The Narrow Corridor of Stable Solutions in an Extended Osipov--Lanchester Model with Constant Total Population

This paper considers a modification of the classical Osipov--Lanchester model in which the total population of the two forces $N=R+B$ is preserved over time. It is shown that the dynamics of the ratio $y=R/B$ reduce to the Riccati equation $\dot y=αy^2-β$, which admits a complete analytical study. The main result is that asymptotically stable invariant sets in the positive quadrant $R,B\ge 0$ exist exactly in three sign cases of $(α,β)$: (i) $α<0,β<0$ (stable interior equilibrium), (ii) $α=0,β<0$ (the face $B=0$ is stable), (iii) $α<0,β=0$ (the face $R=0$ is stable). For $α>0$ or $β>0$ the solutions reach the boundaries of applicability of the model in finite time. Moreover, $α<0,β<0$ corresponds to exponential growth of solutions in the original system. Passing to a model perturbed in $α(t),β(t)$ requires buffer dynamics repelling from the axes to preserve stability of the solution.