Theories of fixed point index and variational inequalities, systems of differential equations and applications to population models

不动点指数和变分不等式理论、微分方程组及其在总体模型中的应用

• 批准号: RGPIN-2018-04177
• 负责人: Lan, Kunquan
• 金额: $ 1.31万
• 依托单位: Ryerson University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758537
• 关键词: Theories; fixed; point; index; variational

This proposed research program mainly deals with systems of parabolic partial differential equations (SPPDEs) or parabolic partial differential inequalities (SPPDIs) involving uniformly elliptic operators (UEOs) with first boundary operators (FBOs). These systems are often used to model various population densities in population dynamics. The UEOs and FBOs contain the Laplacian operators, and the Dirichlet and Neumann boundary operators, respectively, as special cases. The UEOs include the diffusion terms and the terms representing the drift rates of the population due to wind, current or environmental gradients. The nonlinearities (or reaction terms) are either nonnegative or change signs. These models play important roles in modern applicable mathematics. One of the major concerns in population dynamics is to understand the spatial and temporal behaviors of interacting species in ecological systems. Some important topics of the problem are to investigate under what circumstances the species either coexist or become extinct, and to determine whether the species in the system can persist at a coexistence state. Mathematically, these topics lead to study the existence, co-existence, nonexistence and uniqueness of the positive steady-state (classic or weak) solutions of the SPPDE (or SPPDIs) models, and the large time behaviors of positive solutions for these models. There are many population models such as various Volterra-Lotka competition models and predator-prey models incorporating harvesting rates, Allee effect or prey refuge, which have been widely studied in the literature including my own research. But due to the restriction of the existing theoretical tools, the existing results on the existence, co-existence, nonexistence and uniqueness of the positive steady-state solutions, and the large time behaviors of positive solutions for the SPPDE models provide insufficient understanding of the spatial and temporal behaviors of interacting species. Also, there are some important population models governed by difference equations such as discrete population models with Ricker-or Hassell-type functions and their generalizations such as Ricker functions with Allee effect, which have been widely studied in the literature. But there is little study on these populations modeled by the SPPDEs. The objectives of the proposed research program are (1) to search for new ideas and approaches to improve the existing theories such as fixed point index theories and apply the new theoretical results to study the SPPDEs or SPPDIs, and a variety of population models mentioned above; and (2) to generalize the difference equation population models to the SPPDE population models, which is new. The proposed research program will enrich and develop the theories of both modern partial differential equations or inequalities, nonlinear analysis and their applications to population dynamics.

该拟议的研究计划主要涉及抛物线偏微分方程（SPPDES）或抛物线部分差分不平等（SPPDIS）的系统，该系统涉及与第一边界运营商（FBO）均匀椭圆运算符（UEOS）的系统。这些系统通常用于对种群动态的各种人口密度进行建模。 UEOS和FBO包含Laplacian操作员，分别是Dirichlet和Neumann边界运营商作为特殊情况。 UEO包括由于风，当前或环境梯度而导致人口漂移率的扩散项和术语。非线性（或反应术语）是无负或变化符号。这些模型在现代适用数学中起着重要作用。 人口动态的主要关注点之一是了解生态系统中相互作用物种的空间和时间行为。该问题的一些重要主题是在哪种情况下进行研究，该物种可以共存或灭绝，并确定系统中的物种是否可以持续在共存状态下。从数学上讲，这些主题导致研究了SPPDE（或SPPDIS）模型的积极稳态（经典或弱）解决方案的存在，共存，不存在和独特性，以及这些模型积极解决方案的较大时间行为。 有许多人口模型，例如各种Volterra-Lotka竞争模型和结合收获率，Allee效果或猎物避难所的Predator-Prey模型，这些模型在文献中已广泛研究，包括我自己的研究。但是，由于对现有的理论工具的限制，现有的结果，对积极稳态解决方案的存在，不存在和唯一性以及SPPDE模型积极解决方案的巨大时间行为提供了对相互作用物种的空间和时间行为的不足。此外，还有一些重要的人群模型，这些模型受差异方程式控制的，例如具有ricker-or hassell型功能的离散人群模型及其概括，例如带有Alee效应的Ricker功能，在文献中已广泛研究。但是几乎没有关于由SPPDE建模的人群的研究。 拟议的研究计划的目标是（1）寻找改善现有理论（例如固定点索引理论）的新思想和方法，并应用新的理论结果来研究SPPDES或SPPDIS，以及上面提到的各种人群模型； （2）将差异方程人群模型推广到SPDE人群模型，这是新的。 拟议的研究计划将丰富并发展现代部分差分方程或不平等，非线性分析及其对人口动态的应用。