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

Significance

We show that the variation of plant and animal populations through time can provide basic insights into how populations interact with their environment, even when environmental covariates are unobserved. We contrasted the effects of two types of temporal variation on population dynamics showing that fluctuations in the strength of competition among individuals can change a well-known scaling relation between the population variance and population abundances. The scaling of the population variance is explored using a large database of time series, as well as two well-studied ungulate populations. Our results suggest that higher-order variance scaling may be present in many animal populations, both reducing the long-run population variance relative to the standard model and providing important information on how populations are regulated by their environment.

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.

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 $R_t=\ln \left(N_t/N_{t-1}\right)$. 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 additive model may not accurately capture impacts of environmental variables that affect the strength of density dependence. Factors such as the amount or quality of available habitat may drive fluctuations in the strength of competition among individuals. This variability is multiplicative with abundance in the per-capita growth rate, instead of additive, and can cause fluctuations with a higher-order scaling in the population variance (7, 8).

Early theoretical work on stochastic population dynamics considered both additive and multiplicative models of environmental variability, although most recent efforts have ignored multiplicative models. Levins (1) showed that when the strength of density dependence in the logistic model fluctuates through time, the median population abundance is a weighted harmonic mean of the carrying capacities. In addition, he showed that increasing a population's reproductive rate will tend to lower abundances in this multiplicative model. Feldman and Roughgarden (7) followed this up by deriving the population variance under variation in the strength of density dependence. They show that this multiplicative model leads to a population variance scaling that is quartic, instead of the standard quadratic scaling that emerges when environmental variation occurs additively.

A more recent derivation using discrete-time models has shown similar variance scaling to earlier work in continuous-time models (8). The recurrence properties of discrete-time models with both additive and multiplicative environmental variation have been studied (9–11). This work has illustrated how multiplicative environmental variation can lead to growth–catastrophe behavior, characterized by periods of exponential growth followed by a switch back to density-dependent dynamics. Growth–catastrophe dynamics can lead to extreme population declines due to overcompensatory responses that occur when the strength of density dependence varies dramatically, such that periods of strong regulation follow periods of low regulation. Despite a long history of applying models of environmental variation to observations of animal populations, tests of the scaling of the environmental variance have been rare.

We are aware of only two empirical examples that have considered whether the environmental variance affects the per-capita growth rate additively or multiplicatively. Hanski and Woiwod (12) tested which model was a better description of moth and aphid population variation, concluding that the standard additive model was better for these populations. A more recent study by Fowler and Pease (13) on the perennial grass, Bouteloua rigidiseta, found that treating the carrying capacity as a random variable was better than the standard model of environmental variation. In contrast with these few empirical tests of the environmental variance scaling, a number of studies have linked environmental covariates to multiplicative interactions with abundances (e.g., refs. 14–20). This empirical work is suggestive that the additive model of environmental variance scaling may not always capture the important stochastic properties of animal populations.

Multiplicative environmental variation has likely been overlooked in the ecological literature due to several historical factors. Important early work on parameter estimation and testing (21) followed earlier theoretical developments in diffusion processes that relied on modeling environmental variability as an additive contribution to the per-capita growth rate (22). In addition, standard linear regression tools can be used to account for additive environmental variation (23) making this model available to most ecologists. However, the ongoing adoption of hierarchical modeling tools in ecology means that the estimation and testing of models with multiplicative environmental variation is now accessible to many practitioners.

Here, we show that multiplicative environmental variation is a common property of animal populations and that it can lead to more strongly bounded populations (sensu ref. 24) than otherwise predicted. We first derived a model of demographic and environmental variation for discrete-time unstructured populations with a general form of density dependence and compared the behavior of the population due to additive and multiplicative environmental variation. This is followed by a test of the empirical support for both environmental variance models in a large number of datasets from the Global Population Dynamics Database (GPDD) (25). We then used these model fits to examine the stochastic dynamics of populations with additive or multiplicative environmental variation through simulation. To test these models in scenarios where the population–environment interaction is well known, we performed an analysis of long-term time series of Alpine ibex (Capra ibex) and Soay sheep (Ovis aries). Previous studies (17, 20) have found that these populations interact with climate variation through the strength of density dependence. We hypothesized that the ibex and sheep populations would display higher-order variance scaling consistent with this climate–abundance interaction.

