A variational model of irrigation patterns

Irrigation and draining systems, plants and trees together with their root systems, lungs and cardiovascular systems have a common morphology which seems to derive from topological constraints together with energy saving requirements. All of these systems look like spatial trees and succeed in spreading out a fluid from a source onto a volume. The associated morphology is a tree of bifurcating vessels. Their intuitive explanation is that transport energy is saved by using broad vessels as long as possible rather than thin spread out vessels. In this paper, we define a general formalism dealing with irrigation patterns. Related to martingale theory, this formalism permits one to define irrigation trees and their vessels, to give a generic form to their energy, and to show compactness for the irrigation patterns with bounded energy as well as a lower semicontinuity result for the cost functional. As a consequence, we show that a variety of source to volume irrigation problems are well posed.