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组(ODE)来描述，它可以描述种群增长、疾病传播、反复感染等复杂现象，这些动力学过程往往表现出质的变化，称为分叉，如种群爆炸和疾病暴发。分叉理论是研究分叉现象的有力工具，因为它解释了系统参数的微小变化如何导致这些质变。分叉理论的方法使我们能够识别发生分叉的参数值，并预测分岔点附近系统行为的质变。在各种分叉中，Hopf分叉和Bogdanov-Takens(B-T)分叉是两种主要的分叉，它们导致了一种特殊类型的周期解--极限环，它是一种自持续的振荡，几乎出现在所有的物理系统中。为了分析这些分叉，需要使用中心流形理论和规范型理论来大大简化系统方程，提取“关键”信息(或项)以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，二十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实际系统的B-T分叉。在物理和生物系统中经常观察到的另一个有趣的现象是“慢-快”运动，例如疾病中反复出现的行为。然而，被广泛应用于研究奇异摄动动力系统中的慢-快运动的几何奇异摄动理论(GSPT)不能识别或分析这种慢-快运动。基于动力系统理论，发展了一种新的识别这种慢-快运动的方法，该方法包含四个条件。此外，与GSPT相比，我们新的、简单的方法可以很容易地应用于研究高维动力系统中的这种慢-快运动。这项拟议的研究计划有三个目标。第一个是发展常微分方程组所描述的一般n维动力系统的SNF和PSNF理论和有效的计算方法/算法。二是发展了一套严格的数学理论，用动力系统的方法来识别特殊的慢-快振荡运动。三是应用新的理论和方法研究更多的生物系统，如捕食者-食饵系统和HIV模型。这些应用不仅对增加科学认识意义重大，而且进步可能会产生实际的临床益处。

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.