The Fisher-KPP (FKPP) equation, was introduced independently by Fisher and Kolmogorov, Petrovskii and Piskunov in 1937 to model the spread of an advantageous allele through space under the effects of natural selection and migration. It predicts that an advantageous allele will propagate through space as a travelling front, which is physically observed.
Skorokhod established in the 60s a fundamental connection (a ``duality relationshop'') between the FKPP equation and branching Brownian motion. Branching Brownian motion is a fundamental model of particles evolving randomly in space and reproducing. Since then this connection has been exploited to study both the long-time behaviour of the FKPP equation and the properties of branching Brownian motion.
We will study both the FKPP equation, branching Brownian motion, and the duality relationship between them. There will then be the choice to focus on either:
Properties of Branching Brownian motion. In particular we will study the position of the ``tip'' of branching Brownian motion, and its spatial configuration at the tip;
The asymptotic position and shape of solutions of the FKPP equation. In this case, we will study the behaviour of the generalised FKPP equation, which models a wide variety of phenomena.
Whilst the first question is a PDE question and the second probabilistic, in both cases the interplay of PDE and probabilistic tools will be fundamental.
Prerequisites/co-requisites:
At least one of Markov Chains II or Probability II is very useful;
At least one of Partial Differential Equations III or Stochastic Processes III. Students with a background in only one will be taught the requisite background in the other.
Stochastic Processes IV is very useful.
Mode of operation and evidence of learning
The first part of the project will be directed reading on Brownian motion, the heat equation, and their relationship.
We will then engage in directed reading on both branching Brownian motion and the FKPP equation. There will be the option to focus on either of these, and the student will be able to demonstrate a thorough understanding of their chosen area of focus.
There will be the possibility for stronger students to develop novel stochastic representations for different PDEs through applications of the methodology they will learn.