Population dynamics models can be used to evaluate the effectiveness of predator control. On a restricted area, one key process is the rate at which removed individuals are replaced by immigration. Since this rate is difficult and costly to estimate by field study, we develop an analytical method to approximate immigration rate that makes use of data obtained through the removal process itself. In Britain, red fox Vulpes vulpes control is undertaken by gamekeepers on privately-owned shooting estates. The fox cull on each estate derives from both local reproduction and immigration. The proportional contribution of immigration to the cull can be expected to be greater on smaller estates. We describe a mechanism by which the average annual cull per unit estate area on a very small estate approximates the annual rate of immigration. We used fox culling records from 534 estates across seven different landscape types and a Bayesian hierarchical model to relate the density of foxes culled to estate area, with immigration rate assumed to be equal to the model intercept. The posterior predictive distribution of annual immigration rate was lognormal with a median of 2.41 fox km-2 year-1 and a CV of 0.84. Posterior median estimates of immigration rate varied between landscapes, ranging from 0.86 to 4.13 fox km-2 year1. Immigration rate was higher in arable and pastural landscapes compared to upland landscapes. Variation in immigration rate broadly matched differences in fox density characteristic of the regional landscape type. This study presents a widely applicable method for quantifying immigration rate in populations that are subject to depletion, e.g. through culling. The use of the fox immigration rate estimate as an informative prior distribution in population dynamics models could help in evaluating effects of control on local fox populations and lead to improved control strategies.
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