Modelling with and analysis of dynamical systems

动力系统建模和分析

• 批准号: 298446-2010
• 负责人: Willms, Allan
• 金额: $ 0.87万
• 依托单位: University of Guelph
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=526035
• 关键词: Modelling; analysis; dynamical; systems

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.

微分方程用于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研究复杂系统的行为，而不必直接观察真实的系统，并使我们能够改变系统并提出这样的问题：“如果这方面发生变化，会发生什么？”“，这在真实的系统本身上可能是不可能做到的。 本研究计划是研究基于常微分方程系统的模型，特别是研究这些模型与从真实的系统获得的数据之间的关系。 为了使数学模型有用，它们必须尽可能接近真实的系统进行校准。 通常，这意味着基于来自真实的世界的可观察数据来确定某些模型参数的适当值。 然而，该拟合过程可能是极其困难的并且计算密集的。 此外，始终存在这样一个问题，即模型是否过于简单，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为复杂。 本研究的一部分旨在开发计算算法，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。 这项研究将应用于至少两个具体领域。 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合适的模型。 该研究还旨在确定神经科学家可以用来收集不同模型之间更具歧视性的数据的不同实验方案。 食物网动力学模型有助于回答这样的生态问题，是否顶级捕食者，通常跨越较大的地理区域有一个稳定或不稳定的影响，他们的猎物种群，以及是否不对称的食物网增强持久性。