Discrete age-structured models:
The Leslie matrices
Didier Gonze
October 22, 2015
Abstract
Matrix models are useful tools in population dynamics to predict population growth. They are particularly suitable to describe the evolution of age
structured populations. These models can be used to determine best fishing
strategies, strategies to rescue endangered species or to determine the evolution of (human) age pyramids. In the present notes, we briefly describe the
Leslie matrices, their generalization and how they are build and analyzed.
Some applications and exercises are also proposed.
Historical background
Mathematical ecology - including (human) demography - is not a recent field of science.
It is a very old science, as attested by the works from Thomas Malthus (around 1798) or
Pierre-François Verhulst (around 1830-1840). Even Leonardo of Pisa (Fibonacci, around
1230) addressed problems of demography.
The use of matrix models in this field is however rather recent. Although the use of
matrix models in population dynamics already appeared around 1940 (Bernardelli 1941,
Lewis 1942), nowadays we generally cite the work of Leslie (1945, 1948). Patrick Leslie
(1900-1974) recognized a strong limitation in previous population models: the fecundity
and death rates were constant among all individues in a population. In reality it is
obviously not the case. For example, the eggs, the larvae and adults do not have the same
reproductive and death rates. Leslie overcomes this limitation by considering structured
populations in which fecundity and death rates depend on the age or on the developmental
stage.
We often refer to the “models of Leslie” or even to the “theory of Leslie”, when we study
discrete age-structured models in population dynamics. Today recognized as a funder of
the mathematical ecology, Leslie was in fact not really renowned by the ecologists because
the matrix models where to difficult to manipulate. Today the situation is different.
Thanks to computer simulation, the numerical analysis of the matrix model is easy, fast,
and can be done for large models.
Another important contribution to the development of matrix models in ecology is due to
Leonard Lefkovitch (1965), who introduced a more general structuration of the populations. Shortly later, Usher (1966) further developed the idea of size-structured populations
and Rogers (1966) applied matrix models to human populations.
These models are still applied and extended today. They can be used to determine optimal
fishing strategies, to design strategies to rescue endangered species, or to determine the
evolution of (human) age pyramids.
2
Leslie model
The Leslie model (Leslie 1945, 1948) describes the dynamics of an age-structured population and is based on 3 hypotheses:
• The age (here-after denoted x) is a continuous variable starting from 0 and subdivided into discrete age classes, from 0 to w: the age class i thus corresponds to the
ensemble of individues whose age satisfies i − 1 ≤ x < i for i = 1, 2, ...w.
• Time is a discrete variable. We will denote by t the time step (also called the
projection interval).
• The time step is exactly equal to the duration of each age class, meaning that from
t to t + 1, all individues go from class i to class i + 1.
Figure 1: Age classes
If we denote by ni (t) the number of individues of class i at time t and by Nt the population
vector at time t, then
ni (t + 1) = f (ni (t))
(1)
or, using the vectorial notation
Nt+1 = MNt
(2)
The matrix M is called the Leslie matrix. Let’s see how to determine its coefficients.
The life cycle can be schematized by representing each age class by a node and the transfert
of the individues form one class to another by an arrow. We then obtain graphes like the
one shown in Fig. 2 which contains 3 age classes. The age classes 2 and 3 can reproduce
and 3 is the maxium age of the individues.
Figure 2: Life cycle
3
If we write Pi the probability that an individue survives from the class i to the class i + 1
and Fi the number of individues produced by a female from class i and which survive till
class 1, then we obtain the following equations describing the dynamics of the population:

 n1 (t + 1) = F2 n2 (t) + F3 n3 (t)
n2 (t + 1) = P1 n1 (t)

n3 (t + 1) = P2 n2 (t)
(3)
These equations can be written in the matricial form:





n1
0 F2 F3
n1
 n2  =  P1 0 0   n2 
n3 t+1
0 P2 0
n3 t
(4)
which is equivalent to eq. (2).
We can easily generalize to any number w of age classes:




