Mean and variance of densities of stochastic, density-dependent, stage-structured populations

Background/Question/Methods

The relationship between the mean and variance of population density were investigated using stochastic, density-dependent, stage-structured population models. Understanding the relationship between the mean and variance of population density is important for at least two reasons. First, temporal Taylor’s Law states that the natural logs of the mean and variance are linearly related, and this has been demonstrated by some theoretical studies and supported by some empirical observations. However, it has not been investigated with stochastic, density-dependent, stage-structured population models. Second, it is common in ecological studies to use the coefficient of variation (CV) as proxies for population fluctuation and instability. However, the validity of these uses is questionable unless we understand the relationship between the mean and variance of population density.

The models included either compensatory or over-compensatory density dependence affecting either fertility or juvenile survival. Stochasticity affected either juvenile survival (when fertility is experiencing density dependence) or fertility (when juvenile survival is experiencing density dependence). Wide ranges of life history parameters were investigated, but under all cases, the variance of the stochastic term was kept constant.  

Results/Conclusions

The results suggest the relationship between the mean and variance of population density is very complex. First, the relationship between the natural logs of mean and variance can be non-linear. The tendency for non-linearity is high when density dependence is over-compensatory. Second, the relationship is affected by the choice of the stage that we observe. For example, the variance of adult density may increase with its mean while the variance of juvenile density of the same population may decline with its mean, or vice versa.  Third, the order of stochasticity and density dependence that individuals experience matters. Because under all cases, the equilibrium points of equivalent deterministic models are stable, the stochasticity is the cause of fluctuation. However, density dependence can attenuate (common) or magnify (under some parameters with over-compensatory density dependence) the fluctuation. I conclude that (1) the temporal Taylor’s Law is supported under some situations, but it is not always supported and (2) the CV of population density is neither a measure of fluctuation nor the instability of a population.