Highly Variable Spread Rates in Replicated Biological Invasions: Fundamental Limits to Predictability

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Science  18 Sep 2009:
Vol. 325, Issue 5947, pp. 1536-1539
DOI: 10.1126/science.1176138


Although mean rates of spread for invasive species have been intensively studied, variance in spread rates has been neglected. Variance in spread rates can be driven exogenously by environmental variability or endogenously by demographic or genetic stochasticity in reproduction, survival, and dispersal. Endogenous variability is likely to be important in spread but has not been studied empirically. We show that endogenously generated variance in spread rates is remarkably high between replicated invasions of the flour beetle Tribolium castaneum in laboratory microcosms. The observed variation between replicate invasions cannot be explained by demographic stochasticity alone, which indicates inherent limitations to predictability in even the simplest ecological settings.

Spatial spread has been the subject of intense study by ecologists (1), beginning with classic cases like introduced muskrats in Europe and oaks in the United Kingdom (2). This work has provided the basis for attempts to control invasive species ranging from zebra mussels to many agricultural pests (3). In all cases, prediction of spread rates is a key question. There are two fundamental ways of predicting spread (1), on the basis of either estimating parameters in a model (4, 5) or extrapolating from initial observations of spread or spread in another location (6). In both cases, obviously, questions of uncertainty or randomness are key aspects of prediction. These questions of the role of uncertainty in ecological systems and its implication for prediction are of fundamental importance not only for the specific case of spatial spread, but for more general issues of management (7, 8). By focusing on the role of uncertainty in understanding spatial spread, we obtain insights into general questions of importance for management of ecological systems.

Stochastic models are needed to assess uncertainty in the rate of spread because traditional deterministic models provide no information about variance. Analyses have shown that linear stochastic models have the same rate of spread as the equivalent deterministic model, whereas stochasticity slows spread in nonlinear (density-dependent) models (9, 10). Investigations of the variance of spread rate have largely relied on numerical simulations and suggest that variance between realizations can be high, especially when birth rates are low, variation in birth rates between individuals is high, or dispersal includes infrequent long-distance events (1115). This work suggests that great uncertainty in an invasion forecast can be expected because of inherent biological variability.

Empirical documentation of variance in the rate of spread is scarce, hampered by the difficulty of replicating an invasion: Any invasion in nature is only one replicate, or realization, of the stochastic process. Some insights can be gained from natural pseudoreplicates in space or time. For example, many studies have found considerable variance in the rate of spread in different directions from a single point of release, such as for rabies or muskrats (4, 16). Similarly, childhood diseases such as measles show much variation in the rate of spread between successive outbreaks in the same location (17). However, in purely observational cases such as these, it is problematic to determine whether variance in spread rate is driven by inherent biological stochasticity or instead exogenously, such as by environmental factors that differ between the pseudoreplicates. To properly document variance due to biological processes, replicates need to experience identical external conditions. Tantalizing results from experimental epidemics of a fungal disease on radish seedlings in replicated laboratory microcosms suggest that variance in spread rate driven by biological stochasticity can be high (18, 19), although in these experiments replication was low and systematic differences between replicates occurred and were important despite tight control of the ambient environment (13).

We studied variance in spatial spread in highly replicated experimental microcosms (20) using the red flour beetle, Tribolium castaneum. Each microcosm was a one-dimensional landscape consisting of discrete habitat patches linked together by holes drilled into the sides of adjacent patches. We introduced beetles to one end of the landscape and documented spread over multiple nonoverlapping generations with discrete growth and dispersal phases. This is an ideal system for studying variance in spread. First, the Tribolium model system has a long history in ecology, so the processes underlying growth and dispersal are well understood, mathematical models are well developed, and experimental protocols are well established (2126). Second, conditions are tightly controlled in environmental chambers; this constraint minimizes the effect of external (environmental) factors and isolates the effect of biological stochasticity on the variance between replicates. Third, populations were censused completely, with negligible measurement error, which allowed us to investigate demographic stochasticity (23). Finally, both types of prediction can be done. We either forecast by using independent estimates of model parameters from separate experiments to establish growth and dispersal parameters, or by extrapolation from either the initial data from a particular experimental landscape or another experimental landscape.

