ODE - Solving Parameter Dependent on Variable [Matlab]

Question

Lets say I have a function file of the ODE that goes like this

function xprime = RabbitTemp(t,X) 

% Model of Rabbit Population
% where,
%       Xo = Initial Population of Rabbits
%       X(1) = Population density of Rabbit
%       X(2) = Temperature T (that varies with time) 
%       a = test parameter

%%% ODE
a  = 10;


dx(1) = (X(1))*(1 - a*X(1) - 3*(X(2))));
dx(2) = sin(t);
%%%%%%%

xprime = [dx(1) dx(2)]';

end

But what if I would like parameter a to vary as temperature X(2) varies, as the ODE solver calculates.

I understand I would first have to create some data between a and X(2) and interpolate it. But after that I'm not very sure what's next. Could any one point me in the right direction?

Or is there any other way?

Answer 1

It really depends on the function of a, a=f(T). If you can take the derivative of f(T) I suggest you include a as another state, for instance X(3) and then you can access it from the X argument in xprime easily, plus it will help Matlab determine the proper step size during integration:

function xprime = RabbitTemp(t,X) 

% Model of Rabbit Population
% where,
%       Xo = Initial Population of Rabbits
%       X(1) = Population density of Rabbit
%       X(2) = Temperature T (that varies with time) 
%       X(3) = test parameter

%%% ODE
a  = 10;
Yo = 0.4; %Just some Constant

dx(1) = (X(1))*(1 - X(1)^(a) - Yo*(X(2))));
dx(2) = sin(t);
dx(3) = .... 
%%%%%%%

xprime = [dx(1) dx(2) dx(3)]';

end

Second method is to create a function(-handle) for a=f(T) and call it from within xprime:

function xprime = RabbitTemp(t,X) 

% Model of Rabbit Population
% where,
%       Xo = Initial Population of Rabbits
%       X(1) = Population density of Rabbit
%       X(2) = Temperature T (that varies with time) 
%       a = test parameter

%%% ODE
a  = f_a(t,X);
Yo = 0.4; %Just some Constant

dx(1) = (X(1))*(1 - X(1)^(a) - Yo*(X(2))));
dx(2) = sin(t);
%%%%%%%

xprime = [dx(1) dx(2)]';

end

function a = f_a(t,X)
  return 10;  % You might want to change this. 
end

If possible go with the first variant which is creating another state for a.