University of Sydney Solve the Equations Questions

6. In [7], the authors “…report on a simple population dynamic model that indicates that fish
predators could in theory help regulate COTS population levels…” They consider a system
of differential equations that models the interactions between coral, crown-of-thorns starfish
(COTS), and some fish that are predators of the COTS. Here is a simplified version of the
model:
C’
C = s₁C (1) –
– 83CA
A’ = 84A
84A (1-4) –
-86AF
(1)
(2)
F
F-SP (1-4)
F87F
විය.
In the model: t represents time in years; C, A and F represent the normalised population
density of corals, COTS and fish that prey on COTS, respectively; and s₁,…, Sg are constants
to be determined from data. The population densities can be thought of as a proportion of a
theoretical maximum, so C, A, F are unitless and must be between 0 and 1. The constants $2,
55 and 58 are unitless, and the constants 51, 53, 54, 56, and s7 have units year-¹.
(a) A reader of the paper interprets Equation (3) of the model as follows:
The model predicts logistic growth for the population density of predator fish, but
with a carrying capacity that is proportional to the population density of COTS.
Give a similar interpretation of Equation (2) of the model.
(2 marks)(D) Use two iterations of Euler’s method, with a step size of 0.1 years, applied to the model to
predict the population densities of coral and COTS when t=0.2 years. You should use the
parameter values
8₁ = 12 year ¹,
85 = 1,
82 = 1,
86=
= 25 year¹,
83 = 37 year-1,
87 = 0.3 year¹,
84 = 2 year-¹,
88 = 1.
and initial conditions
C(0) = 0.60,
A(0) = 0.10,
F(0) = 0.00.
(Note that the condition F(0) =0.00 is equivalent to assuming that there are no predator
fish).
(3 marks)(c) The following graphs were produced by a Python program that approximates the functions
C(t), A(t) and F(t) described in the model. The program uses Euler’s method with a step
size of 0.1 years, and the parameter values described in (b). The first graph is produced
using the initial conditions
C(0) = 0.60, A(0) = 0.10,
and the second using the initial conditions
F(0) = 0.00,
C(0) = 0.60, A(0) = 0.10,
F(0) = 0.10.
Population predictions when there are no predator fish
Population predictions when there are predator fish
10
10
Coral
COTS
0.8
0.8
Predator fish
Population density of species
0.6
0.4
0.2
0.0
4
time (in years)
Population density of species
0.6
0.4
0.2
0.0
10
time (in years)
Coral
COTS
Predator fish
10
Based on these graphs, and any other information available in this problem, briefly describe
the scientific work that is done in [7]. Your audience is a UQ student who is about to begin
SCIE1000.
(2 marks)