1. Wildlife Population Analysis
Matrix Models for Population Biology
Lecture 12

2. Resources:
Caswell, H Matrix population models. 2nd ed. Sinauer Associates, Inc. Sunderland, Mass.
Tuljapurkar, S., and H. Caswell (eds.) Structured-population models in marine, terrestrial, and freshwater systems. Chapman & Hall, New York.
Manley, B.F.J Stage-structured population: sampling, analysis, and simulation. Chapman & Hall, New York.

3. Matrix addition and subtraction
Simply addition (subtraction) of the corresponding elements in the matrices.
Matrices must be of the same rank (dimension).

4. Matrix multiplication

5. Matrix multiplication
Matrices:
do not have to be of the same rank;
must have the same “inner” dimension;
dimensions are specified as rows x columns.
Thus a 4x3 matrix can be multiplied by a 3x1 matrix,
But a 1x3 matrix cannot be multiplied by a 4x3 matrix.

6. Example:

7. The identity matrix.
For any square matrix, the identity is a diagonal matrix of equal rank with all of the diagonal elements = 1.

8. Leslie and Lefkovitch Projection Matrices

9. History
Application of age-specific survival and fertility rates dates back to the late 19th century
Use of matrix models was developed independently by Bernardelli (1941), Lewis (1942), and Leslie (1945).
Leslie’s works published in 1945, 1948, 1959, and 1966 were apparently the most influential.
Even so matrix models were not mentioned in many notable ecology texts or in ecological research prior to the 1970s.
Lefkovitch worked on the dynamics of agricultural pests published a series of influential papers using the matrix models described by Leslie in the early 1960s. Among these was a 1965 paper that introduced the idea of stage- structured models that classified insects by life-stage rather than age.

10. Age-structured models
The goal of population modeling is to develop equations that allow us to understand the processes that govern population dynamics.
Consider the equation: where Nt is the population size in year t and Nt+1.

11. Age-structured models
In the absence of emigration and immigration, the population growth rate, , subsumes the processes of mortality and recruitment. Thus, one could more explicitly write this equation as where F is fertility, the number of offspring recruited per adult and P is the probability of surviving from year t until year t+1.

12. Age-structured models
Now consider a population of size N with 3 1-year age classes where ni is the number of individuals in age-class i and age class one is the youngest age class. The dynamics of this population could be expressed as three separate equations: and since individuals in this population do not live beyond age 3, all of the n3s die before the next time step (year).

13. Age-structured models
The age-structured transition matrix model representing this system of equations is a square matrix with one column for each age-class: The population N, composed of individuals of three age classes n1-3 is represented by the vector:

14. Age-structured models
This population is projected through time using matrix multiplication by the equation: (Note that the inner dimensions of the matrices (3x3;3x1) agree.)

15. Life-cycle graph F1 F3 F2 P1 P2
Life-cycle diagram, where each node represents an age class, the straight lines connecting the nodes represent the survival (transition) probabilities (P) and the curved lines extending back to the first node represent the fertilities (F). F1 F3 F2 1 2 3 P1 P2

16. Example:
Birth and Survival Rates for Female New Zealand Sheep [from G. Caughley, "Parameters for Seasonally Breeding Populations," Ecology 48(1967) ]

17. Life-cycle diagram:

18. The Leslie matrix:

19. Projection
10-years starting with year olds: Note the rapidly (exponential) increasing population and the initial fluctuations in  due to starting conditions (age distribution).

20. Age distributions
10-year projection starting with year olds.

21. Assumptions of age structured models:
Individuals progress through the life-cycle by discrete time-steps (e.g., years)
Age-specific fertility
Age-specific survival
Single sex models

22. Stage-based models
Lefkovitch relaxed the assumptions of the age-structured models described by Leslie
Useful for animals that had stage-dependent vital rates.
Discrete time-steps (e.g., years)
Individuals allowed to remain in life-stages longer than one year
Stage-specific fertility
Stage-specific survival
Single sex models

23. Stage-based matrix model (3 stages):
F i is still the fertility, the number of offspring recruited per adult;
Pi is the probability of surviving from year t until year t+1 and remaining in stage i ; and
Gi is the probability of growing to stage i during the next time step.

24. Life cycle graph
Generalized 4-stage population:

25. Examples: Arthropods with discrete developmental stages.
Plants, crustaceans, and fish with size-dependent ages of maturity.
Angiosperms, kelp, molluscs, decapods, insects, isopods, amphibians, and reptiles with reproductive rates that vary with adult body size.
Plants with mortality rates that vary with size.
Plants and animals with size related sex changes.

26. Another example
Brault, S. and Caswell, H Pod-Specific Demography of Killer Whales (Orcinus orca). Ecology, 74:
Classified the population into 4 stages of females: yearlings (1 year olds), juveniles (up to 18 yrs), reproductive (up to 45 yrs), and post-reproductive. Thus the life-cycle graph looks like:

27. Matrix

28. Projection

29. Projection stacked

30. Pre-breeding vs. post-breeding
State vector represents population structure immediately prior to breeding season
Post-breeding
State vector represents population structure after reproduction occurs
Transition rates change by adjusting the fertilities and survivals.

