The models introduced in this paper describe a uniform distribution of plant stems competing for sunlight. The shape of each stem, and the density of leaves, are designed in order to maximize the captured sunlight, subject to a cost for transporting water and nutrients from the root to all the leaves. Given the intensity of light, depending on the height above ground, we first solve the optimization problem determining the best possible shape for a single stem. We then study a competitive equilibrium among a large number of similar plants, where the shape of each stem is optimal given the shade produced by all others. Uniqueness of equilibria is proved by analyzing the two-point boundary value problem for a system of ODEs derived from the necessary conditions for optimality.

中文翻译：

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植物茎的竞争模型

本文中介绍的模型描述了竞争阳光的植物茎的均匀分布。每个茎的形状和叶子的密度都是为了最大限度地吸收阳光而设计的，但要考虑将水和养分从根部输送到所有叶子的成本。给定光照强度，取决于离地高度，我们首先解决优化问题，确定单个茎的最佳可能形状。然后，我们研究了大量相似植物之间的竞争平衡，考虑到所有其他植物产生的阴影，每个茎的形状都是最佳的。通过分析从最优性的必要条件导出的 ODE 系统的两点边值问题，证明了平衡的唯一性。