Nonlocal and Anisotropic Partial Differential Equations in Mathematical Biology

在这项研究中，我考虑作为生物过程数学模型出现的非局部和各向异性的微分方程式。这些模型是著名的反应-对流-扩散模型的推广，丰富的定性理论有待探索。我希望找到新的模式形成形式，包括单一解决方案、全球模式和加速入侵前沿。我将把这些多才多艺的模型用于几个特定的生物学应用。例如，我使用这些模型来分析狼如何利用森林环境中的线性特征来改变它们的狩猎策略，以及我们如何理解这种变化对狼-蹄动物动态的影响。另一个例子是海龟，它们使用内部罗盘在海洋中航行很长一段距离，以找到它们的繁殖海滩。数学模型通过量化方向线索(磁性、趋化性)帮助了解它们的导航能力，并用它们来制定保护策略。非局部模型也适用于森林火灾蔓延，它们被用来估计火灾突破障碍进入人类住区的概率。最后，这类模型的另一个应用在于癌症研究。虽然癌症研究不直接是NSERC拨款的一部分，但这里开发的结果和方法将影响癌症建模，并为开发更好的治疗策略提供框架。空间运动的非局部和各向异性连续介质模型从丰富的数学理论和广泛的应用中获得了它们的美。

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.