Grant: Discovery Grants Program - Individual

Differential equations have been extensively used to describe many biological processes. With the emergence of new biological processes and better understanding of some old ones, some existing mathematical models need to be refined and novel ones have to be proposed. This may result in systems that cannot be dealt with existing theories/techniques, which motivates the following studies of this proposal. The Lyapunov direct method is very effective in showing global stability. To apply it, one has to find a suitable Lyapunov function/functional and show the negative (semi-)definiteness of its derivative along the solutions of the system. Usually it is very challenging to do so. In this proposal, on the one hand, we want to find new types of functions for constructing Lyapunov functions/functionals. One the other hand, we look for new ways to rearrange terms of derivatives along solutions so that negative (semi-)definiteness can be easily determined. The proposed approach will be employed to not only improve some existing results on global stability but also obtain some new and sharper ones. Malignant tumor (i.e. cancer) has long been the leading cause of death in the world. The interaction between tumor cells and the immune system is far from being completely understood. Some new mathematical models will be built to explain observed clinical phenomena and to investigate the impacts of affecting factors. We will also construct models to understand the effect of oncolytic immunotherapy on tumor growth and explore the interaction mechanisms among oncolytic viruses, tumor cells, and the immune system. The theoretical results obtained can help us assess the efficacy of treatments, identify the most significant factors affecting the development of cancer, among other publications. Heterogeneity is also an important factor in many biological processes. In this proposal, we concentrate on spatial heterogeneity, which is usually described by reaction-diffusion equations. We mainly study the effect of advection on competition and predation among species. We aim to solve problems including successful species invasion, coexistence of species, and patterns of coexistence. The results will provide guidelines on resources management. To summarize, the proposed studies are expected to advance and enrich the theory of applied dynamics. The interdisciplinary collaboration and the application to the study of cancer will shed light on understanding underlying mechanisms and possible treatments. The studies on population dynamics will provide control strategies for resource sustainability. Moreover, the proposal will provide HQP with opportunities for training and collaboration.