Modelling with and analysis of dynamical systems

微分方程被用来模拟来自许多不同领域的许多不同现象。我研究的两个例子是生态学中的食物网和神经生物学中的离子通道。数学模型使我们能够在不需要直接观察真实系统的情况下研究复杂系统的行为，并允许我们改变系统并提出这样的问题：“如果这一方面发生变化会发生什么？”这可能是在真实系统本身上不可能做到的事情。这项研究建议研究基于常微分方程组的模型，特别是查看这些模型与实际系统中可用数据之间的关系。为了使数学模型有用，它们必须尽可能地与真实系统进行校准。通常，这意味着根据真实世界中的可观测数据来确定某些模型参数的适当值。但是，这种拟合过程可能非常困难，而且计算量很大。此外，始终存在这样的问题：模型是否过于简单，无法捕捉数据中的重要特征，或者模型是否比现有数据所保证的更加复杂。这项研究的部分目的是开发计算算法，通过从时间序列数据中提取信息来限制这些参数值的范围。如果范围仍然过大，这表明更简单的模型可能同样适合数据。这项研究将至少应用于两个特定领域。将研究离子通道激活的各种模型，比较它们的定性和定量特征，并开发方法，供神经科学家根据现有数据选择适当的模型。这项研究还旨在确定神经科学家可以用来收集数据的不同实验方案，这些方案在不同模型之间更具区分性。食物网动力学模型有助于回答这样的生态学问题，如通常横跨较大地理区域的顶级捕食者对其猎物种群是稳定还是不稳定的影响，以及食物网的不对称是否增强了持久性。

Differential equations are used to model many different phenomena from many different areas. Two examples that I study are food webs in ecology and ion channels in neurobiology. Mathematical models allow us to investigate the behaviour of complex systems without necessarily observing the real system directly, and allow us to alter the system and ask the question, "What would happen if this aspect changed?", something that may be impossible to do on the real system itself. This research proposal is to study models based on systems of ordinary differential equations and particularly to look at the relationship between these models and the data available from the real system. In order for mathematical models to be useful they must be calibrated to the real systems as closely as possible. Typically this means determining the appropriate values of some model parameters based on observable data from the real world. However, this fitting procedure can be extremely difficult and computationally intensive. In addition there is always the question of whether the model is too simplistic to capture the important features in the data or whether it is more complicated than is warranted by the data available. Part of this research aims to develop computational algorithms which can confine the ranges of these parameter values by extracting information from time series data. If the ranges remain excessively large, this is an indication that a simpler model might fit the data equally well. This research will be applied to at least two specific areas. Various models of ion channel activation will be investigated, comparing their qualitative and quantitative features and developing methods for neuroscientists to use in choosing appropriate models based on the data they have available. The research also aims to identify different experimental protocols that neuroscientists could use to gather data which are more discriminatory between different models. Models of food web dynamics help answer such ecological questions as whether top predators which typically span larger geographic areas have a stabilizing or destabilizing influence on their prey populations, and whether asymmetry in food webs enhances persistence.