Vacation Scholarships tenable at the
University of Queensland
MASCOS Qld offers supervision to
UQ Summer Research Scholars (Vacation scholars)
with interests in Complex Systems.
Supervision. Vacation
scholars will work under the supervision
of Prof. Phil Pollett and one of the full-time Centre research staff
or PhD students.
The following specific projects are currently available
(details below):
- Patch Occupancy Models for Ecological Systems
- Parameter Estimation in Population Models
- Construction and Classification of Markov Chains
- Estimating Population Persistence via Importance Sampling
- Invariant Measures for Pure-Jump Markov Processes
- Quasi-stationary Distributions in Markovian Models
Or, candidates may wish to pursue a suitable project of their own choice
cognate to any of the Centre's research themes:
critical phenomena, risk modelling, dynamical systems and complex networks.
Projects 1, 2, 4 and 6 have a computational component, and familiarity with
MATLAB will be assumed.
Additional support.
Students that undertake to enrol in an honours degree with a
supervisor from MASCOS Qld will be provided with a portable computer
for their own use for the period of the scholarship and the subsequent
honours year (provided on a loan basis). Support for conference travel
(during the honours year) will also be available.
Application process.
Simply apply for a UQ Summer Research Scholarship
nominating Prof. Pollett as your
supervisor.
Further information.
For further information, please contact:
Prof. Phil Pollett
ARC Centre of Excellence for
Mathematics and Statistics of
Complex Systems
Discipline of Mathematics
The University of Queensland
Queensland 4072
AUSTRALIA
Email P.Pollett@complex.org.au
or pkp@maths.uq.edu.au
Some project
details.
-
Patch Occupancy Models for Ecological Systems
We will examine models for populations that occupy several geographically
separated regions of habitat (patches). Although the individual patches
may become empty through "local" extinction, they may be recolonized
through migration from other patches. There is considerable empirical
evidence which suggests that a balance between migration and extinction
is reached that enables population networks of this kind to persist for
long periods. We will develop methods which account for the persistence
of these populations and which provide an effective means of studying
their long-term behaviour before extinction occurs. Our models will be
adjusted to account for environmental effects on patch suitability.
-
Parameter Estimation in Population Models
Markovian models have been proposed as models for an array
of biological systems, but their application has been limited, partly
due to a lack of clear statistical procedures for model fitting. We
will look at methods that address these statistical limitations. We
first study a general likelihood based approach where the process is
observed at successive, but not necessarily equally spaced, time points
(for example, sets of abundancy data collected at various times). We
then look at an approach which is simpler in terms of computational
implementation, and which is suitable for parameter estimation in
density-dependent population models, where the rates of
transition are a function of the population density.
- Construction and Classification of Markov Chains
This project is concerned with the abstract theory of continuous-time Markov
chains. In attempting to extend analytical results on invariant measures
to Q-processes (that is, processes with transition rates
Q) other than the minimal process, we realize that m-invariant measures for a Q-process
are m-subinvariant for Q, but
may not be strictly m-invariant, and,
moreover, that a m-subinvariant measure for
Q may not be m-invariant for any
Q-process. We start with a subinvariant measure m
for a stable, conservative, single exit q-matrix Q. The aim
is to provide necessary and sufficient conditions for the existence,
and then the uniqueness, of a Q-process for which m
is invariant. The important special case concerning the existence of
a unique, honest Q-process for which m is invariant,
is an important special case. This may prove to be a significant advance
in the theory of Markov chains, for it is hoped that the solution to
this problem will shed some light on the so-called "Modern Construction
Problem", where Q is assumed to be an arbitrary q-matrix.
- Estimating Population Persistence via Importance Sampling
We develop and implement simulation methods for estimating (i)
extinction probabilities and (ii) the expected time to extinction for a
range of population models, being two key measures of population
viability. We will exploit recent advances in simulation technology,
which, in the present context, identify a related model that is easier
to simulate, but which provide more efficient estimators.
- Invariant Measures for Pure-Jump Markov Processes
The aim is to develop a theory of invariant measures for pure-jump
Markov processes by exploiting the construction of certain dual
processes. The first step is to identify an appropriate (and suitably
general) topology for the state space, needed to effect these
constructions. It is expected that the analysis of pure-jump Markov
processes will be more delicate than for Markov chains on a
countable state space. Several applications will be considered,
including the equilibrium analysis of simple stress-release models for
seismicity.
- Quasi-stationary Distributions in Markovian Models
There are many stochastic systems, arising in areas as diverse as wildlife
management, chemical kinetics and reliability theory, which eventually
"die out", yet appear to be stationary over any reasonable time scale.
The notion of a quasi-stationary distribution has proved to
be a potent tool in modelling this behaviour. Our aim is to establish
workable analytical conditions for the existence of quasi-stationary
distributions for Markovian models in terms of their transition rates,
as well as develop and implement efficient computational procedures for
evaluating them.
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