M=



F1 F2 F3
P1 0 0
0 P2 0
0 0 0
:
:
:
0 0 0

... Fw−1 Fw
...
0
0 

...
0
0 

...
0
0 

...
:
: 
... Pw−1 0
(5)
A Leslie matrix is thus characterized by a first line with the fecundity of the different
classes and a sub-diagonal with contains the survival probabilities from one class to the
next one. A Leslie matrix is non-negative, i.e. if mi,j denote the element (i, j) of M, then
mi,j ≥ 0 ∀ i, j.
Estimation of the demographic parameters
The estimation of the coefficients of the Leslie matrix relies on two measurements: the
survival function l(x) and fecundity function m(x).
The survival function l(x) (Fig. 3) is defined as the probability that an new-born individue
survives till age x.
Depending on the species, we can distinguish 3 types of survival function: the K strategy
(or type I) with a low infantile death rate (i.e. a few young individues in the population cf. big mammals), the r strategy (or type III) with a high infantile death rate (i.e. many
young individues in the population - cf. rodents) and the intermediary strategies (type
II).1
1
The “K” and “r” strategies refer to the optimization of either the carrying capacity K or the reproduction rate r used in the logistic growth rate.
4
Figure 3: Survival function (for different strategies)
The fecundity function m(x) (Fig. 4) is defined as the number of new-born individues
appearing by time unit and by individues. In the case of sexual reproduction, we generally
consider only females, implying that the sex-ratio is implicitely contained in m(x).
Figure 4: Fecundity function
Functions l(x) and m(x) are then used to compute the Leslie coefficients Pi and Fi :
l(i + 1)
l(i)
≈ l(1)m(i)
Pi ≈
(6)
Fi
(7)
5
Example and extensions
We illustrate here the use of the Leslie matrices with some numerical simulations.
Consider the following Leslie matrix:


0
1 5
M =  0.3 0 0 
0 0.5 0
(8)
with the following initial condition:


1
N0 =  0 
0
(9)
We first consider that matrix M is constant. We neglect possible time-dependent variations of the demographic parameters. Thus the fecundity and survival rate are constant
for a given age class.
The results of numerical simulation of the system (8) with the initial condition (9) are
shown in Fig. 5. During the first 15 time steps, the abundance of the total population
as well as each population fluctuate in an irregular manner. A slight tendancy for the
population to increase can be detected. If we look at a longer time period, we observe
that each class (and thus also the total population) increases exponentially. Finally, when
we observe the proportion of each class, we see that they converge to constant values.
Simulations run with the same Leslie matrix but with various initial conditions show that
the long-term evolutions of the different populations are similar (exponential growth) and
the fraction of each population converges towards the same values (Fig. 6). Thus the
initial condition does not influence the asymptotic behavior of the system.
Now, we still consider that the Leslie matrix is constant but we run simulations for
different values of parameters. More specifically, in each simulation, we reduce the value
of one of the parameter by 10%. The results, given in Fig. 7, show that changing the
value of these parameters can drastically affect the long-term behavior of the system. In
particular, if the survival probabilities P1 or P2 or the fecundity F2 are reduced, the whole
population tends to vanish, with P2 having the most drastic effect.
6
1
1.5
1
N
tot
0.6
N
tot
0.8
0.4
0.5
0.2
0
0
5
10
0
15
0
10
20
time t
30
40
50
60
time t
1.4
1
1.2
0.8
N
f = N /N
tot
1
0.6
i
0.8
N
i
f
1
1
0.4
i
0.6
0.4
N
f
2
2
0.2
0.2
N
0
3
0
10
20
30
40
50
0
60
f
3
0
10
20
time t
30
40
50
60
time t
Figure 5: Computer simulation of system (8) with initial condition (9). Shown
are the time evolution of the total population size Ntot = N1 + N2 + N3 , the time
evolution of the population of each subclass (N1 , N2 , N3 ), and the time evolution of
the fraction of each subclass (N1 /N , N2 /N , N3 /N ).
7
1
6
0.8
f
1
f = N /N
0.6
i
4
3
0.4
i
Ntot
tot
5
2
f
2
0.2
1
0
0
10
20
30
40
50
0
60
time t
f3
0
10
20
30
time t
Figure 6: Effect of the initial conditions.
7
40
50
60
1.8
1.6
original model
1.4
Ntot
1.2
1
F2 −> F2−10%
0.8
0.6
F3 −> F3−10%
0.4
P2 −> P2−10%
0.2
P1 −> P1−10%
0
0
10
20
30
40
50
60
time t
Figure 7: Effect of parameter variation.
In reality, populations can not grow undefinitely. When a high density (i.e. carrying
capacity) is reached, the population should stabilize. We can take into account such
density-dependance of the growth by assuming that the fecundity decreases as the density
increases.
The Leslie matrix can then be modified as followed:


