Joe Chipperfield Department of Biogeography

1. Novel methods for the assessment of connectivity conservation in large-scale protected area networks Joe Chipperfield Department of Biogeography

2. Connectivity • The term ‘connectivity’ is an attempt to assess the degree to which the spatial configuration of habitat interacts with a species dispersal abilities to hinder or promote the persistence of its populations residing within that habitat • Two main subcategories: • Structural connectivity: Relates only to the spatial configuration of patches • Functional connectivity: Includes biological information in the calculation of connectivity

3. Structural versus Functional Connectivity Structural Connectivity Functional Connectivity Matrix

4. Calculation of Connectivity Two stages required in the calculation of connectivity: • Determine your ‘patches’: • A priori determination • Determination through assessment of species habitat preferences • Calculate the degree of potential patch utilisation within the network ? ? ? ?

5. Scaling Up • These definitions of connectivity might not make sense when scaling-up to larger spatial scales • Measurements made on ‘patch’ centroids or edges make increasingly less sense • Functional responses based off small movement dynamics may not scale to situations where the habitat patches are described in terms of spatial constructs with coarse resolution • We already have methods available for assessing habitat suitability at macroecological scales in the form of species distribution models • Is the concept of the patch still useful at macroecological scales?

6. Species Distribution Models Observation Data • Inputs • Environmental variables • Habitat variables • Spatial filters • Output • Probability of Occurrence • Intensity Environmental Data 1 Environmental Data 1 Model Environmental Data 1 Output

7. Problems with SDMs • The niche concept doesn’t actually map very well onto the output of a SDM • Finding environmental covariates that describe distributional data is not quite the same as calculating the niche of the species • We often treat observations as realisations of a species realised niche but then project the model in a way that suggest we have derived the fundamental niche of a species • The observation records are only partly dependant upon environmental covariates • Biotic interaction • Heterogeneity of sampling effort • Dispersal abilities • Habitat fidelity • Territoriality Generate extra residual autocorrelation

8. Basic Model • We start with a simple probit regression model: β: Vector of regression coefficients c: Intercept term Probit Link Function: Inverse of the cumulative normal distribution function p: Vector of probabilities of occurrence x: Matrix of covariates with each row holding the covariate values for a given location

9. Spatial Autoregression • To account for sources of extraneous autocorrelation we add an autoregressive error term ϕ • ϕ is a vector of random variables with values drawn from a Markov Random Field • The vector has a correlation structure governed by two parameters: • τ: a parameter governing the magnitude of the deviations from zero • α: a parameter controlling the spatial dependency present in the Markov Random Field • We also require a weights matrix that describes the neighbourhood structure of the Markov random field

10. Imperfect Detection • The observation of individuals across the species range is far from perfect • We define two new types of error • ɛ- : The probability of recording an ‘absence’ at a cell when the species really is present • ɛ+ : The probability of recording a ‘presence’ at a cell when the species really is absent • The simplest type of observation error simply assumes the following:

11. Relation to Niche Theory • The mapping to the concept of the ‘niche’ is still not perfect using this method but is substantially improved compared to previous methods Neanderthal Distributions: Last Glacial Maximum Climate Only Prediction: Closer to the fundamental niche of the species Full Prediction: Closer to the realised niche of the species

12. PLMMRF R Package • The ProbitLinear Model with Markov Random Fields has now been developed into an R package • Look out for PLMMRF on CRAN soon • Interface is simple: very similar to the glm function gPLMMRF(observations ~ covar1 + covar2 + factor1 * factor2, autoweights = list(spatialWeightsMatrix, temporalWeightsMatrix), obsModel = “binomial”, ...)

13. PLMMRF and Connectivity • PLMMRF produces an occurrence map that takes into account non-climatic range determinants • Unfortunately both the amount of occupied habitat in the reserve and the amount outside the reserve are random variables • Develop a metric of ‘occupation conservation’: E(Proportion of occupied habitat in the reserve)

14. Further Extensions • The binary presence / absence case can be extended to the case where the landscape can be classified into multiple types by changing the error function to a multinomial distribution and using a polychotomous link function • New observation model possibilities • Allow for a linear observation submodel • Allow observation error to vary between landscape types and sampler effort • Observation can have its own autoregressive component • Interaction terms between spatial autoregression weight and covariates

15. Advantages • The model described here has many advantages over many commonly applied species distribution models • Incorporates extraneous spatial (and even temporal) autocorrelation • Incorporates observation uncertainty • Parameterised using Bayesian methods and so predictions take into account uncertainty surrounding parameter values and predictions • Model component are numerically tractable meaning that there is no need to rely on psuedo-likelihoods: autologistic regression • Seperation of niche concepts in predictions • Connectivity defined on a much more appropriate scale

16. Acknowledgements • Universität Trier: Stefan Lötters, Michael Veith, Katharina Filz, Jessica Weyer, Axel Hochkirch, Thomas Schmitt • Universität Bonn: Dennis Rödder, Jan Engler • University of York: Calvin Dytham, Chris Thomas • Funding: ForschungsinitiativeRheinland-PfalzMinisteriumfürBildung, Wissenschaft, Jugend und Kultur