Q1. Consider the Ricker population model presented in class,
Rt+1 = F(Rt) = aRte−bRt,
where Rt is the total mass of the recruits in time t.
(a) Verify that a is ‘proliferation’ in the model, that is, verify a = F′(0).
(b) What are the equilibrium in this model? Derive conditions for when they are
stable and unstable.
(c) Re-write the model, so that it only has parameters a and k, where a = F′(0)
and k is the positive equilibrium.
(d) Using your new model, in part (c), and R, plot time series (Rt vs. t) given
R0 = 2, a = 2, and k = 100, from t = 0 to t = 25. Repeat for a = 10, a = 13
and a = 18. How do your calculations in part (b) help explain the behaviour
in these graphs?
(e) Generate an orbit diagram for your model in (c), in other words plot Rt, for
all t ∈[1,000 , 1,500], vs. a, from a = 0 to a = 35.
1
Q2. Consider the production model,
̇x = x(1 −x)(x −1/2) −hx, h ≥0
where x(t) is population size, as a proportion of carrying capacity, at time t.
(a) When h = 0, what are the equilibria in this model? Show which equilibria
are stable and which are unstable.
(b) Given h = 0, on the same graph, plot x(t) vs. t for x(0) = 0,0.1,0.4,0.5,0.6,0.9,1.
You many use Ror another programming language to help.
(c) What are the equilibria in this model when h > 0? Derive conditions for
when each equilibrium is biologically meaningful, and for when each equilib-
rium is stable and unstable.
(d) Describe in words what biological process(es) are being accounted for in this
model that are not accounted for in the standard logistic production model.
(e) Derive an expression for the optimal h, the h used to achieve MSY.
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Q3. Consider the dynamic programming total harvest maximization problem
discussed in Section 2 of the week 2 lecture notes, with precisely the same notation
and assumptions. In particular, recall the equations
V (x,T −1) =
{
F(x), if x 6SM
x −SM + F(SM ), if x > SM , (1)
which implied that optimal harvest at the second to last stage is:
hT −1 =
{
0, if x 6SM
x −SM , if x > SM . . (2)
(a) Show that
∂V (x,T −1)
∂x =
{
F′(x) >1, if x 6SM
1, if x > SM . (3)
(b) Recall from lectures that changing the maximization variable from harvest
h to the escapement variable s := x −h we showed that at stage T −2 the
optimality equation becomes
V (x,T−2) = max06h6x
[
h+V (F(x−h),T−1)]
= max06s6x
[
V (F(s),T−1)−s
]
+x. (OP)
Defining ν(s) :=
[
V (F(s),T −1)−s
]
, exploit equation (3) to show that
ν′(s) = ∂V (F(s),T −1)
∂x F′(s) −1 =
{
>F′(s) −1 >0, if s 6SM
= F′(s) −1 < 0,, if s > SM .
(4)
(c) From the above show that optimal escapement is SM or x, whichever one is
smaller and hence that at stage T −2, like T −1, optimal harvest is
hT −2 =
{
0, if x 6SM
x −SM , if x > SM . . (5)
(d) Appropriately combine (5),(OP),(1) to show that V (x,T −2) = V (x,T −
1), ∀x. What can you conclude about optimal harvest strategy at stages
T −3,T −4,…,2,1,0?
3
Q4. Consider an invasive plant species with population growth rate r, growing
according to the linear ordinary differential equation,
̇x = rx(t) −w(t), with x(0) = x0,
where x(t) and w(t) are population size and pesticide application rate (weeding),
at time t, respectively. The manager is hired to control the invasive species for
T year. They’ve decided they would like to solve the following optimal control
problem,
minw(t)
{∫ T
0
w(t)2 + x(t) dt
}
.
(a) In words, describe the biological interpretation of the objective function, and
justify its choice.
(b) Solve for the optimal weeding rate, w∗(t) as a function of t, T, and r.
(c) Solve for the population size, x∗(t), given w∗(t), as a function of t, r, T, and
x0.
(d) Explore your solutions. Plot x∗(t), given w∗(t) vs. t for different values of r,
T, and x0.
(e) Which types of species mean the manager should weed more, high r or low
r? How does the length of weeding, T, affect optimal weeding rates? Refer
to graphs and/or derivations to support your claim. Are there any cases
where the model and optimal control problem, as specified, lead to solutions
that do not make sense? How would you modify the problem statement, in
these cases, to provide better management recommendation
