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General

I welcome inquiries from prospective PhD or Research Masters students who have research interests cognate to my own. I am also happy to offer students in other disciplines access to my expertise through joint supervision with an advisor in another discipline.

Research Areas

I am prepared to supervise students in the areas listed below. To indicate the kinds of projects available, I have given details of four projects. Of course if you are interested in a general area, then I will be able to suggest other projects which you can work on in that area. You will see that my research interests range from the very applied (mathematical modelling of ecological systems) to fundamental mathematics (for example, the spectra of operators on Riemannian manifolds).

• Mathematical Biology

  [Ecological modelling, theoretical population biology]

  Project   Estimating Persistence in Populations that are Subject to Catastrophes

  There are many natural systems which eventually die out, yet over any reasonable time scale appear to settle down to a stable equilibrium. For example, it is not unusual for animal populations to be subject to large-scale mortality or emigration. This can occur when disease, such as a new virus, affects the population, or when food shortages occur, such as those induced by over browsing or dramatic changes in climatic conditions. However, although these populations may eventually become extinct, they can survive for long periods in an apparently stable state. Our aim is to model this behaviour. It is anticipated that this work will be potentially very useful in wildlife management, for it may allow the prediction of persistence times and the distribution of population size.

  Project   Models for Spatially Structured Metapopulations

  A metapopulation is a population that occupies several geographically separated 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 metapopulations to persist for long periods We would expect that, unless (exceptionally) there is a population explosion, the metapopulation would become extinct in finite-mean time. We will develop methods that account for the persistence of these populations and which provide an effective means of studying the long-term behaviour before extinction occurs.

  Project   Stochastic Models for Epidemics in Population Networks

  Small world networks achieved popularity in social sciences, modelling the phenomenon of "six degrees of separation". In 1998, Watts and Strogatz introduced a mathematical model for small world networks, and this work has received considerable attention in recent years. Our starting point is version of the model studied by Barbour and Reinert, where the network is made up of many local links and fewer long range "shortcuts". Barbour and Reinert investigated the distribution of the inter-point network distances, their results being framed in terms of approximations whose accuracy increases with the size of the network. We will study models for the spread of epidemics on these networks, and propose to use approximation methods (which become more accurate as the network grows in size) to address several questions.

  Tools   Differential equations, Markov processes, scientific computation.

  Background   These projects will suit students with broad scientific interests and a strong background in mathematical modelling.

  Recent work   Some of my recent publications in this area are listed below:

  Pollett, P.K. (1996) Modelling the long-term behaviour of evanescent ecological systems. Ecological Modelling 86, 135-139.

  Pollett, P.K. (1997) Limiting conditional distributions for stochastic metapopulation models. In (Eds A.D. McDonald and M. McAleer) Proceedings of the International Congress on Modelling and Simulation, Vol. 2, Modelling and Simulation Society of Australia, Hobart, Australia, pp. 807-812.

  Pollett, P.K. (2001) Quasistationarity in populations that are subject to large-scale mortality or emigration. Environment International 27, 231-236.

  Pollett, P.K. (2001) Diffusion approximations for ecological models. In (Ed. Fred Ghassemi) Proceedings of the International Congress on Modelling and Simulation, Vol.2, Modelling and Simulation Society of Australia and New Zealand, Canberra, Australia, pp. 843-848.

  Cairns, B.J. and Pollett, P.K. (2003) Evaluating persistence times in populations that are subject to local catastrophes. In (Ed. David A. Post) Proceedings of the International Congress on Modelling and Simulation, Vol. 2, Modelling and Simulation Society of Australia and New Zealand, Townsville, Australia, pp. 747-752.

  Cairns, B.J. and Pollett, P.K. (2004) Extinction times for a general birth, death and catastrophe process. Journal of Applied Probability 41, 1211-1218.

  Cairns, B.J. and Pollett, P.K. (2005) Approximating measures of persistence in a general class of population processes. Theoretical Population Biology 68, 77-90.

  Pollett, P.K. and Ross, J.V. (2005) Costs and decisions in population management: koalas on Kangaroo Island. In (Eds. Zerger, A. and Argent, R.M.) Proceedings of the 16th Biennial Congress on Modelling and Simulation (MODSIM05), Modelling and Simulation Society of Australia and New Zealand, December 2005, pp. 2082-2088.

  Ross, J.V. and Pollett, P.K. (2006) Extinction times for a birth-death process with two phases. Mathematical Biosciences 202, 310-322.

