Stochastic modelling of interacting branching systems

Dr Chen, Anyue   (Principal Investigator (PI))

Professor Renshaw Eric   (Co-Investigator)

Professor Pollett Philip Keith   (Co-Investigator)

36

2006-08-01

250000

Stochastic modelling of interacting branching systems

branching system, stochastic modelling

Others - Mathematics,Ecology

HKU 7010/06P

General Research Fund (GRF)

2006

Completed

(1) The main objects of this proposal are as follows: a To validate an appropriate methodology that provides new techniques for investigating interacting branching systems. In particular, (i). the powerful probability approaches, such as the random change technique,will be developed in order to analyze properties of interacting branching systems; (2) (ii). the useful (one-dimensional) Resolvent Decomposition Theorem, refined in Chen and Renshaw (1990, 1993), will be generalized to the multi- dimensional case and facilitate the analysis of jointly interacting branching systems; (iii).the analytic approach, particularly methods in differential equations and special functions, will be further developed in order to analyze the partial differential equations arising from the study, especially for weakly and pair-wise interacting branching systems. (3)b. To investigate the basic properties of interacting branching systems. We expect to find solutions to the following open problems in the course of this project, which will greatly increase our understanding of interacting branching systems. (iv). We intend to obtain the extinction probability, the mean extinction time and the conditional mean extinction time of strongly interacting branching systems. (4)(v). We intend to obtain the explosion probability and the mean explosion time for strongly interacting branching systems. (vi). We intend to investigate and resolve the effect of immigration and emigration on the extinction and explosion properties for weakly, pair-wise, and strongly interacting branching systems. (vii).The limiting-conditional and quasi-stationary distributions for all types of interacting branching systems are unexplored at present. Both will be studied in detail. We expect to make considerable progress in this respect. (5)c. To establish and extend research collaboration, both nationally and internationally, in the study of interacting branching systems. Analyzing interacting branching systems is a challenging task and thus research collaboration is essential in order to achieve the aims of this ambitious project.(6) The results of the study will have a major long-term impact on mathematics and science, and in particular to biologists. It will promote interest, in particular, across the academic community within HK and Mainland China. Given that all the investigators of this proposal are leaders in their research fields and have strong track records, the aims of the study, while certainly ambitious, are readily achievable as long as funding and other strong support can be provided. We are confident that the project can be brought to fruition within the specified time period. This research will improve our understanding of interacting branching systems and spur further research worldwide in this important area. This in turn will enhance the reputation of HK and Mainland China in the branching systems research and cognate areas.