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

本研究计划主要研究具有第一边界算子的均匀椭圆算子（UEOs）的抛物型偏微分方程或抛物型偏微分不等式系统。这些系统通常用于在种群动力学中模拟不同的种群密度。作为特殊情况，ueo和fbo分别包含拉普拉斯算子和狄利克雷边界算子和诺伊曼边界算子。uoe包括扩散项和表示种群因风、水流或环境梯度而漂移率的项。非线性（或反应项）要么是非负的，要么是变号的。这些模型在现代应用数学中起着重要作用。种群动力学的一个主要问题是了解生态系统中相互作用物种的时空行为。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保持共存状态。在数学上，这些主题导致研究SPPDE（或SPPDIs）模型的正稳态（经典或弱）解的存在性、共存性、非存在性和唯一性，以及这些模型的正解的大时间行为。***有许多种群模型，如各种Volterra-Lotka竞争模型和包含收获率，Allee效应或猎物避难所的捕食者-猎物模型，这些模型在包括我自己的研究在内的文献中得到了广泛的研究。但由于现有理论工具的限制，SPPDE模型关于正稳态解的存在性、共存性、非存在性和唯一性以及正解的大时间行为的现有结果对相互作用物种的时空行为的理解不足。此外，还有一些重要的由差分方程控制的种群模型，如具有Ricker或hassell型函数的离散种群模型及其推广，如具有Allee效应的Ricker函数，在文献中得到了广泛的研究。但SPPDEs对这些种群的建模研究很少。***拟开展的研究项目目标为：(1)寻找新的思路和方法，完善定点指数理论等现有理论，并将新的理论成果应用于sppde或sppdi以及上述各种人口模型的研究；(2)将差分方程种群模型推广到SPPDE种群模型。***提出的研究计划将丰富和发展现代偏微分方程或不等式、非线性分析及其在人口动力学中的应用理论。**

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. **