{"ID":2826840,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.18515","arxiv_id":"2512.18515","title":"The Narrow Corridor of Stable Solutions in an Extended Osipov--Lanchester Model with Constant Total Population","abstract":"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) $α\u003c0,β\u003c0$ (stable interior equilibrium), (ii) $α=0,β\u003c0$ (the face $B=0$ is stable), (iii) $α\u003c0,β=0$ (the face $R=0$ is stable). For $α\u003e0$ or $β\u003e0$ the solutions reach the boundaries of applicability of the model in finite time. Moreover, $α\u003c0,β\u003c0$ 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.","short_abstract":"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 th...","url_abs":"https://arxiv.org/abs/2512.18515","url_pdf":"https://arxiv.org/pdf/2512.18515v1","authors":"[\"Sergey Salishev\"]","published":"2025-12-20T21:55:50Z","proceeding":"math.DS","tasks":"[\"math.DS\",\"econ.GN\",\"math.OC\"]","methods":"[]","has_code":false}