Space & Time matter: spatial ecology in non-euclidean spaces

By Luis Gimenez Noya

Patterns of distribution of organisms have been traditionally interpreted as resulting from the balance of reproduction and mortality. However, populations of many organisms occupy fragmented landscapes; for instance, animals living in forest patches are separated by natural or cultivated grassland; marine benthic invertebrates living in rock shores form populations separated by sandy beaches. For those species, the patterns of distribution can also depend on migration/dispersal among patches. Migration and dispersal are usually distance-dependent, meaning that nearby patches exchange individuals more easily (i.e. they are better connected) than those located far away.

Organisms may connect patches by dispersing in all directions across the landscape (e.g. across cultivated land) or may follow specific migratory routes. In the former case, if the landscape is more or less flat, dispersal pathways may be abstracted as straight lines connecting patches. When movement takes place on such Euclidean (=flat) spaces, the Euclidean distance between patches will be an appropriate metric to quantify distance-dependent migration. However, when paths are curvilinear, Euclidean distances are not applicable and other metrics are needed. The curvilinear nature of the migration pathways are usually the consequence of behavioural responses of migrants to the landscape topography, predators or food; or the consequence of curly hydrodynamic structures (e.g. gyres and eddies) for the case of marine organisms.

There are several examples in the literature showing how to abstract animal movement as nonlinear geometrical objects. For example, animal movement has been abstracted as fractal walks. Here, I propose an alternative, geometrical view to animal movement based on concepts developed around the study of non-euclidean spaces, summarised in the figure depicted in this blog. The figure is an idealisation of the process of upward migration exhibited by larvae of a marine organism which are trapped in an eddie system (the actual process is more complex and “noisy”; numerical particle tracking models, e.g. Robins et. al 2013, will include terms that incorporate particle diffusion). During the migration (upward “moving” spheres) in a “flat” 3D space, organisms follow a curvilinear pathway: they actually appear to move on a non-linear 2D surface or space (depicted with a mesh). One can then define a coordinate system on the non-linear space in order to quantify the curvilinear distances followed by the ascending particles. I apply this idea to organisms moving on a 2D space where the curvilinear space has only one dimension. In real case scenarios however, the issue of finding such space involves concepts of statistical science. Hence, I explore a statistical methods of manifold learning (non-linear PCA) to approximate the curvilinear space and obtain the metrics of distance out of the migration pathways; the type of spaces considered in the article are also called smooth manifolds. Overall, a geometrical view to animal movement may help advance the fields of spatial and landscape ecology; it may provide an alternative perspective to the understanding of the ecological role of dispersal and migration.

References:

Robins, P. E., Neill, S. P., Giménez, L., Jenkins, S. R. and Malham, S. K. 2013. Physical and biological controls on larval dispersal and connectivity in a highly energetic shelf sea. - Limnol Oceanogr 58, 505-524.

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