Models and Methods

We used a simple one-dimensional model of stochastic population growth, similar to past work (e.g., refs. 2, 8, 26). These models are analytically tractable, and provide the building blocks for more complex model structures. We first derived additive and multiplicative population variance models, then examined their properties around the population carrying capacity.

Model Properties.

We assumed that the population is unstructured and contains density-dependent survival following our previous work (27). The derivation uses a model structure that can capture a broad range of population dynamics. The random variable $\mathrm{\Lambda }$ describes fluctuations in the deterministic maximum rate of reproduction through time, whereas B describes fluctuations in the strength of density dependence. The deterministic analogs of these parameters are given by the mean values of λ and b, respectively. We use the convention that capital letters represent random variables and assume that the stochastic analogs of these parameters follow normal distributions,

$$\mathrm{\Lambda }\sim N\left(\ln \left(\lambda \right),{\sigma }_{\mathrm{\Lambda }}^2\right)$$ [1a]

$$B\sim N\left(b,{\sigma }_B^2\right).$$ [1b]

Our model for the abundances and per-capita growth rates is then defined as

$$N_t|\left(N_{t-1}=n_{t-1},\mathrm{\Lambda },B\right)=n_{t-1}\exp \left[\mathrm{\Lambda }-Bf\left(n_{t-1}\right)\right]$$ [2a]

$$R_t|\left(N_{t-1}=n_{t-1},\mathrm{\Lambda },B\right)=\mathrm{\Lambda }-Bf\left(n_{t-1}\right).$$ [2b]

The form of density dependence is determined by $f\left(n_{t-1}\right)$. In this article we examined Ricker $\left(f\left(n_{t-1}\right)=n_{t-1}\right)$, theta-Ricker $\left(f\left(n_{t-1}\right)=n_{t-1}^{\theta }\right)$, and Gompertz models of density dependence $\left(f\left(n_{t-1}\right)=\ln \left(n_{t-1}\right)\right)$, and all mathematical results hold for these forms of density dependence. It can be seen from Eq. 2b that random fluctuations in $\mathrm{\Lambda }$ are additive to $f\left(n_{t-1}\right)$, whereas fluctuations in B are multiplicative with $f\left(n_{t-1}\right)$. For example, when the form of density dependence is Ricker, additive variation in $\mathrm{\Lambda }$ is equivalent to a random intercept model in the per-capita growth rate, whereas the multiplicative variation in B is equivalent to a random slope term in the per-capita growth rate (Fig. 1).

Fig. 1.

Impact of environmental variation in the per-capita growth rate of the Ricker model. Additive variation in $\mathrm{\Lambda }$ (A) is a random intercept in the per-capita growth rate, whereas the impact of multiplicative variation in B (B) is a random slope term.

The expected value of the random variable describing the current population abundance, $N_t$, can be shown to be (SI Appendix)

$$\text{E}\left[N_t|N_{t-1}=n_{t-1}\right]=\lambda n_{t-1}{e}^{-bf\left(n_{t-1}\right)+{\sigma }_{\mathrm{\Lambda }}^2/2+{\sigma }_B^{2}f{\left(n_{t-1}\right)}^2/2}.$$ [3]

The full population variance, $\mathrm{Var}\left[N_t|N_{t-1}=n_{t-1}\right]$, is the demographic variance plus the contributions of the environmental components and is derived in SI Appendix. We use the following notation for convenience:

$$\begin{array}{c}\mathrm{Var}\left[N_t|N_{t-1}=n_{t-1}\right]={\mathrm{Var}}_{\mathrm{dem}}\left(n_{t-1}\right)+{\mathrm{Var}}_{\mathrm{\Lambda }}\left(n_{t-1}\right)+{\mathrm{Var}}_B\left(n_{t-1}\right),\end{array}$$ [4]

where the first term arises due to demographic stochasticity and scales by $n_{t-1}$. This is given by

$${\mathrm{Var}}_{\mathrm{dem}}\left(n_{t-1}\right)=\lambda n_{t-1}\exp \left(-bf\left(n_{t-1}\right)\right)\left(1-\exp \left(-bf\left(n_{t-1}\right)\right)\right)+{\sigma }_{\mathrm{dem}}^2{n}_{t-1}\exp \left(-2bf\left(n_{t-1}\right)\right).$$