0 1 × g(N) 5 × g(N)

0
0
M =  0.3
0
0.5
0
(10)
where N = N1 + N2 + N3 is the total population.
If we assume that the fecundity decrease exponentially with the density, we can then
define g(N) as:
g(N) = re−bN
(11)
where b measures the importance of the density-dependance. Note that since N changes
at each time step, M should also be recomputed at each time step. The results obtained
for r = 1, b = 0.01 are given in Fig. 8.
We observe that after a transitory regime which displays fluctuations, each age class,
and consequently also the total population, converges to a steady state. This state is
independent on the initial condition (not shown).
Note that the system does not necessarily converge to a steady state. More complex dynamics can be observed for example of we increase the value of parameter r, as illustrated
in Fig. 9. This rich and complex dynamics is due to the non-linearity of the model.
8
5
5
4
4
N
Ni
Ntot
1
3
3
2
2
1
1
0
0
N
2
N
0
100
200
300
400
3
0
100
time t
200
300
400
time t
Figure 8: Effect of density-dependent fecundities.
100
1000
r=20
800
60
600
Ntot
80
N
tot
r=2
40
400
20
200
0
0
20
40
60
0
80
0
20
time t
40
60
80
time t
4
3500
2
r=500
r=100
3000
1.5
2500
tot
2000
N
Ntot
x 10
1500
1000
1
0.5
500
0
0
20
40
60
0
80
time t
0
20
40
60
80
time t
Figure 9: Examples of dynamics obtained in presence of density-dependent fecundities.
9
Finally, stochastic variations can also be considered. We can indeed assume that in the
course of time, depending on the environmental conditions, the fecundities vary. This
leads to a stochastic Leslie matrix:


0 1 × h(t) 5 × h(t)

