Understanding how Nonlocal Diffusion Shapes Patterns in Biological Systems

许多生物和物理系统都以展示有趣的周期性结构而闻名，比如条纹和斑点模式。在许多情况下，就像化学反应一样，这些结构是由两个相互竞争的过程相互作用产生的，这些过程导致了不稳定性。粗略地说，第一种机制与系统的内在行为有关，而第二种机制对应于传递或扩散过程。虽然在大多数情况下，扩散是可以考虑的正确机制，但在许多生物学应用中，它并不能捕捉到观察到的行为。在模拟种群或植物种子的分散时，或者在观察某些动物的狩猎或觅食行为时，这一点尤其正确。在这个项目中，研究者研究了这些非局部形式的扩散如何影响物理和生物应用中的模式形成。主要的动机来自干旱的生态系统，众所周知，这些生态系统表现出带状植被的模式。虽然大多数数学模型使用扩散来表示植物种子的传播，但数值实验表明，这些方程虽然更容易处理，但忽略了关键信息。例如，对这些“局部”模式的分析表明，由于降雨减少而导致的植被带间距的变化通常预示着向完全沙漠状态的可能的不可逆转的转变。然而，当考虑到非本地种子传播时，模拟表明这些生态系统对这些变化的恢复能力有所增加。这两种模型之间的差异突出了对远距离扩散效应进行更深入研究的必要性。为了研究这些影响，将使用非局部Gray-Scott模型作为测试案例。这组特定的方程也为研究生的有趣项目和本科生的暑期研究经验提供了基础。由于非局部过程通常使用卷积算子建模，因此研究这些系统的主要障碍来自于可用来分析这些映射的有限的数学和数值工具集。事实上，大多数研究使用卷积核来表示这些算子，卷积核具有便于分析的性质。同时，数值模拟往往忽略了卷积映射需要计算域外未知信息的事实，从而错误地实现了边界约束。该项目的目标之一是通过选择具有代数衰减的特定函数类作为其域，为这些映射提供证明Fredholm属性的方法。这项工作允许研究者使用隐函数定理来证明模式的存在。由于所考虑的函数类还提供了关于无穷远处解的衰减水平的信息，因此这些信息可以用于构造具有适当边界约束的数值格式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many biological and physical systems are known for exhibiting interesting periodic structures, like striped and spot patterns. In many instances, like in the case of chemical reactions, these structures emerge from the interaction of two competing processes that lead to an instability. Roughly, the first mechanism relates to the intrinsic behavior of the system, while the second one corresponds to a transport, or diffusion process. While in most cases diffusion is the correct mechanism to consider, in many biological applications it does not capture the observed behavior. This is particularly true when modeling dispersion of populations or plant seeds, or when looking at hunting or foraging behavior of certain animals. In this project, the investigator studies how these nonlocal forms of dispersal affect pattern formation in physical and biological applications. The main motivation comes from arid ecosystems, which are known to exhibit patterns of banded vegetation. While most mathematical models use diffusion to represent dispersal of plant seeds, numerical experiments suggest that these equations, although more tractable, ignore critical information. For example, the analysis of these 'local' models shows that changes in the spacing between bands of vegetation due to reduced rainfall often signal a possible irreversible transition to a fully desert state. However, when nonlocal seed dispersal is considered, simulations indicate that the resilience of these ecosystems with respect to such changes is increased. The discrepancy between these two models highlights the need for more in-depth studies of long-range dispersal effects. To study these effects, a nonlocal Gray-Scott model will be used as a test case. This specific set of equations also provides a basis for interesting projects for graduate students and for summer research experiences for undergraduates. Because nonlocal processes are often modeled using convolution operators, the main obstacle for studying these systems comes from the limited set of mathematical and numerical tools available to analyze these maps. Indeed, most studies represent these operators using convolution kernels that have properties which facilitate their analysis. At the same time, numerical simulations often ignore the fact that convolution maps require information about the unknown outside the computational domain, and thus incorrectly implement boundary constraints. One of the project's goals is to provide methods for proving Fredholm properties for these maps by choosing as their domain a specific class of functions with algebraic decay. This work allows investigators to use the implicit function theorem to prove the existence of patterns. Since the class of functions being considered also provides information about the level of decay of solutions at infinity, this information can then be used to construct numerical schemes with appropriate boundary constraints.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.