31. Pre-breeding vs. post-breeding
Generally, speaking Pis in age-based and Gis in stage-based models are annual (single time step) rates and will not vary.
However
Fis in a pre-breeding census model include productivity and survival of offspring until the end of the first time step (e.g., year).
Fis in post-breeding census model are discounted for annual survival of adults
Also P1s in a prebreeding census reflect survival of individuals between the first and second time step.
P1s in the postbreeding model are survival from postbreeding until the next postbreeding census.

32. Example: Fi = cs*sr*ns*gs*S1+ = 7*0.5*0.35*0.45*0.76 = 0.42
Hypothetical bird population
Postbreeding age-structured matrix
Fi = cs*sr*ns*gs*S1+ = 7*0.5*0.35*0.45*0.76 = 0.42

33. Example: Hypothetical bird population
Prebreeding age-structured matrix
Fi = cs*ns*gs*S0 = 7*0.5*0.35*0.45*0.60 = 0.33

34. Four questions from Caswell (2001)
Asymptotic behavior—Is long-term behavior
Ergodicity—Is the behavior of the model dependent upon the initial state vector
Transient behavior—What are the short term dynamics of the model?
Perturbation analysis—How does the model respond to changes in the vital rates (i.e., what are the relative sensitivities)?

35. Four questions from Caswell (2001)
Asymptotic behavior—
What happens if model processes operate for a very long time?
Does it grow or decline?
Does it persist or go extinct?
Does it converge to an equilibrium, oscillate, or behave chaotically?
Most deterministic matrices reach an asymptotic growth rate and stable age distribution.

36. Population growth rate ()
Will the population grow or decline?
Project population for >20 years, then calculate rate of change (Nt+1/Nt).
Project population for >20 years, then calculate average rate of change (Heyde, C. C., and J. E. Cohen. 1985)

37. Population growth rate ()
Calculate dominant eigenvalue
No. of eigenvalues = no. of stages
Fortunately, most single population matrices have only one real, positive (dominant) eigenvalue.

38. Stable age (stage) distribution (SAD)
What is the predicted structure of the population?
Project the population for >20 years, determine the percentage of the population in each age (stage) class.
Calculate the right eigenvector of the dominant eigenvalue and normalize.
Age/stage structure
R eigenvector
Final SAD
36.4%
19.8%
43.9%

39. Ergodicity
Is the behavior of the model dependent upon the initial state vector?
Project model >20 years with different initial age distributions. Does the population reach (or approach) the expected  and SAD?

40. Transient behavior What are the short term dynamics of the model?
Does it grow or decline?
How rapidly does it converge to an equilibrium?
Does it oscillate, or behave chaotically?
Can be very useful in understanding population responses to perturbations.

41. Example Spectacled Eider population on the Y-K Delta at Kashunuk River Study site
Age 1
Age 2
Age 3
Nest success
0.47
Clutch size (females hatched)
2.15
Breeding propensity
0.56 1
Duckling survival
0.34
Survival of immature
0.49
Survival of adults
- exposed to lead
0.44
- not exposed
0.82
- lead exposure
0.1764
0.315
weighted average
0.75
0.70

42. Example 1 2 3+ A = 0.00 0.094 0.168 0.82 0.75 0.70 Eigenvalues
0.75
0.70 1 2 3+
Eigenvalues
Eigenvectors (R&L)
Real
Imaginary
Age/stage struct
Reprod val
0.86
15.3%
0.95
-0.08
-0.23
14.7%
0.99
0.23
70.0%
1.012

43. Example numerical projection with SAD

44. Example transient after breeding failure

45. Example numerical projection w/loss of 80% breeders

46. Example transient w/loss of 80% breeders

47. Transient dynamics
The rate of convergence on a stable population growth rate is governed by the relative size of the subdominant eigenvalues.
That is, the larger 1 is in relation to i>1 the more rapidly the population will converge on stability. This property often referred to as the damping ratio is defined as:

48. Example:
Hypothetical population matrix with high Fi and low annual survival, similar to a small mammal or a passerine bird: 3 4
A =
0.2
0.4
Eigenvalues
Eigenvectors (R&L)
Real
Imaginary
Age/stage struct
Reprod val
76.4%
14.6%
9.0%

49. Example – transient dynamics

50. Lower F3
A = 3
0.1
0.2
0.4
Eigenvalues
Eigenvectors (R&L)
Real
Imaginary
Age/stage struct
Reprod val
0.7878
66.0%
16.7%
17.3%

51. Lower F3

52. Perturbation analysis
How does the model respond to changes in the vital rates?
What are the relative sensitivities?
Estimates of vital rates always are subject to uncertainty.
Conclusions dependent upon exact values are always suspect.

53. Perturbation analysis - sensitivity of matrix models
Prospective analysis – forward looking.
What could happen to the population if changes occur in vital rates?
Retrospective analyses – examining the past.
How has variation in vital rates affected population growth?

