# Mechanistic models community assembly

(These notes added mainly by James.)

### Overview

With Jess Green I am developing a theoretical and mathematical underpinning for certain spatial patterns in ecology. In the first instance, these patterns include species or taxa-area relationships (how the observed number of species increases with sample area), species/taxa abundance distributions (how many rare vs how many abundant species), how these patterns scale with sample size and how they are connected to other measures of diversity.

My approach is a kind of bottom-up approach, so that individuals in a community go about their lives according to some set of rules, and we calculate what kind of macroscopic patterns emerge. So I am combining the processes of birth, death, speciation and spatial dispersal to develop a model of community assembly, and I will use the predictions of my model to compare with spatial patterns in metagenomic data. Even though I am focussing on taxonomic diversity to begin with (because this is easier from both the theory and data perspectives), my intuition is that the taxonomic framework will be an essential building block for a more general model of phylogenetic diversity.

### Background: Size-Structure

The methods I am using are developed from the PNAS paper here, where I developed a model of stochastic community assembly structured by the size of individuals. This breaks the assumption of neutrality by allowing individuals of different sizes to have different rates of mortality, birth and ontogenetic growth, and we were able to make new predictions for the Species Abundance Distribution and the Species Body-size Distribution.

Our starting point in the spatial case is a spatially-discrete process, where individuals of a given species live at discrete sites, labelled by an index, $i$, and undergo stochastic birth, death and dispersal processes. Individuals die with a per capita mortality rate, $d$, and produce new individuals at a per capita rate, $b$. These new individuals may be dispersed to a different spatial site from their parents, where the probability of dispersal from site $i$ to site $j$ is given by the matrix $Q_{ij}$. We also make an assumption of neutrality, so that the same demographic rates apply across all species, and we specify that $d$ is slightly greater than $b$, so that every extant species will eventually suffer extinction. This extinction process is then balanced at the community level by the introduction of new species.

### Connections to Josh

This is a mechanistic framework---there are some input parameters, like average birth and death rates and dispersal length-scales, and the model (if we can solve it) will make a series of spatial predictions dependent on both the structure of the model and the values of these parameters.

This is a little different from what Josh is doing, which is to make some (reasonable) assumptions about the scaling of diversity, and use these assumptions to create a link between two different macroecological metrics, the species area relationship (SAR) and distance decay of community similarity. However, I think there may be some connections between our approaches. Two of my aims (see below) are to predict the SAR and the two-point correlation function, and so we will be able to see explicitly how these functions respetively depend on those input parameters. What would be interesting is if the same functional form as Josh is using comes out, or the same kinds of connections between the SAR and DD---but of course the story could end up being more complicated than this.

### Progress

more to add here. tune in later. I will derive the central equations in the framework, and show how I have reduced them down to a simple pde for the generating function of the Species ABundance Distribution (which contains the information contained in the SAR, plus more)

1. Moment generating functional is Z[H(m)], whereas moment generating function is Z(h)