The demographic variance has two pieces; the first corresponds to the binomial variation associated with survival and the second corresponds to variation in fecundity. A common assumption is that ${\sigma }_{\mathrm{dem}}^2=\lambda$, corresponding to a Poisson model for the demographic variance. The second term in Eq. 4 is the standard model of environmental variation that arises due to temporal fluctuations in $\mathrm{\Lambda }$. This contribution is additive to the per-capita growth rate and gives the following population variance:

$${\mathrm{Var}}_{\mathrm{\Lambda }}\left(n_{t-1}\right)\approx {\sigma }_{\text{Λ}}^2{\lambda }^2{n}_{t-1}^2\exp \left(-2bf\left(n_{t-1}\right)\right).$$ [5]

Finally, the third term in Eq. 4 arises due to environmental fluctuations in B, which yield an interaction between the environmental variance and the population abundance. This multiplicative environmental variation in the per-capita growth rate gives the following population variance:

$${\mathrm{Var}}_B\left(n_{t-1}\right)\approx {\sigma }_B^2{\lambda }^2{n}_{t-1}^{2}f{\left(n_{t-1}\right)}^2\exp \left(-2bf\left(n_{t-1}\right)\right),$$ [6]

which has the additional scaling of $f{\left(n_{t-1}\right)}^2$ compared with Eq. 5. For the commonly used Ricker model this scales by $n_{t-1}^4$ instead of the standard $n_{t-1}^2$ scaling given in Eq. 5. The environmental variance terms given in Eqs. 5 and 6 are approximations and the full expressions for these terms are given in SI Appendix. We present the approximations as they provide a better intuition for the scaling of the environmental variance than the full expressions. All inferences in the article were performed using the full variance expressions, although we found that the approximations in Eqs. 5 and 6 were consistent with the full expressions for the handful of datasets examined.

To compare the stability properties of the environmental variance models we examined the behavior of a population with no demographic stochasticity and with either additive or multiplicative environmental variation. Then at some value $n\text{*}$, the environmental variance in both models will be equivalent, such that ${\mathrm{Var}}_{\mathrm{\Lambda }}\left(n\text{*}\right)={\mathrm{Var}}_B\left(n\text{*}\right)$. This leads to ${\sigma }_{\mathrm{\Lambda }}^2={\sigma }_B^{2}f{\left(n\text{*}\right)}^2$ from Eqs. 5 and 6. The ratio of the variances can then be rewritten in terms of $n\text{*}$ as

$$\begin{array}{c}\frac{{\mathrm{Var}}_{\mathrm{\Lambda }}\left(n_{t-1}\right)}{{\mathrm{Var}}_B\left(n_{t-1}\right)}=\frac{{\sigma }_{\mathrm{\Lambda }}^2{n}_{t-1}^2\exp \left(-2bf\left(n_{t-1}\right)\right)}{{\sigma }_B^{2}f{\left(n_{t-1}\right)}^2{n}_{t-1}^2\exp \left(-2bf\left(n_{t-1}\right)\right)}\\ =\frac{f{\left(n^{\text{*}}\right)}^2}{f{\left(n_{t-1}\right)}^2}.\end{array}$$ [7]

Eq. 7 suggests that if $n\text{*}$ is greater than the median, then populations will spend much of their time in low variance states relative to predictions by the standard model of environmental variation. If instead the population spends much time above $n\text{*}$, or occasionally reaches extremely high population sizes, then multiplicative variation may be destabilizing. This potential for destabilization has been described in previous work, which showed that populations with multiplicative environmental variation can go extinct from large fluctuations in B due to these growth–catastrophe dynamics (9, 10). The potentially stabilizing effect of multiplicative environmental variation has not been discussed in the literature, to the best of our knowledge.

Data and Time Series Analysis.

