**By its very nature, it is not easy to define
subdivisions of the field of complex systems. However, for the purpose of
describing my research, I would like to distinguish between two different ways
in which stochastic effects (also called noise) may influence the system and be
analysed. The first is where the noise may be thought of as an extra aspect of
the system to be modelled, and can be investigated using the techniques of
the theory of stochastic processes. Of particular interest are novel mechanisms
not seen in deterministic systems. The other way in which stochastic effects
may be important is when the effects of noise are "creative", in the sense
of evolutionary dynamics. Examples of the creative effects of noise usually
stem from the production of "errors" (e.g. mutations) which may give rise to
new variants and eventually lead to complex structures. This naturally finds
most applications in the fields of biology, economics and social science, but
is being used more and more in other areas, for example in computer science
(genetic algorithms).**

**In the first case, the noise may be internal or
external. An example of internal (or intrinsic) noise is demographic
stochasticity in biological systems (noise due to the discreteness of
individuals who comprise the system) and of external noise is environmental
stochasticity, which has its origin in external factors not included in the
model description. The nature of the model may also vary depending on the type
of noise. If the model is individual-based, then the starting point might be
a Markov chain or master equation (continuous time Markov chain) whereas if
the noise is external, the starting point might be a phenomenological equation
of the Langevin type. These factors will also govern the techniques that are
used to analyse the model. If it is formulated as a master equation, a
powerful technique that I frequently use is van Kampen's system size
expansion. At leading order in the system size, it gives the deterministic
equation(s) for the process and at next-to-leading order stochastic
corrections to this. If the model is formulated as a Langevin equation, the
path-integral is a useful calculational tool. The discussion on amplified stochastic cycles is mainly focused on individual
based models described as master equations and analysed using the system-size
expansion. The discussion on stochastic dynamics
also covers systems of this type, but in addition those described as Langevin
equations and analysed using path-integral methods are discussed.**

**The second case mentioned above relates to
evolutionary dynamics. Although, in the standard evolutionary paradigm,
mutations may give rise to individuals who are fitter, and so are selected,
as explained in the section on neutral theories
this need not be the case. Examples of neutral theories from the fields of
population genetics, ecology and language evolution are described in that
section. I have also been interested in applying a more standard evolutionary
dynamics to construct networks, in particular
food webs, where webs containing a large number of species are "grown"
starting with only a very small number of species.**

**While I believe those working in the area of
statistical mechanics have many skills to bring to modelling such systems,
I also believe that close collaboration with experts in the appropriate
fields is very necessary. So for instance, I have been working with
Bill Croft from the Department of
Linguistics, University of New Mexico in constructing a mathematical model of the evolution of language and studying it both
analytically and using computer simulations. In addition, I have been involved
in other ways in trying to forge links between researchers in those disciplines
who have interests in the theory of complex systems. For example, I have in
the past organised a course entitled "Complex Adaptive Systems" which was
jointly given by members of the Departments/Schools of Biological Sciences,
Computer Science, Economic Studies and Physics, and have run a "Complex Systems
Discussion Group" for 10 years. I am also the principal investigator of a
linked grant which is part of the NANIA collaboration involving individuals from universities in Edinburgh and
Manchester. This is funded under the EPSRC "Novel computation: coping with
complexity" initiative.**

**Interdisciplinary research of this kind is now one
of the most active areas of theoretical science. I believe that the trend
whereby knowledge generated by those working in statistical mechanics is
applied to problems in other disciplines is one which is set to grow and will
be an important feature of scientific activity in the 21st century.**