Environmental variability and density dependence in the temporal Taylor's law

Taylor's law (TL) is an empirical rule describing the approximate relationship between the variance and the mean of population density: log10(variance) ≈ log10(a) + b × log10(mean). Although TL has been verified in various ecological systems, essential questions remain unanswered. Why is TL so widely observed? What mechanisms or processes generate TL? Why do most observed slopes b fall in the limited range 1 < b < 2? Density-dependent movement of individuals among populations has been proposed as a mechanism that leads to TL with slopes 1 < b < 2. We used the Gompertz model (a second-order autoregressive model of the logarithm of population density) to analyze the temporal TL of gray-sided vole populations. Our extensive simulations using various combinations of model parameters for environmental variability and density dependence demonstrated that sustainable populations could obey TL in the absence of density-dependent movement among populations, and identified the parameter combinations that produced slopes 1 < b < 2. When environmental variability was low and density dependence was intermediate, simulated data sets showed higher probabilities for 1 < b < 2, but the probability was not very high. In general, slopes became steeper (b increased) as environmental variability increased and as density dependence coefficients decreased. In the Gompertz model, both environmental variability and density dependence cause population density to vary, and on the logarithmic scale of population density, those effects are symmetric above and below the equilibrium density. However, effects of the variability are higher above the equilibrium density on the natural scale of population density, and thus the mean of population density increases with increasing population variability. Therefore, the temporal TL can be formed when population density is measured in the natural scale. In sustainable populations well described by the Gompertz model, the slope b can be determined by the interplay of environmental variability and density dependence.