%%%% Problem 1%%%%%

function dx = nuPop(t,x)
global q u V K r

dx = zeros(size(x));

dx(1) = (q*u)-(q*x(1)) - x(2)*((V*x(1))/(x(1)+K));
dx(2) = -(q*x(2))+(r*x(2))*((V*x(1))/(x(1)+K));

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%% Parts A-C %%%%

clear all
global q u V K r
q = 1.0;
u = 2.0;
V = 5.0;
K = 7.0;
r = 2.0; 

tt = [0.0:0.01:10];
x0 = [0;1];

[t,x] = ode45('nuPop',tt,x0);

plot(t,x)

hold on 

q = 1.0;
u = 2.0;
V = 5.0;
K = 7.0;
r = 2.0; 

tt = [0.0:0.01:10];
x0 = [1;1];

[t,x] = ode45('nuPop',tt,x0);

plot(t,x)

hold on 

q = 1.0;
u = 2.0;
V = 5.0;
K = 7.0;
r = 2.0; 

tt = [0.0:0.01:10];
x0 = [2;1];

[t,x] = ode45('nuPop',tt,x0);

plot(t,x)

hold off
xlabel('time')
ylabel('abundance')
title('Population and food source over time')

q = 1.0;
u = 1.0;
V = 1.0;
K = 1.0;
r = 1.0; 

tt = [0.0:0.01:10];
x0 = [1;1];

[t,x] = ode45('nuPop',tt,x0);

plot(t,x)

hold on

q = 2.0;
u = 1.0;
V = 1.0;
K = 1.0;
r = 1.0; 

tt = [0.0:0.01:10];
x0 = [1;1];

[t,x] = ode45('nuPop',tt,x0);

plot(t,x)

hold on

q = 1.0;
u = 2.0;
V = 1.0;
K = 1.0;
r = 1.0; 

tt = [0.0:0.01:10];
x0 = [1;1];

[t,x] = ode45('nuPop',tt,x0);

plot(t,x)

hold on

q = 1.0;
u = 1.0;
V = 2.0;
K = 1.0;
r = 1.0; 

tt = [0.0:0.01:10];
x0 = [1;1];

[t,x] = ode45('nuPop',tt,x0);

plot(t,x)

hold on

q = 1.0;
u = 1.0;
V = 1.0;
K = 2.0;
r = 1.0; 

tt = [0.0:0.01:10];
x0 = [1;1];

[t,x] = ode45('nuPop',tt,x0);

plot(t,x)

hold on

q = 1.0;
u = 1.0;
V = 1.0;
K = 1.0;
r = 2.0; 

tt = [0.0:0.01:10];
x0 = [1;1];

[t,x] = ode45('nuPop',tt,x0);

plot(t,x)

hold off 

xlabel('time')
ylabel('abundance')
title('Population and food source over time')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Problem 2 %%%%%

function dx = nuPop(t,x)
global q u V K r m

dx = zeros(size(x));

dx(1) = (q*u)-(q*x(1)) - x(2)*((V*x(1))/(x(1)+K));
dx(2) = -(q*x(2))+((r)*x(2))*((V*x(1))/(x(1)+K))-(m*x(2));

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Parts A-C %%%%

clear all
global q u V K r m
q = 1.0;
u = 2.0;
V = 5.0;
K = 7.0;
r = 2.0;
m = 1.0; 

tt = [0.0:0.01:10];
x0 = [0;1];

[t,x] = ode45('mort',tt,x0);

plot(t,x)

hold on 
q = 1.0;
u = 2.0;
V = 5.0;
K = 7.0;
r = 2.0;
m = 0.5; 

tt = [0.0:0.01:10];
x0 = [0;1];

[t,x] = ode45('mort',tt,x0);

plot(t,x)

hold on

q = 1.0;
u = 2.0;
V = 5.0;
K = 7.0;
r = 2.0;
m = 0.1; 

tt = [0.0:0.01:10];
x0 = [0;1];

[t,x] = ode45('mort',tt,x0);

plot(t,x)

hold off

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Problem 3 %%%%

function dx = nuPop(t,x)
global q u V K r m

dx = zeros(size(x));

dx(1) = (q*u)-(q*x(1)) - x(2)*((V*(x(1)+x(3)))/(x(1)+x(3)+K));
dx(2) = -(q*x(2))+((r)*x(2))*((V*x(1))/(x(1)+K));
dx(3) = (q*u)-(q*x(3)) - x(2)*((V*(x(1)+x(3)))/(x(1)+x(3)+K));

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all
global q u V K r
q = 1.0;
u = 2.0;
V = 5.0;
K = 7.0;
r = 2.0; 

tt = [0.0:0.01:10];
x0 = [0;1;1];

[t,x] = ode45('threenu',tt,x0);

plot(t,x)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%