# Growth Models for Tree Stems and Vines

## TL;DR

This paper develops a PDE model for the growth of tree stems and vines, incorporating effects of gravity, obstacles, and curling behavior, with theoretical existence results and numerical simulations.

## Contribution

It introduces a novel PDE model with differential inclusions for plant growth, including obstacle interaction and curling, and proves existence of solutions.

## Key findings

- Existence of local and global solutions under certain conditions
- Numerical simulations illustrating model behavior
- Identification of breakdown configurations in growth modeling

## Abstract

The paper introduces a PDE model for the growth of a tree stem or a vine. The equations describe the elongation due to cell growth, and the response to gravity and to external obstacles. An additional term accounts for the tendency of a vine to curl around branches of other plants. When obstacles are present, the model takes the form of a differential inclusion with unilateral constraints. At each time t, a cone of admissible reactions is determined by the minimization of an elastic deformation energy. The main theorem shows that local solutions exist and can be prolonged globally in time, except when a specific "breakdown configuration" is reached. Approximate solutions are constructed by an operator-splitting technique. Some numerical simulations are provided at the end of the paper.