System of nonlinear differential equations $x^\prime(t) =x(t)(1 - x(t)) -2x(t) y(t)$, $y^\prime (t) =-y(t) +7x(t) y(t)$ Let the system of two nonlinear differential equations
$$
\begin{cases}
x^\prime(t) =x(t) - x^2(t) -2x(t) y(t)  \\
y^\prime (t)  =-y(t)  +7x(t)  y(t)
\end{cases} \text{ and }
\begin{cases}
x(0)  =100\\
y(0)  =10
\end{cases}
$$
Using Matlab I need to plot the graph of $x(t)$ and $y(t)$ when $t\in[0,1]$. After some efforts I write the
following M-file code
function dydt = vdp1(t,y)
dydt = [y(1)-y(1)*y(1)-2*y(1)*y(2); -y(1)+7*y(1)*y(2)];

and the commands
xvalues=0:.1:1; 
[t,y] = ode45(@vdp1,xvalues,[100; 10]);
plot(t,y(:,1),'-o',t,y(:,2),'-o')
title('Solution with ODE45');
xlabel('Time t');
ylabel('Solution y');
legend('y_1','y_2')

I think this is the right solution. Now, I don't see which way can I follow to aproximate $x(0.5)+2y(0.5)$ and how can I interpret this solution via predator-prey model...?
 A: You've got a problem in your equation in that $x^{\prime}(t)<0$ unless $x(t)+2y(2)<1$, so the population of prey just dies off immediately. The population of predators goes up fast unless the prey population $x(t)<\frac17$. So I believe you should think about fixing your equations if you want to see some nifty oscillations.  
EDIT: As @Did pointed out, there are $3$ stationary points: saddle points at $(0,0)$ and at $(1,0)$ and a stable spiral at $(1/7,3/7)$. As it turns out, the equation isn't in itself so bad because the numbers are scaled and so $x=1$ could represent $1.0e6$ rabbits, for example. In some contexts the numbers would always represent integral numbers of animals and so populations of less than $1$ would be interpreted as extinction.  
The initial values are really screwy, though. On the first loop around with these values $x$ gets so small that the ODE45 solver sets $x<0$ due to roundoff error. Setting tolerances quite small or switching to stiff solvers doesn't help. By way of apology for my rejection of your equations by imposing my own prejudices on your problem, I tried setting some initial values that were more reasonable and plotting the resulting trajectory in phase space. As you can see, $t\in[0,1]$ isn't enough of a time span to see even a single loop around the attractor.
% nonlinear.m

dydt = @(t,y) [y(1)-y(1)*y(1)-2*y(1)*y(2); -y(2)+7*y(1)*y(2)];

xvalues=[0 20];
options = odeset('RelTol',1.0e-11);
%[t,y] = ode45(dydt,xvalues,[100; 10],options);
[t,y] = ode45(dydt,xvalues,[0.15; 0.43],options);
plot(y(:,1),y(:,2));
title('Predator-Prey Oscillations');
xlabel('x(t)');
ylabel('y(t)');

Here is the phase space trajectory:

There is a bug in your Matlab code; your formula for $y^{\prime}(t)$ is typed in incorrectly.
You should probably make two plots of results: one where you let Matlab decide what time values is likes with xvalues=[0 1]; and then plot with lines and no markers (the default) and another with xvalues=0:0.1:1 as you have so you can get $x(0.5)$ and $y(0.0)$, as [t(6) y(6,1) y(6,2) y(6,1)+2*y(6,2)].
