Testing for density-dependence from population time series in the presence of observation error

Background/Question/Methods

Population regulation has been one of the most heavily debated topics in ecology. An emerging consensus that density-dependence is ubiquitous in nature has been recently challenged by analyses that explicitly account for both observation error and natural fluctuations in  time-series of population size. Here, we re-examine the evidence for density dependence in population time-series by fitting a common population dynamics model to data from the Global Population Dynamics Database (GPDD), and to simulated data with known intensity of density-dependence. We fit the model using four different formulations of the model’s likelihood, each of which has been used in previous studies. One likelihood makes a point estimate of the initial population size (point method); one uses a “diffuse” initial population size with large variance (large variance method); and two use an initial population size based on the stationary (long-run equilibrium) distribution. Of the two likelihoods assuming stationarity, one treats observed log-abundances as the observations (abundance-based stationarity method), whereas the other treats annual changes in log-abundance as the observations (first differences method). These four likelihoods should be asymptotically equivalent, producing the same estimates when time series are infinitely long. However, their relative performance for realistic, short time series is unknown.

Results/Conclusions

Of the four likelihoods, the point and large-variance methods yielded very strong evidence for density-dependence in the GPDD, whereas the two stationarity approaches yielded much weaker evidence. The simulation study indicated that, for ecologically realistic time series lengths, the point and large-variance methods were severely biased towards finding density dependence, when both observation error and process noise were present. The abundance-based stationarity method was slightly biased towards density-dependence, but the first-differences method exhibited negligible bias. Detailed inspection of model fits revealed that, for realistic time series lengths, all methods produced multi-modal, ridged likelihoods whose global maxima were often far from the true parameter values. With replicate observations, however, the first-difference method performs acceptably well. We conclude that unreplicated ecological time series, such as those in the GPDD, are simply too short to provide conclusive evidence for density dependence, at least given currently available statistical methods, and that replicated observations are essential to underpin reliable estimation of population parameters.