BIO 357 Evolutionary Ecology
Fall 2000
Handout 1
Population Modeling
First, we would like to know how many individuals are in each age class of a population. The
simplest model assumes non-overlapping generations (and therefore, no age class structure).
1)
2)
Nt+1 =  Nt
Number at time t+1 is equal to number at time t
multiplied by the finite rate of increase, , a
constant.
Nt+2 =  Nt+1
=  ( Nt)
= 2 Nt
We can use this to find the number of individuals
at any time (and eventually we will be able to
use this for age classes).
Nt+x = x Nt
Generalization of the above.
To make this age class specific, we need to add the probability of surviving from the previous age
class as well as the new young being produced.
Nt+1 = p Nt +  Nt
where p = probability of survival
*A Leslie Matrix can simplify all of this!
Use this diagram to help you visualize the age structure of a population.
P0
P1
m3 m2 m1
P2
m1
P3 = 0
m2
m3
N0
N1
N2
N3
age class 0
age class 1
age class 2
age class 3
Px = lx+1/lx
We must assume that mortality precedes reproduction.
N0(t+1) = N0(t) P0 m1 + N1(t) P1 m2 + N2(t) P2 m3 + N3(t) P3 m4
N1(t+1) = N0(t) P0
N2(t+1) =
N3(t+1) =
N1(t) P1
N2(t) P2
The above equations are taken directly from the diagram. These equations make up the basis of the
Leslie Matrix when rewritten:
N0(t+1)
P0m1
N1(t+1)
P1m2 P2m3 0
N0(t)
P0
0
0
0
N2(t+1)
0
P1
0
0
N2(t)
N3(t+1)
0
0
P2
0
N3(t)

=
Vector
=
N1(t)
 Vector of Variables
Leslie Matrix (constant)
To make the notation a little less cumbersome, let:
N0(t+1)
P0m1
N1(t+1)
P0
P1m2 P2m3 0
0
0
N0(t)
0
= Nt+1
N1(t)
=L
= Nt
N2(t+1)
0
P1
0
0
N2(t)
N3(t+1)
0
0
P2
0
N3(t)
Now, as before, let’s look at time t + 1 and time t + 2 to find a useful generalization.
3)
= L N
N
t+1
N
t+2 =
Note the similarity to equation 1
t
L N
t+1
= L ( L N t)
4)
N
= L2 N
t
Lx N
t
t+x =
Generalization – note the similarity to equation 2
Using this information, we can predict the number of individuals in all age classes at any time in the
future. Equation 4 can be used to find the population trajectory for all age classes. Note, the size of
n
the whole population is the sum of all of the age classes (  Nx (t ) ).
x 0
If the population is not already at its stable age structure, it will converge on this stable structure
through a series of damped oscillations. The stable age structure is reached when the proportion of the
population in each of the age classes stays constant over time. This will occur whether the population
size is increasing (a), decreasing (b), or not changing (c).
N0
N1
N2
N0
N1
ln Nt
N0
N2
N1
N2
(a)
t
(b)
(c)
Matrix Algebra
Here are some basic rules of matrix algebra to aid you in using the Leslie Matrix.
Matrix – vector multiplication
a b c
d e f
g h i
j
k
l

3  3 matrix
aj + bk + cl
dj + ek + fl
gj + hk + il
=
1  3 vector
1  3 vector
Scalar –vector multiplication
a
d b
c
=
da
db
dc
Matrix – matrix multiplication
a b c
d e f
g h i
j k l
m n o
p q r
=
(aj + bm + cp) (ak + bn + cq) (al + bo + cr)
(dj + em + fp) (dk + en + fq) (dl + eo + fr)
(gj + hm + ip) (gk + hn + iq) (gl + ho + ir)
Notice that with matrix-matrix multiplication:
A  B  B  A , unless A = B
Relationship between r and 
Both r (the intrinsic rate of increase) and  ( the finite rate of increase) correspond to the growth rate
of a population. The primary difference between them is that r is used in differential equations
whereas  is used in discrete equations.
Nt+x = x Nt
dN/dt = rN
solve
Nt+x = N0 er(t+x)
set t = 0 and x = 1
N1 = N0 
N1 = N0 er
So,
N1/N0 = er = 
r = ln 
So, the intrinsic rate of increase of a Leslie Matrix (r) is the natural log of the finite rate of increase ()
which is, in turn, the dominant eigenvalue of the Leslie Matrix.
An important side note
The proportion of a population in age class j is given as
cj =
l j e  rj
n
n
 l x e rx
and
cj = 1
j 1
x 0
The Euler Equation
Assuming that we have a stable age distribution, we can use the Euler equation to find r. This process
requires tedious iteration and therefore, you will not be required to do it.
Derivation of the Euler equation
n
N0(t) =  N x (t ) m x
Observation
2)
Nx(t) = N0(t-x)lx
Observation
3)
N0(t) =  N 0(t  x ) l x m x
Combine equations 1 and 2
4)
N0(t) = N0(0)ert
Observation
5)
N0(t-x) = N0(t)e-rx
Observation
6)
N0(t) =  N 0(t ) e rxl x mx
1)
x 0
n
x 0
n
Combine equations 3 and 5
x 0
7)
n
N0(t) = N0(t)  e rxl x mx
Remove constant from summation
x 0
n
8)
1 =  e rxl x mx
x 0
Divide both sides by (N0(t)), to get the Euler equation