Macroscopic consequences of demographic noise in non-equilibrium dynamical systems
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Date
28/11/2013Author
Russell, Dominic Iain
Metadata
Abstract
For systems that are in equilibrium, fluctuations can be understood through
interactions with external heat reservoirs. For this reason these fluctuations are
known as thermal noise, and they usually become vanishingly small in the thermodynamic
limit. However, many systems comprising interacting constituents
studied by physicists in recent years are both far from equilibrium, and sufficiently
small so that they must be considered finite. The finite number of constituents
gives rise to an inherent demographic noise in the system, a source of fluctuations
that is always present in the stochastic dynamics.
This thesis investigates the role of stochastic fluctuations in the macroscopically
observable dynamical behaviour of non-equilibrium, finite systems. To
facilitate such a study, we construct microscopic models using an individual based
modelling approach, allowing the explicit form of the demographic noise to be
identified.
In many physical systems and theoretical models, absorbing states are a
defining feature. Once a system enters one, it cannot leave. We study the dynamics
of a system with two symmetric absorbing states, finding that the amplitude of
the multiplicative noise can induce a transition between two universal modes of
domain coarsening as the system evolves to one of the absorbing states.
In biological and ecological systems, cycles are a ubiquitously observed
phenomenon, but are di cult to predict analytically from stochastic models. We
examine a potential mechanism for cycling behaviour due to the flow of
probability currents, induced by the athermal nature of the demographic noise,
in a single patch population comprising two competing species. We find that such
a current by itself cannot generate macroscopic cycles, but when combined with
deterministic dynamics which constrain the system to a closed circular manifold,
gives rise to global quasicycles in the population densities.
Finally, we examine a spatially extended system comprising many such patch
populations, exploring the emergence of synchronisation between the different
cycles. By a stability analysis of the global synchronised state, we probe the
relationship between the synchronicity of the metapopulation and the magnitude
of the coupling between patches due to species migration.
In all cases, we conclude that the nature of the demographic noise can play a
pivotal role in the macroscopically observed dynamical behaviour of the system.