Chapter 9 Problems
Problem 9.1: In addition to being colonized by crabs, sea anemones are also sometimes
colonized by specialized fish species. Let’s use the variables O(t) and U(t) to denote the number
of occupied and unoccupied anemones in generation t. Suppose that, in each generation, an
empty anemone becomes occupied with probability κ , and an occupied anemone loses its fish
through mortality and becomes unoccupied with probability µ . This results in the discrete-time
model:
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 O(t + 1) 1− µ κ  O(t)

=

.
U(t + 1)  µ 1− κ U(t)
(a) Find the general solution to this model describing the long-term dynamics of occupied and
€ unoccupied anemones (simplify your answer as much as possible).
(b) Describe in words what this general solution tells you about the sea anemone dynamics.
Problem 9.2: Many eukaryotic genes have both exons and introns, where only exons code for
protein sequence. mRNAs transcribed from such genes initially include the introns, which must
be spliced out before the mRNAs can be translated. Let n1(t) equal the number of unspliced
“pre-mRNAs”, which are produced by the cell at a rate, c, per minute, and let n2(t) equal the
number of spliced “processed mRNAs” within a cell at time t. If a is the fraction of the premRNAs that are spliced per minute and if a proportion, d, of processed mRNAs degrade per
minute, the numbers of pre-mRNA and processed mRNA change over time according to the
equations:
n1 ( t + 1) = (1− a) n1 ( t ) + c
n 2 ( t + 1) = a n1 ( t ) + (1− d ) n 2 ( t )
(a) Find the one equilibrium of this model, i.e., the point at which both types of mRNA remain
€ constant in number over time.
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(b) Perform a transformation of this model. Define the new variables ε1 ( t ) = n1 ( t ) − nˆ1 and
ε2 ( t ) = n 2 ( t ) − nˆ 2 . This transformation represents how far each type of mRNA is from its
equilibrium. Determine recursion equations for ε1 ( t + 1) and ε2 ( t + 1) in terms of ε1 ( t ) and ε2 ( t ) .
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(c) Show that the recursions developed in part (b) can be written in matrix form:
 ε1 ( t + 1) 
 ε1 ( t ) 

=M

ε2 ( t + 1)
ε2 ( t )
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and specify the transition matrix, M. [If you cannot write the recursions in this form or if the
€ matrix M looks ugly, then go back and check your answers to (a) and (b).]
(d) Calculate the eigenvalues of M. Given that both a and d represent proportions, can there be
an eigenvalue whose magnitude is greater than one? Whenever there is an eigenvalue greater
than one, the equilibrium is unstable. Conversely, as long as all eigenvalues are less than one in
magnitude, the equilibrium will be stable.
(d) Calculate the eigenvectors of M.
(e) Write a diagonal matrix, D, containing the two eigenvalues along the diagonal. Find Dt by
raising these eigenvalues to the tth power.
(f) Write a transformation matrix, A, containing the eigenvector associated with the first
eigenvalue of D in the first column and the eigenvector associated with the second eigenvalue of
D in the second column.
(g) Find the inverse matrix, A-1.
(h) With these calculations in hand, predict the numbers of pre-mRNA and processed mRNA at
any point in time (the general solution) by multiplying out and simplifying:
ε1 ( t ) 


t −1 ε1 (0)

=A D A 

ε2 ( t )
ε2 (0)
You can check your answer by setting t = 0 to regain the initial condition and t = ∞ to regain the
€ equilibrium.
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(i) Transform the general solution obtained in (h) back into terms of the original variables, n1(t)
and n2(t), to predict how the number of pre-mRNAs and processed mRNAs change over time.
Problem 9.3: You are presented with a series of linear equations that you solve by matrix
manipulation. In your general solution, you let time go to infinity and find that the system
reaches a specific point. Is this point an equilibrium? Is it unstable or stable? If stable, is it
locally or globally stable? Should you do a local stability analysis?
Problem 9.4: Perform a proportional transformation of the logistic model,
 n
dn
= r n 1−  .
 K
dt
dn *
˜
˜
What choices of n and t lead to the simplified differential equation
= n * 1− n * ? What do
dt
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these choices of n˜ and t˜ represent biologically?
(
)
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€ scales using a diploid version of
Problem 9.5: In the text, we performed a separation of time
€ natural selection. Using the haploid version of the model:
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density-dependent
N(t + 1) = W ( N(t), p(t)) N(t)
p(t + 1) =
,
W A ( N(t))
p(t)
W ( N(t), p(t))
where W ( N(t), p(t)) = p(t) W A ( N(t)) + (1− p(t)) W a ( N(t)) , W A (N) = (1+ r)(1− α A N ) , and
W (N) = 1+ r 1− α€N , perform a separation of time scales. As in the text, assume that
a
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(
)(
A
)
ecological changes occur over a faster time scale than allele frequency changes. Compare your
€
answers to (9.37) and (9.39).
Problem 9.6: In the two-locus model of selection, we performed a separation of time scales and
found that the change in frequency of allele A1 is the same as the one-locus recursion,
pA ( t + 1) =
pA W A1
pA W A1 + (1− pA ) W A 2
, to leading order in the selection coefficients. Here,
W A1 = pB w1 + (1− pB ) w 2 and W A 2 = pB w 3 + (1− pB ) w 4 represent the marginal fitnesses of
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alleles A1 and A2, averaged over the genetic states at the B locus. Using this approximation, show
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that the dynamics at locus A do not depend on the allele frequencies at locus B under the
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multiplicative fitness regime (9.48) but not in the case of the additive fitness regime (9.46). This
result is also consistent with the conclusion that it is natural to measure fitness interactions on a
multiplicative scale.
Problem 9.7: Consider the following consumer-resource model with two consumers and two
resources:
 R
dR1
= r1R11− 1  − N1a11R1 − N 2 a21R1
dt
 K1 
 R 
dR2
= r2 R2 1− 2  − N1a12 R2 − N 2 a22 R2
dt
 K2 
.
dN1
= N1 ( a11R1 + a12 R2 − µ1 )
dt
dN 2
= N 2 ( a21R1 + a22 R2 − µ2 )
dt
The resources have abundances R1 and R2 and the consumers have abundances N1 and N 2 . The
€ resources are assumed to grow logistically in the absence of consumption, and the consumers eat
these resources according to a type 1 functional response. This model provides a mechanistic
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description of the dynamics of resource competition between the two consumer species. Use a
separation of timescales in which the dynamics of the resources are assumed to occur much
faster than those of the consumer to demonstrate that the model can be reduced to a two variable
model having the form
 α N + α12 N 2 
dN1
= ρ1N11− 11 1

dt
κ1


 α N + α 22 N 2 
dN 2
= ρ 2 N 2 1− 21 1

dt
κ2


where you should specify how ρ1, ρ 2 ,α 11,α 12 ,α 21,α 22 ,κ 1, and κ 2 relate to the parameters of the
€ original model. This two variable model is identical in form to the Lotka-Volterra competition
equations (see Table 3.1).
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