Stage 1
Stage 2
How will these models differ?
Age-structured:
Age 0
Age 1
Age 2
Age 3
Stage 3
Stage 4
Stage-structured:
Stage 1
Stage 2
To do:
Go back through the previous results
for age-structure and think about how
they will change for stage-structured
populations.
Read Vonesh and de la Cruz (carefully
and deeply) for discussion next time.
We'll also go into more detail about the
analysis of these types of models.