0
0
M =  0.3
(12)
0
0.5
0
where h(t) accounts for the random fluctuation of the fecundity with respect to time. For
simplicity, we will consider here that there are favorable years, characterized by a high
fecundity rate (h(t) = 1.5) and defavorable years (low fecundity, h(t) = 0.75). We also
assume that the good or bad years succeed themselves randomly, with a probability 0.5.
The Leslie matrix must be recomputed at each time step. The stochastic nature of this
model make it difficult to do precise prediction of the model. The behaviour may indeed
change from one run to another (see Fig. 10). Depending on the succession of good and
bad years, the population may explode (more or less rapidely) or vanish.
Despite the stochastic nature of the model, we can nevertheless detect a tendancy. If we
look at the mean over many simulations, we can see that the average behaviour follows
the corresponding deterministic growth curve (Fig. 11).
Of course, density-dependent growth and stochasticity can be combined. In Fig. 12, we
incorporate the cell dependent growth defined by Eq. (11) with some stochasticity on r
(r randomly switches between 1 and 1.2, b is fixed to 0.01).
120
200
100
150
tot
60
N
N
tot
80
100
40
50
20
0
0
20
40
60
0
80
time t
0
20
40
60
80
time t
Figure 10: Effect of stochasticity (Left panel: one realization; right panel: 20 realizations).
10
25
20
Ntot
15
10
5
0
0
50
100
150
time t
Figure 11: Effect of stochasticity: The thick curve is the average over 500 simulations (the vertical lines represent the standard deviation), the dashed curve in the
deterministic growth. Here we used h = 0.5 vs h = 1.5 in stochastic simulation and
h = 1 in deterministic simulations.
20
14
N1
18
12
16
10
14
i
8
N
Ntot
12
10
6
8
6
N2
4
4
2
2
0
0
100
200
300
0
400
time t
N3
0
100
200
300
400
time t
Figure 12: Effect of density-dependent fecundities and stochasticity (Left panel: the
total population; right panel: each class).
11
Generalization: matrix models of population
Matrix models in population dynamics cover a more general range of models than the
Leslie matrices. They do not restrict to the internal structuration into age classes. It
is indeed often useful to consider classes of size or to regroup individues by stage of
development. Thus, generally, matrix models can account for structuration in age, size,
maturity degree, developmental stages, physiological conditions, etc.
An example of such natural structuration comes from the insects. The typical life cycle
of insects would be egg → larvae → pupae → adult, and only the adults can lay eggs.
Figure 13: Insect cycle
Lefkovitch (1965) is among the first to generalize the Leslie theory, allowing to consider a
population with any subdivision (Fig. 14). We can then for example include in the model
the fact that only a proportion of the individues change from one class to another during
a given time step.
Figure 14: Life cycle (generalization).
Thus, during a given time step, an individue can either switch from one class to the
next one with a probability Gi , or remains in the same class with a probability Pi . The
corresponding matrix can be written:

P1 F2 F3 F4
 G1 P 2 0 0 

M=
 0 G2 P 3 0 
0
0 G3 P 4

12
(13)
Example: A model for the orca life cycle
Let’s consider another case, the life cycle of the orca (killer whale) (Fig. 15) (Brault &
Caswell, 1993). Here, the time step is the year and the life cycle is subdivided into 4
classes: yearling → juvenile → adult → postreproductive, and both the juveniles and
adults can breed (Fig. 16).
Figure 15: Killer whale (Orcinus orca). Orca are found in all oceans. Its typical size
is 6-9m and weight is 6-8 tons. Its life span is about 50 years. They live in social
groups (“pods”).
Figure 16: Model for the life cycle of orca.
We define Gi the probability to survive from class i to class i + 1, Pi the probability to
survive and remain in class i. Since the first class lasts one year, we have P1 = 0. In this
case, the matrix reads:


0 F2 F3 0
 G1 P 2 0 0 

(14)
M=
 0 G2 P 3 0 
0
0 G3 P 4
Based on some measured and estimated parameters, Brault & Caswell (1993) obtained
the following matrix:


