NRES 310

Leslie Matrix

8 October 2008

LESLIE MATRIX

Way to project population growth of age-structured populations through time.

Named after population biologist Patrick H. Leslie.

Describes changes in population size due to mortality and reproduction.

If there are k age classes, then the Leslie matrix is represented as a k x k square matrix and has the following form.

P0m1 / P1m1 / P2m2 / P3m3
P0 / 0 / 0 / 0
0 / P1 / 0 / 0
0 / 0 / P2 / 0
n0
n1
n2
n3

Each column of the leslie matrix is the age at time t and each row is the age at time t+1

Each entry in the matrix represents a transition or change in the number of individuals from one age class to another.

In the Leslie Matrix the fertilities are always in the first row; they represent contributions to newborns from reproduction of each age class. The survival probabilities are always in the subdiagonal. They represent transitions from one age class to the next. All other entries are 0 because no other transitions are possible. Individuals cannot remain in the same age class from one year to the next, so the diagonals must equal 0. Also individuals cannot skip or repeat age classes so other entries in the matrix are zero.

If you calculate the dominant eigenvalue = lambda, finite rate of growth.

Dominant eigenvector is the SAD, broken down by age class.

Assumptions:

·  Sample is representative of the population

·  Stable age distribution: proportion of animals in each age class is unchanging.

o  lx and mx are fixed by age class in leslie matrix, which causes a stable-age distribution by definition, even if the population is increasing, decreasing, or stationary.

Nobody projects population growth using life tables, use the Leslie matrix to project population growth into the future.

Use only with organisms that do not exhibit strong density dependence (some waterfowl and upland game birds).

L = square matrix x column vector

ð  moves the population forward 1 increment in time.

This is where we use px = 1-qx (concerned with survivorship, not mortality here).

Remember survivorship and fecundity are rates!

Guts of the leslie matrix come from life-tables

x / qx / mx / px
0 / 0.6 / 0 / 0.4
1 / 0.6 / 1.25 / 0.4
2 / 0.5 / 3.75 / 0.5
3 / 1.0 / 0 / 0

Leslie Matrix

Assumptions:

·  Sample is representative of the population

·  Stable-age distribution: proportion of animals in each age class is unchanging.

o  Stationary-age distribution: (special case of stable-age distribution) neither the proportion or numbers of animals in each age class are changing, r=0, λ=1

·  lx and mx are fixed by age class in leslie matrix, which causes a stable-age distribution whether the population is increasing, stationary, or decreasing.

Nobody projects population growth using life tables, use the leslie matrix.

Use with organisms that do not exhibit strong density dependence (e.g., waterfowl, upland game birds)

Guts of Leslie Matrix come from Life-tables.

L = => move the population forward 1 increment of time.

Age class / Age-specific mortality rate / Age specific fecundity / survivorship
x / qx / mx / px
0 / 0.6 / 0 / 0.4
1 / 0.6 / 1.25 / 0.4
2 / 0.5 / 3.75 / 0.5
3 / 1.0 / 0 / 0

Emilen’s correction P0 · m1

P0m1 / P1m2 / P2m3 / 0
p0 / 0 / 0 / 0
0 / p1 / 0 / 0
0 / 0 / p2 / 0
n0
n1
n2
n3
(0.4)(1.25) / (0.4)(3.75) / (0.5)(0) / 0
0.4 / 0 / 0 / 0
0 / 0.4 / 0 / 0
0 / 0 / 0.5 / 0
0.5 / 1.5 / 0 / 0
0.4 / 0 / 0 / 0
0 / 0.4 / 0 / 0
0 / 0 / 0.5 / 0
0
100
0
0

L =

150
0
40
0

190 animals

0.5 / 1.5 / 0 / 0
0.4 / 0 / 0 / 0
0 / 0.4 / 0 / 0
0 / 0 / 0.5 / 0
150
0
40
0
75
60
0
20

155 animals

The Leslie matrix allows populations to fluctuate over time. It mimics changes in population over time.

Based on life tables so it places the same assumptions, namely stable-age distribution.

Works well for populations that do not show strong density dependence. WHY?

K?

Spectral Decomposition of leslie matrix

Get eigenvalues and vectors.

Dominant eigenvalue is λ (finite rate of growth).

Corresponding eigenvector (or dominant eigenvector) is the stable age distribution broken down by age class.

Matrix defined by number of rows (m) and number of columns (n)

4 x 4 matrix 4x1 matrix => column vector (1x4 matrix row vector)

1 1 1 1 1 4

2 2 2 2 1 8

3 3 3 3 x 1 = 12

4 4 4 4 1 16

1x1 + 1x1 + 1x1 + 1x1 = 4

1x2 + 1x2 + 1x2 +1x2 = 8

1x3 + 1x3 + 1x3 + 1x3 = 12

1x4 + 1x4 + 1x4 + 1x4 = 16