Models of Controlled Biological Growth

植物和动物的组织在生长过程中，形成了清晰可辨的形状：叶、花、骨等；几乎在所有情况下，组织生长似乎都得到了非常精确的控制。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。从这个理论的角度来看，一些关键问题将得到解决：与树干相比，哪些物理参数会产生爬藤的不同行为？大自然是如何控制不同种类植物的叶子或花朵的形状和大小的？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。该项目将侧重于用偏微分方程系统描述的各种组织生长模型。这代表了组织内形态形成因子的扩散和吸收、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设置和分层域中进行研究。所得到的结果将扩展目前关于移动边界域的进化问题的理论。研究还将解决新的类型的平稳问题，有点像特征值问题，但在一个集值框架。对于不同类别的分层域，首席研究员期望发现依赖于有限多个参数的局部不变“形态平稳”构型族。这些将产生丰富多样的新几何形状，这些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.