Revealing the ghost in the machine: Using spectral analysis to understand the influence of noise on population dynamics

To manage populations, whether of threatened, exploited, or pest species, or even of parasites or pathogens in hosts, we must be able to predict their population dynamics, and to do this, we need to understand the underlying processes. An important recent focus is the way that deterministic density-dependent processes interact with random influences (“noise”) to create novel patterns in the dynamics. Such noise typically has two sources. First, environmental noise is driven by changes in extrinsic processes (such as climate and resource availability). Second, demographic noise is the random way individuals progress through their life history and arises through the “discreteness of individuality”: an organism can be male or female, dead or alive, but the “vital rate” (sex ratio, survival, fecundity) may be noninteger. In large populations, demographic stochasticity can be ignored because it is averaged across a large number of individuals. So, at any point in time or space, the population will typically be very close to the average, but in small populations, the deviation between the population average and the realized value at any particular time may be considerable. The “strength” of demographic noise is inversely related to the square root of population size, and a general question that is asked by Reuman et al. (1) in this issue of PNAS is “how does demographic noise of different strengths affect dynamics?”

Why Does Noise Matter?

This is a key question because recent studies have repeatedly indicated that noise can be an important influence on population dynamics and can interact with deterministic density dependence to give rise to population dynamics that are not solely “deterministic dynamics with noise” but that may be qualitatively different (2, 3). This qualitative difference between deterministic and stochastic dynamics may arise from two main causes.

First, noise can cause dynamical systems to move between different nearby dynamical attractors, either in state- or parameter-space. A well known example of this movement occurs if a system is close in parameter space to a bifurcation (such as where the dynamics change from equilibrium to cyclic). In this case, noise can “excite” the system such that the stochastic dynamics are those associated with the cyclic attractor (4, 5), so a deterministically equilibrium system may, with small amounts of environmental variation, show sustained and noisy cycles (5). Where there are nearby attractors that noise can move the system between, the resulting dynamics are likely to be a fusion of the characteristics of the attractors, e.g., with periods of equilibrium dynamics followed by periods of cycles (6, 7). Similarly, noise can move a system toward a point that may be both an attractor and a repellor. In this case, such as a saddle point, the attractor is unstable, and noise will cause the system to move away again (8). If one of the attractors is zero population size, i.e., extinction, then noise can simply cause the system to cross an extinction threshold (3).

Interaction between noise and deterministic processes is complex.

Second, noise also can cause a resonance whereby density-dependent interactions between cohorts introduce delays in the density-dependent regulation, such that long-term, multigenerational trends result (9, 10). The appearance of long-term declines in some populations, such as cod, may therefore need no explanation beyond the interaction of environmental noise and deterministic processes (9).

Because noise, whether environmental or demographic, is the way in which organisms' life histories, and thus a model's parameters, vary, dynamics may depend sensitively on the exact realization of noise. Even in exactly similar environments, demographic stochasticity will tend to cause replicated populations to exhibit different dynamics. In variable environments, the response of any population to a given environmental state will further depend on details of its age, stage, and/or physiological structure (11). Even superficially similarly structured populations may diverge radically over time, explaining why, in a spatially replicated system, the correlation between populations is always less than the correlation in the environmental noise (12, 13).

Understanding Dynamics

The interaction between noise and deterministic processes is therefore complex and of potentially large impact. To understand and predict population dynamics, we need to thoroughly understand this interaction. Simply taking a population model and “adding noise” can be instructive, but without ground-truthing the model or its predictions, one has to treat the model's results with caution. Biological understanding of the processes by which variation in life history, and therefore in demographic rates, translates into the population dynamics is necessary to ensure that models are biologically reasonable. Three broad approaches have been particularly helpful in investigating the biology of noise: (i) highly detailed, long-term, single-species population dynamics [e.g., ungulates such as Soay sheep and red deer (11, 14)]; (ii) analysis of replicated time series from different populations [e.g., diseases, such as measles (15), or small mammals (16)]; and (iii) detailed studies on laboratory model systems, where replicated population experiments are possible, allowing the iterated process of theoretical development and experimental test (see ref. 17 for a recent review).

