1. Demographic matrix models for structured populations
Dominique ALLAINE

2. Structured populations
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

3. Structured populations
Obviously, vital rates differ between individuals. These vital rates may
differ according to age, to sex, to size category …
Each individual is then described by a state at a given time. For example,
an individual may be in the state « yearling female » at time t.
Populations are made of individuals that are different. So, we can
consider the structure of the population according to sex, to age, to
size category, to status …
The structure of a population at a given time is then characterised by
a distribution function expliciting the number of individuals in each state

4. Structured-population models
General characteristics
Models based on the global size of a population does not take into
account the differences between individuals.
It is important to analyse the response of vital rates in structured
populations to take into account differences between individuals
Structured-population models aim at including the differences
between individuals in vital rates. These models are applied to
structured populations.

5. Structured-population models
We will assume that all individuals in a same state experience the
same environment and then respond in the same way. In other words,
all individuals in a same state will have the same values of vital rates.
Structured-population models are mathematical rules that allow to
calculate the change over time of the distribution function of the
number of individuals in each state.
Three types of mathematical approaches:
matrix models
delayed differential equation models
partial differential equation models

6. Matrix population models
General characteristics
The different states are discrete
This means that individuals are classified into categories. For example,
individuals are either males or females, are eitheir juveniles or
yearlings or two-years old …
When considering size, discrete categories have to be identified.
We thus have to recognise categories and at a given time, each
individual may be associated unambiguously to a category.

7. Matrix population models
General characteristics
The change in the population structure is in discrete time
This means that the change of the distribution of the number of
individuals in each state is given in discrete time. If we use, for
example, an annual time scale, the number of individuals in each state
is given for each year.

8. N(t+1) = M N(t) Matrix population models General characteristics
First developped by P.H. Leslie during the 1940’s
The number of individuals in each state at time t is given by a
vector N(t)
This vector is projected from time t to time t+1 by a population
projection matrix M
N(t+1) = M N(t)
(n,1) (n,n) (n,1)

9. Matrix population models
General characteristics
Different steps:
To determine the projection interval (time scale)
To determine the states of importance
To identify the vital rates necessary
To establish the life-cycle graph

10. Matrix population models
Populations structured in age
The projection interval is usually the year in analyses on vertebrates
States of importance are often age, sex
Vital rates refer to:
proportion p of reproductive females per age class
fecundity f per age class
survival rate s per age and sex classes

11. Matrix population models
Populations structured in age
Life cycle in the case of a population structured in age

12. Matrix population models
Populations structured in age
The matrix model corresponding to the previous life-cycle graph is:
 Leslie matrix

13. Matrix population models
Populations structured in stage
Life cycle in the case of a population structured in stage
E J Ad R1 R2

14. Matrix population models
Populations structured in stage
The matrix model corresponding to the previous life-cycle graph is:
 Lefkovitch matrix

15. Matrix population models
General characteristics M
Types of matrix models:
Variable
Constant
Internal
External
Both
Density-
dependent
Deterministic stochastic
Periodic Aperiodic

16. Matrix population models
General characteristics
Types of matrix models:
Linear => constant, time-varying
Non linear => internally generated variability

17. N(t+1) = M N(t) Matrix population models Constant linear models
(n,1) (n,n) (n,1)
In this case, the matrix M has constant coefficients
Example: a two age-classes model

18. Matrix population models
Constant linear models
The Perron-Frobenius theorem
A non negative, irreductible, primitive matrix has 3 properties:
the first eigenvalue 1 is unique, real and positive
the right eigenvector w1 corresponding to 1 is strictly positive
the left eigenvector v1 corresponding to 1 is strictly positive

19. Matrix population models
Constant linear models
The Perron-Frobenius theorem
Non negative
Reductible Irreductibe
primitive
imprimitive
1 > 0
w1 > 0
v1 > 0
1 = |j| j = 2 …d
1 > |j| j> d
1 ≥ 0
w1 ≥ 0
v1 ≥ 0
1 ≥ |2|
1 > 0
w1 > 0
v1 > 0
1 > |2|

20. Matrix population models
Constant linear models
A reductible matrix has some stages that make no contribution to some
other stages
Example:

21. Matrix population models
Constant linear models
A matrix is primitive if the greatest common divisor of the lengths of the
loops in the life-cycle graph is 1
Example:
Primitive
Imprimitive

22. N(t+1) = M N(t) N(t+1) = M N(t) = Mt N(0) Matrix population models
Constant linear models
Leslie matrices are often irreductible and primitive
N(t+1) = M N(t)
N(t+1) = M N(t) = Mt N(0)

23.   M = WW-1 M2 = WW-1 WW-1 M2 = W2W-1 Mt = WtW-1
Matrix population models
Constant linear models
Let  an eigenvalue of the matrix M
Let N(t) the right eigenvector of the matrix M associated to 
Diagonalisation of the matrix M
M = WW-1 
M2 = WW-1 WW-1 
M2 = W2W-1
Mt = WtW-1

