几类生物数学模型的全局结构及分支问题

This project is concerned with the global analysis and bifurcation problems for some mathematical models in biology, including toxin determined functional response models, a predator-prey model with Hassell-Varley type function response, models of antibiotic resistance in hospitals and a parasite-host model with nonlinear incidence. For higher dimensional models, we will investigate the Hopf bifurcation, Bogdanov-Tankens bifurcation and period doubling bifurcation, etc; for 2-dimensional models, the existence and uniqueness of limit cycle will be discussed. On the other hand, we will analyze the global dynamics for some models and formulate new mathematical models which can be better to characterize the phenomenon in biology... In recent decades, theory of ordinary differential equations has been widely applied in ecology and epidemiology. It is important to study mathematical models mentioned above for both biology and theory of ordinary differential equations. Therefore, this project has important academic values and application prospect.

本项目讨论几类生物数学模型的全局结构及分支问题，涉及到的模型有：食草动物与有毒植物相互作用模型、捕食者-被捕食者模型、抗药性模型，以及寄生物-宿主模型。对高维系统，我们将研究这些模型的Hopf分支、Bogdanov-Tankens分支、倍周期分支等；对二维模型，我们将研究极限环的存在性及唯一性；并对某些系统进行全局结构分析；建立一些新的更符合实际意义的生物数学模型并分析其分支现象和全局结构。.. 最近几十年来，常微分方程理论在生态学及流行病学等方面得到广泛应用。对上述生物数学模型的研究，一方面会对生物学产生影响，另一方面又会促进常微分方程理论的发展，有重要的学术价值及应用前景。

本项目研究了几类生物数学模型的全局结构及分支问题，主要结果包括：研究了几类捕食者-被捕食者模型，特别是系统的Hopf分支及小振幅极限环个数问题；研究了三维、四维抗药性模型的分支现象及全局动力学行为；改进了一类三维抗药性模型，发现双稳现象，并就如何减少抗药性给出建议；研究了一类传染病模型的鞍结点分支、Bogdanov-Takens分支及Hopf分支。除此之外，还研究了一类二次不连续系统及一类复数形式的三单项式系统的极限环问题。