Evidence and implications of higher-order scaling in the environmental variation of animal population growth

Abstract

Environmental stochasticity is an important concept in population dynamics, providing a quantitative model of the extrinsic fluctuations driving population abundances. It is typically formulated as a stochastic perturbation to the maximum reproductive rate, leading to a population variance that scales quadratically with abundance. However, environmental fluctuations may also drive changes in the strength of density dependence. Very few studies have examined the consequences of this alternative model formulation while even fewer have tested which model better describes fluctuations in animal populations. Here we use data from the Global Population Dynamics Database to determine the statistical support for this alternative environmental variance model in 165 animal populations and test whether these models can capture known population-environment interactions in two well-studied ungulates. Our results suggest that variation in the density dependence is common and leads to a higher-order scaling of the population variance. This scaling will often stabilize populations although dynamics may also be destabilized under certain conditions. We conclude that higher-order environmental variation is a potentially ubiquitous and consequential property of animal populations. Our results suggest that extinction risk estimates may often be overestimated when not properly taking into account how environmental fluctuations affect population parameters.time series | environmental variance | variance scaling | population viability analysis | stochastic model A key question for ecologists is determining how environmental fluctuations drive population variability. Stochastic models of population dynamics consider environmental fluctuations as temporal perturbations to the mean of the birth and death rates of individuals in a population (1, 2). Because specific information on environmental covariates is not required in these models, the approach allows ecologists to make significant progress understanding how environmental perturbations drive population variability. These models have also proven to be useful in empirical settings where a stochastic process model is combined with an observation model to construct a likelihood of the observed abundances that can then be used to make inferences about the processes underlying the population (3, 4). A potentially important aspect of these models is the specification of how environmental forces drive variation in model parameters.Current practice assumes that environmental variation occurs as an additive term in the log of the per-capita growth rate, defined as R t = ln ðN t =N t−1 Þ. This variation can be derived by assuming that the density-independent reproduction rate is a random variable, which leads to a population variance with the well-known quadratic scaling of the population variance on abundances (e.g., refs. 1, 5). This model will capture the effects of environmental factors, like temperature, that can affect the maximum reproductive rate of individuals (6). However, the additi...

An updated perspective on the role of environmental autocorrelation in animal populations

“…The Gompertz model is given by $$N\left(t\right)=N(t-1)\exp [a+b\ln \left(N(t-1)\right)+E\left(t\right)],$$ and the Ricker model is $$N\left(t\right)=N(t-1)\exp [a+bN(t-1)+E\left(t\right)].$$ Here N ( t ) is the abundance metric at time t , a is the maximum per capita rate of increase, b is the strength of density dependence, and the E ( t ) is the environmental perturbation which is usually assumed to affect the rate of increase and is therefore typically modeled as independent and identically distributed ( iid ) draws from a normal distribution with mean 0 and where σ 2 is the environmental variance in the per capita growth rate (Ferguson and Ponciano, 2015). We have not included the effects of demographic stochasticity in the variance model E ( t ).…”

Integrating abundance and diet data to improve inferences of food web dynamics

“…Populations are subjected to process error at each time step, Z ( t ), drawn from a standard normal distribution scaled by σ, the standard deviation. This error can be interpreted as temporal variation in the reproductive rate (Ferguson & Ponciano, ). The term, , is the probability that an individual in the prey population does not escape consumption, while the efficiency of converting prey to new enemies is given by ε.…”

Ecological change points: The strength of density dependence and the loss of history

“…Another possibility, however, is to assume temporal randomness in the strength of density dependence. This approach assumes that the environmental forces shaping the population dynamics affect the intensity of density-dependence rather than the maximum growth rate and to date, remains a rarely explored model 150 alternative (but see [45,46].…”