Understanding how individuals move in space, what habitats they prefer, and how the environmental features channel or resist movement is central to landscape ecology and wildlife management.  Dramatic improvements in the acquisition, resolution, and extent of two relevant types of data have recently occurred: remotely sensed environmental data and high-resolution animal location (telemetry) data.  These data drive a statistical industry serving wildlife management agencies, private companies, and academia. Improvements in tracking technology are likely to cause a revolution in movement ecology analogous to the impact of gene sequencing on molecular genetics.  This project synthesizes theoretical advances (statistical techniques for estimating movement probability between sites and how environmental resources are selected), existing results (mathematical techniques for rapidly predicting the envelope of future animal positions using mechanistic assumptions) and untapped data (remotely sensed habitat maps and high resolution individual telemetry) to rigorously characterize how landscape features condition population movement and habitat choice.  The research will encompass case studies investigating the movement of mule deer and elk in Utah, harbor seals off southeastern Alaska, and Canada lynx, which have recently been reintroduced in Colorado and are dispersing throughout the Rocky Mountains.   Research students will be cross-trained in mathematics, statistics, and movement ecology; undergraduates will be included in the research process by developing individual-based models to test estimation technologies.  A teaching lab in mathematical biology, illustrating movement models using real biological systems, will also be developed and distributed.Statistical point process models provide well-understood statistical approaches for obtaining inference from individual-based telemetry data, with resource selection functions describing individual habitat preferences and availability functions describing dispersal probability between locations. However, point process models require numerical quadrature for proper normalization, making them slow for large data sets.  Classical availability functions are not constructed to handle major issues like movement constraints, autocorrelation, and landscape resistance, affecting quality of resource selection inference and computational feasibility.  However, a parallel and untapped literature of partial differential equations predicts dispersal likelihood based on mechanistic assumptions about individual movement.  Ecological diffusion and ecological telegrapher's equations provide natural scalings from Lagrangian to Eulerian perspectives. They are fully mechanistic and allow for population-level dynamics, but are not inherently statistical nor automatically suited to handling individual-based telemetry data.  This project will reconcile point process modeling with mechanistic dispersal equations to arrive at a unified method for analyzing telemetry data.  Homogenization techniques, which are well-accepted in physical sciences but not often applied in mathematical biology or statistics, will be used to speed up solutions in heterogeneous environments. Coupled point process models and homogenized partial differential equations will accelerate model fitting, provide resource selection inference and naturally accommodate environmental heterogeneity and barriers/constraints to movement.  The ecological movement equations will be homogenized and simplified using asymptotic approximations suitable for point process models, addressing correlation among position observations and velocity constraints.  Rapid numerical techniques for movement models will be developed to allow facile representation of movement barriers (e.g., shorelines, major rivers or roads) as boundary conditions. To develop efficient computational techniques for resource selection functions and landscape resistance inference, the homogenized ecological movement equations will be dovetailed with point process models in a hierarchical framework.  The integrated approach will be applied to telemetry data from foraging ungulates in Utah, harbor seals in the Gulf of Alaska, and Canada lynx in Colorado.