Stochastic dynamics of evolving populations

这项研究计划的总体目标是发展思想和方法来制定和分析复杂和不断发展的系统的随机模型。当今科学的一个重要目标是为管理大规模系统提供理解和工具。例如计算机通信网络，如万维网，大规模并行计算系统，新兴的全球经济，国际金融公平和债券市场以及地球生态系统。其他例子出现在群体遗传学、免疫系统和神经科学中。这种大规模和高度互动的系统提供了全新的科学挑战。为了了解这些系统的行为，有必要开发新的数学模型、复杂的计算机模拟方法和新的工具来研究大规模复杂的经验数据。所有这些系统的两个潜在特征是高水平的交互性和波动性，这可能导致不可预测性和不稳定性。简单系统的传统数学建模问题导致了概率模型和随机过程。近年来，随机过程的数学理论得到了迅速的发展，现在提供了研究新类型模型的可能性，这些模型有可能在开发解决大规模系统问题的方法方面发挥有用的作用。这些模型结合了人口的家庭结构和大规模系统中出现的相互作用的等级性质。复杂大型系统的一个重要特征是，在许多情况下，分散的“自下而上”管理和概率策略比集中式管理系统和确定性策略表现得更好。进化生物学的思想在理解这些系统中起着关键作用。尽管这是科学史上最成功的理论之一，但要更全面地理解这个庞大的主题，仍然存在许多挑战。例如，这包括更全面地了解物种的发展和共同进化的挑战。

The  overall objective of this research program is to develop ideas and methods to formulate and  analyse stochastic models of complex and evolving systems. An important goal of science today is to provide understanding and tools to manage large scale systems. Examples are computer communications networks such as the World Wide Web, massively parallel computational systems, the emerging global economy, international financial equity and bond markets and the earth's ecosystems. Other examples arise in population genetics, the immune system, and neuroscience. Such large scale and highly interactive systems provide completely new scientific challenges. In order to gain an understanding of the behaviour of these systems it is necessary to develop new classes of mathematical models, sophisticated computer simulation methodology and new tools to study large scale complex empirical data. Two of the underlying features of all these systems are a high level of interactivity and fluctuation which can lead to unpredictability and instability. Traditional mathematical modeling problems for simpler classes of systems have led to probabilistic models and stochastic processes. In recent years the mathematical theory of stochastic processes has developed rapidly and now presents the possibility for the study of new classes of models which have the potential to serve a useful role in developing an approach to the problems of large scale systems. These models incorporate the family structure of populations and the hierarchical nature of the interactions which arise in large scale systems. An important feature of complex large systems is that in many cases decentralized ``bottom-up'' management and probabilistic strategies perform better than centralized management systems and deterministic strategies. Ideas  of evolutionary biology play a key role in understanding many of these systems. Although this is one of the most successful theories in the history of science,  there remain many challenges to a fuller understanding of this vast subject. For example, this includes a challenge to more fully understand the development and coevolution of species.