Models of Controlled Biological Growth

在生长过程中，植物和动物的组织发育成明显可辨认的形状：叶子、花、骨头等；几乎普遍地，组织的生长似乎以惊人的精度被控制。这项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，将解决一些关键问题：与树干相比，什么物理参数产生攀缘藤条的不同行为？？自然如何控制不同物种植物的叶或花的形状和大小？该项目将研究各种偏微分方程组(PDE)，模拟生物组织的生长。这些方程描述了促进生长的化学物质的扩散和吸附、体积膨胀、组织的变形和弹性变形。重点将放在独特形状的出现上，以及如何通过改变影响生长的物理参数来修改这些形状。该项目将专注于由偏微分方程组描述的各种组织生长模型。它们代表了形态原在组织内的扩散和吸收、体积生长和弹性变形。本课程将研究解的存在、唯一性和定性性质，以及在各向异性环境和分层区域中的解。将获得的结果将扩展当前关于具有移动边界的区域上的演化问题的理论。该研究还将解决新类型的静止问题，类似于特征值问题，但在集值框架中。对于不同类别的分层区域，首席调查者期望发现依赖于有限多个参数的局部不变“形态稳定”构型的族。这些将产生丰富多样的新的几何形状，将被分析和数值研究。

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.