Models of Controlled Biological Growth

在生长过程中，动植物的组织发育成明显可识别的形状：叶、花、骨等;几乎普遍地，组织生长似乎以显著的精确度被控制。本研究的主要目的是开发新的数学工具来理解这种控制机制。从这个理论的角度来看，一些关键的问题将得到解决：什么物理参数产生不同的行为的攀援藤，与树干相比？大自然是如何控制不同种类植物的叶子或花的形状和大小的？该项目将研究模拟生物组织生长的各种偏微分方程（PDE）。这些方程描述了生长诱导化学物质的扩散和吸附、体积膨胀和组织的弹性变形之间的平衡。重点将放在独特形状的出现，以及如何通过改变影响生长的物理参数来修改这些形状。该项目将侧重于组织生长的各种模型，由偏微分方程系统描述。这些代表组织内形态发生素的扩散和吸收、体积生长和弹性变形。解的存在性、唯一性和定性性质也将在各向异性和分层域中进行研究。所得到的结果将扩大目前的理论与移动边界域的发展问题。该研究还将解决新类型的固定问题，有点像特征值问题，但在集值框架。 对于各种类别的分层域，主要研究者希望发现家庭的局部不变的“形态稳定”的配置，这取决于许多参数。这些将产生丰富多样的新的几何形状，将进行分析和数值研究。

In the process of their growth, tissues in plants and animals develop into distinctly recognizable shapes: leaves, flowers, bones, etc.; almost universally, the tissue growth appears to be controlled with remarkable accuracy. The principal aim of this research is developing new mathematical tools to understand this control mechanism. From this theoretical perspective, some key questions will be addressed: What physical parameters produce the different behavior of a climbing vine, compared with a tree stem? How does nature control the shape and size of leaves, or flowers, in plants of different species? The project will study various partial differential equations (PDE) modeling the growth of a biological tissue. These equations describe the balance between diffusion and adsorption of growth-inducing chemicals, volume expansion, and elastic deformation of the tissue. The focus will be on the emergence of distinctive shapes, and on how these shapes can be modified by varying the growth-affecting physical parameters. The project will focus on various models of tissue growth, described by systems of partial differential equations. These represent diffusion and absorption for morphogens within the tissue, bulk growth, and elastic deformation. Existence, uniqueness, and qualitative properties of solutions will be studied, also in an anisotropic setting and for stratified domains. The results to be obtained will expand the current theory of evolution problems on domains with moving boundaries. The research will also address new types of stationary problems, somewhat like eigenvalue problems but in a set-valued framework.  For various classes of stratified domains, the Principal Investigator expects to discover families of locally invariant "morpho-stationary" configurations, depending on finitely many parameters. These will generate a rich variety of new geometric shapes, that will be studied both analytically and numerically.