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Abstract Aims To test a key prevision of the dynamic equilibrium theory of island biogeography, namely that changes in species numbers on islands over time (hereafter, species trajectories) are equilibrial, and to characterize aspects of the dynamical properties of species change over time using a model-independent test. Methods We tested for regulatory equilibrium and non-random structure in species numbers through time by comparing observed correlation coefficients at lag-k for species trajectories from four true islands and two habitat islands. First, we estimated the shape of the autocorrelation function for each observed species trajectory by calculating correlation coefficients of the observed data between pairs of values Nt−k and Nt separated by lag-k (k = 1, 2, …, N − 1). Second, we tested the observed correlation coefficients at each lag against a distribution of correlation coefficients generated by randomly ordering observed numbers in the species trajectories. Results The patterns of autocorrelation functions for all but one of the observed species trajectories did not exhibit evidence of regulatory equilibrium, and, in fact, closely matched what would be expected from a non-stationary or ‘random walk’ process. The majority of the correlation coefficients generated from the observed species trajectories did not deviate significantly from correlation coefficients produced by the randomized trajectories. However, there was strong evidence of unusual positive autocorrelation at small time lags for birds on islands measured annually (2- to 4-year lags) and for arthropods on islands measured weekly (7- to 8-week lags), suggesting some degree of structure in change in species richness over time. Main conclusions The autocorrelation function patterns for all but one of the observed species trajectories showed various forms of non-stationarity. These types of patterns suggest that the numbers of species through time gradually wandered away from their initial sizes. Our model-independent test of individual correlation coefficients revealed significant structure in the observed species trajectories. These trajectories appear to be non-random at relatively short lag intervals, indicating a process with short memory.