1. FW364 Ecological Problem Solving Class 15: Stage Structure
October 23, 2013

2. Outline for Today
Continue to make population growth models more realistic
by adding in stage structure
Objectives for Today:
Introduce stage structure
Objectives for Next Class:
Show how to obtain stage matrices from census data
Complete stage structure exercises
Text (optional reading):
Chapter 5

3. Stage Structure Introduction
Distribution of individuals in each age class in a population
Stage structure
Distribution of individuals in defined stages in a population
Works well when:
We can determine the exact age of individuals
Age is important for ecology
Works well when:
We cannot determine age of individuals, but we can define stages
Stage is important for ecology

4. Stage Structure Introduction
Types of stages
Physiological
e.g., stages of pregnancy and/or lactation
Morphological
e.g., changes in defensive traits
Size
e.g., 0-10mm, 11-20mm fish
Developmental
e.g., tadpole, frog
e.g., larva, pupa, adult
…or any combination
(often the case)

5. Stage Structure Introduction
When stage can be more important to ecology than age:
Example (from text): Forest trees
Trees can be defined in stages:
seeds, saplings, and canopy trees
 Stages are important for ecology
Saplings cannot become canopy trees until a canopy opening occurs
Because canopy openings typically occur by chance (i.e., sporadically),
the age of a tree is not a good predictor of its likelihood of growing to the canopy (and reaching reproductive maturity)
Stage structured model better than age for many trees

6. Modeling Stage Structure
Work our way into a stage structured model
by building conceptual diagrams:
Case 1: Simple age-structured model
Case 2: Composite age class model (type of stage model)
Case 3: Stage-structured model

7. Case 1: Age Structure Leslie matrix for age-structured population1S 2S 3S
Leslie matrix for age-structured population
Assume 0F = 0

8. Case 1: Age Structure Leslie matrix for age-structured population1F 2F 3F 4F 0S 1S 2S 3S
Leslie matrix for age-structured population
Assume 0F = 0
Not going to assume 1F = 2F = 3F…
With age structure, there are limits to the types of transitions allowed:
Individuals can only:
Move into next higher age class e.g., Age 1  Age 2
Produce offspring e.g., Age 2  Age 0
With stage structure, more transitions are possible
Some plant species can fill every stage transition in a Leslie matrix!

9. Case 1: Age Structure Leslie matrix for age-structured population1F 2F 3F 4F 0S 1S 2S 3S
Leslie matrix for age-structured population
Assume 0F = 0
Not going to assume 1F = 2F = 3F…
Let’s build a conceptual model
Can use box and arrow diagrams to represent age/stage transitions 0S 1S
Age 0
Age 1
Age 2 2F
Conceptually similar to stock and flow models:
Boxes are age (stage) classes (like “stocks”)
Arrows are transitions between classes (like “flows”)
NOTE: we are NOT doing / assuming “mass balance” or T = S/F

10. Case 1: Age Structure Leslie matrix for age-structured population1F 2F 3F 4F 0S 1S 2S 3S
Leslie matrix for age-structured population
Assume 0F = 0
Not going to assume 1F = 2F = 3F…
Let’s build a box and arrow model that corresponds to the Leslie matrix above
How many boxes (i.e., age classes) will the model have?
 5 age classes
How many arrows (i.e., transitions) will the model have?
 8 transitions 0S 1S 2S 3S
Age 0
Age 1
Age 2
Age 3
Age 4 4F 3F 2F 1F

11. Case 2: Composite Age Class
In some situations, it might be advantageous
to pool multiple age classes into a single
age class (often pool oldest classes)
e.g., the honeyeater 3+ age class
We are justified in pooling age classes
when vital rates for classes do not vary
For many organisms (mammals, birds),
- Individuals do not grow after reaching maturity
- Fecundity and survival may not be very different after reaching maturity
 Can pool age classes into a composite age class to simplify modeling
e.g., Pool age 3 individuals and older into class for age 3+
age 0
age 1
age 2
age 3+

12. Case 2: Composite Age Class
In some situations, it might be advantageous
to pool multiple age classes into a single
age class (often pool oldest classes)
e.g., the honeyeater 3+ age class
We are justified in pooling age classes
when vital rates for classes do not vary
age 0
age 1
age 2
age 3+
Composite age classes (e.g., the honeyeater age 3+ class)
are not really true age classes
Rather, composite age classes are a type of stage class
Let’s build a conceptual model with a composite age class

