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Chemical pollution in a river
创建日期：2024-04-24 16:28:10
标签： equation
Chemical pollution in a river

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Assessment 2A

Please submit your answers as a single pdf.

Question 1: Chemical pollution in a river
In this question, we will consider the river as a one-dimensional space, modelled as
a line of 80 points. There is a factory is at point 30. There is a continuous release of
chemical from the factory at that point only.

Question 1A

First model the dynamics of a single chemical, u(x,t). The chemical is released by
the factory at a constant rate, the chemical diffuses in the river, with diffusion rate,
D=2, the chemical is advected by the river with rate v=2, and the chemical decays at
rate γ =0.2. Therefore the dynamics of the chemical, u(x,t), can be described by the
following equation:


= () −


+
2
2
−

where P(x) is the rate of release of the chemical at position, x, given by

() = {
5 at = 30,
0 for all other .


We assume that initially there is no chemical in the river:
(, ) = 0 at = 0.

We assume that the points at the boundaries of the river (i.e. point 1 and point 80) do
not receive chemical from outside the line of 80 cells.

Please produce a python code to simulate the dynamics of the chemical u(x,t).
Please plot the outputs at t=0, 2, 4, 6, 8, 10.

Please submit both your python code and a figure showing the simulation
output.

Hint: You may find it helpful to consider the practical exercise studied in the
‘Diffusion and Pollution’ session.

[20 marks]

Question 1B

Now let’s extend the model of question 1A, to suppose that instead of decaying, the
chemical u(x,t) is converted (at rate μ=0.2) into a different chemical w(x,t). Chemical
w(x,t) diffuses at rate E=5, and is advected at rate v=2 .



Please turn over.
The dynamics of the chemical u(x,t) and w(x,t) can be described by the following
coupled equations:


= () −


+
2
2
−



= −


+
2
2
+

Please produce a python code the simulate the dynamics of the two chemicals u(x,t)
and w(x,t). Please plot the outputs at t=0,2,4,6,8,10.

Please submit both your python code and figures showing the simulation
outputs for both u(x,t) and w(x,t).

Hint: We looked at code that simulates the dynamics of two substances in a ring of
cells in the Cellular Models session and adapted this in the Diffusion and Patterns
session. You may find it useful to look at these examples.
[40 marks]


Question 2

Describe the differences between the Turing patterning and the patterning by Delta-
Notch signalling. Please illustrate your answer with suitable figures.

Please note that credit will be given to clear, concise answers.

Hint: Consider both the model equations and the model outputs.
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