Empirical analyses were conducted using a one-step time series analysis. We fit models to data assuming that transitions from one time step to the next are log-normally distributed. The mean and variance for the transition distribution were given by Eqs. 3 and 4 with the density dependence form depending on the particular time series analysis, described below. We performed all optimizations in the R statistical environment (28) using the rgenoud package (29) using a population size of ${10}^4$.

We curated 165 high-quality time series from the GPDD to examine the general properties of environmental variance scaling in animal populations. Our criteria for the inclusion of a time series into our analysis were that the dataset had a minimum of 15 observations, no zero abundances, a qualitative reliability rating of 4 or 5 out of 5, and we did not include datasets in the analysis that indicated sampling was due to harvesting as these may not accurately reflect underlying population trends. We provide further details on these series in SI Appendix.

For each time series selected from the GPDD we fit four models per dataset. The models were a Gompertz density dependence model with additive variance, one with multiplicative variance, one with both terms together, and a density-independent growth model. All models were also assumed to have Poisson demographic stochasticity and a moving average term to account for observation error (4). Previous work has shown that density dependence in the GPDD is mostly concave (30, 31), consistent with the Gompertz form. Out of the four models fit to each time series, the best was selected using the Bayesian information criterion (BIC). This analysis of the GPDD allowed us to determine which environmental model, if any, is a suitable default when building stochastic population models.

For all models selected to have density dependence, we calculated the point where the additive and multiplicative variances are equal, denoted as $n\text{*}$ in Eq. 7. This was estimated as $\widehat{n}\text{*}=\exp \left(\sqrt{{\widehat{\sigma }}_{\mathrm{\Lambda }}^2/{\widehat{\sigma }}_B^2}\right)$ using estimates of the environmental variance components from the additive and multiplicative models, respectively. We then tested whether $\widehat{n}\text{*}$ was greater than the observed population median using a binomial exact test. This allowed us to test whether populations with multiplicative variation were spending the majority of their time in relatively low variance states.

To test the strength of regulation of the different variance models we simulated abundances for all populations in the GPDD that were selected to have density dependence. We initiated each simulation at the first observed abundance for each time series and projected future populations with growth, density dependence, and environmental variation using estimated parameters for each model. In the simulations of the multiplicative variance we assumed that random draws of B came from a lognormal distribution with the estimated mean and variance values to constrain draws to the known support of the random variable. Simulated time series were of length ${10}^4$ and for each simulation we calculated the coefficient of variation (CV) as a measure of the long-term boundedness of a population, where lower values correspond to more stochastically stable populations (32). We tested that the CV of data generated with multiplicative variance was lower than the CV of the those generated with additive environmental variation using a binomial exact test.

To further explore the applications of these environmental variance models we used two additional high-quality abundance time series of Alpine ibex and Soay sheep. Previous work on these ungulate populations has linked climate covariates to their carrying capacities; given this interaction with the environment we predicted that each population would display multiplicative environmental variation in B, rather than additive variation in $\mathrm{\Lambda }$. In the Alpine ibex, two previous studies (16, 20) found that mean winter snow depth interacted with abundance in the per-capita growth rate. This was found to occur both in the overall population abundances and in most of the stage and sex classes. We used the same dataset as ref. 20, which was provided by the authors. The Soay sheep dataset for 1995–2010 was obtained by digitizing the time series plot available at soaysheep.biology.ed.ac.uk/population-ecology using Engauge software (33). Previous work on these data has linked the population carrying capacity to the severity of winter weather (17), although many other studies have also examined the interaction between this population and the environment (e.g., refs. 15, 34, 35).

In the analysis of these high-quality time series we followed similar methods as previous work, although we did not incorporate climatic covariates into our analysis. For the Alpine ibex we used the analysis of ref. 20 as a reference, whereas for the Soay sheep we used ref. 17. Previous work (20) assumed that the Alpine ibex followed a Ricker model of density dependence for the total population and for all stage and sex classes. We analyzed the total population abundances between 1956 and 1985, ignoring later dates due to the nonstationary perturbation occurring after 1985. We also present an analysis of total abundances of the full Alpine ibex dataset, 1956–2008, as well as each stage class treated independently in SI Appendix. For the Soay sheep analysis we did not include the data from 1985 to 1994 in the main analysis as previous work has found that a breakpoint in 1994 led to different population dynamics before and after this year (17). Following ref. 17 we used a theta-Ricker model of density dependence for this analysis. We fit models with all combinations of the environmental and demographic variance components to each ungulate time series and used the BIC to perform model selection. In addition, we tested whether including a moving average term improved model fits. In SI Appendix we present the results of an analysis of the Soay sheep dataset using a breakpoint model from 1985 to 2010. Our results from this analysis are consistent with those presented here in the main text.

