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

本文主要研究抛物型偏微分方程组(SPPDE)或抛物型偏微分方程组(SPPDI)的第一边界算子(FBO)的一致椭圆算子(UEO)。这些系统经常被用来对人口动力学中的各种人口密度进行建模。UEO和FBO分别包含拉普拉斯算子和Dirichlet和Neumann边界算子作为特例。UEO包括扩散项和表示由于风、流或环境梯度而导致的人口漂移率的项。非线性(或反应项)要么是非负的，要么是变化的符号。这些模型在现代应用数学中扮演着重要的角色。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。这个问题的一些重要课题是调查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共存状态。在数学上，这些主题导致研究SPPDE(或SPPDI)模型正稳态(经典或弱解)解的存在、共存、不存在和唯一性，以及这些模型正解的大时间性态。有许多种群模型，如各种Volterra-Lotka竞争模型和考虑收获率、Allee效应或食饵避难所的捕食者-食饵模型，这些模型在文献中得到了广泛的研究，包括我自己的研究。但由于现有理论工具的限制，已有的关于SPPDE模型正定态解的存在性、共存性、不存在性和唯一性的结果，以及正解的大时间行为，对相互作用物种的时空行为理解不足。此外，也有一些重要的种群模型由差分方程描述，例如具有Ricker型或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，它们在文献中得到了广泛的研究。但对这些由SPPDE模拟的人群的研究很少。研究的目标是(1)寻求新的思路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SPPDI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。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、非线性分析及其在种群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.