MEI Conference 2012
Application of Transition
Matrices to Investigate
Populations
Kevin Lord
Kevin.lord@mei.org.uk
Modelling Populations with Transition Matrices
Ecologist, zoologist and biologist use modelling to predict population size for several reasons



Conservation – to stop species going extinct
Economics – to ensure there are sustainable resources to support the world population
Health – to control spread of disease in populations
Mathematical modelling offers a cheap and quick method for predict changes. It also enables
scientists to analyse how different factors affect the population without the need for lengthy or
perhaps unethical experiments.
Writing down population equations
Think of a simple model where the population is made up of two groups
Juveniles (J) -
Young individuals not old enough to reproduce
Adults (A)
Older individuals who can give birth
-
How do the number of Juveniles and Adults change over time?
J1
J2
A1
A2
Writing this in words leads to a recurrence relation:
J2
=
A1 x births per adult (i.e. the birth rate)
A2
=
J1 x chance of survival + A1 x chance of survival
And in symbols:
J2
=
A1 x b
A2
=
J1 x p
+
A1 x q
These equations can also be written in matrix notation:
0
and
are column vectors and they contain the number of juveniles and adults for each
generation.
0
is a 2 by 2 Transition Matrix and it contains the information about how the population
changes between each generation.
Exercise 1
Consider an initial population J1 = 100 and A1 = 100.
Investigate what happens to the population for different values of b, p and q.
For example, let b = 2, p = 0.5 and q = 0.5 and see how the population changes for the first few
generations.
Leslie Matrices
A Leslie matrix is a discrete, age structured model of population growth named after Patrick H.
Leslie. It is one of the best known ways to describe the growth of populations (and their projected age
distribution), in which a population is closed to migration and where only one sex, usually the female
is considered.
The population is divided into groups based on age classes. The Leftkovitch matrix is a similar
model which replaces age classes with life stages, whereby individuals can both remain in the same
stage class or move on to the next one.
This age-structured growth model leads to a steady-state, or stable, age-structure and growth rate
which is constant. Regardless of the initial population size or distribution, the population will settle
down to the same age structure and steady growth rate. It also returns to this state following
perturbation.
Solving the characteristic equation of the Leslie matrix gives the eigenvalues and the dominant
eigenvalue is the steady growth rate. The associated eigenvector is the stable age-distribution.
Exercise 2
Find the eigenvalues and eigenvectors for these 2x2 Leslie Matrices and comment on the population
growth over time.
0
2
0.5 0.5
0
6
0.25 0.5
0
2
0.25 0.5
0
2
0.2 0.5
Three Stage Population Distributions
Consider a population that can now be divided into 3 stages
Juveniles (J) -
Young individuals not old enough to reproduce
Adults (A)
-
Older individuals who can give birth
Seniles (S)
-
Older adults that no longer reproduce
Consider how the number of Juveniles, Adults and Seniles change over time?
J1
J2
A1
A2
S1
S2
Exercise 3
If the birth rate is still b and the survival rates for juveniles, adults and seniles are p, q and r
respectively, construct the recurrence relation and associate matrix equation for the diagram above.
Loggerhead Sea Turtle Problem
Description
Age
Eggs & Hatchlings
0-1
Small juveniles
1-7
Large juveniles
8-15
Sub-adults
16-21
Novice Breeders
22
1st Year reimmigrants
23
Mature breeders 24-54
24-54
Transition Matrix
0
0
0
0
127
4
80
Eggs & Hatchlings
0.68
0.74
0
0
0
0
0
Small juveniles
0
0.05
0.66
0
0
0
0
Large juveniles
0
0
0.01
0.69
0
0
0
Sub-adults
0
0
0
0.05
0
0
0
Novice Breeders
0
0
0
0
0.81
0
0
1st Year reimmigrants
0
0
0
0
0
0.81
0.81
Mature breeders 24-54
Threats to Survival: The greatest threat is loss of nesting habitat due to coastal development,
predation of nests, and human disturbances (such as coastal lighting and housing developments) that
cause disorientations during the emergence of hatchlings. Other major threats include incidental
capture in longline fishing, shrimp trawling and pollution. Incidental capture in fisheries is thought to
have played a significant role in the recent population declines observed for the loggerhead.
Population Estimate: 44,560 nesting females (based on monitoring reports and publications from
Source – Sea Turtle Conservancy
2004.)
Current management practices appear to be focused on the least responsive life stage. eggs on
nesting-beaches. Alternative protection efforts for juvenile loggerheads. such as using turtle excluder
devices (TEDs).may be far more effective
Deborah T. Crouse; Larry B. Crowder; Hal Caswell Ecology, Vol. 68, No. 5. (Oct., 1987)