When can environmental variability benefit population growth? Counterintuitive effects of nonlinearities in vital rates

Using models for unstructured populations, we investigate the effect of environmental variability on population growth when the environment affects vital rates through nonlinear functions. We focus here especially on interannual variation in food resources availability, for which sigmoid functions are relevant. Considering first unregulated populations in stochastic environments, we show that classic sigmoid annual growth rates cannot lead to positive effects of increased environmental variability on population growth. This is true even when the temporal average of food availability is low, and Jensen’s inequality predicts an increased arithmetic mean of the annual growth rate. The result is due to the log-concavity of many sigmoid (and other accelerating) functions, as convexity of the logarithm of the annual growth rate is needed for positive effects of variability to appear. Then, separating the effects of a food availability variable on reproduction and survival rates, we show that populations with less sensitive survival rate to food are more likely to benefit from food variability—as opposed to populations that have survival rates accelerating with food availability, which is rather counterintuitive given Jensen’s inequality. Again, this is explained by log-convexity properties of nonlinear functions. We further extend these results to regulated populations, in which similar positive effects of food variability can affect average population size. Positive variability effects seem however more likely to occur in regulated populations. Finally, we extend our results to stage-structured populations. We connect to the previous work showing positive effects of environmental variability with matrix models, and show that these effects are well captured by simpler unstructured models.