Modelling Spatial Substructure in Wildlife Populations using an Approximation to the Shortest Path Voronoi Diagram

In the field of conservation biology, accurate population models are important for managing wildlife. It has been acknowledged that spatial structure should form an integral part of such models. Advancement in the power of desktop computers has allowed biologists to construct individual-based, spatially-explicit simulation models to predict the population dynamics of species. These models take into account local interactions and output emergent trends. However, whilst in many cases such models have been useful, for large landscapes, they may still be too computationally expensive. An alternative approach is to construct a metapopulation model. These are usually less computationally expensive, and whilst they often incorporate less detail than spatially-explicit models, they may still include enough spatial structure to be ecologically realistic. However, the assumption made in metapopulation models, that any individual within a patch can interact with any other individual of the patch may not always be appropriate. For example, in social species where the size of the social group home range is much smaller than the average patch size, individuals in a social group from one part of the patch may not interact with individuals from another social group in another part the patch. In such cases it is useful to consider spatial structure within the patch itself. In this paper we propose using the Shortest Path Voronoi Diagram (SPVD) to model the spatial structure within patches for social species. For such a diagram we use seeds to represent the centres of the social groups in each patch. We then assign each point of the patch to the seed it is closest to (where distance is measured with the shortest path metric). This partitions each patch containing seeds into regions. Together these regions form the SPVD. By linking the seeds of neighbouring Voronoi regions with shortest paths, a network among social groups is created. This can be used to model the dispersal paths of a population. Whilst analytic algorithms exist for the construction of the SPVD, these have often been developed for a polygonal domain. In complex landscapes, the time-complexity of such algorithms may become just as slow as grid-based approximations. Moreover, analytic methods may be less numerically robust and harder to extend to more complex variations of the Voronoi diagram. In this paper we offer a new grid-based approximation for the shortest path Voronoi diagram referred to as the quadtree-grid Voronoi Diagram (or qgrid VD). The construction procedure involves a decomposition of the landscape into a quadtree, and a propagation of circular wavefronts from each seed through a grid that is laid over the quadtree structure. We show how the q-grid VD can be applied to a wildlife population model using a squirrel glider (Petaurus norfolcensis) population in a semi-rural landscape as an example. By averaging outputs across multiple qgrid VDs we generate a time series of density maps. Such maps could be useful for informing wildlife management.