On the optimal shape of tree roots and branches

Abstract: Featured Article: "Optimal Shapes for Tree Roots", SIAM J. Math. Anal. 54(4) (2022), pp. 4757-4784.

Featured Article Authors: Alberto Bressan, Sondre T. Galtung and Qing Su.

Living organisms come in an immense variety of shapes. In many cases one can expect that, through natural selection, a ``best possible" shape will have evolved. From a mathematical perspective, it is thus of interest to understand whether similar geometric shapes can be recovered as minimizers of suitable functionals.

As a step in this direction, we consider two functionals measuring the efficiency of roots and branches in a tree. Namely: (i) a ``sunlight functional", modeling the total amount of sunlight captured by the leaves of a tree, and (ii) a ``harvest functional", modeling the amount of nutrients collected by the roots. The above functionals must be combined with a ``ramified transportation cost", for transporting nutrients from the roots to the base of the trunk, or from the base of the trunk to the leaves.

The talk will address the semicontinuity of these functionals, and the existence and properties of optimal solutions, in a space of measures. Open problems will also be discussed, including computational issues and how these optimal shapes may depend on various parameters.

analysis of PDEs

Audience: researchers in the topic

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Seminar In the Analysis and Methods of PDE (SIAM PDE)

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