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

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  • Sergey Salishev

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 $\dot y=\alpha y^2-\beta$, 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 $(\alpha,\beta)$: (i) $\alpha 0$ or $\beta>0$ the solutions reach the boundaries of applicability of the model in finite time. Moreover, $\alpha

Suggested Citation

  • Sergey Salishev, 2025. "The Narrow Corridor of Stable Solutions in an Extended Osipov--Lanchester Model with Constant Total Population," Papers 2512.18515, arXiv.org.
    • Handle: RePEc:arx:papers:2512.18515
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    References listed on IDEAS

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    1. Kress, Moshe & Caulkins, Jonathan P. & Feichtinger, Gustav & Grass, Dieter & Seidl, Andrea, 2018. "Lanchester model for three-way combat," European Journal of Operational Research, Elsevier, vol. 264(1), pages 46-54.
    2. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, November.
    3. S. J. Deitchman, 1962. "A Lanchester Model of Guerrilla Warfare," Operations Research, INFORMS, vol. 10(6), pages 818-827, December.
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