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包含一致椭圆算子（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。这些系统经常被用来模拟人口动态中的各种人口密度。UEO和FBO分别包含拉普拉斯算子、Dirichlet和Neumann边界算子作为特例。UEO包括扩散项和代表由于风、水流或环境梯度引起的种群漂移率的项。非线性（或反应项）是非负的或变化的符号。这些模型在现代应用数学中起着重要的作用。 种群动力学研究的主要内容之一是了解生态系统中相互作用物种的时空行为。该问题的一些重要课题是研究在什么样的环境下物种是共存还是灭绝，以及系统中的物种是否能持续共存。在数学上，这些主题导致研究SPPDE（或SPPDI）模型的正平衡态（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，如各种Volterra-Lotka竞争模型和捕食者-食饵模型的收获率，Allee效应或猎物避难所，已被广泛研究的文献，包括我自己的研究。但由于现有理论工具的限制，现有的关于SPPDE模型正平衡解的存在性、共存性、不存在性和唯一性以及正解的大时间行为的结果对相互作用种群的时空行为的理解还不够深入.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，这些模型在文献中得到了广泛的研究。但是，很少有研究这些人口的模拟的SPDEs。 本研究计划的目标是：（1）寻找新的思想和方法来改进现有的理论，如不动点指数理论，并将新的理论结果应用于研究SPPDE或SPPDIs，以及上述各种人口模型;（2）将差分方程人口模型推广到SPPDE人口模型，这是新的。 该研究项目将丰富和发展现代偏微分方程或不等式理论、非线性分析及其在种群动力学中的应用。

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.