CAREER: Pattern formation in singularly perturbed partial differential equations

时空格局的形成遍及整个自然界；例如，旱地生态系统中植被格局的动态变化、混合物中的相分离以及化学反应中的格局。基本的理论问题涉及识别普遍的图案形成机制和预测出现的图案的性质和动态。本项目旨在通过研究偏微分方程(PDE)在生物学、生态学和化学反应模型中产生的模式形成不稳定性来解决这些问题。特别是，该项目的重点是开发技术来解决偏微分方程中的图案形成问题，这些方程是在研究组件在长度或时间尺度上大不相同的系统时自然产生的。应用领域包括旱地生态系统中的荒漠化前沿，以及数学生物学和神经科学中的波动现象。该项目包括通过研究生的指导将研究和教育相结合的活动，以及为本科生提供的外展计划和暑期研究机会。该项目集中在三个主要研究领域，目的是深入了解奇异扰动反应扩散偏微分方程组中复杂的时空格局形成现象。受旱地生态系统沙漠化锋面现象的启发，第一个研究领域涉及多组分反应扩散系统中双稳平面界面的图案形成不稳定性。第二个领域旨在建立严格的时间脉冲复制理论，通过该理论，单个行进脉冲(或局部波)可以自我复制，以类似于时空鸭子爆炸的方式顺序地形成额外的脉冲。第三个领域涉及在模式形成入侵过程之后形成的远离开始的模式的性质和性质，即当扰动入侵不稳定的稳定状态时，模式被成核。这些研究领域将通过发展几何奇异摄动方法和空间动力系统工具来发展和发展远未开始的模式形成机制的理论，用于波浪和模式的存在性、稳定性和分叉分析。这个项目中探索的一些现象发生在标准奇异微扰方法失效的地区，需要开发新的技术。研究生和本科生将参与该项目的各个方面。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Spatiotemporal pattern formation appears throughout the natural world; examples include the dynamics of vegetation patterns in dryland ecosystems, phase separation in mixtures, and patterns in chemical reactions. Fundamental theoretical questions concern identifying universal pattern-forming mechanisms and predicting the nature and dynamics of the patterns which appear. This project aims to address these questions through the study of pattern-forming instabilities in the setting of partial differential equations (PDEs) arising in models from biology, ecology, and chemical reactions. In particular, the project focuses on the development of techniques to tackle pattern formation in PDEs which are singularly perturbed; such equations arise naturally in the study of systems where components operate on widely differing length or time scales. Application areas include desertification fronts in dryland ecosystems, as well as wave phenomena in mathematical biology and neuroscience. Included in the project are activities which integrate research and education through the supervision of graduate students as well as outreach programs and summer research opportunities for undergraduate students.This project focuses on three primary research areas, with the goal of providing insight into complex spatiotemporal pattern formation phenomena in singularly perturbed reaction diffusion PDEs. Motivated by the phenomenon of desertification fronts in dryland ecosystems, the first research area concerns pattern-forming instabilities of bistable planar interfaces in multi-component reaction diffusion systems. The second area aims to build a rigorous theory of temporal pulse replication, by which a single traveling pulse (or localized wave) self-replicates, sequentially nucleating additional pulses in a manner resembling a spatiotemporal canard explosion. The third area is concerned with the nature and properties of far-from-onset patterns formed in the wake of pattern-forming invasion processes, whereby a pattern is nucleated when a disturbance invades an unstable steady state. These research areas will grow and advance the theory of far-from-onset pattern-forming mechanisms through the development of geometric singular perturbation methods and spatial dynamical systems tools to be used in the existence, stability, and bifurcation analysis of waves and patterns. Several of the phenomena explored in this project occur in regions where standard singular perturbation methods break down, requiring the development of novel techniques. Graduate students and undergraduate students will participate in various aspects of the project.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.