4H Mathematical Biology Computer Lab #2
Discrete-time Matrix Models.
DUE: 5pm Monday 13th March 2017
The aim of this assignment is to investigate how changes in life-cycle parameters effect long-
term population growth rates.
In this assignment, we shall study an example of an Usher Model Nt+1 = ANt where A
is an n× n Usher matrix and Nt = (N1t , N1t , . . . , Nnt )T is a 1× n column vector, and N it ≥ 0
is the number of individuals in stage i at time t. This is a variation on a Leslie Model, that
has some non-zero entries on the main diagonal of the Usher matrix A, e.g.
A =
P1 F2 F3 F4 F5 F6 F7
G1 P2 0 0 0 0 0
0 G2 P3 0 0 0 0
0 0 G3 P4 0 0 0
0 0 0 G4 P5 0 0
0 0 0 0 G5 P6 0
0 0 0 0 0 G6 P7
.
Fi ≥ 0 is the stage specific fecundity, 0 ≤ Pi ≤ 1 is the probability of surviving and remaining
in the same stage, and 0 ≤ Gi ≤ 1 is the probability of surviving and growing to the next
stage.
Figure 1: Desert tortoise
Leslie and Usher models can be used to help design
intervention strategies to help declining populations
recover. An example of this is the study of desert
tortoises, that are a threatened species in the West-
ern Mojave Desert of California. The tortoises have
seen rapid population decline due the destruction of
vegetation by grazing and off-road vehicle traffic.
An intervention might be designed to affect any
one of the entries in the Usher matrix that models
the population. Because the dominant eigenvalue of
the matrix determines the overall growth rate, it is
necessary to study how changing each matrix entry affects the dominant eigenvalue. Deter-
mining the effect of small changes in each of the entries is called a sensitivity analysis.
The Usher matrix for the desert tortoises is
A =
0 0 0 0 127 4 80
0.67 0.74 0 0 0 0 0
0 0.05 0.66 0 0 0 0
0 0 0.015 0.69 0 0 0
0 0 0 0.052 0 0 0
0 0 0 0 0.81 0 0
0 0 0 0 0 0.81 0.81
The desert tortoise model has 7 stage classes: (1) Eggs, hatchlings, (2) small juveniles, (3)
large juveniles (4) subadults, (5) novice breeder, (6) mature breeders, (7) old breeders. The
corresponding Usher model describes the changes over a time step of 1 year.
4H Mathematical Biology Computer Lab #2
1. Draw a flow diagram illustrating the transitions between age groups.
2. Use the eig and max or related commands in Matlab to calculate the dominant eigen-
value λ of the Usher matrix A and the stable age distribution (v); state their values.
Will the population grow or decline and at what rate? What does the value of v tell
you about the distribution of the population among the age classes?
3. We now investigate how changes to the matrix entries affect the growth rate. As an
example, we’ll replace the a22 entry in A by another non-negative constant a. For each
value of a between 0 and 1, in increments of 0.05, calculate the dominant eigenvalue.
You should do the calculation inside a for loop and store your output in a vector. Then
plot the graph of the dominant eigenvalue against a. What does it tell you about how
the a22 entry of A effects the decline or growth of the tortoise population?
4. One disadvantage of experimenting with changes in parameters in the way we have just
done is that the results are dependent on the size of the perturbation, so it’s difficult to
compare the effects of changing different parameters, such as fecundity versus survival.
To avoid this difficulty, we calculate the proportional sensitivity (or “elasticity”) of
λ, the dominant eigenvalue. Elasticity is the proportional change in λ caused by a
proportional change in one of the life cycle parameters aij. Because these elasticities
sum to one, the relative contribution of the matrix elements (Fi, Pi and Gi) to λ can
be compared.
The proportional sensitivity (elasticity) of λ to changes in a matrix element aij is given
by
∂ lnλ
∂ ln aij
=
aij
λ
∂λ
∂aij
=
aij
λ
(
wivj∑
k wkvk
)
, (1)
where λ is the dominant eigenvalue of A, v the corresponding eigenvector and w is the
corresponding left eigenvector (wA = λw). The vk and wk are the k-th entries in the
vectors v and w respectively.
Now you will derive the formula given above for elasticity for a general n × n matrix
A.
(a) Use the definition of right eigenvector Av = λv to show:
∂A
∂aij
v + A
∂v
∂aij
=
∂λ
∂aij
v + λ
∂v
∂aij
(∗).
Here you take the partial derivative of a matrix by taking the partial derivative
of each entry. For example, in computing ∂A
∂aij
, you have to compute the partial
of every entry. So ∂A
∂aij
is a matrix with zeros in every place except the ith row
and jth column.
(b) Using the equation (*) above together with the fact that wA = λw (w is referred
to as a left eigenvector) to prove
w · v
∂λ
∂aij
= w ·
∂A
∂aij
v .
4H Mathematical Biology Computer Lab #2
(c) This is almost the formula we want; use the discussion above about differentiating
a matrix to show:
w ·
∂A
∂aij
v = wivj .
Now finish proving the formula in equation (1) (Note: You only need to show
∂λ
∂aij
=
wivj∑
k wkvk
in your answer).
5. Using matlab find w, the dominant left eigenvector, corresponding to the original
matrix A for the tortoise model. In part (2), you found λ which may help.
6. Write a Matlab programme which includes your matlab code for calculating w, v and
λ which you worked out in parts (2) and (5) which calculates the elasticity of λ to
changes in each value of Pi. So for each stage i = 1, …, 7 calculate the elasticity of Pi
Recall the formula for elasticity you need is the right hand side of following expression:
∂ lnλ
∂ ln aij
=
aij
λ
(wivj
w.v
)
(2)
(Note w.v = w ∗ v in Matlab).
7. Produce a plot of elasticity of λ to changes in Pi. The x-axis contains stage i and the y-
axis will have the corresponding value of the elasticity. Now modify the programme to
plot elasticity of λ to changes in Gi and plot this on the same graph. Your plot should
now contain two lines, one corresponding to the effects of Gi and one corresponding
to the effects of Pi. Change the programme again so that this time it plots elasticity
of λ to changes in Fi. You may want to use different symbols to distinguish between
the 3 lines in the plot. (Include a print out of your commented code for the case where
you plot elasticity of λ to changes in Fi and Gi in your report together with your final
plot.)
8. Examine the plot that you have produced. Nearly all conservation efforts focus on the
egg life-stage. Tortoise nests are readily accessible and can be protected, and losses
and protection successes are easily monitored. But, given our poor understanding
of tortoise population dynamics, it is not clear whether egg protection efforts will
ultimately prevent extinction. From the results of your elasticity plot, state with
reasons whether the model supports egg protection, identifying which parameters are
affected by egg protection.
9. Explain why P7, although making a large contribution to λ is unlikely to be a useful
target for conservation.
10. Some authors have suggested that reduction in juvenile and/or adult (breeder) mor-
tality may be important to the enhancement of dwindling tortoise populations. From
the results of your elasticity plot state with reasons whether the model supports this
suggestion and identify which parameters would be effected. State your recommenda-
tion for tortoise protection, based on the model and provide any necessary evidence to
support your recommendation.