1. Matrix modeling of age- and stage- structured populations in R
Chris Free

2. What is a population model?
A model is a representation of reality.
A population model is no different!
A population model is a simplification of a biological system.
Complexity can be incorporated with additional data and programming skills.

3. Why use population models?
Estimate r (intrinsic growth rate) or k (carrying capacity) and other parameters
Predict consequences of conservation or management strategies
Understand the spread of parasites, viruses, and diseases
Compare populations in different locations

4. A population can be described by its demographic rates
Nt+1 = Nt + Bt – Dt + It - Et
N = Abundance
B = Birth rate
D = Death rate
I = Immigration rate (individuals entering the population)
E = Emigration rate (individuals leaving the population)
These rates often vary between age classes and its important to account for these variations in population modeling. This can be done using matrix models

5. Demographic rates are age-specific

6. What information is needed to build an age-structured matrix model?
Age of females in population
Age-specific survivorship (lx, proportion of females surviving to age x)
Age-specific fecundity (mx or bx, number of female offspring born to females of age x)
Easier for some taxa then others!
Must be able to age individuals, follow survival, and measure reproductive output

7. How to build a life table
Age (x) nx lx sx mx
lxmx
1000
1.0
0.5
0.0 1
500
0.2 2
100
0.1
5.0 3 50
0.05
9.0
4.5 4 5
---
nx = number of female of age x
lx = probability a female newborn attains age x
(calculated as nx / n0)
sx = age-specific survival - survival between age x to x+1
(calculated as lx+1 / lx)
mx = number of female progeny per female
lxmx= age-specific fecundity – per capita average number of female progency from a female of age x (calculated as Ix * Mx)

8. What can you learn from a life table?
Age (x) nx lx sx mx
lxmx
1000
1.0
0.5
0.0 1
500
0.2 2
100
0.1
5.0 3 50
0.05
9.0
4.5 4 5
---
Net reproductive rate (R0, net female replacement rate): average lifetime number of offspring produced per female
Generation time (G): average age of the parents of newborn individuals
Population growth rate (r):
ln(R0) / G puts you within 10% of the true r
OR use the Euler equation:

9. Life tables help setup matrix models
Age (x) nx lx sx mx
lxmx
1000
1.0
0.5
0.0 1
500
0.2 2
100
0.1
5.0 3 50
0.05
9.0
4.5 4 5
---
initial
abundance
(Nx)
age-specific
survivorship
(Px)
age-specific
fecundity
(Fx)
Age-structuctured matrix models are powered by Leslie matrices

10. Leslie matrices
Example 1: A Leslie matrix for an organism living 5 years
with 2-5 year olds reproducing:
Life cycle diagram P1 F2 P2 F3 P3 F4 P4 F5
Age-specific fertilities in the first row
Age-specific survival probabilities in the diagonal

11. Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: 6 P5 F6
How should we modify the life cycle diagram?
removed P1 P2 F3 P3 F4 P4 F5 P5 F6
What should the Leslie matrix look like?

12. How does matrix multiplication work?
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing: 6 P5 F6
Abundancet+1
Leslie matrix
Abundancet
N1,t+1
N2,t+1
N3,t+1
N4,t+1
N5,t+1
N6,t+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Note: we have to use matrix multiplication!

13. Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing:
Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Leslie matrix
Abundancet

14. N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6
Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing:
Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Leslie matrix
Abundancet
N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6

15. Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing:
Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Leslie matrix
Abundancet
N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6
N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0

16. Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing:
Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Leslie matrix
Abundancet
N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6
N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0

17. Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing:
Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Leslie matrix
Abundancet
N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6
N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0

18. Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing:
Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Leslie matrix
Abundancet
N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6
N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*0 + N4,t*P4 + N5,t*0 + N6,t*0

19. Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing:
Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Leslie matrix
Abundancet
N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6
N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*0 + N4,t*P4 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*P5 + N6,t*0

20. Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing:
Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Leslie matrix
Abundancet
N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6
N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*0 + N4,t*P4 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*P5 + N6,t*0
If we do this over and over, we can track age-specific abundance over time.
This will allows us to learn things like the finite rate of increase (λ), stable age distribution (SAD), and the reproductive value, sensitivity, and elasticity of each age class.

21. Leslie matrices
Example 2: A Leslie matrix for an organism living 6 years with 3-6 year olds reproducing:
Abundancet+1 P1 P2 F3 P3 F5 P5 F4 P4 F6
N1,t
N2,t
N3,t
N4,t
N5,t
N6,t
Leslie matrix
Abundancet
N1,t*0 + N2,t*0 + N3,t*F3 + N4,t*F4 + N5,t*F5 + N6,t*F6
N1,t*P1 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*P2 + N3,t*0 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*P3 + N4,t*0 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*0 + N4,t*P4 + N5,t*0 + N6,t*0
N1,t*0 + N2,t*0 + N3,t*0 + N4,t*0 + N5,t*P5 + N6,t*0
If we do this over and over, we can track age-specific abundance over time.
This will allows us to learn things like the finite rate of increase (λ), stable age distribution (SAD), and the reproductive value, sensitivity, and elasticity of each age class.

22. Leslie matrices Simulate age-specific abundance over time
Evaluate population trends under different management strategies
Measure important characteristic about the populations dynamics

23. What can we learn about the population from Leslie matrices?
λ, finite rate of increase: the direction and rate of population growth, λ < 1 = decreasing and λ > 1 = increasing population
The dominant eigenvalue of the Leslie matrix
Stable age distribution: the population structure that would emerge if demographic rates (F, P) don’t change
The eigenvector of the dominant eigenvalue of the Leslie matrix
Reproductive value: the expected contribution of an individual of each age to present and future reproduction
The dominant left eigenvector of the Leslie matrix
Sensitivities and elasticities: sensitivities indicate the direct contribution of each transition to λ; elasticities are sensitivities weighted by transition probabilities; more complicated to measure

24. Advantages/disadvantages of age-structured matrix models
Easy to implement
Fewer assumptions than models without age structure
Don’t need to assume stable age distribution
Sensitivity analyses
Can be modified to incorporate stochasticity, density dependence, stage structure, etc.
Disadvantages
Assumptions
Sex ration is 1:1
Individuals can be aged
Demographic rates are constant in age class
No density dependence
Survivorship and fecundity can be difficult to derive

25. Let’s look at an example…
The authors wanted to know which of the following stage classes is both an important driver of population growth and responsive to management actions.

26. They use a stage-structured model driven by Lefkovitch matrices1 3 2 4 5 6 7 G1 G2 G3 G4 G5 G6 P1 P2 P3 P4 P5 P6 P7 F5 F6 F7
Eggs/
hatchlings
Small
juveniles
Large
Sub-
adults
Novice
breeders
1st-year
remigrants
Mature

27. Matrix modeling exercise
Task #1 Which stage classes should management target?
If management actions could increase the survivorship of the following stage categories by 20%, which set of management actions would be most beneficial?
eggs, hatchlings (1)
small/large juveniles, sub-adults (2-4)
novice breeders, 1st-yr remigrants, mature breeders (5-7)
Hint: Make a model for each management action where you increase survivorship for the targeted categories by 20% of their current values. Which action best promotes population recovery and how do you know?
Task #2 Incorporate stochastic survivorship into your population model.
The survival values in the current projection matrix are constant. Incorporate stochasticity (randomness) into your best model from the task above.
Hint: This can be done using the rnorm() function to randomly generate a value using a normal distribution with a mean of the constant survival and a variance of 0.03, 0.05, and 0.1. You will have to enforce against negative values using pmax().
How do your results change as survival becomes more variable?