Results

We found that 26% of the GPDD time series had minimum BIC values for the density-independent model. These results are not too surprising given the difficulty past work has found in statistically detecting density dependence in the GPDD (36). Although we expect that all populations display density dependence over some range of abundances, samples over a limited range may not statistically detect it. This can be due to low sample size, influences of delayed density dependence, convexity in population growth rates, and measurement error.

Due to the potential difficulties in selecting between the environmental variance models (37), we used a simulation study of the model selection procedure, similar to previous work (26), and found that models with a $\mathrm{\Delta }$ BIC $\ge 2$ between the top two ranked models led to much more reliable model selections than simply using the best overall model (SI Appendix). Out of the 45 (27% of the 165 series) density-dependent time series that met this criterion 53% were better explained with additive environmental variation in $\mathrm{\Lambda }$, 40% were best explained by multiplicative variation in B, and 7% were selected to have variation in both parameters (Fig. 2). In most populations it is expected that both forms of environmental variation are likely to occur. However, when populations are not observed over a sufficient range of abundances and time series are not long enough it will be difficult to statistically distinguish these terms.

Fig. 2.

BIC selections of the best environmental variance models in the GPDD time series that displayed density dependence. Additive environmental variation corresponds to variation in $\mathrm{\Lambda }$, multiplicative environmental variance corresponds to variation in B, and additive + multiplicative variation corresponds to both models. The height of the black bars are the number of selections made with $\mathrm{\Delta }\mathrm{BIC}<2$ and the blue region corresponds to the number of selections made with $\mathrm{\Delta }\mathrm{BIC}\ge 2$.

We also performed simulations to determine the robustness of our analysis to the form of density dependence (SI Appendix) and found that the proportion of models selected with multiplicative variance is sensitive to the form of density dependence used. Our simulation found that as the concavity of the density dependence increases, the proportion of datasets selected as having multiplicative variance increases. Because we used the Gompertz model for our analysis, a concave function that previous work has identified as the most common pattern in the GPDD (31), our estimates are likely to be accurate despite this sensitivity to model structure.

In our estimate of $n\text{*}$, the point at which multiplicative and additive variation are equal, we found that a majority (67%, P value $=0.002$, $n=122$) of populations had an $\widehat{n}\text{*}$ greater than the estimated population median and the ratio of $\widehat{n}\text{*}$ to the estimated population median had a median value of 1.14 (all values presented in SI Appendix). Out of populations where the multiplicative model had the minimum BIC, the proportion went up to 80% (P value $<{10}^{-6}$, $n=66$) and the ratio of $\widehat{n}\text{*}$ to the estimated population median had a median value of 1.54. This suggests that populations with multiplicative environmental variation will spend the majority of their time in states that have lower variance than predicted by the standard model. A likely consequence of this is that many populations will be more strongly bounded than previously predicted.

In our comparison of the CV of GPDD time series with multiplicative or additive variability, we found that 70% (P value = $9.7\times {10}^{-7}$, $n=122$) of the populations generated with multiplicative environmental variation in B had the lower CV and the median of the ratio of the CVs (${\mathrm{CV}}_B/{\mathrm{CV}}_{\mathrm{\Lambda }}$) was 0.93 (Fig. 3). Therefore, populations with variation in B are predicted to be more strongly bounded than populations with additive variation in $\mathrm{\Lambda }$. However, we also found that several of the time series generated with environmental variation in B had significantly higher CVs than the complementary series generated with environmental variation in $\mathrm{\Lambda }$ (Fig. 3). Simulations of these time series showed dynamics qualitatively similar to the growth–catastrophe phenomenon described in previous work (9–11). This suggests that is possible to achieve growth–catastrophe dynamics, although they may be a relatively rare phenomenon.

