1. Matrix Population Models
Life tables
Intro to matrix multiplication
Examples of age and stage structured models
Elasticity analysis

2. Life Tables and Matrices: Accounting demographic parameters
Age lx mx 2
1.000
0.5147 3
0.6703
1.3618 4
0.4493
1.6819 5
0.3102
1.8816 6
0.2019
2.0257 7
0.1353
2.1358 8
0.0907
2.2347 9
0.0608
2.2686 10
0.0408 11
0.0273 12
0.0183
Stage 1
Stage 2
Stage 3
Stage 4
0.0043
0.1132
0.9775
0.9111
0.0736
0.9534
0.0542
.9804
Killer Whale Lefkovich matrix from Brault, S. and H. Caswell (1993)
Sardine life table(Sardinops sagax) from Murphy (1967)

3. Matrix multiplication
Scalar Multiplication - each element in a matrix is multiplied by a constant.

4. Matrix multiplication
Multiply rows times columns.
You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix.

5. 2 x 3 3 x 2 Matrix multiplication Multiply rows times columns.
You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix.
They must match.
Dimensions:
3 x 2
2 x 3
The dimensions of your answer.

6. 2 x 2 2 x 2 *Answer should be dimension ? 0(4) + (-1)(-2)
0(-3) + (-1)(5)
1(4) + 0(-2)
1(-3) +0(5)

7. Matrix Multiplication (Population Model):=
Answer should be dimension ?
Explain
Matrix
Vector

8. Matrix Multiplication (Population Model):=
N1 * a1,1 + N2* a1,2
N1 * a2,1 + N2* a2,2
Explain
Matrix
Vector

9. Introduction to Matrix Models
Vital rates describe the development of individuals through their life cycle (Caswell 1989)
Vital rates are : birth, growth, development, reproductive, mortality rates
The response of these rates to the environment determines:
population dynamics in ecological time
the evolution of life histories in evolutionary time

10. The general form of an age-structured Leslie Matrix models “Projection Matrix”:f3 f4 p1 P2 p3

11. The general form of an age-structured Leslie Matrix models “Projection Matrix”:f3 f4 p1 P2 p3

12. (Size class at first reproduction)
Age based matrix population model
Fecundity fx
Survivorship sx
Age Class 4
Age Class 3
Age Class 1
Age Class 2
Size class 1
Size class 2
Size class 3
(Size class at first reproduction)
Size class 4
Size based matrix models are useful when
Demographic parameters, such as, fecundity and survivorship are best expressed as funciions of size and not age
Relationship of age and size are not clear
Size class 1
Size class 2
Size class 3
Size class 4 f3 f4 s1 s2 s3

13. The general form of Lefkovitch Matrix model Stage – structured Projection Matrix
g1*s1 f3 f4
g2,1*s1
g2,2*s2
g3,1*s1
g3,2*s2
g3,3*s3
g4,3*s3
g4,4*s4

14. (Size class at first reproduction)
The general form of Lefkovitch Matrix model Stage – structured Projection Matrix
Size class 1
Size class 2
Size class 3
(Size class at first reproduction)
Size class 4
Size class 1
Size class 2
Size class 3
Size class 4
g1*s1 f3 f4
g2,1*s1
g2,2*s2
g3,1*s1
g3,2*s2
g3,3*s3
g4,3*s3
g4,4*s4

15. (Size class at first reproduction)
Stage-based matrix population model
Fecundity fx
Growth gx and Survivorship sx
Size Class 4
Size Class 3
Size Class 1
Size Class 2
Size class 1
Size class 2
Size class 3
(Size class at first reproduction)
Size class 4
Size based matrix models are useful when
Demographic parameters, such as, fecundity and survivorship are best expressed as funciions of size and not age
Relationship of age and size are not clear
Size class 1
Size class 2
Size class 3
Size class 4
g1*s1 f3 f4
g2,1*s1
g2,2*s2
g3,1*s1
g3,2*s2
g3,3*s3
g4,3*s3
g4,4*s4

16. Some of the utility of matrix population models
Population projection – deterministic and stochastic
Elasticity Analysis
Conservation
Management
Meta population dynamics

17. Caswell
Types of model variability

18. Population projection Example: What is the population at t1?
Juvenile Adult t0 fx px
Njuvenile
Nadult fx
NJuvenile
NAdult px

19. fx px Njuvenile, t0 Nadult, t0fx px
Njuvenile, t0
Nadult, t0 x =
Njuvenile, t1 = 0*(Njuvenile, t0) + fx*(Nadult, t0)
Nadult, t1 = px*(Njuvenile, t0) + 0 *(Nadult, t0) fx
Juvenile
Adult px

20. Projecting this sample matrix indefinitely will result in the finite population growth rate: λ
Age1
Age 2
Age 3 t0 4
0.8
0.5 20 x =

21. Look at the y-axis
λ is on the natural log scale…

22. Stable age distribution

23. Stable age distribution. Expected in a static environment…

24. Population trajectories with process and observation errorf3 f4 p1 P2 p3

25. Elasticity: Proportional sensitivity (of λ ) to matrix element perturbations
Age1
Age 2
Age 3 4
0.8
0.5
λ = 2.00
A 25% decrease in Age 1
survivorship results in a
12% decrease in population
growth.
Age1
Age 2
Age 3 4
0.6
0.5
λ = 1.76

26. Elasticity: Proportional sensitivity (of λ ) to matrix element perturbations
Age1
Age 2
Age 3 4
0.8
0.5
λ = 2.00
A 25% increase in Age 2
fecundity results in a
9% increase in population
growth.
Age1
Age 2
Age 3 5 4
0.8
0.5
λ = 2.18

27. Brault and Caswell

28. Elasticity A type of “perturbation” analysis
The elasticity eij indicates the relative impact on  of a modification of the value of the parameter aij
Scaled, therefore The elasticity is independent on the metric of the parameter aij
and

29. Stage-specific survival and reproduction
G, P, F

30. Stage-specific survival and reproduction
G, P, F

31. Stage-specific survival and reproduction
Initialize matrix A