Bifurcation theory and applications in mathematical biology

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Hilbert's 16th problem is the most difficult one proposed by Hilbert 100 year ago but still challenges our wisdom. Canadian mathematicians have been leading and contributing to the research on the problem. As an excellent mathematician working in this field for over 20 years, the applicant proposed to continue to tackle the finiteness part of the problem by showing that the number of periodic solutions which can be born from a degenerate solution is finite. The proposed research will enable the applicant to maintain the leading and active position in this field. The mathematical knowledge and skills for solving Hilbert's 16th problem can also help to understand the mechanisms and complexity of the ecology of our ecosystems. The applicant will study the birth and death of the fast-slow alternating periodic phenomena in the predator-prey type of systems. Amazingly, such study not only helps the understanding of the biology of predator-prey but also can help to solve Hilbert's 16th problem. Using the similar tools of dynamical systems, the applicant proposes to study the transmission and spread of mosquito-borne diseases, such as West Nile virus. To prepare and respond to emerging mosquito-borne threat, the applicant will build new predictive models to forecast the mosquito-abundance, triggering factors and mechanisms of an outbreak, and reason of recurrence of the outbreak of the mosquito-borne virus. The advanced research based on the monitoring and surveillance program data and real-time forecasting tools development considering climate change will enable us to know early the risk so that people will get prepared and protected from the biting of virus-carrying mosquitoes. The applicant will also consider the factors including daily temperature and rainfall, community and urban variations; wildlife species diversity and distribution. Canadians will for sure benefit directly and indirectly from his research.