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. [40 marks]