13. Case 2: Composite Age Class
How would our conceptual model below change if we pooled ages 3 and 4? 0S 1S 2S 3S
Age 0
Age 1
Age 2
Age 3
Age 4 4F 3F 2F 1F
We would have:
Box for Age 3+
and age 0, 1, and 2
Fecundity for Age 3+
and age 1 and 2
Survival within Age 3+
and other between-class survivals
“Retention”
in age 3+
3+S 0S 1S 2S
Age 0
Age 1
Age 2
Age 3+
3+F 2F 1F

14. Case 2: Composite Age Class
3+F 2F 1F
There are two possible transitions to the age 3+ class:
- From age 2 to age 3+ (2S)
- Age 3+ remaining in age 3+ (3+S)
To make the transitions more explicit, we’ll use a new notation for survivals:
33S
01S
12S
23S
Age 0
Age 1
Age 2
Age 3+
3+F 2F 1F

15. Case 2: Composite Age Class
With the new notation:
01S : Survival from age 0 to age 1
12S : Survival from age 1 to age  NOT survival for age 12!
23S : Survival from age 2 to age  NOT survival for age 23!
33S : Survival within age  NOT survival for age 33!
Flexible notation for stages; also may help with remembering transitions 0S 1S 2S
Age 0
Age 1
Age 2
Age 3+
3+F 2F 1F
There are two possible transitions to the age 3+ class:
- From age 2 to age 3+ (2S)
- Age 3+ remaining in age 3+ (3+S)
To make the transitions more explicit, we’ll use a new notation for survivals:
33S
01S
12S
23S
Age 0
Age 1
Age 2
Age 3+
3+F 2F 1F

16. Case 2: Composite Age Class
3+F 2F 1F
In-Class Challenge:
What would a Leslie matrix look like with a composite age class?
See if you can figure out the (general) Leslie matrix for the model above
Some hints / guiding questions:
How many rows and columns will the matrix have?
Where will 33S go?
 Talking through the transitions for each matrix element will help

17. Case 2: Composite Age Class Leslie matrix with composite age class:
3+F 2F 1F
Leslie matrix with composite age class: X
How many rows and columns?
Where does 33S go?

18. Case 2: Composite Age Class Leslie matrix with composite age class:
3+F 2F 1F
Leslie matrix with composite age class:
Key Features:
New notation for survivals
33S goes on the diagonal
All “within class” rates go on diagonal
Fecundities do not look very different 1F 2F
3+F
01S
12S
23S
33S
 We can now take this general form and build a full stage-structure model

19. Stage structure example: Leopard Frogs – 4 stages
Case 3: Stage Structure
Stage structure example: Leopard Frogs – 4 stages
Tadpoles and frogs can remain in a stage for more than one time step:
 We’ll have transitions within Stages 2, 3, and 4
Stage 1: Eggs
Stage 2: Tadpole 1
Stage 3: Tadpole 2 (tadpole with legs, but still aquatic)
Stage 4: Frog (adult; only mature stage)

20. Leopard frog conceptual model: And corresponding Leslie matrix:
Case 3: Stage Structure
Leopard frog conceptual model:
11S
22S
33S
01S
12S
23S
Egg (0)
Tadpole 1
Tadpole 2
Frog (3) 3F
And corresponding Leslie matrix:
Key Features:
Fecundity for frogs only (3F)
Within stage survivals on the diagonal: 11S, 22S, 33S
Between stage survivals on sub-diagonal: 01S, 12S, 23S
All other transitions are zeros (not possible for leopard frogs) 3F
01S
11S
12S
22S
23S
33S

21. Stage Structure – Forecasting Growth
As we did for age structure,
we can forecast stage-structured population growth using:
- Stage structured Leslie matrix
- Vector of stage sizes (xNt)
(number of individuals, N, at stage class, x, at time, t)
Leopard frog Leslie matrix
xNt 3F
01S
11S
12S
22S
23S
33S
0Nt
1Nt
2Nt
3Nt
Each element is the number of individuals in each stage
Let’s do some examples

22. Stage Structure – Forecasting Growth
Let’s start with a population of solely 1000 stage 1 tadpoles:
xNt
1000
Stage 0 : Eggs
Stage 1 : Tadpole 1
Stage 2 : Tadpole 2
Stage 3 : Adult
We need to put some numbers in the Leopard frog Leslie matrix 3F
01S
11S
12S
22S
23S
33S
100
0.3
0.1
0.4
0.05
0.2 

23. Stage Structure – Forecasting Growth
In-Class Exercise: Forecasting to time t+1
Leopard frog Leslie matrix
xNt
xNt+1
100
0.3
0.1
0.4
0.05
0.2
1000 ? = *
Nt = 1000
Nt+1 = ?