  Ross, J.V., Taimre, T. and Pollett, P.K. (2006) On parameter estimation in population models. Theoretical Population Biology 70, 498-510.

  Pollett, P.K., Zhang, H. and Cairns, B.J. (2007) A note on extinction times for the general birth, death and catastrophe process. Journal of Applied Probability 44, 565-568.

  Ross, J.V. and Pollett, P.K. (2007) On costs and decisions in population management. Ecological Modelling Modelling 201, 60-66.

  Sani, A., Kroese, D.P. and Pollett, P.K. (2007) Stochastic models for the spread of HIV in a mobile heterosexual population. Mathematical Biosciences 208, 98-124.

  Ross, J.V., Sirl, D.J., Pollett, P.K. and Possingham H.P. (2008) Metapopulation persistence in a dynamic landscape: more habitat or better stewardship? Ecological Applications (to appear).

  
  
  
• Teletraffic Theory

  [Telecommunications modelling, network optimisation]

  Project   Fixed Point Methods for Loss Networks

  We aim to improve procedures for evaluating the performance of telecommunications networks by proposing several new models that account for dependencies between neighbouring communications links, and by developing algorithms for estimating performance measures. A variety of networks will be studied, including local-area networks, broadband packet networks and mobile radio systems. Our parallel algorithms will be implemented on a small-scale multiprocessing system, thus providing the teletraffic engineer with a convenient tool for assessing network performance far more quickly than was previously possible.

  Tools   Mathematical modelling, Markov processes, algorithms, optimisation, graph theory.

  Background   This project will suit students with technological interests and a strong background in mathematical modelling and scientific computation.

  Recent work   Some of my recent publications in this area are listed below:

  Bebbington, M., Pollett, P.K. and Ziedins, I. (1997) Improved fixed point methods for loss networks with linear structure. In (Ed. W.J. Lavery) Proceedings of the 4th International Conference on Telecommunications, Vol. 3, Office of Continuing Education, Monash University, Clayton, Victoria, Australia, pp. 1411-1416.

  Pollett, P.K. (1997) Optimal allocation of effort in packet switching networks with generally distributed transmission times. In (Eds J. Hullett and D. Sampson) Proceedings of the 3rd Asia-Pacific Conference on Communications, Vol. 3, IREE Society of Australia, Sydney, Australia, pp. 1489-1493.

  Bebbington, M., Pollett, P.K. and Ziedins, I. (1998) Two-link approximation schemes for linear loss networks without controls. Journal of the Korean Mathematical Society 35, 539-557.

  Bebbington, M., Pollett, P.K. and Ziedins, I. (1999) Product form approximations for highly linear loss networks with trunk reservation. In (Eds Richard J. Wilson, Shunji Osaki and M.J. Faddy) Proceedings of the 1st Western Pacific and 3rd Australia-Japan Workshop on Stochastic Models in Engineering Technology and Management, Centre for Statistics, University of Queensland, pp. 45-54.

  Pollett, P.K. (1999) Bottlenecks in Markovian queueing networks. In (Ed. E. Kozan) Proceedings of the 15th National Conference of the Australian Society for Operations Research, Vol. 2, Queensland University of Technology, pp. 1047-1056.

  Modelling congestion in closed queueing networks. International Transactions in Operational Research 7, 319-330.

  Pollett, P.K. (2000) Resource allocation in general queueing networks. In (Ed. A. Ohuchi) Putting OR/MS Theory into Real Life: Proceedings of the 2nd Joint International Workshop of the Operations Research Society of Japan the Australian Society for Operations Research, Operations Research Society of Japan, Sapporo, Japan, pp. 221-228.

  Thompson, M.R. and Pollett, P.K. (2001) A reduced load approximation accounting for link interactions in a loss network. In Operations Research in the Third Millennium, Proceedings of the 16th National Conference of the Australian Society for Operations Research, The University of Adelaide (Available on CD).

  Bebbington, M., Pollett, P.K. and Ziedins, I. (2002) Two-link approximation schemes for loss networks with linear structure and trunk reservation. Telecommunication Systems 19, 187-207.

  Bebbington, M., Pollett, P.K. and Ziedins, I. (2003) Product form approximations for highly linear loss networks with trunk reservation. Mathematical and Computer Modelling 38, 1147-1156.

  Thompson, M.R. and Pollett, P.K. (2003) A reduced load approximation accounting for link interactions in a loss network. Journal of Applied Mathematics and Decision Sciences 7, 229-248.