0
0.0043 0.1132
0

 0.9775 0.9111
0
0

(15)
M=


0
0.0736 0.9534
0
0
0
0.0452 0.9804
13
Solving a matrix model - eigen values/eigen vectors
As we have seen previously (cf. Edelstein-Keshet, Chap 1), the matrices equations can
be solved analytically by computing the eigen values λi and eigen vectors vi :
Mv = λv
(16)
The eigen values are obtained by solving the characteristic equation:
Det(M − λI) = 0
(17)
There are as many eigen values as lines (rows) in the matrix: λ1 , λ2 , λ3 ...λk (where k in
the matix size = number of classes).
Perron-Frobenius theorem
Theorem2 :
simple root
eigenvalue,
eigenvector
For any Leslie matrix M, it exists a real positive eigenvalue λ1 that is a
of the characteristic equation. This eigenvalue, which is called the dominant
is strictly greater in magnitude than any other eigenvalue. The associated
v1 is real.
Interpretation: The dominant eigenvalue (λ1 ) determines the asymptotic properties of the
population:
• When λ1 = 1 the population is stationary,
• When λ1 > 1 the population will increase.
• When λ1 < 1 the population will decline (and tend to vanish).
The corresponding eigenvector v1 is proportional to the stable age distribution. It can be
rescaled to give either the proportion or the percentage of individuals in each age class in
the asymptotical regime.
Asymptotic behaviour and intrinsic growth rate
As t becomes large,
N(t) ∼ λt1 = ert
(18)
The intrinsic growth rate r is thus defined by
r = ln λ
(19)
Thus, r > 0 means an increase while r < 0 means a decline of the population.
2
The theorem is in fact more general and applies not only to the Leslie and generalized Leslie matrices,
but also to a more general class of matrices.
14
Application to the toy example shown previously
Let’s illustrate these properties on the toy example described above:


−λ 1
5
Det  0.3 −λ 0  = 0
0 0.5 −λ
−λ3 + 0.3λ + 0.75 = 0
(20)
(21)
This is a cubic equation. It can be solved by trials-errors or with the computer. We then
find that the largest eigen value is λ1 = 1.018, implying that the population will grow. This
is indeed what we observed in our simulation (Fig. 5). The two other values are complex
conjugates (λ2,3 =-0.509 ± 0.691i), explaining why we observe transient oscillations at the
beginning of the dynamics.
The eigen vector associated to λ1 is v1 = (0.6948, 0.2047, 0.1005), which corresponds to
the asymptotic fractions of each class.
15
Application: the loggerhead turtles
We illustrate here the use of matrices models on a concrete example (from Crouse et al,
1987, see also Krebs, pp. 158-159). Crouse et al analyzed the dynamics of the loggerhead
sea turtle, an endangered species (Fig. 17).
Figure 17: Loggerhead Turtle (Carreta carreta). The loggerhead sea turtle is an
oceanic turtle living in the Atlantic, Pacific, and Indian oceans as well as the Mediterranean Sea. The average loggerhead turtle measures around 90 cm and weights
around 135 kg, although some specimens can reach 270 cm and 454 kg. It reaches
sexual maturity within 17-33 years and has a lifespan of 47-67 years. Loggerhead turtles are considered an endangered species and must be protected (Source: wikipedia).
These turtles have a long life span (around 30 years) that can be subdivided into seven
stages based on the size. Figure 18 lists these stages together with the size and approximate age range of the turtles in each stage. Survivorship varies with size and only
individues over 87 cm are sexually mature.
Figure 18: Data (Crouse et al 1987).
The challenge here is to build the corresponding matrix, in

