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Evolutionary game theory and population dynamics

School of Mathematics

Applications accepted all year round Competition Funded PhD Project (European/UK Students Only)

About the Project

Evolutionary Game Theory (EGT) [MSP73] has proven to be a powerful and versatile mathematical framework for the modelling and analysis of complex biological (and economical) systems whenever there is frequency dependent selection. At the core of evolutionary game dynamics lies the replicator-mutator equation, which is a set of differential equations that describe the evolution of the frequencies of the different types in the population. In many practical applications, uncertainty is inevitable due to the incompleteness of information, for instance when the environment fluctuates so rapidly and frequently that one cannot describe the payoffs of their inhabitants’ interactions. It is thus vital to study random evolutionary games, where payoff entries are random variables.

The aim of this project is to study equilibrium and non-equilibrium properties of the replicator-mutator equation taking into account additional effects such as mutation and environment feedbacks. The key idea would be to relate the study of equilibria in EGT to the study of roots of random polynomials [TV14] (for static problems) and to develop variational techniques capturing underlying physical structures exhibited in the models [CMRS19] (for dynamical problems).

The project will be based at the School of Mathematics at University of Birmingham and will be supervised by Dr. Hong Duong (possibly with another member staff). The School of Mathematics at the University of Birmingham is an internationally leading centre for mathematical research, with particular strengths in mathematical analysis, biological mathematics, combinatorics and optimization. The PhD candidate will have many opportunities to interact with leading scientists at the school of mathematics and other departments.

Recent collaborative works of Dr Hong Duong in the direction of research of the project include [DH16, DTH19, DH19b, DH19a].

Applications before 31 January is strongly encouraged. Please contact for any questions.


[CMRS19] F. A. C. C. Chalub, L. Monsaingeon, A. M. Ribeiro, and M. O. Souza. Gradient ow formulations of discrete and continuous evolutionary models: a unifying perspective. arXiv: 1907.01681, 2019.

[DH16] M. H. Duong and T. A. Han. Analysis of the expected density of internal equilibria in random evolutionary multi-player multi-strategy games. Journal of Mathematical Biology, 73(6):1727-1760, 2016.

[DH19a] M. H. Duong and T. A. Han. On equilibrium properties of the mutator-replicator equation in deterministic and random games. Submitted for journal publication, arXiv: 1904. 09805, 2019.

[DH19b] M. H. Duong and T. A. Han. On the expected number and distribution of equilibria in multi-player evolutionary games. Proceedings of the Artificial Life Conference 2019, MIT Press, 2019.

[DTH19] M. H. Duong, H. M. Tran, and T. A. Han. On the distribution of the number of internal equilibria in random evolutionary games. Journal of Mathematical Biology, 78(1):331-371, 2019.

[MSP73] J. Maynard Smith and G. R Price. Nature, (246):15-18, 1973.

[TV14] T. Tao and V. Vu. Local Universality of Zeroes of Random Polynomials. International Mathematics Research Notices, 2015(13): 5053-5139, 2014.

Funding Notes

For UK and EU candidates: funding may be available through a college or EPSRC scholarship in competition with all other PhD applications; the scholarship will cover tuition fees, training support, and a stipend at standard rates for 3-3.5 years;

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