## Transcribed Text

Biological Movement
This assignment consists of two parts The first part requires you to
formulate system of PDE equations from word description of a
problem. In the second part you will investigate the behaviour of a
PDE model of cell migration due to self-generated gradients
Part Formulating model
You will formulate mathematical model of spatial predator prey
system involving birds and insect larvae. Your model should incorpo
rate the following facts
Insec larvae move randomly and are also advected passively by
the wind with velocity U.
The growth larvae logistic
Larvae are consumed by birds ata rate that depends linearly on
larval density at low density but which saturates at high larval
density (this known asa Holling's type functional response).
Birds move randomly
The birth rate of birds proportional to larval consumptio
Birds have constant death rate.
Write down mathematic tal model for this situation. You should
formulate your model in terms of larval density 1(x,1) and bird
density b(x,1). Here denotes time and denotes position in a
spatial domain which may be taken to be one-dimensional. You must
carefully justify each term in your model. You need not supply initial
or boundary conditions
Part Cell migration and self-generated gradients.
Consider the simple example of reaction-diffusion equation with
linear population growth.
Duxx 7u,
(1)
where D, are positive constants. At time zero all individuals are
concentrated one point, say x: then u(x,0) 5(x),
Dirac
delta distribution.
a) Show that by applying the change of variables u(x,f) n(x,f)er
equation (1) becomes the diffusion equation for n. namely,
Dnzx-
(2)
Dictyostolium
b) The diffusion equation (2) with initial condition n(0,x)
solution
n(x,1)
4Di
(3)
Show that this is in fact : solution to equation (2).
c)
Write matlab programme that uses the PDE solver pdepe to
numerically solve the PDE given in qquation (2). Use the following
initial conditions and boundary conditions
n(x,0) = 0,forx<-landx>1,
an
ax
(100,1)
for between -100 and 100 steps of 0.1. and between 0 and 200
in steps of 20. Use the parameter value 0.5. Plot the solution
to the PDE at each point in time between and 200 in steps of 20
d) The root mean squared distance (RMSD) travelled by the cells in
time can be calculated using the equation
RMSD
n(x,f)x2
(4)
1-100
where Nr is the total number of cells and given by
(5)
I--100
Plot the RMSD for values of between between Dand 200 ir steps
of 20. Equation provides theoretica value the RMSD, plot
this value along with your result cal culated from Equation 4.
e) Next we will introduce chemical substance which the cells are
attracted to and also consume The full model is now,
(1am
(6)
ar 22x2
(7)
where x 0.05 Solve this model using pdepe with the following
initial conditions and zero flux boundary conditions
n(x,0)
c(x,0)
Use values of between -200 and 200 stepso 0.1 and between
O and 200 in steps 20.
f)
On single figure, plot the concentrations of both cells and the
chemical asa function of at each time between 0 and 200 steps
20. Explain the results you observe.
g) Calculate the RMSD travelled by the cells teach time point as you
did in part (d). Plot this quantity along with the predicted value
obtained from equation in the absence of chemical attractant
Discuss the differeno between the twocurves

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Question 2(c)

D=0.5; r=0.2; m=0;

x=-100:0.1:100; t=0:1:200;

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

% Extract the first solution component as u.

u = sol;

figure('Name','Pure Diffusion: Spatio-Temporal cell count

plot','NumberTitle','off','units','normalized','outerposition',[0 0 1

1]);

surf(x,t,u,'FaceColor','interp','EdgeColor','none');

title('Pure Diffusion: Cell count computed for -100 to 100 in time

0-20')

xlabel('Distance x')

ylabel('Time t')

zlabel('Cell count n');

figure('Name','Pure Diffusion: Cell count

spatial variation snapshots at different

times','NumberTitle','off','units','normalized','outerposition',[0 0

1 1]);

hold on;

for i=1:length(t)

str=['Solution@t=',num2str(t(i))];

plot(x,u(i,:),'LineWidth',2,'DisplayName',str);

end

legend('show');

xlabel('Distance x');

ylabel('Cell count n');

title('Pure Diffusion: Cell count spatial variation snapshots at

different times');...