Complex Dynamics in Biological Systems: A Bifurcation Theory Approach

生物系统中出现的许多问题都可以用非线性常微分方程（ODE）来建模，它可以描述诸如人口增长、疾病传播、复发性感染等复杂现象。这种动态过程经常表现出质变，称为分岔，如人口爆炸和疾病爆发。分岔理论是研究分岔现象的有力工具，因为它解释了系统参数的微小变化如何导致这些质变。分岔理论的方法使我们能够识别分岔发生的参数值，并预测分岔点附近系统行为的质变。在各种分岔中，Hopf分岔和Bogdanov-Takens （B-T）分岔是两种主要的分岔，它们导致了一种特殊类型的周期解——极限环，它是一种自持续振荡，几乎出现在所有物理系统中。为了分析这些分岔，需要使用中心流形理论和范式理论，通过提取“关键”信息（或项）来大大简化系统方程，形成一个简单的系统，同时保持系统在分岔点附近的定性行为不变。此外，20多年前发展起来的最简单范式（SNF）理论和参数最简单范式（PNSF）理论最终被应用于解决现实世界系统的B-T分岔问题。在物理和生物系统中经常观察到的另一个有趣现象是“慢-快”运动，如疾病中的反复行为。然而，这种慢速运动无法用众所周知的几何奇异摄动理论（GSPT）来识别和分析，而GSPT已被广泛应用于研究奇异摄动系统中的慢速运动。提出了一种基于动力系统理论的新方法，该方法包含四个条件，可以方便地识别这类慢速运动。此外，与GSPT相比，我们的新颖，简单的方法可以很容易地应用于研究高维动力系统中的慢速运动。这个拟议的研究计划有三个目标。第一个目标是发展一般n维动态系统的单SNF和PSNF理论和有效的计算方法/算法。二是建立了用动力系统方法识别特殊的慢速振荡运动的严格数学理论。三是将新的理论和方法应用于研究更多的生物系统，如捕食者-猎物系统和HIV模型。这些应用不仅对提高科学认识意义重大，而且进展可能产生实际的临床效益。

Many problems arising in biological systems can be modelled by nonlinear ordinary differential equations (ODE), which can describe complex phenomena such as population growth, spread of diseases, recurrent infection, etc. Such dynamical processes often exhibit qualitative changes, called bifurcations like population explosions and disease outbreaks. Bifurcation theory is a powerful tool in studying bifurcation phenomena as it explains how small changes in the system parameters can lead to these qualitative changes. The methods of bifurcation theory enable us to identify the parameter values where bifurcations occur, and to predict the qualitative changes in behaviour of the system near the bifurcation points. Among various bifurcations, Hopf bifurcation and Bogdanov-Takens (B-T) bifurcation are two main bifurcations, leading to a special type of periodic solutions - limit cycles, which is a self-sustained oscillation and appears in almost all physical systems. In order to analyze these bifurcations, centre manifold theory and normal form theory need be used to greatly simplify system equations by extracting the "key'' information (or terms) to form a simple system, while keeping the qualitative behaviour of the system unchanged near bifurcation points. Further, the simplest normal form (SNF) theory and parametric simplest normal (PNSF) theory, developed two decades ago were eventually applied to solve the B-T bifurcation of real world systems. Another interesting phenomenon often observed in physical and biological systems is the "slow-fast" motion such as recurrent behaviour in diseases. However, such slow-fast motions cannot be identified or analyzed by the well-known geometric singular perturbation theory (GSPT), which has been widely applied to study slow-fast motions in singular perturbed dynamical systems. A new method based on dynamical systems theory has been developed, which contains four conditions, to easily identify such slow-fast motions. Moreover, compared to the GSPT, our novel, simple approach can be easily applied to study such slow-fast motions in higher dimensional dynamical systems. This proposed research program has three objectives. The first one is to develop the SNF and PSNF theories and efficient computation methods/algorithms for general n-dimensional dynamical systems described by ODEs. The second one is to develop a rigorous mathematical theory for the dynamical system approach to identify the special oscillating slow-fast motions. The third one is to apply the new theory and methods to investigate more biological systems such as predator-prey systems and HIV models. These applications are not only significant for increasing scientific understanding, but progress may yield practical, clinical benefits.