Parameter Scan (Non-Linear system)

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Joseph Hughes
Joseph Hughes on 23 Mar 2011
So my coursework for university involves a predator prey model:
?x = ax(1-x) - xy - p(1 - exp(-qx))
?y = -y + bxy(-y/c)
We are given the variables a, b, c and q (q is generated by an unseen m-file called GeneratePar, this makes everybody's coursework individual)
The first task we were given was to write a function that would find the Jacobian of any matrix(called MyJacobian.m).... I have done this successfully.
The second task was to use newton iteration to write a function (called MySolve.m) that incorporated MyJacobain, the task involved writing a function that would solve a non linear system of equations of the form f(x)=0...... I have done this successfully (with the help of some people here)
so the task I am now stuck with is writing a function called ParameterScan.m:
The aim of this function is to find the equilibria of the predator prey system. the equilibria exist only in a range of parameters p from 0 to p(max). we are told to use a for-loop for parameter p which calls MySolve for each value of p and use the solution for the previous parameter p as the initial guess. then to plot the results in the (p, x) plane and the (p, y) plane
Here is the code i have so far which obviously is not working,what i have written so far is based upon some examples we have been shown.
x y=@(x,a,b,c,p,q)...
[((a*x(1))*(1-x(1)))-(x(1)*x(2))-(p*(1-(exp(-q*x(1)))));...
-x(2)+((b*x(1)*x(2))*(exp(-x(2)/c)))];
a=5;
b=5;
c=5;
q=GeneratePar(414378);
pvec=0:5e-3:0.2;
x0=[0,0];
na=length(pvec);
xvec=zeros(length(x0),na);
for i=1:na
f=@(x)xy(x,a,b,c,pvec(i),q);
[x,converged]=MySolve(f,x0,1e-10,10);
if converged>0
xvec(:,i)=x;
x0=x;
fprintf('i=%d, a=%g, |x|=%g\n',i,pvec(i),norm(x));
else
fprintf('i=%d, a=%g, no convergence\n',i,pvec(i));
xvec(:,i)=NaN;
end
end
plot(pvec,xvec,'.');
grid on
I am aware that this may seem like i am asking someone to 'do' my coursework for me, i am not i'm just trying to get some tips as i am now stuck :(
  1 Comment
Walter Roberson
Walter Roberson on 12 Apr 2011
Sorry, what do the ? represent in the model ?

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