Dynamical systems: Analysis, Modelling, and Numerics

这项研究计划将以几种不同的方式研究生物过程的数学模型。 通常，生物过程模型中的参数是无法测量的，或者在很大范围内变化。从记录的实验数据中确定这些参数的正确值可能是一项非常困难的任务。这项研究将继续开发和实施一种数值方法，以快速消除与数据不一致的大部分参数范围。这将允许更准确的模型用于预测目的。 一些生物系统表现出一种行为，即它们迅速从初始状态崩溃到“稳定状态”，然后随着系统元素的演变而缓慢地沿着这种稳定状态移动。如果可以在模型中识别出稳态，并且如果可以采用适当的方法将整个系统投影到稳态，则可以简化这类系统的数学模型。这项研究将调查具有特定约束的模型，寻找这些约束对稳态施加的可识别特征，这将允许更容易地识别这些状态并减少整个系统。 离子通道的数学模型是神经生物学家描述这些蛋白质的有用工具。这项研究将增强软件Neurofit，在该软件中实现额外类型的数学离子通道模型，并使用上述参数范围缩减方案将这些模型与实验数据进行拟合。 许多系统具有某种对称性，例如，相同的神经元网络以相同的方式相互耦合。这项研究将研究这类系统，特别是那些具有振荡行为的系统，以便确定和分类这些系统在这些振荡第一次出现的点附近的所有可能的动力学行为。 这项研究还将建立各种生物过程的数学模型。例如，农民需要知道在作物上施肥料和收获之间应该等待的适当时间长度，以确保粪便中的任何病原体不会转移到食物中。为了回答这个问题，我们将模拟粪便中各种病原体在不同类型和不同天气条件下传播后的死亡率。

This research program will investigate mathematical models of biological processes in several different ways. Often the parameters in models of biological processes are unmeasurable or vary over wide ranges. Identifying correct values for these parameters from recorded experimental data can be a very difficult task. This research will continue the development and implementation of a numerical method for quickly eliminating large portions of the parameter ranges that are inconsistent with the data. This will allow for more accurate models that can be used for predictive purposes. Some biological systems exhibit a behaviour whereby they rapidly collapse from an initial state to a "steady state", and then slowly move along this steady state as elements of the system evolve. Mathematical models of such systems can be simplified if the steady state can be identified in the model and if appropriate methods for projecting the full system onto the steady state can be employed. This research will investigate models with specific constraints looking for identifiable characteristics that these constraints impose on the steady state, which will allow for easier identification of these states and reduction of the full system. Mathematical models of ion channels are a useful tool for neurobiologists to characterize these proteins. This research will enhance the software NEUROFIT by implementing additional types of mathematical ion channel models into this software and using the parameter range reduction scheme mentioned above to fit these models to experimental data. Many systems have some sort of symmetry, for example, networks of identical neurons that are coupled to one another in the same way. This research will investigate such systems, especially those with oscillating behaviour, in order to determine and classify all possible dynamic behaviour of such systems near the point where these oscillations first appear. This research will also build mathematical models for various biological processes. For example, farmers need to know the appropriate length of time one should wait between manure application to crops and harvest, in order to be sure that any pathogens in the manure are not transferred to the food. We will model the die-off rates of various pathogens in manure after it is spread on fields of different types and under different weather conditions in order to answer this question.