Combat modelling with partial differential equations

The limitations of the classic work of Lanchester on non-spatial ordinary differential equations for modelling combat are well known. We present work seeking to more realistically represent troop dynamics and to enable a deeper understanding of the nature of conflict. We extend Lanchesters ODEs, constructing a new physically meaningful system of partial differential equations. Spatial force movement and troop interaction components are represented with both local and non-local terms, expanding upon the swarming behaviour of fish and birds proposed by Mogilner et al. We are able to reproduce crucial behaviour such as the emergence of cohesive density profiles and troop regrouping after suffering losses in both one and two dimensions.