Nonlocal and Anisotropic Partial Differential Equations in Mathematical Biology

数学生物学中的非局部和各向异性偏微分方程

• 批准号: RGPIN-2017-04158
• 负责人: Hillen, Thomas
• 金额: $ 3.13万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710486
• 关键词: Nonlocal; Anisotropic; Partial; Differential; Equations

In this research I consider nonlocal and anisotropic differential equations that arise as mathematical models for biological processes. These models are generalizations of well known reaction-advection-diffusion models and a rich qualitative theory is waiting to be explored. I expect to find new forms of pattern formation that include singular solutions, global patterns and accelerated invasion fronts. I will use these versatile models for several specific biological applications. For example, I use the models to analyse how wolves use linear features in the forest environment to change their hunting strategies, and how we can understand the impact of this change on wolf-ungulate dynamics. Another example involves sea turtles, who use an internal compass to navigate long distances across the ocean to find their breeding beaches. Mathematical models help to understand their navigational abilities by quantifying directional cues (magnetic, chemotactic), and they are used to develop protection strategies. The nonlocal models also apply to forest fire spread, and they are used to estimate the probability that a fire breaches an obstacle to enter human settlements. Finally, another application of these class of models lies in cancer research. While cancer research is not directly part of this NSERC grant, results and methods developed here will impact cancer modelling and provide a framework to develop better treatment strategies. Nonlocal and anisotropic continuum models for spatial movement obtain their beauty from a rich mathematical theory and wide reaching applications.

在这项研究中，我认为作为生物学过程的数学模型出现的非局部和各向异性微分方程。这些模型是众所周知的反应 - 接地扩散模型的概括，并且正在等待探索丰富的定性理论。我希望找到新形式的模式形成，包括奇异解决方案，全球模式和加速入侵前沿。我将将这些多功能模型用于几种特定的生物学应用。例如，我使用这些模型来分析狼如何在森林环境中使用线性特征来改变其狩猎策略，以及我们如何理解这种变化对狼牙动态的影响。另一个例子涉及海龟，他们使用内部指南针在整个海洋上长距离航行以找到自己的繁殖海滩。数学模型通过量化方向提示（磁性，趋化性）来帮助理解其导航能力，并用于制定保护策略。非局部模型也适用于森林火灾的传播，它们用于估计大火违反进入人类定居点的障碍的可能性。最后，这些类别模型的另一种应用在于癌症研究。尽管癌症研究并不是该NSERC赠款的直接一部分，但此处开发的结果和方法将影响癌症的建模，并为制定更好的治疗策略提供框架。空间运动的非本地和各向异性连续模型从丰富的数学理论和广泛的应用程序中获得了美丽。