Fig. 3.

CV under multiplicative and additive environmental variation for each dataset with density dependence. Relatively high CVs for the multiplicative models correspond to growth–catastrophe dynamics.

Our analysis of the Alpine ibex and Soay sheep time series found positive evidence that both populations were better modeled with multiplicative environmental variation in B rather than the standard model of environmental variation in $\mathrm{\Lambda }$ (Table 1). This supports our predictions about the scaling of the environmental variance, based on previous work that linked fluctuations in these populations to climatic variables. Furthermore, it indicates that basic variance models can be used as a diagnostic tool to determine how populations interact with their environment, an important consideration for researchers trying to link population growth to environmental factors.

Table 1.

BIC values for model selections of Soay sheep and Alpine ibex datasets

Population	Years	Autocorrelation	${\mathrm{Var}}_{\mathrm{\Lambda }}\left(n_{t-1}\right)$	${\mathrm{Var}}_B\left(n_{t-1}\right)$	${\mathrm{Var}}_{\mathrm{\Lambda }}\left(n_{t-1}\right)+{\mathrm{Var}}_B\left(n_{t-1}\right)$
Alpine ibex	1956–1985	MA(0)	429.14	425.14	428.54
Alpine ibex	1956–1985	MA(1)	431.82	437.54	436.64
Soay sheep	1995–2010	MA(0)	234.57	230.56	234.14
Soay sheep	1995–2010	MA(1)	235.90	233.88	235.51

Bold values are the minimum for each dataset.

The estimated population variance in the ibex and sheep populations is given in Fig. 4. The predicted variance when the environment interacts multiplicatively with abundance variance is lower than the additive environmental model for the majority of observed population states, suggesting that the multiplicative model tends to stabilize these populations as in the majority of the GPDD populations.

Fig. 4.

Predicted population variance for the Alpine ibex and Soay sheep under alternative environmental variance models. The black line corresponds to additive environmental variation that occurs in $\mathrm{\Lambda }$, whereas the red line corresponds to multiplicative environmental variation that occurs in B. Horizontal dashes on the x axis denote the observed abundances and the horizontal dashed lines represent the estimated carrying capacities under the different variance models. Following the predictions of Eq. 7, the multiplicative variance is less than the additive variance at low abundances and higher at high abundances.

Discussion

Our analysis has shown that higher-order variance scaling can have consequences that affect population predictions, often promoting stability by constraining populations to spend more time in lower variance states than predicted by the standard model of environmental variation. However, populations may also be destabilized with this model through previously described growth–catastrophe dynamics (9–11). Our empirical results suggest that growth–catastrophe dynamics are rare and that multiplicative environmental variance will most often stabilize dynamics.

We have treated these variance models as arising from particular ecological mechanisms. The addition of complexity not accounted for in our models can lead to residual patterns that may also be captured by these variance models. For instance, the effects of age structure on the population variance will likely depend on whether generations are overlapping or not. Nonoverlapping generations will lead to cycles that will be accounted for by multiplicative variation. Using the wrong model will lead to these dynamics being mistaken for temporal changes in the strength of density dependence similar to growth–catastrophe dynamics. In general, periodic dynamics may be treated as multiplicative variation due to the qualitative similarities between cycles and changes in the strength of density dependence. The effects of nonoverlapping generations are often accounted for by introducing higher-order lag terms into the density dependence (38) and this case may favor the traditional model of environmental variance, as lag terms that are not accounted for will appear as additive terms in the per-capita growth rate. Whether the population variation occurs due to basic mechanisms described in our model derivation or through other population processes must be dealt with by taking into account what is known about the population’s ecology.

Despite the difficulties of identifying biological mechanisms with observational data, we showed that these variance models can serve as tests of the population–environment interactions without specific knowledge about the state of the environment. There is reason to think that fluctuating environmental resources will often lead to variation in the strength of density dependence; therefore, higher-order variance scaling should be a consideration for practitioners when performing population viability assessments. This can lead to improvements in predictions of population abundance and increased understanding of the role of population–environment interactions on regulation.