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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676959
• 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. **

该研究计划主要涉及涉及均匀椭圆算子（UEO）和第一边界算子（FBO）的抛物型偏微分方程（SPPDE）或抛物型偏微分不等式（SPPDI）系统。这些系统通常用于对人口动态中的各种人口密度进行建模。作为特殊情况，UEO 和 FBO 分别包含拉普拉斯算子以及狄利克雷和诺伊曼边界算子。 UEO 包括扩散项和代表由于风、水流或环境梯度而导致的种群漂移率的项。非线性（或反应项）要么是非负的，要么是变化符号。这些模型在现代应用数学中发挥着重要作用。*** 种群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研究物种在什么情况下共存或灭绝，并确定系统中的物种是否能够持续处于共存状态。在数学上，这些主题导致研究 SPPDE（或 SPPDI）模型的正稳态（经典或弱）解的存在性、共存性、不存在性和唯一性，以及这些模型的正解的大时间行为。 *** 有许多种群模型，例如各种 Volterra-Lotka 竞争模型和结合收获率、Allee 效应或猎物避难所的捕食者-猎物模型，这些模型已在包括我自己的研究在内的文献中得到广泛研究。但由于现有理论工具的限制，现有的SPPDE模型关于正稳态解的存在、共存、不存在和唯一性以及正解的大时间行为的结果对相互作用物种的时空行为的理解还不够。此外，还有一些由差分方程控制的重要种群模型，例如具有 Ricker 或 Hassell 型函数的离散种群模型及其推广，例如具有 Allee 效应的 Ricker 函数，这些模型已在文献中得到广泛研究。但对这些由SPPDEs建模的群体的研究却很少。***本研究计划的目标是（1）寻找新的思路和方法来改进现有的理论，如不动点指数理论，并应用新的理论成果来研究SPPDEs或SPPDIs以及上述各种群体模型； (2) 将差分方程总体模型推广到 SPPDE 总体模型，这是一项新的研究。*** 所提出的研究计划将丰富和发展现代偏微分方程或不等式、非线性分析及其在总体动力学中的应用的理论。 **