Stochastic dynamics of evolving populations

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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.