Species distribution models (or SDM's) are used to explore how the occurrence of a species is related to the environment, and how a species might respond to changes in its environment. This can help find new locations where a rare species might be found, or understand the potential threats to a species due to urban encroachment, climate change, or other causes.

For example, here is a model of the distribution of the Sydney Red Gum (*Angophora costata*) in the Sydney basin, as a function of climate variables and fire history. The model was constructed using a Poisson point process model, which models the "intensity" \( \lambda \) of *A. costata* as a function of environmental variables \( (X_1, X_2, ..., X_p) \): \[ \log(\lambda) = \beta_0 + \beta_{11} X_1 + \beta_{12} X_1^2 + \beta_{21} X_2 + \beta_{22} X_2^2 + \cdots + \beta_{p1} X_p + \beta_{p2} X_p^2. \]

We fit this model by taking a set of locations where *A. costata* is found and comparing it to a map of environmental

variables, to see what environments the species tends to be associated with. The parameters that describe this relationship, namely \( \boldsymbol { \beta } =( \beta_0,\beta_{11}, \cdots, \beta_{p2} ) \) are estimated via a common statistical procedure known as maximum likelihood, meaning that we need to find the vector \( \boldsymbol{ \beta }\) that maximise the following function:

\[ l( \boldsymbol { \beta } ) = \sum_i \log( \lambda_i) - \int \lambda \mathrm{d} y. \]

This function involves an integral which doesn't have a closed-form solution in the general case, so we estimate it via numerical integration (like using the trapezoidal rule, or Simpson's rule). This can get tricky in practice, and in the *A. costata* example we needed a total of 86,800 function evaluations to get a good estimate of the integral: sounds hard, but a modern computer can do this almost instantly. As you can see, fitting a species distribution model involves an interesting mix of different mathematical tools, and a bit of computing!

Methodology for constructing species distribution models is an exciting interdisciplinary research front in which contributions are being made by statisticians, computer scientists and ecologists. The Eco-Stats group, together with collaborators at the Australian Museum and elsewhere around Australia, are developing methods that can predict species distribution at a spatial resolution finer than ever before and with greater functionality than previously.

#### Further Information