Modeling drone swarms with a PDE extension of Lanchester’s laws
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Authors
Charles Randow
Ryan H Allaire
Antonio Mastroberardino
Issue Date
2026-04-03
Type
Article
Language
en_US
Keywords
Alternative Title
Abstract
We extend the Lanchester equations of combat modeling by constructing a coupled system of partial differential equations with the goal of representing engagements between an attacking drone swarm and a layered defense. Our model includes an attacking force of drones traversing the battlefield (domain) in an attempt to reach its target. The defending force, which engages the incoming attackers, is made up of a static defense and a mobile drone force capable of intercepting and tracking the attackers. Drone swarm motion is modeled with advection–diffusion terms and engagement at range by nonlocal reaction terms. We solve the underlying model using a finite difference method and investigate how interaction range, swarm dispersion, and the allocation between static and mobile defenses affect attacker survivability. To account for expendable or single-use countermeasure platforms (e.g. kamikaze–style interceptors), we also examine how expendability affects the final engagement outcome. This study provides a foundation for modeling spatially distributed combat dynamics reflecting current and future combat scenarios.
Description
We extend the Lanchester equations of combat modeling by constructing a coupled system of partial differential equations with the goal of representing engagements between an attacking drone swarm and a layered defense. Our model includes an attacking force of drones traversing the battlefield (domain) in an attempt to reach its target. The defending force, which engages the incoming attackers, is made up of a static defense and a mobile drone force capable of intercepting and tracking the attackers. Drone swarm motion is modeled with advection–diffusion terms and engagement at range by nonlocal reaction terms. We solve the underlying model using a finite difference method and investigate how interaction range, swarm dispersion, and the allocation between static and mobile defenses affect attacker survivability. To account for expendable or single-use countermeasure platforms (e.g. kamikaze–style interceptors), we also examine how expendability affects the final engagement outcome. This study provides a foundation for modeling spatially distributed combat dynamics reflecting current and future combat scenarios.
Citation
Randow, C., Allaire, R. H., & Luo, R. (2026). Modeling drone swarms with a PDE extension of Lanchester’s laws. Advance online publication. https://doi.org/10.1177/15485129261436073
Publisher
The Journal of Defense Modeling and Simulation: Applications, Methodology, Technology