A modell for two competing species is the non-linear system of differentialequations
\begin{align} y'_1(t) &= s_1 y_1(t)\left(1-\frac{y_1(t)}{N_1}\right)-a_1y_1(t)y_2(t)\\ y'_2(t) &= s_2 y_2(t)\left(1-\frac{y_2(t)}{N_2}\right)-a_2y_2(t)y_1(t) \end{align}
where $s_i,N_i,a_i>0, \ i=1,2$ are positive parameters and $y_i(t)$ are the populationsizes of the respective species.
I want to show that the maximal existence interval for solutions to the system above with $t=0, \ y_1(0)>0, \ y_2(0)>0$ is the positive real numbers.
I don't know how to attack this problem since all the examples in my book only deal with single ODE:s where the initial value is given, like
$$y'(t)=y(t)^2, \quad y(0)=1,$$
where they simply solve this by separating variables. But how should I apply this to a system of non linear ODEs that has no analytical solution?
Hint: the trajectory stays in the region $0 < y_i \le \max(N_i, y_i(0))$.