Complex Dynamics in Biological Systems: A Bifurcation Theory Approach

生物系统中的复杂动力学：分岔理论方法

• 批准号: RGPIN-2020-06414
• 负责人: Yu, Pei
• 金额: $ 1.97万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741195
• 关键词: Complex; Dynamics; Biological; Systems; Bifurcation

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.

在生物系统中引起的许多问题可以通过非线性普通微分方程（ODE）来建模，这些方程（ODE）可以描述复杂的现象，例如人口增长，疾病的传播，复发感染等。这种可扩展的定性变化的动态过程称为人口爆炸和疾病爆发等分歧。分叉理论是研究分叉现象的强大工具，因为它解释了系统参数的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确定发生分叉的参数值，并预测分叉点附近系统行为的定性变化。在各种分叉中，Hopf分叉和Bogdanov-Takens（B-T）分叉是两个主要分叉，导致了一种特殊类型的定期解决方案 - 极限循环，这是一种自我维持的振荡，几乎在所有物理系统中出现。为了分析这些分歧，需要使用中心流形理论和正常形式理论来极大地简化系统方程，通过提取“关键”信息（或术语）以形成一个简单的系统，同时保持系统的定性行为不变，进一步的正常（SNF）理论（pnf）最简单（pns fordies decond），最终是最简单的（pns fordies decond），以预期（pns fordic）（pns for），以前现实世界中的另一个有趣的现象。其中包含四个条件，可以轻松识别这种缓慢的运动。此外，与GSPT相比，我们的新颖，简单的方法可以轻松地应用于在较高维度的动力学系统中研究这种缓慢快速运动。该建议的研究计划有三个目标。第一个是为ODES所描述的一般N维动力系统开发SNF和PSNF理论以及有效的计算方法/算法。第一个是为ODES所描述的一般N维动力系统开发SNF和PSNF理论以及有效的计算方法/算法。第二个是为动态系统方法开发严格的数学理论，以识别特殊的振荡慢速运动。第三个是应用新的理论和方法来研究更多的生物系统，例如捕食者 - 捕集系统和艾滋病毒模型。这些应用不仅对于提高科学理解的意义很重要，而且进步可能会带来实际的临床益处。