Coexistence of plant populations in a spatial continuum

In a recent paper of Etheridge et al. (Etheridge, et al. 2023), a new model for the dynamics of a plant population in a spatially heterogeneous environment is introduced. A particular feature of this model is that it separates juvenile and establishment phases of reproduction, leading to novel scaling limits. There are many potential applications of this model to the understanding of the evolution of populations. The particular focus of this project will be on plant populations in which distinct varieties produce seeds of different sizes, resulting in separate dispersal mechanisms. An obvious question, and the primary objective of this project, is whether these different varieties can coexist without any form of natural selection acting on the population.
As a first step, we will formulate a haploid model of reproduction in which total population size is regulated through a nonlocal logistic growth mechanism, and dispersal of seed is determined by seed size. We will then examine, for this toy model, whether or not coexistence is possible under neutrality. This stage will involve both numerical and analytical work. There is some preexisting code available for simulating the models of Etheridge et al. (Etheridge, et al. 2023), based on a powerful software called SLiM, but it requires modification for this scenario. In this part of the project, we shall seek input from Peter Ralph (University of Oregon). The analysis will draw on experience of models of competing species
in which heterogeneity is arises through interaction rather than migration (Blath, Etheridge and Meredith 2007). Natural questions we expect to follow from this initial analysis include
how many different seed sizes can coexist? What if there is a continuum of seed sizes?
Of course, this is not a realistic model and will require more complex modification. The next
step will be to introduce the dispersal of pollen, which one would expect to follow another type of mechanism (often long-range dispersal...