Bifurcation theory and applications in mathematical biology

希尔伯特的第16个问题是希尔伯特100年前提出的最难的问题，但仍然挑战着我们的智慧。加拿大数学家一直在领导和贡献对这个问题的研究。作为一名在这一领域工作了20多年的优秀数学家，申请人提出通过证明简并解可以产生的周期解的数量是有限的，来继续解决问题的有限部分。所提出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个问题的数学知识和技能也有助于理解我们生态系统的机制和生态的复杂性。申请人将研究捕食者-猎物型系统中快-慢交替周期现象的生与死。令人惊讶的是，这样的研究不仅有助于理解捕食者-猎物的生物学，而且有助于解决希尔伯特的第16个问题。申请人建议使用类似的动力系统工具，研究蚊媒疾病（如西尼罗病毒）的传播和传播。为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，预测蚊媒病毒暴发的蚊量、爆发的触发因素和机制，以及再次爆发的原因。基于监测和监测项目数据和考虑气候变化的实时预测工具开发的先进研究将使我们能够及早了解风险，以便人们做好准备并免受携带病毒的蚊子的叮咬。申请人亦须考虑的因素包括每日气温及雨量、社区及市区的变化；野生动物物种的多样性和分布。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.