# Up-scaling biodiversity: the mathematics and statistics of spatial ecology

• Applications accepted all year round
• Competition Funded PhD Project (European/UK Students Only)

## Project Description

Up-scaling biodiversity: the mathematics and statistics of spatial ecology - with Dr Sandro Azaele (School of Maths) & Prof William Kunin (School of Biology)

Species richness is intrinsically non-additive, which makes it difficult to estimate the number of species in a large area from the numbers found in small subsamples within it. For some time it has been known that in principle it should be possible to estimate the species-area relationship (SAR) of a region from the spatial turnover in local samples, practical methods have proved elusive. Novel methods based on pair-correlation functions have recently been developed by the supervisory team. The methodology used so far is able to predict the SAR across scales and also the number of rare species which are present outside the sampled area and therefore have not been observed yet. However, in the present formulation it needs species abundances, which strongly limits the applicability of the approach. This project would extend the methods to include other data types (e.g. presence/absence, non-count abundance) and statistical properties (e.g. the derivation of confidence intervals, optimized sampling designs). This is crucial for conservation strategies and poorly understood taxa. Depending on the candidate’s interests, the project might also include a field or database component, applying the methods to datasets of management or conservation interest.

Background and rationale
Harte and colleagues (1999) first suggested that it should be possible to estimate the SAR for a region by examining scattered point survey data. The mean species richness of the samples would provide an estimate of the height of the SAR curve at the scale of an individual sample, while the turnover of species (“beta diversity”) across space would provide information about the SAR’s scale-specific slope. Unfortunately, the original method was constrained by very restrictive assumptions, but a number of other methods have subsequently been proposed. Most of these are relatively inflexible in form, and thus cannot be easily applied to managed systems or monitoring. Recently, however, we have developed a novel method based on pair correlation functions which appears both mathematically sound and sufficiently flexible to allow widespread application. This project would be devoted to developing, extending and applying the method.

Key research questions
Some of the points to be explored include:
• Extending the approach to cover different data formats, such as presence/absence data or non-count abundance data
• Developing alternative up-scaling methods if desired
• Developing new spatial patterns that can be linked to the SAR
• Developing methods of assessing confidence intervals for upscaled species-richness estimates
• Incorporating environmental information alongside spatial information in analyzing richness patterns
• Optimizing sampling designs for assessing species richness within a region
• Testing methods on well-studied systems (e.g. UK plant and butterfly datasets)

Methodologies
The core of the method is based on the empirical estimation of the pair-correlation function (PCF) and the mean density of individuals per species (δ) when abundances of species are available. The PCF can be empirically estimated from fine scale samples distributed across a spatial area. When the PCF has a known analytical form, we can in principle calculate the variance in species abundances. This latter index provides us with a link with another important ecological measure: the spatial species abundance distribution (sSAD), which gives the number of species with n individuals present within an area. By imposing that the variance in abundance obtained from the PCF equals the one given by the sSAD one can therefore link the spatial turnover of species with their relative abundance distribution across scales, which is one of the key patterns in community ecology. The sSAD can finally be used to calculate the SAR at different spatial scales. This latter informs us on the mean number of species expected in a sampled area. The methodology works for any form of the PCF and sSAD as long as the PCF provides a good fit to the data.

Requirements
Students should have a good knowledge of stochastic processes, statistics and probability theory in general with good analytical skills. A good knowledge of statistical software (R or SPSS) and/or GIS software would also be desirable. A good background in spatial ecology would also be highly desirable.

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