Consistent multiscale modelling of movement and habitat selection

In spatial ecology, the concept of resource selection expresses the idea that for many animals, the distribution of an individual's location is not uniform over the region available to them; instead, they spend time preferentially in some locations compared to others, in a way that can often be related to spatial covariates. The related concept of step selection describes the variation in an individual's tendency to move to particular locations in the short term, taking into account both spatial covariates and the constraints of their process of movement. Consistent modelling of resource selection and step selection is necessary to understand animals' distribution in space and to interpret movement, telemetry, and spatial survey data in a meaningful way. In this paper, I take advantage of recent developments in stochastic processes and statistical algorithms to develop a range of new stochastic models in which both the dynamics and the long-term behaviour are tractable and described parametrically, and which are flexible enough to represent a wide range of patterns of movement and space use encountered in reality. I extend the mathematical analogy between movement modelling and Markov chain Monte Carlo algorithms, first proposed by Michelot, Blackwell & Matthiopoulos (2019; Ecology 100, e02452), to a wide range of continuous-time stochastic processes, including both diffusion processes and velocity-jump models, that in different ways are motivated by the simple discrete-time step-and-turn models widely used in practice. Particular cases include a diffusion process where the dynamics are defined in terms of speed and direction of movement, and a velocity-jump process in d dimensions, generalizing the 'bouncy particle sampler' used in Bayesian inference, in which the distribution of velocity after a so-called 'bounce' event has support over a region which itself has dimension d. I also show how this mathematical approach can be extended to models incorporating distinct behavioural states and to higher dimensional models representing the joint movement of interacting individuals.