The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH3063: Nonlinear ODEs & Applications Semester 1, 2022
Web Page: http://www.maths.usyd.edu.au/u/UG/SM/MATH3063/
Lecturer: Nathan Duignan
Due Sun 20 Mar 2022, 11:59PM.
To be submitted as a pdf online via gradescope on Canvas.
The implementation of a good harvesting policy for fisheries is crucial to ensure a sustainable population of
fish. If N(t) is the population of fish at some time t, a reasonable differential equation that models the change
in population over time is
dN
dt
= RN
(
1 N
K
)
H(N),
where H is the harvesting function, R > 0 is the principle rate of growth, and K > 0 the carrying capacity.
Note that, if no human effects were considered (H = 0) the fish are assumed to behave logistically.
Using bifurcation analysis, we can study the effect of different harvesting policies on the fish population. In
lecture 4, we studied the effect of constant quota harvesting (H(N) = h) in fisheries. In this assignment we will
consider a more refined harvesting function given by
H(N) =
BN
A+N
,
where B > 0 and A > 0 are constants.
1. (a) Use a plot of the harvesting function H(N) to argue why policy based on this harvesting function
may be an improvement over constant quota harvesting.
(b) Show that the model can be non-dimensionalised to the form
dx
dτ
= x(1 x) bx
a+ x
.
Give an expression for x, τ, a, b in terms of the original variables N, t and constants R,K,A,B.
We want to analyse the effect of the choice of constants in the harvesting function. Hence, for the next questions,
we will consider the non-dimensionalised model from Q1 part (b). Only consider biologically relevant values of
x (so non-negative values) and parameter values a > 0 and b > 0.
2. (a) Show that the system has a fixed point at x = 0 for any positive value of a and b. Use linear
stability analysis to determine the stability of x = 0 depending on the values of a and b.
(b) In addition to the fixed point at zero, show that there can be zero, one or two more biologically
relevant fixed points depending on the values of a and b.
3. (a) For each of the fixed values a = 1/2, 1, 2, draw the bifurcation diagram with b as parameter. Identify
the bifurcations and the bifurcation points bc.
(b) A bifurcation occurs whenever a = b. Determine what type of bifurcation it is.
(c) Show that another bifurcation occurs when b = 14 (a + 1)
2 and that it is biologically relevant when
a ≤ 1. Classify this bifurcation.
4. (a) There is a region in parameter space (a, b) where the model displays hysteresis. Identify this region.
[Hint: You might find it useful to draw a stability diagram for (a, b)]
(b) Which values of a and b must be avoided in policy making to ensure there is no catastrophic collapse
of the fish population
(c) Which values of a and b ensure the possibility of sustainable harvesting
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