Complex Dynamics in Biological Systems: A Bifurcation Theory Approach

生物系统中的许多问题都可以用非线性常微分方程（ODE）来描述，它可以描述复杂的现象，如人口增长、疾病传播、反复感染等。这些动力学过程往往表现出质的变化，称为分叉，如人口爆炸和疾病爆发。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定性变化。分叉理论的方法使我们能够识别分叉发生的参数值，并预测系统在分叉点附近的行为的定性变化。在众多的分支中，Hopf分支和Bogdanov-Takens（B-T）分支是两种主要的分支，它们导致了一类特殊的周期解-极限环。极限环是一种自激振荡，几乎存在于所有的物理系统中。为了分析这些分叉，需要使用中心流形理论和规范形理论，通过提取“关键”信息（或项）来大大简化系统方程，以形成简单的系统，同时保持系统在分叉点附近的定性行为不变。此外，20年前发展起来的最简正规形（SNF）理论和参数最简正规形（PNSF）理论最终被应用于解决真实的世界系统的B-T分岔。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，例如疾病中的复发行为。然而，这样的慢-快运动不能识别或分析著名的几何奇异摄动理论（GSPT），这已被广泛应用于研究奇异摄动动力系统中的慢-快运动。一种新的方法，基于动力系统理论，它包含四个条件，很容易识别这样的慢，快的运动。此外，相比GSPT，我们的新的，简单的方法可以很容易地应用到研究这种慢-快运动在高维动力系统。这项研究计划有三个目标。第一个是发展一般n维动力系统的SNF和PSNF理论和有效的计算方法/算法。二是建立了一个严格的数学理论，用于动力系统方法来识别特殊的振荡慢-快运动。三是将新的理论和方法应用于更多的生物系统，如捕食系统和HIV模型。这些应用不仅对提高科学认识有重要意义，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.