  We present a model for mechanically-induced pattern formation in growing
biological tissues and discuss its application to the development of leaf
venation networks. Drawing an analogy with phase transitions in solids, we use
a phase field method to describe the transition between two states of the
tissue, e.g. the differentiation of leaf veins, and consider a layered system
where mechanical stresses are generated by differential growth. We present
analytical and numerical results for one-dimensional systems, showing that a
combination of growth and irreversibility gives rise to hierarchical patterns.
Two-dimensional simulations suggest that such a mechanism could account for the
hierarchical, reticulate structure of leaf venation networks, yet point to the
need for a more detailed treatment of the coupling between growth and
mechanical stresses.