We collected data for 30 replicate landscapes, composed of identical patches with standard medium, maintained under identical environmental conditions, and inoculated with 20 adult T. castaneum from the same stock culture. Even though the landscapes were putatively identical replicates, there was considerable variance in the distance spread among landscapes (Fig. 1). Furthermore, there was considerable variance in the spatiotemporal dynamics of abundance among landscapes (Fig. 2 and movie S1). We found that the variance, and hence uncertainty, in the distance spread increased with time (Fig. 3): By the end of the 13-generation experiment, the distance spread ranged from 10 to 31 patches, a threefold difference (Fig. 1). This demonstrates that intrinsic biological variability makes an extremely large contribution to the variance in spatial spread. We next examine the consequences of this intrinsic variability for the uncertainty in ecological forecasts using different approaches to prediction.

Fig. 1

Spatial spread of T. castaneum in 30 replicate landscapes through 13 generations. Landscapes were started with 20 adult beetles in patch 1, the leftmost end of the landscape. Distance spread is the number of patches spanning from (and including) patch 1 to the farthest forward occupied patch. Points on the y axis are perturbed slightly from integer values to see overlapping points.

Fig. 2

Spatiotemporal dynamics of abundance and spatial spread of T. castaneum in 30 replicate landscapes. Each panel is a separate landscape. Landscapes were started with 20 adult beetles in patch 1, the leftmost end of the landscape. Patches are numbered from 1 (leftmost) to 31 (rightmost). Lines show the dynamics of the advancing invasion front, moving from left to right through 13 generations, and show the number of beetles in each patch in each generation. The red line is the final state of the invasion front (generation 13). A time-series animation of these data is available in movie S1.

Fig. 3

Uncertainty in distance spread for the ensemble of replicate landscapes and for different forecast methods. Uncertainty was measured as the 95% interval width, calculated as the difference between the 2.5% and 97.5% quantiles. (Key) Data, the experimental data. Stochastic model, parameterized using independent experimental data on growth and dispersal. Pairwise, extrapolation from one landscape to another; interval width is for the prediction residuals (see fig. S5). Extrapolation, linear extrapolation from the first seven generations; interval width is for the prediction residuals (see fig. S4).

One approach to ecological forecasting often applied to spatial spread is to estimate the parameters of population models using demographic observations separate from the dynamical (or spread) data, for example, observations of births, survival frequencies, and dispersal distances from mark-recapture studies (4, 5). To emulate this approach, we first derived models for the Tribolium microcosms and estimated parameters in independent experiments.

Spread models generally consist of two submodel components: local population growth and dispersal (1). We previously derived models for within-patch population growth, which comprehensively account for sources of stochasticity in the life cycle, including demographic stochasticity (in births, density-dependent mortality, and density-independent mortality), environmental stochasticity (variation in the growth rate between patches and in time), demographic heterogeneity (variation in the birth rate between individuals within a patch), and stochasticity in the sex of offspring (27). Biological and stochastic parameters of these models were estimated by using data from a growth experiment and show that demographic sources of stochasticity make an important contribution to variance in the number of offspring in the next generation (27).

Here, we derive models for dispersal that are closely matched to the experimental system (20). These include a standard discrete space diffusion model and variants allowing for stochasticity in dispersal rates between pairs of patches and between individual beetles, the latter giving rise to dispersal kernels with “fatter tails.” We fitted the dispersal models to data from a replicated dispersal experiment (20). These results show that variation in dispersal rates between pairs of patches was important, which enhanced dispersal variability, whereas there was little evidence for an important role of between-individual variability in dispersal rates (table S1, fig. S1).

We used the full stochastic spatial model, combining the best-fitting, independently estimated growth and dispersal submodels, to forecast population spread. This standard approach to ecological forecasting slightly overestimated the mean rate of spread observed in the data but, more important, dramatically underestimated the variance in spread rate among landscapes (Fig. 3 and fig. S2). This is surprising, as the model was designed to be comprehensive in accounting for sources of stochasticity. Thus, the experimental data suggest that an important source of stochasticity is missing from the standard spread models.

