MAST90011 Mathematical Biology 2022
Assignment 2: Submit solutions via the LMS by 5pm Friday 23 Septem-
ber 2022. A penalty of 10% of the assignment weight per day will be
applied to late submissions.
For all questions, you are expected to make some sensible biologically-sound remarks.
That is, to obtain full marks you must do more than perform the mathematical analyses
correctly.
Question 1: Certain ant species use pheromones as a signal for danger. The one-
dimensional diffusion equation is a good model for the spread of pheromones in a very
long tube. Suppose u(x, t) is the pheromone concentration. It satisfies
∂u
∂t
=
∂2u
∂x2
, −∞ < x <∞ , (1)
where the diffusion coefficient here is D = 1 for simplicity.
In experiments in the 1960s, Bossert and Wilson released ants in a long tube and
stimulated one ant until it released a pheromone. Assume that at time t = 0 a signal of
strength α is released at x = 0, giving u(x, 0) = αδ(x) as the initial condition. Other ants
react to the stimulus if the concentration they perceive is 10% of α or higher.
(a) For each t > 0, find the region in the tube −X(t) ≤ x ≤ X(t) where the ants would
react to the stimulus. (This is what Bossert and Wilson measured experimentally.) The
interval is called the region of influence. Comment on the result.
(b) Sketch the time evolution of X(t).
(c) Find the time T ∗ such that the region of influence is empty for all t > T ∗.
Hint: You can just state the solution of the PDE. Please note that this question should
really only take one page.
Question 2: A model for signal propagation in an axon is given by
∂u
∂t
=
∂2u
∂x2
+ u(1− u)(u− 1
2
) , 0 ≤ x ≤ L , (2)
with homogeneous Neumann boundary conditions, namely
∂u
∂x
(0, t) = 0 ,
∂u
∂x
(L, t) = 0 , (3)
where u represents the membrane potential.
For this question, consider the steady state solution of Eqs. (2)–(3); that is,
solutions which are independent of time.
(a) Determine the system of two ordinary differential equations which describe the steady
state solutions.
(b) Find the equilibrium points (that is, fixed points) of the system in (a) and determine
their type and stability.
(c) Using software of your choice, sketch the phase plane in the (u, du/dx) plane.
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(d) On the phase plane, illustrate one of the steady state solutions that satisfies the
boundary conditions. Mark u(0), u(L/2) and u(L) on your sketch.
(e) From part (d), deduce a schematic sketch of the solution u as a function of x. Mark
u(0), u(L/2) and u(L) on your sketch. Make some comments of course!
Question 3: Travelling bands of micro-organisms, chemotactically directed and diffusing
move into a food source. They consume food as they go. A (dimensionless) model is given
by
∂b
∂t
=
∂
∂x
(
∂b
∂x
− 2 b
a
∂a
∂x
)
,
∂a
∂t
= −b , (4)
where b(x, t) and a(x, t) are the bacteria and nutrient concentrations respectively. The
boundary conditions are
b(x, t)→ 0 , a(x, t)→ 0 as x→ −∞ , b(x, t)→ 0 , a(x, t)→ 1 as x→∞ .
(a) Briefly interpret each of the terms in the system given by Eqs. (4).
(b) Look for travelling wave solutions b(x, t) = B(z) and a(x, t) = A(z), where the travel-
ling wave coordinate is z = x− ct and c is the positive wave speed.
(c) Determine equations for dB/dz and dA/dz.
Ensure that you determine and state the appropriate boundary conditions.
(d) Solve for B(z) and A(z). (Note: you will have an arbitrary constant in your solution.)
(e) Using software of your choice, sketch the solutions B(z) and A(z) and briefly explain
what is happening biologically.
(f) Plot representative trajectories in the phase plane and make some biologically-informed
comments.
The end
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