How to vary a variable in a system of ODE's?

Hello everybody,

I want to make a plot where I from where I can find and show the best value for a parameter. In order to do this I have to vary a variable in my system of ODE's (here a simplified example from the Matlab site). In the lines of codes below I show the set of ODE's, here Tau is the variable that I want to vary. In other words, I want solutions for a system where Tau = 1, Tau = 2, Tau = 3, etc. I tried to do this using a for-loop but it doesn't seems to be working as only the first two columns are calculated and the last ones remain their inital values. Does somebody have a smart solution?

function dpdt = Tester(t,p,val)
dpdt = zeros(2*val,1);
delta = 0.02;
beta = 0.01;
    for tau = 1:1:val
    dpdt(1) = tau * p(1) .* (1 - beta*p(2));
    dpdt(2) = tau * p(2) .* (-1 + delta*p(1));
    end
end

And the solution script is given below:

tspan = [0 15]
val = 2;

[t,p] = ode45(@Tester,tspan,[5 5 50 50],[],val);

plot(t,p)
title('Predator/Prey Populations Over Time')
xlabel('t')
ylabel('Population')
legend('Prey','Predators')

Thanks in advance for your help.

Answer 1

[Star Strider]

Try this slightly changed version of your code:

function dpdt = Tester(t,p,tau)
dpdt = zeros(2,1);
delta = 0.02;
beta = 0.01;

dpdt(1) = tau * p(1) .* (1 - beta*p(2));
dpdt(2) = tau * p(2) .* (-1 + delta*p(1));

end


tspan = linspace(0, 15, 150);
Tau = 1:5;
for k = 1:numel(Tau)
    [t,p{k}] = ode45(@(t,p)Tester(t,p,Tau(k)),tspan,[5 50]);
end

Np = numel(Tau);
figure
for k = 1:Np
    subplot(Np,1,k)
    plot(t,p{k})
    title(sprintf('Predator/Prey Populations Over Time \\tau = %d',Tau(k)))
    xlabel('t')
    ylabel('Population')
    legend('Prey','Predators', 'Location','EastOutside')
end

pos = get(gcf,'Position');
set(gcf,'Position',pos+[0 -200 0 200])

.

[Danny Helwegen]

Thanks for your answer, thats exactly what I needed. If it's okay I have one follow up question. After using your answer I convert the obtained p values into an array:

P_array = cell2mat(p)

As I only need the values in the last row I do:

P_last = P_array(end,:);

Next, I only want to keep the values of the second column (so here columns 2, 4, 6, 8 and 10) and I want to pick them out using a systemetic way. I known I can just use the command x = P_last(:,[2 4 6 8 10]) in this case, but for larger arrays this wont work. Do you known a method to extract these columns automatically? I tried using a for loop (see below) but that won't work.

for i = 2:2:10
    x = u(:,[i]);
end

[Star Strider]

My pleasure!

I would do something like this:

P_array = cell2mat(p);
x = P_array(end,2:2:end)

producing:

x =
          136.440841693042          350.022691010316          86.7936097755792        8.55109347731735          9.06889302716956

That appears to produce the correct results, and should do what you want.

[Danny Helwegen]

Ah yes, didn't know that was possible. This is exactly what I needed, thank you very much for all you help.

[Star Strider]

As always, my pleasure!