The second approach to forecasting is extrapolating from initial observations of spread, or more generally from observations at another time or place. Apart from some special cases, such as Allee effects (28) and fat-tailed dispersal kernels (29), most biological scenarios lead to a linear increase in the distance spread (30). For each experimental landscape, we fitted a linear regression model for the first 7 generations to forecast the distance spread in generations 8 to 13. This approach seriously underestimated the uncertainty in the forecasts: The observed trajectories of many experimental landscapes exceeded the 95% prediction interval (fig. S3). In contrast to the naïve regression estimates of uncertainty, the forecast error directly measured from the ensemble of experimental landscapes was very high (Fig. 3 and fig. S4). By generation 13, the 95% interval width was 16, which is three-quarters of the ensemble mean distance spread (Fig. 3). Even higher forecast error was observed when the estimated spread rate from one landscape was used to predict the distance spread in another (Fig. 3 and fig. S5). Clearly, forecast uncertainty due to biologically generated variability is remarkably high, and it was only possible to measure this with replicated experimental invasions.

What, then, is the source of the unaccounted for and biologically generated variance in spread rates? The extra variability observed in the experimental ensemble might be accounted for by systematic biological differences between landscapes, such as those caused by founder effects. A hierarchical model fitted to the full spatiotemporal abundance data from the spread experiment (20) suggests that systematic variation in growth, cannibalism, and dispersal between landscapes was moderately high (Table 1). The hypothesis that all landscapes had the same (or similar) growth, cannibalism, and dispersal rates was rejected, because lower 95% confidence limits for the variation parameters (σR, σα, σD) were above zero (Table 1). This variation is unlikely to have been caused by external factors, such as environmental differences between landscapes, because these were tightly controlled. One explanation is that the random draw of the small founding population (20 individuals) from the stock beetle culture resulted in essentially different and randomly determined initial conditions for each landscape. Although further experimental work is needed to verify these model-based conclusions, variability in founders is clearly relevant not just to our experiments, but to almost all natural introductions as well, especially those repeated or where spread is driven by long-range dispersal and establishment of small outlying populations (14, 31).

Table 1

Fit of spatiotemporal model to test founder-effect hypothesis. The estimated parameters were Embedded Image, Embedded Image, and Embedded Image, respectively, the mean density-independent per capita growth rate (per 48 hours), mean density-dependent parameter, and mean diffusion rate among landscapes; σR, σα, and σD, respectively, the standard deviation of R, α, and D among landscapes; kNB, the dispersion parameter of the negative binomial distribution. Confidence intervals were obtained from quantiles of the posterior simulations.

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Our results show that understanding and predicting spatial spread, as well as prediction in ecology more generally, must take into account considerable uncertainties. Prediction in ecology is becoming increasingly important, partly because of anthropogenic changes (8). Using insights only possible from replicated experiments under controlled conditions, we have suggested that even without exogenous sources of variability, fundamental uncertainties arise on account of inherent differences among individuals and small population sizes. These issues are ones that are central to questions of spread, more generally to invasive species or range shifts under environmental change, and still more generally to ecological prediction. Given the need for prediction both as a tool for ecological understanding and in public-policy decisions and management, it is important to understand both the magnitude and cause of uncertainty. These uncertainties are large, but our work here and by others (11, 13, 32) provides ways to estimate the magnitude of these uncertainties, which allows for estimates of errors in prediction.

Supporting Online Material

Materials and Methods

SOM Text

Figs S1 to S6

Table S1


Movie S1

References and Notes

  1. Analytic solutions for some special cases are provided in (14).
  2. Materials and methods are available as supporting material on Science Online.
  3. We thank M. Gibson, D. Hodgkiss, C. Koenig, T. McCabe, D. Paulus, D. Smith, N. Tcheou, R. Villalobos, and M. Wu for assistance. This study was funded by National Science Foundation grant DEB 0516150 to A.H. and B.A.M. and by NSF Biological Invasions Integrative Graduate Education and Research Traineeship (IGERT) DGE 0114432.
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