Asymptotic expansion approximation for spatial structure arising from directionally biased movement

TL;DR

This paper introduces an asymptotic expansion method to efficiently approximate equilibrium solutions of spatial moment dynamics in models with neighbor-dependent directional bias, improving computational feasibility for parameter inference.

Contribution

It develops a novel asymptotic expansion approach for spatial moment dynamics under weak bias, enabling faster equilibrium approximations compared to traditional numerical methods.

Findings

The asymptotic expansion accurately approximates equilibrium solutions.

The method significantly reduces computational cost.

Applicable for parameter inference in spatial models.

Abstract

Spatial structure can arise in spatial point process models via a range of mechanisms, including neighbour-dependent directionally biased movement. This spatial structure is neglected by mean-field models, but can have important effects on population dynamics. Spatial moment dynamics are one way to obtain a deterministic approximation of a dynamic spatial point process that retains some information about spatial structure. However, the applicability of this approach is limited by the computational cost of numerically solving spatial moment dynamic equations at a sufficient resolution. We present an asymptotic expansion for the equilibrium solution to the spatial moment dynamics equations in the presence of neighbour-dependent directional bias. We show that the asymptotic expansion provides a highly efficient scheme for obtaining approximate equilibrium solutions to the spatial moment dynamics equations when bias is weak. This scheme will be particularly useful for performing parameter inference on spatial moment models.