The size of animal home ranges often varies inversely with population density among populations of a species. This fact has implications for population monitoring using spatially explicit capture–recapture (SECR) models, in which both the scale of home-range movements σ and population density D usually appear as parameters, and both may vary among populations. It will often be appropriate to model a structural relationship between population-specific values of these parameters, rather than to assume independence. We suggest re-parameterizing the SECR model using k_p = σ_p √D_p, where k_p relates to the degree of overlap between home ranges and the subscript p distinguishes populations. We observe that k_p is often nearly constant for populations spanning a range of densities. This justifies fitting a model in which the separate k_p are replaced by the single parameter k and σ_p is a density-dependent derived parameter. Continuous density-dependent spatial variation in σ may also be modelled, using a scaled non-Euclidean distance between detectors and the locations of animals. We illustrate these methods with data from automatic photography of tigers (Panthera tigris) across India, in which the variation is among populations, from mist-netting of ovenbirds (Seiurus aurocapilla) in Maryland, USA, in which the variation is within a single population over time, and from live-trapping of brushtail possums (Trichosurus vulpecula) in New Zealand, modelling spatial variation within one population. Possible applications and limitations of the methods are discussed. A model in which k_p is constant, while density varies, provides a parsimonious null model for SECR. The parameter k of the null model is a concise summary of the empirical relationship between home-range size and density that is useful in comparative studies. We expect deviations from this model, particularly the dependence of k_p on covariates, to be biologically interesting.