Modelling general mechanisms driving evolution in natural populations
Dr. Mike Evans (SoM), Dr. Sandro Azaele (SoM)
Contact email: r.m.l.evans@leeds.ac.ukUnderstanding the mechanisms of evolution is vital, both for enhancing our knowledge of existing ecosystems, and for modelling how natural populations will adapt to environmental change. Although Darwin discovered the fundamental rules of evolution, the application of those rules, within large, interacting and fluctuating populations, gives rise to a number of emergent mechanisms, such as convergent and parallel evolution, speciation, mimicry and altruism. Such processes are universal, in that they are generic to many different species and ecosystems. Identifying the universal processes of evolution is an active research area, often involving the analysis of idealized mathematical models.
The modelling of large numbers of interacting particles is commonplace also in theoretical physics, where the mathematical tools of thermodynamics and statistical mechanics have been developed to model the interactions of molecules within solids, liquids and gases. For instance, the concepts of entropy and critical points are central to our understanding of those systems. Those tools are beginning to be applied across several levels of organization, from genes (Wagner A., 2008) to large ecosystems (Borile et al., 2012), with interesting results for specific experiments and models.
More generally, there are recent theoretical developments which show that living systems could operate in the vicinity of a critical point (Hidalgo et al., 2014), because this confers them more robustness and flexibility when facing environmental variability. Even speciesrich ecosystems, which are made up of a large number of interacting individuals and species, could exploit criticality as a source of evolutionary innovations which enable them to enhance robustness, adaptability and plasticity. This could be particularly important when complex ecosystems have to cope with and adapt to a changing environment. Meanwhile, universal evolutionary mechanisms are studied using comprehensibly simplified models (Page et al., 2000), mostly without reference to statistical mechanics, and the effects of disorder.
Figure 1: Output from an agentbased stochastic simulation of “The Ultimatum Game”, a fundamental model of Evolutionary Games Theory. Introducing some disorder into the model is found to stabilize altruistic behaviour in some cases. In this image altruistic agents are shown with redder colours than selfish agents. (Evans, R. M. L. (2012) private communication).
This project will involve analyzing some models of Darwinian evolution that are important when understanding evolving ecosystems. They will be sufficiently simple to be both analytically tractable and suitable for agentbased simulation (such as the simulation shown in Fig.1). Studying such models will allow a link to be established between the concept of genotypic entropy, which is welldefined in the stochastic numerical models, and a Hamiltonian formulation of the models (e.g., Plank M., 1995), which enable one to develop a statistical mechanical approach to the theory. The development of such links would allow the investigation of general principles by means of powerful tools borrowed from statistical mechanics. This has the potential to reveal general mechanisms which can act as driving forces for some universal evolutionary processes.
Objectives:
In particular, according to your particular research interests, the studentship could involve:

Numerical simulation of some models of Darwinian evolution;

Study the analytical properties of the deterministic and stochastic formulations of such models;

Introducing a Hamiltonian formulation with the aim of developing a statistic mechanical approach to the underlying models;

Developing a sound formulation of entropy in the genotypic space;

Investigating the emergence of general patterns and mechanisms which could play a fundamental role in evolutionary processes;

Charting the various stable states of the model ecosystem;

Collecting numerical data on the typical features of the probability distributions that characterise the evolutionary processes.
Potential for high impact outcome
This project will try to link subjects which are apparently far from each other. It has the potential to uncover new tools and methods, as well as to explain quantitatively important processes, which could have important implications for our understanding of how species and ecosystems adapt and evolve. Therefore, there is substantial potential for high impact publications.
Training
The student will work under the supervision of Dr. Mike Evans and Dr. Sandro Azaele within the Department of Applied Mathematics. This project provides a high level of specialist and interdisciplinary scientific training in: (i) numerical simulation, ii) stochastic and deterministic modelling, iii) statistical mechanics, iv) interpretation of data and critical evaluation of models.
The successful PhD student will have access to a broad spectrum of training workshops and lectures, including training in analytical methods, numerical modelling, managing your research degree, writing a CV, and preparing for your viva (http://www.emeskillstraining.leeds.ac.uk/).
References

Wagner, A. (2008) Robustness and evolvability: a paradox resolved, Proc. R. Soc. B vol. 275 no. 1630: 91100.

Borile, C.; Muñoz, M. A.; Azaele, S.; Banavar, J. R.; Maritan, A (2012) Spontaneously broken neutral symmetry in an ecological system. Physical Review Letters, 109, 038102.

Hidalgo, J; Grilli, J; Suweis, S; Munoz, MA; Banavar, JR; Maritan, A (2014) Informationbased fitness and the emergence of criticality in living systems, Proceedings of the National Academy of Sciences of the USA, 111, 1009510100.

Page, K. M., Nowak, M. A., and Sigmund, K. (2000) The spatial ultimatum game, Proc. R. Soc. Lond. B 267: 21772182.
 Plank M., Hamiltonian structures for the n‐dimensional Lotka–Volterra equations (1995), J. Math. Phys. 36, 3520.
Related undergraduate subjects:
 Computer science
 Mathematics
 Physics