Mechanistic home range models are important tools in modeling animal dynamics in spatially-complex environments. We introduce a class of stochastic models for animal movement in a habitat of varying preference. Such models interpolate between spatially-implicit resource selection analysis (RSA) and advection-diffusion models, possessing these two models as limiting cases. We find a closed-form solution for the steady-state (equilibrium) probability distribution u* using a factorization of the redistribution operator into symmetric and diagonal parts. How space use is controlled by the preference function w then depends on the characteristic width of the redistribution kernel: when w changes rapidly compared to this width, u* ~ w, whereas on global scales large compared to this width, u* ~ w^2. We analyse the behavior at discontinuities in w which occur at habitat type boundaries. We simulate the dynamics of space use given two-dimensional prey-availability data and explore the effect of the redistribution kernel width. Our factorization allows such numerical simulations to be done extremely fast; we expect this to aid the computationally-intensive task of model parameter fitting and inverse modeling.