Length-scale selection in adaptive transport networks

TL;DR

This paper presents a continuum model for adaptive transport networks, revealing a pattern-forming instability that determines network density and spacing, advancing understanding of their hierarchical organization.

Contribution

It introduces a continuum framework and analytical instability analysis that explain scale selection in adaptive transport networks, extending beyond lattice models.

Findings

Identifies a finite-wavelength instability with a preferred scale.

Simulations confirm analytical predictions of pattern formation.

Predicts intrinsic network density and spatial scale of resource delivery.

Abstract

Adaptive transport networks in biological and physical systems exhibit hierarchical organization, characteristic channel spacing, and robust scaling relations. Existing adaptive network models, formulated on a lattice, successfully reproduce many observed topologies and conduit scaling laws; however, the mechanism that selects network density and spatial spacing remains unclear. We address this in a continuum formulation where conductivity evolves as a tensor field coupled to pressure-driven flow. Linearizing about a homogeneous conducting state, we identify a finite-wavelength instability with a $-1/4$ preferred wavelength scaling in the control parameter. Simulations of the full equations confirm the analytical predictions and demonstrate the formation of anisotropic conducting structures above threshold. These results establish a scale-selection principle for adaptive transport network formation which arises from a pattern-forming instability rather than solely from relaxation within a nonconvex energy landscape. The instability mechanism places adaptive transport systems within a broader class of nonequilibrium pattern-forming media in which constitutive transport feedback generates spatial organization. Beyond reproducing hierarchical scaling laws, the theory additionally predicts the intrinsic density of transport networks and the spatial scale of resource delivery.