24. Matrix population models
Constant linear models
Mt = WtW-1
Mt = WtV*

25. Matrix population models
Constant linear models
N(t) = Mt N(0)
Remember:
wi is a column vector and vi* is raw vector so their product is a matrix

26. Matrix population models
Constant linear models
It follows that:

27. Matrix population models
Constant linear models
Asymptotic results
And consequently:
The dynamic is driven by the dominant eigenvalue and associated eigenvectors

28. N(t+1) = M N(t) =  N(t) Matrix population models
Constant linear models
Asymptotic results
It results that asymptotically:
N(t+1) = M N(t) =  N(t)

29.  = er where r is the population growth rate
Matrix population models
Constant linear models
Asymptotic results
Asymptotically, the dominant eigenvalue  corresponds to the annual
multiplication rate
n(t+1) =  n(t)
 = er where r is the population growth rate
Asymptotically, the right eigenvector associated to the dominant eigenvalue
 gives the stable state structure
M W =  W

30. Matrix population models
Constant linear models
Asymptotic results
Asymptotically, the left eigenvector associated to the dominant eigenvalue
 gives the stable reproductive value, i.e. the contribution of each state to
the population size
VM =  V
The annual multiplication rate, the stable state structure and the reproductive
values depend on the values of vital rates but are independent of initial
conditions (ergodicity)

31. Matrix population models
Constant linear models
Asymptotic results
The damping ratio  expresses the rate of convergence to the stable
population structure.
In other words, the convergence will be more rapid when the dominant
eigenvalue is large relative to the other eigenvalues.

32. N(t+1) = Mt N(t) Matrix population models Matrix population models
Stochastic linear models
N(t+1) = Mt N(t)
(n,1) (n,n) (n,1)
In this case, the matrix M has coefficients varying randomly with time
Example: a two age-classes model

33. Matrix population models
Stochastic linear models
Growth rate
The simple estimation of the multiplication rate, the stable state structure
and the reproductive value calculated from eigenvalues and eigenvectors are
no longer valid in stochastic linear models
The population size at time t is: 

34. Matrix population models
Stochastic linear models
Growth rate
It has been shown (Furstenberg & Kesten 1960) that:
where Lns is the stochastic growth rate

35. Matrix population models
Stochastic linear models
Growth rate

36. Matrix population models
Stochastic linear models
Growth rate
The analytical calculation of Ln s is not easy
The stochastic growth rate can be found by
Simulation
Approximation

37. å Matrix population models Stochastic linear models Growth rate = - Ln
1. Simulation
The stochastic growth rate can be estimated from the average growth rate
over a long simulation from the maximum likelihood estimator :
An estimation of the stochastic growth rate from a simulation is: å = - i t s Ln n l 1 ) (

38. Matrix population models
Stochastic linear models
Growth rate
1. Simulation
Usually, the stochastic growth rate is calculated as the average of
estimations obtained from several simulations

39. Stochastic linear models
Matrix population models
Stochastic linear models
Growth rate
1. Simulation: environmental stochasticity
P(1)
B1(s0,s0)
B1(p1,p1)
B1(s1,s1)
B1(s2,s2)

40. Matrix population models
Stochastic linear models
Growth rate
1. Simulation: demographic stochasticity

41. Matrix population models
Stochastic linear models
Growth rate
1. Simulation: demographic stochasticity
Number of reproductive females :
Number of young produced :

42. Matrix population models
Stochastic linear models
Growth rate
2. Approximation
The stochastic growth rate can be analytically approximated when vital rates
vary in a small way that is, their coefficients of variation are much less than 1
If we assume that all distributions are identical and independent, then, the
stochastic growth rate is approximated by:
where is the mean growth rate

43. Matrix population models
Perturbation analysis
From a biological point of view, it is important to know which factor has the
greatest effect on 
Perturbation analyses allow to predict the consequences of changes in the
value of one (or more) vital rate on the value of 
Two concepts:
Sensitivity
Elasticity

44. Matrix population models
Perturbation analysis
1. Sensitivity
The sensitivity sij indicates how the value of  is impacted by a modification of the value of the parameter aij 
The sensitivity is dependent on the metric of the parameter aij

45. Matrix population models
Perturbation analysis
2. Elasticity
The elasticity eij indicates the relative impact on  of a modification of the value of the parameter aij
and
The elasticity is independent on the metric of the parameter aij

46. Matrix population models
Perturbation analysis
3. Prospective analyses
The prospective analyses are done by calculating sensitivities or elasticities

47. Matrix population models
Perturbation analysis
4. Retrospective analyses
The objective of a retrospective analysis is to quantify the contribution
of each vital rate to the variation in 
The variability of  may be due to variations in vital rates in space (between
populations) or in time (between years) for example
The impact of a given component aij of the projection matrix on the variation of  depends on both the variation of the component and the sensitivity of  to this component

