Interacting particle systems in biology, chemistry and the social sciences are
traditionally described by ordinary or partial differential equations. These
are purely deterministic and based on the assumption that effects of noise can
safely be neglected. Such approaches are no doubt chosen for their mathematical
simplicity, the theory of differential equations is well developed, whereas
existing theories of non-equilibrium stochastic dynamics are more complex and
largely incomplete. In this talk I will consider systems of a finite number of
interacting particles, their intrinsic stochasticity can then no longer be
ignored. I will discuss noise-induced phenomena such as quasi-cyles,
quasi-Turing patterns and travelling quasi waves, and explain how they can be
characterised analytically within the so-called linear noise approximation.
Examples where this is relevant include systems in evolutionary dynamics, game
theory and chemical reaction systems. One focus of the talk will be on models
with delay dynamics, important for example in gene regulatory systems or in
epidemiology. I will show how path-integral approaches can be used to derive
general results for their chemical Langevin equation and linear-noise
approximation.