Models for Branching Networks in Two Dimensions

In this paper mathematical models for branching networks growing in one and two dimensions are described. Continuum equations are formulated to represent evolving spatial distributions of density given a variety of assumptions about branching and crosslinking kinetics. In accommodating the influenceof angular branch distributions, a set of integropartial differential equations are obtained. It is found that crosslinking (which eliminates apical growth) acts as a density-regulating mechanism. If the branching angle φ is small, this mechanism further leads to a phenomenon of orientation selection: It is shown analytically that for small φ a spatially homogeneous network with an initial uniform distribution of branch orientations will align along a single axis as a result of instability of the uniform steady state to small perturbations that are nonuniform in the angular variable. Because instability first occurs to a mode ei/θ for which $l = 2$, such networks eventually contain only parallel branches. Similar phenomena lead to a transition in properties of spatially propagating networks, whose angular distributions are diffuse for φ > 0.29 and sharply defined for φ < 0.24. Other properties of propagating networks, including traveling wave behavior, are described analytically and illustrated by numerical simulations.