Bifurcation theory and applications in mathematical biology

希尔伯特第16问题是希尔伯特100年前提出的最困难的问题，但至今仍在挑战我们的智慧。加拿大数学家一直在领导和促进对这个问题的研究。作为在该领域工作了20多年的优秀数学家，申请人提出通过证明可以从退化解产生的周期解的数量是有限的来继续解决问题的有限性部分。拟议的研究将使申请人在该领域保持领先和积极的地位。解决希尔伯特第16问题的数学知识和技能也可以帮助理解我们生态系统的生态机制和复杂性。申请人将研究捕食者-被捕食者类型系统中的快-慢交替周期现象的诞生和死亡。令人惊讶的是，这样的研究不仅有助于理解捕食者-被捕食者的生物学，而且有助于解决希尔伯特第16问题。使用动力系统的类似工具，申请人提出研究蚊子传播的疾病（例如西尼罗河病毒）的传播和扩散。为了准备和应对新出现的蚊媒威胁，申请人将建立新的预测模型，以预测蚊子的丰度、爆发的触发因素和机制以及蚊媒病毒爆发复发的原因。基于监测和监视计划数据的先进研究以及考虑气候变化的实时预测工具的开发将使我们能够及早了解风险，以便人们做好准备，免受携带病毒的蚊子的叮咬。申请人亦会考虑其他因素，包括每日气温及雨量、社区及城市的变化、野生生物物种的多样性及分布。加拿大人肯定会直接或间接地从他的研究中受益。

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.