Jonathan R. Potts, Ulrike E. Schlägel

• Mathematical analysis of partial differential equations (PDEs) has led to many insights regarding the effect of organism movements on spatial population dynamics. However, their use has mainly been confined to the community of mathematical biologists, with less attention from statistical and empirical ecologists. We conjecture that this is principally due to the inherent difficulties in fitting PDEs to data. To help remedy this situation, in the context of movement ecology, we show how the popular technique of step selection analysis (SSA) can be used to parametrize a class of PDEs, calleddiffusion-taxismodels, from an animal's trajectory. We examine the accuracy of our technique on simulated data, then demonstrate the utility of diffusion-taxis models in two ways. First, for non-interacting animals, we derive the steady-state utilization distribution in a closed analytic form. Second, we give a recipe for deriving spatial pattern formation properties that emerge from interacting animals: specifically, do those interactions causeMathematical analysis of partial differential equations (PDEs) has led to many insights regarding the effect of organism movements on spatial population dynamics. However, their use has mainly been confined to the community of mathematical biologists, with less attention from statistical and empirical ecologists. We conjecture that this is principally due to the inherent difficulties in fitting PDEs to data. To help remedy this situation, in the context of movement ecology, we show how the popular technique of step selection analysis (SSA) can be used to parametrize a class of PDEs, calleddiffusion-taxismodels, from an animal's trajectory. We examine the accuracy of our technique on simulated data, then demonstrate the utility of diffusion-taxis models in two ways. First, for non-interacting animals, we derive the steady-state utilization distribution in a closed analytic form. Second, we give a recipe for deriving spatial pattern formation properties that emerge from interacting animals: specifically, do those interactions cause heterogeneous spatial distributions to emerge and if so, do these distributions oscillate at short times or emerge without oscillations? The second question is applied to data on concurrently tracked bank volesMyodes glareolus. Our results show that SSA can accurately parametrize diffusion-taxis equations from location data, providing the frequency of the data is not too low. We show that the steady-state distribution of our diffusion-taxis model, where it is derived, has an identical functional form to the utilization distribution given by resource selection analysis (RSA), thus formally linking (fine scale) SSA with (broad scale) RSA. For the bank vole data, we show how our SSA-PDE approach can give predictions regarding the spatial aggregation and segregation of different individuals, which are difficult to predict purely by examining results of SSA. Our methods provide a user-friendly way into the world of PDEs, via a well-used statistical technique, which should lead to tighter links between the findings of mathematical ecology and observations from empirical ecology. By providing a non-speculative link between observed movement behaviours and space use patterns on larger spatio-temporal scales, our findings will also aid integration of movement ecology into understanding spatial species distributions.…