function [sol0,sol1] = prob1
% prob1, prob1f of tutorial.

history = [80; 30];
tspan = [0, 100];
opts = ddeset('RelTol',1e-5,'AbsTol',1e-8);
a =  0.25;
b = -0.01;
c = -1.00;
d =  0.01;

% Solve the ODEs that arise when there is no delay.
sol0 = dde23(@prob1f,[],history,tspan,opts,a,b,c,d);
   
% Solve the DDEs that arise when there is a delay of tau.
tau = 1;
sol1 = dde23(@prob1f,tau,history,tspan,opts,a,b,c,d);

plot(sol0.y(1,:),sol0.y(2,:),sol1.y(1,:),sol1.y(2,:))
title('Problem 1. Solution with and without delay.')
xlabel('y_1(t)')
ylabel('y_2(t)')
legend('No delay',['Delay \tau = ',num2str(tau)],2)

%=============================================
function v = prob1f(t,y,Z,a,b,c,d)
v = zeros(2,1);
if isempty(Z)     % ODEs
   v(1) = a * y(1) + b * y(1) * y(2);
   v(2) = c * y(2) + d * y(1) * y(2);
else              % DDEs
   m = 200;
   ylag = Z(:,1);
   v(1) = a * y(1) * (1 - y(1) / m) + b * y(1)    * y(2);
   v(2) = c * y(2)                  + d * ylag(1) * ylag(2);
end