% Lab 5, Oct. 20, 2016

% Recall the forward Euler method for approximating the solution 
% curve x(t) from the initial value problem dx/dt=f(x), x(t_0)=x0 with a 
% uniform step size h:
% dx/dt~[x(t+h)-x(t)]/h
% [x(t+h)-x(t)]/h=f(x)
% x(t+h)=x(t)+h*f(x)

% Consider the linear univariate continuous deterministic model:
% dx/dt=rx, x(t_0)=x0

% Example: Take r=2, x0=0.2
% a) Use the forward Euler method with h=0.05 to estimate x(1)

% First we need to calculate how many steps of size h it will take to get
% from x(0) to x(1)
% h=(b-a)/N; The step size is the length of the time interval divided by 
% the number of steps.  Rearrange this to solve for N
%%
h=0.05;
N=(1-0)/h;
%%
% Now do the usual setup of initial conditions
Xnext=0; %will replace these values with the appropriate value from the loop
tnext=0;
Xcurr=0.2;
tcurr=0;
r=2;
f=@(x) (2*x);
%%

for(i=1:N)
    tnext=tcurr+h;
    Xnext=Xcurr+h*f(Xcurr);
    tcurr=tnext;
    Xcurr=Xnext;
end

%%

disp('The approximate value of x(1) is:')
disp(Xcurr);
%%

% b) Now calculate x(t) explicitly
% dx/dt=r*x
% dx/x=r*dt
% ln(x)=r*t+C
% x(t)=A*e^(rt)
% sub initial condition: x(0)=0.2
% 0.2=A
% explicit solution: x(t)=0.2e^(2*t)

x1=0.2*exp(2*1);
disp('The exact value of x(1) is:')
disp(x1);

%%

% Example 2: Logistic Equation
% dx/dt=r*x*(1-x/K)

% What are the fixed points?
% x*=0 and x*=K

% What is the stability of the fixed points?
% f(x)=r*x*(1-x/K)
% Take the derivative of f(x) and evaluate at the fixed points
% f'(x)=r-2*r*x/K=r*(1-2*x/K)
% f'(0)=r  Therefore x*=0 is stable if r<0
% f'(K)=-r  and x*=K is stable if r>0