Stability and Dynamics of Invasion Fronts in Spatially Extended Systems

该项目涉及数学技术的发展，以预测和解释在物理，生物和化学中应用的不稳定状态的前沿传播的特征。在实际问题中，当系统参数发生变化或引入新物种时，经常会出现不稳定状态。随后的动力学通常由称为入侵前沿的移动界面的形成所主导，这些界面以固定速度传播到不稳定状态，并在其尾流中选择一些次级状态。该提案是围绕扩展这些过程的知识而建立的，其具体目标是阐明导致入侵前沿出现的机制，然后利用这些知识在实际系统中进行预测。 这项工作的一部分将开发新的数学方法来研究局部化强迫对偏微分方程（PDE）模型中界面扩展速度的影响。第二部分将发展预测和一般原则的动态不稳定性蔓延在复杂的网络。 提出了两种互补的研究途径。 建议的工作领域之一涉及的稳定性，重点是了解外部强迫的作用，在选择的界面速度和随后的动态的入侵战线。由于入侵前沿传播到不稳定状态，稳定性分析需要这些前沿的扰动充分本地化。该建议将研究强迫函数也局部化，但不足以局部化以适应齐次方程的稳定性框架的问题。成功地实施所描述的项目将导致新的数学方法来研究行波的稳定性，并将澄清和解释一些新的现象，这将是感兴趣的研究人员在应用问题。 建议的工作的第二个领域涉及入侵战线的空间是一个复杂的网络与当地的动态发生在每个节点之间的耦合通过图形拉普拉斯算子。这里的目标是利用PDE设置中的前沿传播知识来做出与网络上的不稳定性传播相关的具体预测。 应用包括全球流行病的传播和离散环境中入侵物种的传播。这项工作的成功完成将为这一领域的研究人员提供新的工具来估计到达时间，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的评估来支持。影响审查标准。

This project concerns the development of mathematical techniques to predict and explain features of front propagation into unstable states with applications in physics, biology and chemistry. Unstable states arise frequently in applied problems as system parameters are changed or new species are introduced into the problem. The subsequent dynamics are often dominated by the formation of moving interfaces called invasion fronts that propagate into the unstable state at fixed speed and select some secondary state in their wake. This proposal is built around expanding knowledge of these processes with a specific goal of elucidating mechanisms leading to the emergence of invasion fronts and then leveraging this knowledge to make predictions in actual systems of interest. One portion of this work will develop new mathematical approaches to study the effect of localized forcing on the spreading speed of interfaces in partial differential equation (PDE) models. The second portion will develop predictions and general principles for the dynamics of instabilities spreading over complex networks. Two complementary avenues of research are proposed. One area of proposed work concerns stability of invasion fronts with a focus on understanding the role of external forcing in the selection of the inter-facial velocity and subsequent dynamics. Since invasion fronts propagate into unstable states, stability analysis requires perturbations of these fronts to be sufficiently localized. This proposal will study problems where the forcing function is also localized, but not sufficiently localized to fit into the stability framework of the homogeneous equation. Success in carrying out the projects described will lead to new mathematical approaches to the study of stability of traveling waves and will clarify and explain some novel phenomena that will be of interest to researchers working in applied problems. A second area of proposed work concerns invasion fronts where space is taken to be a complex network with local dynamics occurring at each node with coupling between nodes via a graph Laplacian. The goal here is to leverage knowledge of front propagation in the PDE setting to make concrete predictions related to the spread of instabilities over networks. Applications include the spread of global epidemics and that of invasive species in discrete environments. Successful completion of this work will provide researchers working in this area with new tools to estimate arrival times, provide a rigorous theoretical framework by which to understand these spreading phenomena and provide insight into the way features of complex networks influence arrival times of invasions.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.