Analytical approximations of Lotka-Volterra integrals

In this note we derive simple analytical bounds for solutions of $x-\ln x=y-\ln y$ and use them for estimating trajectories following Lotka-Volterra-type integrals. We show how our results imply estimates for the Lambert $W$ function as well as for trajectories of general predator-prey systems, including e.g. Rosenzweig-MacArthur equations.