Research Interests |
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AREAS OF SPECIALIZATION:
- Differential Equations
- Nonlinear Analysis
- Mathematical Biology
My research program mainly deals with systems of parabolic partial differential equations (PPDEs) involving uniformly elliptic operators (UEOs) with first boundary operators (FBOs).
These systems are often used to model various population densities in population dynamics. The nonlinearities (or reaction terms)
either are nonnegative or change sign. 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, nonexistence and
uniqueness of the positive steady-state (classic or weak) solutions of the PPDE models and the large time behaviors
of positive (classic or weak) solutions for the PPDE models.
My current research program is to search for new ideas and approaches to improve the existing theories such as
fixed point index theories and variational inequality theories, and to apply the new theoretical results to study
systems of PPDEs with the UEOs and FBOs, and a variety of population models in order to provide
better understanding of the spatial and temporal behaviors of interacting species.
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