  Pollett, P.K. (2008) Optimal capacity assignment in general queueing networks. In (Ed. C.E.M. Pearce) Optimization: Structure and Applications, Springer Optimization and its Applications Series (to appear).

  
  
  
• Markov Processes and Operator Theory

  [Markov chain theory, Q-processes, construction theory, diffusions]

  Project   Convergence Rates for Markov Processes

  This project concerns convergence rates for strongly ergodic Markov processes. This work will provide the most general formulation of convergence rates for Markov Chains, for diffusions on the real line, and for diffusions on Riemannian manifolds. It will open up the possibility of studying the principal eigenvalue of Laplace-Beltrami operator on Riemannian manifolds.

  Tools   Markov processes, ergodic theory, functional analysis

  Background   This project will suit students with a strong background in Analysis, and with interests in Markov processes

  Recent work   Some of my recent publications in this area are listed below:

  Pollett, P.K. (1991) Invariant measures for Q-processes when Q is not regular. Advances in Applied Probability 23, 277-292.

  Pollett, P.K. (1991) On the construction problem for single-exit Markov chains. Bulletin of the Australian Mathematical Society 43, 439-450.

  Nair, M.G. and Pollett, P.K. (1993) On the relationship between m-invariant measures and quasistationary distributions for continuous-time Markov chains. Advances in Applied Probability 25, 82-102.

  Pollett, P.K. (1994) On the identification of continuous-time Markov chains with a given invariant measure. Journal of Applied Probability 31, 897-910.

  Pollett, P.K. (2001) Similar Markov chains. Journal of Applied Probability 38A, 53-65.

  Pollett, P.K. (2002) Identifying Q-processes with a given finite m-invariant measure. In (Eds Zhenting Hou, Jerzy A. Filar and Anyue Chen) Markov Processes and Controlled Markov Chains, Kluwer, pp. 41-55.

  Pollett, P.K. and Stefanov, V.T. (2002) Path integrals for continuous-time Markov chains. Journal of Applied Probability 39, 901-904.

  Chen, A., Pollett, P.K., Zhang, H. and Li, J. (2004) The collision branching process. Journal of Applied Probability 41, 1033-1048.

  Pollett, P.K. and Zhang, H. (2004) Existence and uniqueness of Q-processes with a given finite m-invariant measure. In (Eds. Philip K. Pollett and Peter G. Taylor) Festschrift in Honour of Daryl Daley, Australian & New Zealand Journal of Statistics 46, 113-120.

  Chen, A., Pollett, P.K., Zhang, H. and Cairns, B. (2005) Uniqueness criteria for continuous-time Markov chains with general transition structures. Advances in Applied Probability 37, 1056-1074.

  Gray, B., Pollett, P.K. and Zhang, H. (2005) On the existence of uni-instantaneous Q-processes with a given finite m-invariant measure. Journal of Applied Probability 42, 713-725.

  Chen, A., Pollett, P.K., Li, J. and Zhang, H. (2007) A remark on the uniqueness of weighted Markov branching processes. Journal of Applied Probability 44, 279-283.

  Sirl, D., Zhang, H. and Pollett, P.K. (2007) Computable bounds for the decay parameter of a birth-death process. Journal of Applied Probability 44, 476-491.

Publications

A complete list of my published work as well as general information about my research can be found here.

Research Funding

My work is currently supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS), of which I am Chief Investigator and Director (Qld).

Support

Whilst I will not usually provide fully-funded postgraduate scholarships, I do offer support (through MASCOS Qld) in the form of a topup of your scholarship to $27,000pa, and/or financial support for (i) conference travel or short study trips in Australia or abroad ($3,000pa) and (ii) computer equipment.

(See below for details on how to apply for a postgraduate scholarship.)

How to contact me

Please e-mail me (pkp@maths.uq.edu.au) to make an appointment. Or, if you want to drop in, the best time to catch me is Thursday mornings. My office is Room 652 of the Priestley Building. Further contact details and a list of my office hours can be found here.

International students

Special information for international students is located here.

Postgraduate scholarships

Make sure to apply for a postgraduate scholarship and a MASCOS PhD topup Scholarship!

The closing date for Australian Postgraduate Awards (APA) and UQ Research Scholarships (UQRS) is effectively 13 October 2008, but you can apply for admission and a scholarship at any time. Please refer to Timing of Scholarship Applications.

Applications for International Postgraduate Research Scholarships (IPRS) and University of Queensland International International Research Tuition Awards (UQIRTA) close at the School on 15 September 2008.

My postgraduate students: past and present