Centre of Excellence for Mathematics
and Statistics of Complex Systems

University of Queensland site
The Vortex

PhD Scholarships
Vacation Scholarships
Stochastic Coffee
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):

  1. Patch Occupancy Models for Ecological Systems
  2. Parameter Estimation in Population Models
  3. Construction and Classification of Markov Chains
  4. Estimating Population Persistence via Importance Sampling
  5. Invariant Measures for Pure-Jump Markov Processes
  6. 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

Some project details.

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

The Centre of Excellence for Mathematics and Statistics
of Complex Systems is funded by the Australian Research
Council, with additional support from the Queensland
State Government and the University of Queensland