Long range jumps in locally regulated populations

This project falls within the EPSRC Mathematical sciences research area.
We will be describing and analysing properties of a mathematical model derived from population genetics. This model describes the evolution of a particle system over space and time, where the particles are supposed to represent any biological organism. When creating such models, we want to ensure that these models are both realistic, in that they accurately describe the evolution of some or many organism(s) in this world while also being mathematically tractable, in that we can evaluate qualitative and quantitative features of such a model. The main model studied in the literature for the evolution of one species is the so-called Dawson-Watanabe superprocess.

The problem with the model above is the advent of a phenomenon called clumping, which is when many individuals in a population aggregate in a small area leading to high population density. This is a problem as it is not biologically realistic, as competition for finite resources in an area ensures that there is an upper limit on the population density that could happen, therefore we need to adapt our model. This problem arises as population density is not a factor considered in the models mentioned above.

In the novel model considered by Alison Etheridge et al, there are 3 parameters for individuals: birth rate, establishment rate of their offspring and lifetime. Individuals reproduce with a certain rate, leaving one offspring which gets established with a certain probability. Importantly offspring are born close to their parents. Individuals die at a certain rate, and do not move as the only movement that happens is when offspring are born and establish away from their parents. The strength of this model compared to previous ones is that these parameters have an explicit dependence on the population density to incorporate the fact that high population density means competition. In this paper, they take appropriate limits as the population tends to infinity (rescaled appropriately) and as the density gets localised on smaller areas. Taking limits is useful to get overall behaviour of the process.

My aim, in collaboration with my supervisor, is to introduce long range jumps from parents to offspring, so offspring do not necessarily need to establish close to where their parents are. This would be useful to model species which have births far away from the parents such as Dandelions which are flown away by the wind. Concretely we would like to incorporate this in the current model, reproduce similar results, but also identify differences in behaviours between the 2 models.
Another project would be to perform more careful analysis on features of the model above called patches that develop in the limiting distribution. These are clumps of organisms that gather in an area and stay there in a stable configuration without expanding or dying. I would like to analyse the size and frequency of the patches, why the patches occur and if there is migration between different patches.