A minimalist model for coevolving supply and drainage networks

Abstract

Numerous complex systems, both natural and artificial, are characterized by the presence of intertwined supply and/or drainage networks. Here, we present a minimalist model of such coevolving networks in a spatially continuous domain, where the obtained networks can be interpreted as a part of either the counter-flowing drainage or co-flowing supply and drainage mechanisms. The model consists of three coupled, nonlinear partial differential equations that describe spatial density patterns of input and output materials by modifying a mediating scalar field, on which supply and drainage networks are carved. In the two-dimensional case, the scalar field can be viewed as the elevation of a hypothetical landscape, of which supply and drainage networks are ridges and valleys, respectively. In the three-dimensional case, the scalar field serves the role of a chemical signal, according to which vascularization of the supply and drainage networks occurs above a critical ‘erosion’ strength. The steady-state solutions are presented as a function of non-dimensional channelization indices for both materials. The spatial patterns of the emerging networks are classified within the branched and congested extreme regimes, within which the resulting networks are characterized based on the absolute as well as the relative values of two non-dimensional indices.

Transient emergence of ramified river channels: simulations of geographical cycle by Erosion-Diffusion Model (EDM)

“…The absence of the tectonic uplift term is a remarkable feature in Eq. (7), which differentiates the EDM from other existing LEMs [13,14,15,16,17]. Equation ( 7) is analyzed by means of the Runge-Kutta approximation method.…”

“…At first, we assume uniform precipitation and constant water velocity for the horizontal direction over the whole area, which are proper assumptions on dealing with geographic processes of large spatiotemporal scales [16,17]. The slope line is a continuous curve along the direction of the steepest descent.…”

“…Among LEMs, widespread models consist of two state variables, which are the land elevation z and the upstream catchment area A [4,5,11,12,13,14,15,16,17]. Both of them are two-dimensional functions on Cartesian coordinates (𝑥, 𝑦).…”