The data contains three variables, that is, 1)YEAR, 2) estimates of ADULT DENSITY from line transect counts of willow grouse in central Sweden between 1964 and 2019, and 3) the standardized brood size in the same period. Note that data on brood size is missing between 1995 and 2003.

Abstract of article

Decisions on management policies require insights in how populations respond to different actions. The form and strength of negative density feedback is central in understanding population response, but identifying the proper function can be challenging when the relative magnitude of stochastic variation is high. Observation error in addition to natural process error will bias population variability and estimate of density dependence. Hierarchical state-space models can be used to separate process and observation errors, and recent advances in Bayesian framework and MCMC methods have increased their popularity. Here we develop a hierarchical state-space model, where the process equation is a Gompertz model with per capita variation in breeding success added as a stochastic process. We use data from a 56 year line transect monitoring program of a willow grouse (Lagopus lagopus) population in south central Sweden. The model fit resulted in a carrying capacity of 8.608 adults per km2 and a λ_max of 3.20, which is close to the 4.5 based on maximum survival and fecundity values. Accounting for a stochastic density-independent per captita breeding success resulted in a substantially reduced processes error (standard deviation), 0.2410 compared to 0.1638. Maximum growth is expected to occur at 2.208 adults per km2. Combining both process and observation error resulted in a CV-value of 0.163 at carrying capacity, which is similar to previously reported range of CV-values of many bird populations. However, only using the reduced process error result in a CV of 0.076. A bootstrap test for monotonic trend was statistically insignificant, probably due to a steady increase in density the last six years of the time series. We conclude that state-space models to separate observation and process error can provide important information on population dynamics, but that effort should be made to estimate measurement error independently. There is a lack of data on population dynamics at low densities, and we suggest that additional experimental harvest should be considered to improve the understanding of negative density feedback in relation to stochastic processes at low densities.