1. FW364 Ecological Problem Solving Class 16: Stage Structure
October 28, 2013

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

3. We considered a stage-structured population of leopard frogs:
Recap from Last Class
We considered a stage-structured population of leopard frogs:
Stage 0 : Eggs
Four stages:
Stage 1 : Tadpole 1
Can remain in a stage for more than one time step
Stage 2 : Tadpole 2
Only adult stage reproduces
Stage 3 : Adult frog
We created a box and arrow diagram (i.e., conceptual model) for leopard frogs:
11S
22S
33S
01S
12S
23S
Egg (0)
Tadpole 1
Tadpole 2
Frog (3) 3F

4. Recap from Last Class 11S 22S 33S We created a general Leslie matrix:
And also filled in specific numbers: 3F
01S
11S
12S
22S
23S
33S
100
0.3
0.1
0.4
0.05
0.2 
We created a box and arrow diagram (i.e., conceptual model) for leopard frogs:
11S
22S
33S
01S
12S
23S
Egg (0)
Tadpole 1
Tadpole 2
Frog (3) 3F

5. Recap from Last Class
Saw examples of forecasting population size using
a vector of stage sizes:
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
One item we did not address:
How do we get data to build a stage structured Leslie matrix?

6. Building a Stage Matrix
How do we get data to build a stage structured Leslie matrix?
First, need to decide on stages for a population
For our leopard frog model:
Stage 0 : Eggs
Stage 1 : Tadpole 1
Stage 2 : Tadpole 2
Stage 3 : Adult frogs
Then follow individuals through at least two time steps (time t to t+1)
Can be tricky for some organisms!
Record stage of all individuals at each time step
Let’s look at an example
Need to record mortality as well

7. Building a Stage Matrix
stage at time t 1 2 3 1 2 3 9 2 8 1
Table of census data that could be collected for a frog population across two years
stage at time t+1
Deaths 21 16 11 8
Total 30 20 10
Stage 0 : Eggs
Stage 1 : Tadpole 1
Stage 2 : Tadpole 2
Stage 3 : Adult frogs

8. Building a Stage Matrix
stage at time t
Columns for stage of individuals at time t
(starting stage-structured population) 1 2 3 1 2 3 9 2 8 1
Rows for stage of individuals at time t+1
(ending stage-structured population,
only considering survival, not fecundity)
stage at time t+1
Deaths 21 16 11 8
Number of deaths for each stage at time t
Total 30 20 10
Total number for each stage at time t
Data in the table represent number of individuals
making each transition, for example:
9 individuals transitioned from stage 0 to stage 1
2 individuals remained in stage 1
2 individuals transitioned from stage 1 to stage 2
Stage 0 : Eggs
Stage 1 : Tadpole 1
Stage 2 : Tadpole 2
Stage 3 : Adult frogs

9. Building a Stage Matrix Leopard frog Leslie matrix
stage at time t
Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1
stage at time t+1
0.3
0.1
Deaths 21 16 11 8
Total 30 20 10
To construct the Leslie matrix:
Take each number in the table and divide by the total number for each stage
Result goes in corresponding survival rate position in the Leslie matrix
E.g., 9 / 30 = 0.3 = 01S
2 / 20 = 0.1 = 11S

10. Building a Stage Matrix
stage at time t
Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1 X
0.3
0.1
0.4
0.05
0.2
stage at time t+1
Deaths 21 16 11 8
Total 30 20 10
To construct the Leslie matrix:
Take each number in the table and divide by the total number for each stage
Result goes in corresponding survival rate position in the Leslie matrix
E.g., 9 / 30 = 0.3 = 01S
2 / 20 = 0.1 = 11S
That’s how we get survivals…
… now we need fecundities

11. Building a Stage Matrix Leopard frog Leslie matrix
stage at time t
Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1 X
0.3
0.1
0.4
0.05
0.2
stage at time t+1
Deaths 21 16 11 8
Total 30 20 10
To obtain fecundities:
Determine the reproductive stages
 For leopard frogs, only adults reproduce

12. Building a Stage Matrix Leopard frog Leslie matrix
stage at time t
Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1 X
0.3
0.1
0.4
0.05
0.2
stage at time t+1
Deaths 21 16 11 8
Total 30 20 10
To obtain fecundities:
Determine the reproductive stages
Count # stage-0 individuals at time t+1
Divide # stage-0 individuals at time t+1
by # adults at time t
 For leopard frogs, only adults reproduce
 We’ll say there are 1000 eggs
 1000 eggs / 10 adults = 100

13. Building a Stage Matrix Leopard frog Leslie matrix
stage at time t
Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1
100
0.3
0.1
0.4
0.05
0.2
stage at time t+1
Deaths 21 16 11 8
Total 30 20 10
To obtain fecundities:
Determine the reproductive stages
Count # stage-0 individuals at time t+1
Divide # stage-0 individuals at time t+1
by # adults at time t
 For leopard frogs, only adults reproduce
 We’ll say there are 1000 eggs
 1000 eggs / 10 adults = 100
 3F = 100

