I am a civil engineer with basic mathematics skills and need help for the following - perhaps simple - problem.
Consider the following autonomous system of two non-linear ordinary differential equations for the unknown functions $y_1$ and $y_2$: $$dy_1/dt = P(y_2) * ( Q(y_2)-y_1 )$$ $$dy_2/dt = y_1$$ with the initial conditions
$$y_1(0) = y_2(0) = 0$$
Assume that the functions $P(y_2)$ and $Q(y_2)$ have the following properties:
$$P(y_2) > 0, dP/dy_2 <0, \lim_{y_2\rightarrow ∞} P = P_∞ = constant >> 0 $$ $$Q(y_2) > 0, dQ/dy_2 >0, \lim_{y_2\rightarrow ∞} Q = Q_∞ = constant >> 0 $$
I "feel" that
$$\lim_{t\rightarrow ∞} y_1 = Q_∞ $$
Is that correct? And, if yes, how can I prove that?