THE UNIVERSITY OF SYDNEY
MATH3888
Semester 2 Interdisciplinary Project (Planaria) 2020
WEEK 6 REPORT GUIDELINES
Submission:
As outlined in the information sheet of this interdisciplinary project course, you will create reports using
the (maths) editing software LaTeX:
https://en.wikibooks.org/wiki/LaTeX
Use the following basic setup for your LaTex file:
\documentclass[11pt]{article}
\usepackage{fullpage,amsmath}
. . .
\begin{document}
. . .
\end{document}
Submission of the corresponding pdf file is via turnitin (where it will be checked for plagariasm).
As outlined in the course info sheet, this report is worth 5% of your final mark.
Deadline is Thursday, week 7 (October 15th), 23:59. No late submission will be accepted!
Constraints:
The final submitted pdf document shall consist of no more than 4 pages (including figures,....). The
‘fontsize’ is strictly 11 points and the margins of the document are automatically set by the ‘fullpage’
package (as instructed above).
(The other package (‘amsmath’) might be needed for the mathematical editing. Add any other packages,
if needed.)
A mutual activation model
Consider the following system of ODEs:
(1)
with non-negative variables x, y ≥ 0 and non-negative parameters ri (i = 1, . . . 4), dj (j = 1, 2), nk ≥ 1
(k = 1, 2).
1. Explain the motif of ‘mutual activation’ in this model (1) in no more than 75 words.
r1 = 0.1, r2 = 1.0, r3 = 0.2, r4 = 1.5,
d1 = 1, d2 = 1, n1 = 3.0, n2 = 4.0, k1 = 0.75, k2 = 0.5 .
(2)
Implement system (1) in pplane and identify its equilibria. How many are there? Provide their
coordinates.
3. Continue any equilibrium in the parameter r2 in order to plot a bifurcation diagram in the (r2, x)−plane.
What bifurcation(s) can you identify? Describe which equilibrium branches are stable. Based on
this bifurcation diagram you should be able to identify a bistability regime with two possible stable
equilibrium states. For which r2 values does bistability occur?
4. Choose either limit point of the bistability regime as an initial point. Use MatCont to perform
a 2-parameter continuation of the limit point curve, plotting your findings in the (r2, r4)−plane.
Identify a codimension-2 bifurcation point. What is the relationship between this diagram and
bistability? Explain in no more than 75 words.
5. Finally, reset your parameter values to the one given in (2) but set r1 = r3 = 0. Continue this time in
the parameter r4 in order to plot a bifurcation diagram in the (r4, x)−plane. What bifurcation(s)
can you identify? Is there bistability in this system? If yes, provide r4 values where bistability
occurs.

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