48. Matrix population models
Perturbation analysis
4. Retrospective analyses
A component aij will have a small contribution to the variation in  if this
component does not change much or if  is not very sensitive to aij or if both
are true.
The contribution of aij to the variation in  involves the products of sij and the
observed variation in aij
Two approaches are possible:
random effects
fixed effects

49. Matrix population models
Perturbation analysis
4. Retrospective analyses
a. Random effects
We want to identify the contribution of vital rates to the variance in 
during a time period for example (where time is considered as a random factor)
Covariances between parameters are calculated directly from observed
matrices
Sensitivities are calculated from the mean matrix

50. Matrix population models
Perturbation analysis
4. Retrospective analyses
a. Random effects
The contribution of the vital rate aij can be measured as:
It is the sum of contributions including aij

51. Matrix population models
Perturbation analysis
4. Retrospective analyses
a. Random effects
Tuljapurkar showed that:
The contribution of the component aij can be measured as:

52. Matrix population models
Perturbation analysis
4. Retrospective analyses
a. Random effects
These formula can also be rewritten as:
The contribution of the component aij can be measured as:

53. Matrix population models
Perturbation analysis
4. Retrospective analyses
b. Fixed effects
We want to identify the contribution of vital rates to the difference in
multiplication rates between traitments.
To simplify, consider two populations in different environments. These
populations then vary according to the values of vital rates and,
consequently in projection matrices.
To these projection matrices A1 and A2 correspond two multiplication rates
respectively equal to 1 and 2

54. Matrix population models
Perturbation analysis
4. Retrospective analyses
b. Fixed effects
It can be shown that:

55. Matrix population models
Perturbation analysis
4. Retrospective analyses
b. Fixed effects
The contribution of the vital rate aij to the difference in  is given by:
This may be generalised to more than two traitments

56. Matrix population models
Perturbation analysis
4. Retrospective analyses
b. Fixed effects
This may be generalised to more than two traitments
where r is a reference traitment
where Cm is the vector of contributions
Cm gives the contributions of vital rates to the difference between the
traitment m and the reference traitment r

57. Metapopulation
Concept of Metapopulation

58. Metapopulation
Definition
A metapopulation is a population of local populations
that are susceptible to extinction and that are
inter connected by migration.
The persistance of the metapopulation depends on
a stochastic balance between local extinctions
and recolonisation of empty sites.

59. Metapopulation

60. Population of populations: diffrent types
Metapopulation
Population of populations: diffrent types
« island/mainland »
« source/sink »
Metapopulation
Isolated

61. Metapopulation
Extinction
Stochastic causes
Demographic stochasticity
Genetic stochasticity
Environmental stochasticity
Deterministic causes

62. Metapopulation Extinction General message
The risk of local extinction depends on :
the patch size
the mean growth rate
the variability of the growth rate

63. Metapopulation
Migration - Colonisation
Two processes:
Migration (dispersal)
Colonisation

64. Migration - Colonisation
Metapopulation
Migration - Colonisation
General message
The success of dispersal depends on :
the distance between patches
the realistion of particular conditions (corridors)

65. Effect of the presence of corridors
Metapopulation
Migration - Colonisation
Effet of the distance of dispersal
Effect of the presence of corridors

66. Migration - Colonisation
Metapopulation
Migration - Colonisation
General message
The success of colonisation depends on :
the number of immigrants

67. Metapopulation dynamics
The Levins model (continuous time) t1 ti tn

68. Metapopulation dynamics
The Levins model (continuous time) 
The metapopulation will persist if c > e
Analogy with the logistic model

69. Metapopulation dynamics
The island/mainland model t1 ti tn

70. The metapopulation will persist as long as
Metapopulation dynamics
The island/mainland model 
The metapopulation will persist as long as
colonisation occurs

71. Metapopulation dynamics
Spatially explicit models t
1-pi pi
t+1
Ci(t) ei

72. Metapopulation dynamics
Spatially explicit models

73. Structured populations: constant models
Metapopulation dynamics
Structured populations: constant models
Life cycle in the case of several populations j
Sites k

74. Fragmented populations: deterministic models
Metapopulation dynamics
Fragmented populations: deterministic models
number of newborn females in site j
born to a female of age i in site k

75. Fragmented populations: deterministic models
Metapopulation dynamics
Fragmented populations: deterministic models
proportion of surviving females in site j
from those of age i-1 in site k

76. Fragmented populations: deterministic models
Metapopulation dynamics
Fragmented populations: deterministic models

77. Fragmented populations: deterministic models
Metapopulation dynamics
Fragmented populations: deterministic models
N*(t+1) = M N*(t)
(np,1) (np,np) (np,1)
Det (M – I) = 0   annual multiplication rate of the metapopulation
N*(t) right eigenvector  stable age structure per site