P1 F2 F3 F4 F5 F6
 G1 P 2 0 0
0
0

 0 G2 P 3 0
0
0


0 0 P4 0
0
M= 0
 0
0
0
G
P
0
4
5

 0
0 0 0 G5 P 6
0
0 0 0
0 G6
the form

F7
0 

0 

0 

0 

0 
P7
(22)
where Fi , is the stage-specific fecundity, and Pi , and Gi are, resp. the probability of
surviving and remaining in the same stage vs. the probability of surviving and growing
to the next stage.
16
The transition probabilities Gi , and Pi , can be estimated from the stage-specific survival
probabilities pi , and stage duration di . Because we know little about the variability of
survival and growth rates within a stage, we will assume that all individuals within a
stage are subject to the same survival probability and stage duration. As more precise
data on the growth rates and survival of turtles of various sizes become available they can
be readily incorporated into the model.
Within each stage there are individuals who have been in that stage for 1,2,...d years.
By setting the proportion of individuals alive in the first cohort of stage class i to 1 and
the probability of turtles in that cohort surviving to the next year top (annual survival
probability for the entire sire = stage class), the probability of those individuals surviving
d years become pdi . Assuming that the population is stationary and the age distribution
within stages is stable, the relative abundance of these groups of individuals then becomes
1, pi , p2i , ...pd−1
. In the interval from t to t + 1, the oldest individuals in this stage will
i
move to the next stage, if they survive. All the younger individuals will remain in the
stage. Thus the proportion remaining, and surviving, is given by
Pi =
1 + pi + p2i + ... + pdi i −2
1 + pi + p2i + ... + pdi i −1
!
pi
(23)
Thus, the number of individuals in any cohort within a stage class declines through time
as a function of the stage-specific annual survival probability and the number of years
spent in that stage.
Using the geometric series
1 + p + p2 + ... + pd−1 =
we can rewrite Pi as:
Pi =
1 − pdi i −1
1 − pdi i
!
1 − pd
1−p
(24)
(25)
pi
The proportion of the population that grows into the next stage class and survives (Gi )
is similarly given by the proportion of individuals in the oldest cohort of the stage times
the annual survival for the stage, or
Gi =
pdi i −1
1 + pi + p2i + ... + pdi i −1
!
pi
(26)
which can be rewritten, in the same manner as before, as
Gi =
pdi i (1 − pi )
1 − pdi i
!
pi
The resulting stage class population matrix takes the form:
17
(27)
Figure 19: Matrix (Crouse et al 1987).
The eigenvalue and intrinsic rate of increase for this matrix are λ = 0.9450 and r = ln λ =
−0.0565. These values indicate a decline of the population.
Now the question is to determine the optimal management practices to reduce the decline
of the population. What would happen if we eliminate mortality in any of these stages?
Of course, no management practice can promise zero mortality for any period of time,
but such a simulation should help identify the life stage(s) on which management efforts
would be most efficiently spent. The results of elimination of mortality for each stage
class respectively are presented in Fig. 20. Also included is a simulation of a doubling in
fecundity, which is within the range of possibilities.
Figure 20: Results (Crouse et al 1987).
These results suggest that the juvenile and subadult stages are most responsive to such a
change. In fact, an increase in survival to 1.0 in any one of stages 2, 3. or 4 (or that of
the suddenly immortal mature females) was sufficient to reverse the decline of the model
population. More importantly, the simulation indicates that no matter how much effort
was put into protecting eggs on the beach, this alone could not prevent the eventual
extinction of the model population. Similarly, the turtles could not reverse their decline
via increases in fecundity unless they could more than double egg production, which seems
unlikely.
18
Matrix models and age pyramids
Matrix models can be used to predict the evolution of human population in a country.
Age pyramids can be very important for public policy decisions. For example, if you
compare the percentage of the population aged 70 and over in the year 2000 with the
same percentage in the year 2025 you will see immediately why people are worried about
the Social Security and Medicare...
Keyfitz and Flieger (1971) devised a matrix with age structuration for the USA population. Based on a census dating back from 1966, these authors estimate an asymptotic
