Age Structure

1
Age Structure

• A population can be broken up into an age vector
  representing the numbers in each age class N
• Can multiply this by a Leslie matrix (L) which
  summarizes vital rates

2
Age Structure

• Can use the Leslie matrix to calculate the
  numbers in each age class at the next time step
• Nt1LNt
• Nt LtN0
• To calculate the population after t years you
  would have to multiply the initial population
  (N0) by the Leslie matrix (L) t times

3
Age Structure

• The stable age distribution refers to the ratio
  of individuals in a given age class (Not the
  number of individuals)
• If we assume ratio of year classes is constant,
  than growth rates must affect year classes
  equally.
• LN?N

4
Age Structure

• LN?N

N
L
5
Eigenvalues and Eigenvectors

• If you know the Leslie matrix and Age vector you
  can summarize population growth rate with
  eigenvalues and eigenvectors

6
Age Structure

• Start with LN?N
• LN-?N0
• LN-?IN0 where I represents the identity
• matrix

7
Identity matrix

• If we multiply any number by 1 the answer is just
  the number we started with. The number 1 is
  called the multiplicative identity. The matrix
  that plays the same role is called the identity
  matrix and is denoted by I, so INN for any
  matrix or vector N of the appropriate size. The
  identity matrix with two rows and columns is

I
8
Age Structure

• Start with LN?N
• LN-?N0
• LN-?IN0
• (L-?I)N0

9
Age Structure

• A fact from matrix algebra is the final equation
  is solved if and only if the determinant of
  (L-?I)0

10
Determinant of a matrix

• If A

Then det(A) AD - BC
11
Age Structure

• So say L

?I
L-?I
12
Age Structure
det(L-?I)0
det(L-?I)(a11- ?)(a22- ?)-a12a22
Can solve for ? with the quadratic formula
13
Quadratic formula
ax2bxc0
14
Eigenvector

• Now that we have the eigenvalues, we can find the
  corresponding eigenvectors
• NtLtN0
• Let ?0 and ?1 be the two eigenvalues and v0 and
  v1 be the corresponding eigenvectors

15
Eigenvectors

• We will first express initial population size
  (N0) as the sum of the two eigenvectors
• N0a0v0 a1v1
• NtLtN0Lt(a0v0a1v1)
• a0

v0
a1
v1
If ?0 gt ?1 then Nt?a0 ?0tv0
16
Solving for eigenvector

• L
• Find the eigenvector v0 that corresponds to the
  dominant eigenvalue ?0 such that Lv0 ?0v0. Try
  vector of the form

17
Solving for eigenvector

• Thus,
• a11a12?0
• a21 a22b ?0b
• In general each equation will have the same
  solution for b. The resulting vector represents
  the stable age distribution.

18
Eigenvector

• Typically the age distribution vector is
  expressed so that the abundance of each age class
  is given as a proportion of the total population
  (i.e. nx sum to 1)

19
EXAMPLES

• By Hand
• Excel
• R
• Matlab example from TJ Case

20
Summary

• Age specific vital rates can be expressed in a
  square matrix called a Leslie Matrix (L).
• Can use this matrix and an initial age vector N0
  to calculate the total population size at time t
• Eventually population will approach a stable age
  structure and an ultimate geometric growth rate
• Can determine ultimate geometric growth rate by
  finding the dominant eigenvalue of L.
• The corresponding eigenvector represents the
  stable age distribution