14. Building a Stage Matrix
stage at time t
Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1
100
0.3
0.1
0.4
0.05
0.2
stage at time t+1
Deaths 21 16 11 8
Total 30 20 10
Note:
I want you to be aware of how a Leslie matrix
for stage-structured data can be obtained,
but I will not test you on how to create a stage-structured
Leslie matrix from census data

15. Age and Stage Structure Summary
Age structure is just a special case of stage structure
where all individuals transition between stages in exactly one time step
The result: no within stage survivals for age structure
i.e., the diagonal (except for fecundity) is always 0 for age structure 0F 1F 2F 3F 4F 0S 1S 2S 3S
Age and stage structure are sources of deterministic variation
i.e., predictable variation (not stochastic)
(although we can add stochasticity, if desired)

16. Age and Stage Structure Summary
We’ve been making a number of ASSUMPTIONS:
Age structure:
Vital rates for individuals (fertilities and survival chances) are related to their age
Among individuals of the same age, there is little variation in the vital rates
(relative to variation between ages)
Stage structure:
Vital rates for individuals (fertilities and survival chances) are related to their stage
Among individuals of the same stage, there is little variation in the vital rates (relative to variation between stages)
For both age and stage structure:
Working in closed populations  could include immigration or emigration if desired
No environmental or demographic stochasticity  could include if desired (Ramas)
No density dependence of vital rates  could include if desired (Ramas)

17. (note: all of these problems will be posted on the website)
Let’s do some
in-class exercises
(note: all of these problems will be posted on the website)

18. Many insect populations have 4 stages: egg, larvae, pupae, and adult
Exercises
Many insect populations have 4 stages: egg, larvae, pupae, and adult
Gypsy moth
life cycle

19. Exercises
Consider a gypsy moth population with the following vital rates:
Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die
(i.e., no eggs ever stay an egg after one time step)
Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage
Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce)
Exercises
Draw a box and arrow diagrams that illustrate:
A. All of these stages and transitions conceptually
(just use symbols for S and F; use E, L, P, and A for stage notations)
B. All of these stages and transitions using numbers from above

20. Exercises ELS LPS PAS LLS PPS AAS AF
Consider a gypsy moth population with the following vital rates:
Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die
(i.e., no eggs ever stay an egg after one time step)
Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage
Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce)
Exercise A: Conceptual model with symbols
Egg
Larvae
Pupae
Adult
ELS
LPS
PAS
LLS
PPS
AAS AF

21. Exercises
Consider a gypsy moth population with the following vital rates:
Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die
(i.e., no eggs ever stay an egg after one time step)
Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage
Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce)
Exercise B: Conceptual model with numbers
Egg
Larvae
Pupae
Adult
0.3
0.5
0.2
0.1
0.4
100

22. Exercises Exercises Con’t
Consider a gypsy moth population with the following vital rates:
Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die
(i.e., no eggs ever stay an egg after one time step)
Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage
Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce)
Exercises Con’t
C. Construct the Leslie matrix for this gypsy moth population
D. If there are currently 2000 eggs, 200 larvae, 100 pupae, and 100 adults,
how many individuals of each stage will there be next year (i.e., in the next
time step)?

23. Exercises Exercises C: Gypsy moth Leslie matrix
Consider a gypsy moth population with the following vital rates:
Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die
(i.e., no eggs ever stay an egg after one time step)
Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage
Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce)
Exercises C: Gypsy moth Leslie matrix
100
0.3
0.2
0.5
0.1
0.4

24. Exercises Exercises D: Forecasting growth
Consider a gypsy moth population with the following vital rates:
Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die
(i.e., no eggs ever stay an egg after one time step)
Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage
Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce)
Exercises D: Forecasting growth
Starting with:
2000 eggs
200 larvae
100 pupae
100 adults
100
0.3
0.2
0.5
0.1
0.4
2000
200
100
10000
640
110 90 * =

25. Exercises
In the stage-structured Leslie matrix below, there is one unknown transition (labeled with an X): 75
0.2 X
0.4
0.3
0.5
Which of the following could be the value of that transition (more than one answer may be correct)?
a. 0
b. 1.2
c. 0.3
d. 0.8

26. Exercises
In the stage-structured Leslie matrix below, there is one unknown transition (labeled with an X): 75
0.2 X
0.4
0.3
0.5
Which of the following could be the value of that transition (more than one answer may be correct)?
a. 0
b. 1.2
c. 0.3
d. 0.8

27. (i.e., spatial structure)
Looking Ahead
Next Two Classes:
Metapopulations
(i.e., spatial structure)
Lab Tomorrow