54. Motivation
Determining which rate(s) have the greatest affect on population growth
Predicting results of future changes in vital rates
Quantifying the effects of past changes in vital rates
Predicting the actions of natural selection (if changes in phenotypes result in changes to vital rates)
Designing sampling schemes. (i.e. choosing which vital rates are the most important to measure accurately)

55. Prospective Analysis Two Metrics:
Sensitivities - the effect on population growth rate, 1, of unit changes in the vital rates.
e.g., 1 egg increase in mean clutch size
Elasticities – the relative effect of vital rates on population growth rate, 1.
e.g., 1% increase in mean clutch size.

56. Sensitivity
Sensitivity refers to the effect on population growth rate, 1, of unit changes in the vital rates.
Rate of change in  for a unit change in aij while holding all other vital rates constant.

57. Example – desert tortoise model
Doak, D. P. Kareiva, and B. Klepetka Modeling population viability for the desert tortoise in the western Mojave Desert. Ecological Applications 4:
Stage (size) based matrix model 1 2 3 4 5 6 7 8

58. Example – desert tortoise model1 2 3 4 5 6 7 8

59. Example – eigen analysis

60. Sensitivity
Sensitivity refers to the effect on population growth rate, 1, of unit changes in the vital rates.
Rate of change in  for a unit change in aij while holding all other vital rates constant.

61. Example
a88 – survival rate of individuals in the largest size class

62. Sensitivity
Measure of the effect of a change in aij holding all else constant.
The slope of  as a function of aij.
0.98
0.981
0.982
0.983
0.984
0.985
0.986
0.987
0.988
0.989
0.82
0.84
0.86
0.88
0.90
0.92
0.94
a(8,8) l
Actual
Observed values
Slope

63. Sensitivity matrix

64. Elasticities
Relative effect on population growth rate, 1, of small changes in the vital rates.
Sum to 1.
Interpreted as the relative contributions of the vital rates to .

65. Elasticities
Relative effect on population growth rate, 1, of small changes in the vital rates.
Slope of ln() as a function of ln(aij)

66. Elasticities Elasticities can be calculated from projections as:
where * is the population growth rate after a proportionate change in aij , and p is the change in aij..
Since elasticities are scaled with respect to  they sum to 1.0 and thus are directly comparable.
Elasticities also can be summed to determine the relative contributions of more than one vital rate.

67. Desert Tortoise example:
P7 the probability of surviving and remaining in stage 7 has nearly 2.25 times as much of an effect on  as does P6
Elasticity of transition probabilities (Ps and Gs) = 0.95,
Elasticity of Fs =.05
Population is 19 times as sensitive to growth and survival as productivity.

68. Lower-level elasticities
Relative sensitivities of  to parameters contributing to the aij.
Example:
Fertility( females recruited per female)
Product of:
clutch size
nest survival
hatchling survival
sex ratio

69. Lower-level elasticities
Relative sensitivities of parameters (xk) contributing to the aij. or

70. Lower-level elasticities
Do not sum to 1, so they can not be interpreted as contributions to .
Values are relative
If lle(x1) = 0.4 and lle(x2) = 0.8, then
lle(x2) has twice the influence on 

71. Lower-level elasticities – example
Desert tortoise F8 = 4.38 (females/female)
Identical because F8 is the product of parameters
Estimate
Sensitivity
Elasticity
Clutch size
32.00
0.0002
0.005
Sex ratio
0.50
0.0107
Nest survival
0.61
0.0087
Breeding propensity
0.80
0.0067
Offspring survival
0.56
0.0096
Product
4.38

72. Retrospective analysis
Life Table Response Experiments (LTRE)
Set of vital rates (matrix) is the response variable in an experimental design.
Treatments affect the various vital rates
 most frequently used statistic to evaluate the effect of the treatments.
Often used to examine the effect of past variation in vital rates on population growth rates.

73. LTRE designs Analogous to analysis of variance
one-way, two-way, or factorial
Random effects
Regression analysis.

74. Example – One-way fixed design One treatment (t) One control (c)
Vital rates are used to populate the matrices:

75. Calculate the mean matrix

76. Calculate the sensitivities of Am

77. Calculate the difference matrix (D)
Difference between At and Ac

78. Calculate the difference matrix (D)
Multiply by the sensitivities
The result is the contributions of the differences in the vital rates to the change in the population growth rate.

79. Prospective versus Retrospective
Prospective analysis – forward looking.
What could happen to the population if changes occur in vital rates?
Effect of future management actions
Retrospective analyses – examining the past.
How has variation in vital rates affected population growth?
Effect of environmental variation or past actions
Frequently don’t have information for retrospective

80. Prospective versus Retrospective
Not a panacea
Parameters with greatest elasticities will have the greatest relative impact on 
May not be “manageable”
Parameters with greatest contribution to past variation can be indicative of management opportunity
Valuable to look at both when possible
Neither incorporates cost or risk