Reuman et al. (1) present a study based on an empirical–theoretical investigation of that iconic laboratory model system for understanding nonlinear dynamics, the flour beetle, Tribolium castaneum (see chapter in ref. 17). The dynamics of this model system are well understood and described by a basic nonlinear model called the larvae–pupae–adult (LPA) model. By varying parameters in the laboratory (notably adult survival rates, which are easy to manipulate by the addition or subtraction of animals), it has been possible to test the LPA model's predictions and therefore demonstrate both theoretically and empirically a range of dynamical behaviors and phenomena. Much of the Tribolium work has concentrated on deterministic behavior (explaining and identifying the biological basis of cycling and chaotic dynamics, for example) by using experiments to inform and test the LPA model. More recently, the relationship between stochastic processes and the realized dynamics have been investigated by using a discrete version of the LPA (the lattice stochastic demographic version of the LPA; LSD-LPA) to acknowledge that populations are inherently based on integer arithmetic (18).

Spectral Analysis

An important tool in describing noisy population dynamics is spectral analysis. The power spectrum of a time series shows the frequencies at which the population fluctuates. Populations that fluctuate equally at all frequencies have a flat spectrum (termed a “white” spectrum by analogy with optics). Populations that fluctuate with predominantly low frequency have spectra dominated by low-frequency peaks (“red” dynamics), and vice versa for those dominated by high-frequency fluctuations (“blue” dynamics). The pattern of peaks and troughs across the spectral graph therefore is a complete description of any arbitrarily complex stationary time series. Spectral analysis of time series has recently become more common because of the growing recognition that a model may describe the data well in the time domain (e.g., one-step-ahead forecasting may work well) but may not fit the data well in the frequency domain (i.e., the models' output may not create time series with the same shaped spectra as the data), suggesting that models should be examined for fit in both domains. In addition, a common method of analyzing the behavior of deterministic models with noise is to assume that the noise creates only small perturbations, allowing linear approximation around the attractor (9, 19, 20). Making such small-noise assumptions and linearizing allows the analytical investigation of power spectra and therefore predictions of the way the deterministic dynamics will depend on characteristics of the noise and how it impacts the life history (10, 21). Although linear theory has proved very useful to our understanding of stochastic population dynamics, it has rarely, if ever, been tested in any meaningful way. This lack of test is partly because most ecological time series are necessarily rather short, so the spectrum is estimated with low statistical power and therefore has little resolution, making the comparison between theoretically derived and empirically derived spectra difficult.

The article by Reuman et al. (1) is important for three reasons. First, they develop the statistical tools to assess the frequency-domain fit between a parameterized model and experimental time series. When one has a well fitting model in both the time and frequency domains, the model can be used to generate many new time series of greater length than the empirical time series, allowing higher-resolution prediction of the spectrum. They demonstrate the “fitting and spectral enhancement” process by taking long time series, truncating them, fitting the model to the short time series in both domains, and then using the model to generate spectra based on time series of the original length and finally comparing the predicted and empirical spectra. Obviously, this technique depends crucially on initially having a mechanistic model that fits well, and this study benefits from having model-based understanding developed over several years. Second, having developed a model and tested its ability to fit and predict in the frequency domain, Reuman et al. (1) can, for the first time with confidence, vary properties of the noise and therefore look at the way that the spectrum varies. In this way, they have been able to describe how demographic noise alters the dynamics of a real system as the noise varies in “strength.” In low-noise systems, spectra tend to be more complex, with many small peaks in addition to a major peak. As the noise increases (as population size decreases), the number of subsidiary peaks changes, with some fading away or shifting location or strengthening (see figure 4 in ref. 1). The third advance is that, having developed a well fitting model, Reuman et al. (1) can use linear theory to see whether the analysis of the model predicts the spectral peaks observed in the data (or in the model's output); in this way, they can test whether linear theory works. They examine two cases and show in one case that linear theory predicts the properties of the time series well. In the other case, it does not. This latter case occurs because of stochastic jumping between different parts of the attractor, which changes the dominant frequency of the population fluctuations (and therefore the peak location).

There are two important “take home” messages from this article. First, as many modelers acknowledge, theory is good only as long as approximations are valid. This message is sometimes ignored because simplifying assumptions by definition simplify and often allow the use of analytical techniques that would otherwise not apply. Obviously, the utility of analytical models in providing general results is enormous, and we should not throw out the baby with the bathwater by being too cautious to use simplifying assumptions; however, we should remain aware that assumptions, often untested, underlie much theory. Second, this article is an example of an approach that advances our understanding by jointly involving statistical models, mechanistic analytical models, and experiments. Developing the models hand in hand with data means that the models can be tested and their results and predictions relied upon. This approach is therefore a productive route to understanding the highly complex drivers of population dynamics when noise and determinism interact.