24. Stage Structure – Forecasting Growth
In-Class Exercise: Forecasting to time t+1
Leopard frog Leslie matrix
xNt
xNt+1
100
0.3
0.1
0.4
0.05
0.2
1000
100 = *
Nt = 1000
Nt+1 = 200
80% population decline

25. Stage Structure – Forecasting Growth Leopard frog Leslie matrix
Forecasting to time t+2
Leopard frog Leslie matrix
xNt+1
xNt+2
100
0.3
0.1
0.4
0.05
0.2
100 10 50 5 = *
Nt+1 = 200
Nt+2 = 65
68% population decline

26. Stage Structure – Forecasting Growth Leopard frog Leslie matrix
Forecasting to time t+3
Leopard frog Leslie matrix
xNt+2
xNt+3
100
0.3
0.1
0.4
0.05
0.2 10 50 5
500 1 21 4 = *
Nt+2 = 65
Nt+3 = 526
709% population increase

27. Stage Structure – Forecasting Growth Leopard frog Leslie matrix
Forecasting to time t+4
Leopard frog Leslie matrix
xNt+3
xNt+4
100
0.3
0.1
0.4
0.05
0.2
500 1 21 4
400
150 9 2 = *
Nt+3 = 526
Nt+4 = 561
If we kept going, we’d find out the population starts steadily shrinking!
 Try on your own
7% population increase

28. Stage Structure – Forecasting Growth
Let’s look a bit closer at the realism of the leopard frog Leslie matrix
Leopard frog Leslie matrix
These were just
made-up data…
… let’s consider the ecological plausibility
100
0.3
0.1
0.4
0.05
0.2
Would we expect 11S (0.1) to be less than 22S (0.4)?
Means “retention” of stage 1 tadpoles as stage 1 is less than “retention” of stage 2 tadpoles as stage 2
(i.e., individuals stay in stage 2 tadpole longer than stage 1 tadpole)
 Depends on ecology we don’t know (but we could research)

29. Stage Structure – Forecasting Growth
Let’s look a bit closer at the realism of the leopard frog Leslie matrix
Leopard frog Leslie matrix
These were just
made-up data…
… let’s consider the ecological plausibility
100
0.3
0.1
0.4
0.05
0.2
Would we expect 12S (0.1) to be greater than 23S (0.05)?
Means survival from stage 1 tadpole to stage 2 tadpole is greater than survival of stage 2 tadpole to adults
 Maybe not too likely; survival is typically better at later stages…
… although for the frog example, 23S represents an important transition from water to land, which could involve high mortality

30. Stage Structure – Forecasting Growth
Let’s look a bit closer at the realism of the leopard frog Leslie matrix
Leopard frog Leslie matrix
These were just
made-up data…
… let’s consider the ecological plausibility
100
0.3
0.1
0.4
0.05
0.2
Are there other general restrictions on the components of a stage-structured Leslie matrix?
All survival rates must be less than 1

31. Stage Structure – Forecasting Growth
Let’s look a bit closer at the realism of the leopard frog Leslie matrix
Leopard frog Leslie matrix
These were just
made-up data…
… let’s consider the ecological plausibility
100
0.3
0.1
0.4
0.05
0.2
Are there other general restrictions on the components of a stage-structured Leslie matrix?
All survival rates must be less than 1
Survival column sums must also be less than 1!
e.g., 11S + 12S + 13S < 1
 < 1

32. Stage Structure – Forecasting Growth
Let’s look a bit closer at the realism of the leopard frog Leslie matrix
Leopard frog Leslie matrix
These were just
made-up data…
… let’s consider the ecological plausibility
100
0.3
0.1
0.4
0.05
0.2
General point:
It’s good to think about the ecological plausibility
of any Leslie matrix you build…
… and more generally, any model you build!

33. Continue with stage structure Practice stage structure exercises
Looking Ahead
Next Class:
Continue with stage structure
Practice stage structure exercises