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

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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. **