The below problem arises in mathematical epidemiology. I consider the system of ode $$\dot u=d u - \alpha uv,\,\,\dot v= \alpha uv-\gamma v,$$ where all parameters $d,\alpha,\gamma$ are positive. This system has two stationary points $(u,v)=(0,0)$ and $(u,v)=(\frac{\gamma}{\alpha},\frac{d_1}{\alpha})$
Phase space is picured below
Let me consider initial conditions: $u(0)=u_0,\,\,\,$ $v_0=\varepsilon\ll u_0;\,\,u_0,\varepsilon>0$ (black point on fig.)
My question is as follows. How to estimate the maximum value of $v$? (red point on fig.) Of course I can try to find the conservation law $I(u,v)=0$, but this is not always possible.
At all extrema of $v(t)$ you get $0=\dot v=v(αu−γ)$. As $v=0$ is excluded, you can insert the value $u=\fracγα$ into $I(u,v)=C$ to get the corresponding values of $v$.
In general you will not get helpful first integrals, then you can still use the condition $\dot v=0$ with a change of positive to negative at a maximum to set an event for a numerical solver.