growth rate of 1.05, meaning that the population increases by 5% every 5 year (the projection interval was 5 years). They also compute the stationnary age distribution.
Figure 21 shows the estimated United States population in the year 2000 and predictions for the year 2025 presented by age and gender. This data was obtained from the
International Data Base3 maintained by the United States Census Bureau.
Figure 21: Age pyramids for USA.
19
Figure 22 shows the same information for India obtained from the same source. The
population pyramid for India in the year 2000 is a typical example of a population pyramid
for an underdeveloped country. Because survival rates are so low, the population drops
rapidly as age increases. The population pyramid for India for the year 2025 is a typical
population pyramid for a developing country. As the public health improves, survival
rates improve and we see smaller population decreases as age increases. You can see this
effect clearly by comparing the population pyramids for India for the years 2000 and 2025.
Note that the population pyramids for the United States and India for the year 2025 are
based on predictions, not on actual facts. So demographers are predicting that over the
next twenty years India will develop rapidly.
Figure 22: Age pyramids for India.
20
Concluding remarks
Projection matrices have become a common modelling approach in population demography because (1) they are relatively easy to formulate, (2) they compile complex data in
a structured and analytically tractable manner, (3) they provide parameters with direct
biological meaning, (4) they allow the investigator to address general or specific, experimental and/or theoretical, and ecological and evolutionary questions (Salguero-Gomez &
de Kroon, 2010).
Since the first version by P. Leslie (1945, 1948) and Lefkovich (1965), numerous developments and extensions have been proposed. Matices models can account for densitydependent growth, for environment-dependent parameters, for the interaction between 2
(or more) populations, or for multi-regional ecosystems.
The last decade has witnessed major advancements in this field, that have brought demographic models much closer to the real world, in particular in the analysis of effects
of spatial and temporal environmental variation on populations (Salguero-Gomez & de
Kroon 2010). Recent theoretical works led to the development of stochastic life table,
stochastic elasticities, analysis of the transient dynamics and phylogenetic analyses.
Applications of these approaches include the preservation of endangered species, the optimization of fishery strategies, the control of animal or plants populations, as well as
human demography (and spreading of diseases).
21
Exercises
Exercise 1
A population of bats living in a cave is studied and the following data are collected:
Age (months)
0 - 6 6 - 12
Initial population 4500 1800
Birth rate
0
1.9
Death rate
0.5
0.2
12 - 18 18 - 24
900
130
1.5
0.7
0.6
1
1) Calculate the survival rates
2) Determine the Leslie matrix
3) What is the total population after 6, 12, 18 months and after 5 years?
4) In the long term, what is the percentage of increase in the population every 6 months?
5) How does this growth rate change when the initial condition changes?
Exercise 2
A population of kangoroos is studied and the following data are collected:
Age (years)
Initial population
Breeding rate
Survival rate
0 - 2 2 - 4 4 - 6 6 - 8 8 - 10
3400 2500 2300 1750 650
0
0
3.9
2.7
0.9
0.5
0.8
0.7
0.4
0
Suggest a suitable culling rate to maintain a stable population if the culling is carried out
every 2 years.
Exercise 3
A population of buffalo in a certain area is famed as a resource. The following data are
available:
Age (years)
Initial population
Breeding rate
Survival rate
0-1 1-2 2-3 3-4 4-5
2350 2000 2000 1450 825
0
0
1.5
1.7
1.2
0.6
0.7
0.9
0.5
0.3
400
0.4
0
Investigate the effect on the population of different harvesting rates. Only buffalo aged
2-4 years would be slaughtered for the meat and the skin.
22
Exercise 4
Plant growth: cf. Edelstein-Keshet, p. 34 ex. 19.
Exercise 5
Calculate (with a computer) the (larger) eigen value and its associated eigen vector for
the orca matrix (Eq. 15) and determine if the population will increas or decrease and
what will be the asymptotic proportion in each class.
Exercise 6
Age pyramid. Let’s consider the population of a (fictive) country in 2012 divided into 5
age groups: 0-19, 20-39, 40-59, 60-79 and 80+:


