Testing the Levy Hypothesis of Biological Foraging

The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing either in September 2016 (funded students) or at any point in the academic year (self-funded students).

Movement patterns of biological organisms are incredibly rich and complex. While animals make persistent steps on small spatio-temporal scales, on larger scales their foraging paths resemble random walks. Correspondingly their mathematical analysis is highly non-trivial and interesting. A characteristic feature of random walks is that they generate normal diffusion in the long time limit, a concept well-known from statistical physics and the theory of stochastic processes, which means that the mean squared displacement of an ensemble of foragers grows linearly in time. But over the past two decades evidence has accumulated that animals may perform anomalous diffusion, where the mean squared displacement grows nonlinearly in the long time limit. In particular, it was claimed that many animals make local random movements interspersed with rare long-distance jumps in order to optimize foraging success. When the probability distribution for the step size follows a power law the resulting stochastic model is called a Levy walk. These findings led to the Levy hypothesis, which infers optimality of this movement strategy in terms of a minimal search time for sparse, randomly distributed, immobile, replenishing food sources in unbounded domains. Curiously, the Levy hypothesis has never been proven. It is discussed controversially in the literature.

The project will start with the easy task to simulate a stochastic Levy walk on the computer. Numerical results should be compared to analytical calculations. This simple model should be generalised step by step, both numerically and analytically, in order to reproduce, and check, the findings reported elsewhere. A related important goal should be to derive a so-called fractional diffusion equation modeling a Levy search problem and to solve it analytically. Search times should be calculated both numerically and analytically by matching the results to each other. The project may also involve the analysis of real biological data.

This project will be supervised by Dr Ranier Klages. For full details, see the project abstract: http://www.maths.qmul.ac.uk/sites/default/files/phd%20projects%202015/DSSP/klages1.pdf

The application procedure is described on the School website. For further enquiries please contact Dr Rainer Klages ([Email Address Removed]).

This project is eligible for several sources of full funding for the 2016/17 academic year, including support for 3.5 years’ study, additional funds for conference and research visits and funding for relevant IT needs. Applicants interested in the full funding will have to participate in a highly competitive selection process. The best candidates will be eligible to receive a prestigious Ian Macdonald Postgraduate Award of £1000, for which you will be considered alongside your application. The application deadline for full funding is January 31st 2016.

There is also 50% funding scheme available for students who are able to find the matching 50 % of the cost of their studies. Competition for these half-funded slots will be less intensive, and eligible students should mention their willingness to be considered for them in their application. The application deadline for 50 % funding is January 31st 2016.

This project can be also undertaken as a self-funded project, either through your own funds or through a body external to Queen Mary University of London. Self-funded applications are accepted year-round.