10000000
 9800000 



(28)
N0 = 
 9600000 
 9000000 
8400000
and the following matrix, using a 5-year projection interval:

0.20 0.40 0.30 0.10 0
 0.95 0
0
0
0


0
0
M =  0 0.95 0
 0
0 0.90 0
0
0
0
0 0.80 0.50






(29)
1) Using a computer to predict the population size in years 2022, 2042, and 2082. For each
year, determine the percentage of the population in each age group (i.e. draw the age
pyramids). Compute the eigen values and eigen vectors to determine the (theoretical)
asymptotic behavior (proportion of poeple in each class) and compare the theoretical
results with the results of the simulations.
2) What is the problem with this matrix? How would you correct it?
23
References
Books
• Caswell H (2001) Matrix Population Models: Construction, analysis, and interpretation. Sinauer Associates.
• Edelstein-Keshet (2005) Mathematical Models in Biology, SIAM.
• Krebs CJ (2008) Ecology: The Experimental Analysis of Distribution and Abundance, Ed. Pearson.
Lecture notes
• Anonymous (undated) Les modèles matriciels de population.
• Mosimanegape Irvin Montshiwa (2007) Leslie Matrix Model in Population Dynamics.
• Gerlach J (2012) Mathematical Analysis and Modeling.
Papers
• Leslie, P.H. (1945) The use of matrices in certain population mathematics. Biometrika
33:183-212.
• Leslie PH (1948) Some further notes on the use of matrices in population mathematics. Biometrika 35:213-245.
• Lefkovitch LP (1965) The study of population growth in organisms grouped by
stages. Biometrika 35:183-212.
• Usher MB (1966) A matrix approach to the management of renewable resources,
with special reference to selection forests. Journal of Applied Ecology 3:355-367.
• Pennycuick CJ, Compton RM, Beckingham L (1968) A computer model for simulating the growth of a population, or of two interacting populations, J. Theor. Biol.
18:316-29.
• Usher MB (1969) A matrix model for forest management. Biometrics 25:309-315.
• Caswell H (1978). A general formula for the sensitivity of population growth rate
to changes in life history parameters. Theoretical Population Biology 14:215-230.
• Ziebur AD (1984) Age-dependent models of population growth. Theor Popul Biol.
26:315-9.
• de Kroon et al (1986) Elasticity: the relative contribution of demographic parameters to population growth rate, Ecology 67:1427-1431.
24
• Crouse DT, Crowder LB, Caswell H (1987) A Stage-Based Population Model for
Loggerhead Sea Turtles and Implications for Conservation Ecology 68:1412-1423.
• Allen LJ (1989) A density-dependent Leslie matrix model. Math Biosci. 95:179-87.
• Clarke RP, Yoshimoto SS (1990) Application of the Leslie Model to Commercial
Catch and Effort of the Slipper Lobster, Scyllarides squammosus, Fishery in the
Northwestern Hawaiian Islands, Marine Fisheries Review, 52:1-7.
• Brault S, Caswell H (1993) Pod-Specific Demography of Killer Whales (Orcinus
Orca), Ecology 74:1444-54.
• Smith GC, Trout RC (1994) Using Leslie matrices to determine wild rabbit population growth and the potential for control. J. Appl. Ecology 31:223-30.
• Jensen AL (1995) Simple density-dependent matrix model for population projection,
Ecological modelling 77:43-48
• Fifas S, Goujon M, Antoine L (1998) Application of Leslie’s model on a population
of common dolphins (Delphinus delphis): Sensitivity study, Aquat. Living Resour.
II(6):359-369.
• Li CK, Schneider H (2002) Applications of Perron-Frobenius theory to population
dynamics. J Math Biol. 44:450-62.
• Gerber, L. and S.S. Heppell (2004) The use of demographic sensitivity analysis in
marine species conservation planning. Biological Conservation 120:121-128
• Miethe T, Dytham C, Dieckmann U, Pitchford JW (2010) Marine reserves and
the evolutionary effects of fishing on size at maturation. ICES J Marine Science
67:412-425,
• Salguero-Gomez (2010) Matrix projection models meet variation in the real world,
J Ecology 98:250-254.
• Caceres MO, Caceres-Saez I (2011) Random Leslie matrices in population dynamics.
J Math Biol. 63:519-56.
• Thomas JR, Clark SJ (2011) More on the cohort-component model of population
projection in the context of HIV/AIDS: A Leslie matrix representation and new
estimates. Demogr Res. 25:39-102.
• Skalski JR, Millspaugh JJ, Clawson MV (2012) Comparison of statistical population
reconstruction using full and pooled adult age-class data. PLoS One. 7:e33910.
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