Autocorrelated Environmental Variation and the Establishment of Invasive Species

Autocorrelated environmental variation and the establishment of invasive species

Kim Cuddington and Alan Hastings

K. Cuddington (http://orcid.org/0000-0003-1191-6973)([email protected]), Dept of Biology, Univ. of Waterloo, Waterloo, ON N2L 3G1, Canada. – A. Hastings, Dept of Environmental Science and Policy, Univ. of California – Davis, Davis, CA 95616, USA.

Environmental parameters such as temperature and rainfall have a positively autocorrelated variance structure which makes it likely that runs of good or bad conditions will occur. It has previously been demonstrated that such autocorrelated environmental variance can increase the probability of extinction in small populations, in much the same way that increased variance without autocorrelation can increase extinction risk. As a result, it has also been suggested that positive autocorrelation will decrease the probability that a species will establish in a novel location. We suggest that describing the probability of invasion success as the probability of indefinite persistence may be an inappropriate definition of risk. Economic or ecological damage may be associated with a population that initially reaches high densities before going extinct in the new location. In addition, such populations may spread to new locations before extirpation. We use a modeling approach to examine the effect of positively autocorrelated conditions on the probability that small populations will reach large size before extinction. We find that where variance is high and the geometric mean of the population growth rate is low, autocorrelation increases the risk that a population will pass a an upper threshold density, even when extinction probability is unaffected. Therefore species classified as having low probability of invasion risk on the basis of population growth rates measured in low variance environments may actually have quite a substantial probability of establishing a large population for a period of time. The mechanism behind the effect is the disproportionate influence of short runs of good conditions initially following introduction.

Abiotic conditions in a new environment fundamentally the sudden increase in availability of a limiting resource. influence the establishment probability and spread rates of Johnstone (1986) proposes a similar “invasion window” non-native species. More common approaches to determin- theory: opportunities for invasion change in time in response ing the effect of abiotic conditions on potentially invasive to removal of obstacles. Data both support (Davis and Pelsor species depend on matching average climate conditions in 2001, Foster and Dickson 2004) and refute these claims the native range to those in the new location (Drake and (Gilbert and Lechowicz 2005, Walker et al. 2005). For exam- Bossenbroek 2004, Roura-Pascual et al. 2006). Typically, ple, according to Davis and coauthors, periodic increased factors such as minimum average winter temperatures or availability of water should lead to increased invasion suc- minimum average precipitation are targeted as important cess in dry climates. In support of this theory, Hobbs and predictors (Evans and Gregoire 2007). However, the Mooney (1991) found that the probability of Bromus mollis temporal variability of abiotic conditions is also likely to establishment in California serpentine grasslands increased have a large impact on invaders. We use the term temporal in the presence of local gopher disturbance and in years environmental variation to refer to the year to year or day following heavy rains, but it decreased dramatically after two to day variability in environmental conditions that occurs consecutive years of drought. In contrast, Brown and Archer separately from the effects of predictable cyclic variation (1999) found that recruitment of the invasive honey mes- such as seasonal or El Niño events, and which is also dis- quite, Propsis glandulosa, did not depend on variable rainfall tinct from trended phenomena such as global change. Using events in a semi-arid Texas savannah. a modeling approach, we show that characteristics of tempo- The probability that a species will persist in anew ral environmental variation which alter population growth location is intimately related to extinction probabilities. rates, such as the variance and degree of autocorrelation, can Where environmental fluctuations are assumed to be inde- alter the invasion process. pendent, increases in environmental variation increase It has been previously suggested that temporal variation extinction risk (Lewontin and Cohen 1969, Lande 1993). in environmental factors can affect the establishment of Haccou and Iwasa (1996) were the first to realize this non-native species (Gabler and Siemann 2012). Davis et al. significance of the finding for the modeling of invasive (2000) propose that plant invasion success depends on species. They used an inhomogeneous Markov process to

1027 study the combined effects of environmental and demo- Moreover, individuals in dense populations may be more graphic stochasticity on establishment probability, and likely to disperse and to reach locations with different abiotic found lower establishment for greater variance. Drake and and biotic risk factors. Highly dense, widely spread popula- Lodge (2004) tested this claim using Daphnia magna, where tions can also be expected to have greater impact than those variance was introduced by providing varying amounts of species that successfully establish a small persistent popula- food on each day to 281 replicate populations. In agreement tion in a given location (Parker at al. 1999). Consequently, with theory, the authors found reduced establishment while one may be tempted to argue that invasion probability probability with increased variance. is merely the flip side of extinction probability, an alternate The empirical studies make it clear that environmental measure of invasion risk, the probability that a population variation may affect the success of invasive species and the will surpass a given threshold density before crossing an modeling efforts suggest that increased variation reduces extinction threshold, may be of importance when calculat- the probability of permanently colonizing a new area. These ing the risk that a novel species will have a large impact. model predictions regarding environmental variation may We note that when variation in environmental conditions not hold, however, if the underlying description of environ- is autocorrelated, the population dynamics in finite time can mental variation is inaccurate. It may seem that environ- be very different than the asymptotic behavior. The extreme mental variation is easily quantified with a single measure, case, where autocorrelation is approaching its maximum variance, and thus is simply incorporated into predictive value, provides intuition. Consider the most extreme exam- models of invasive dynamics. With this conceptual model, ple of almost complete correlation first. Only one environ- the variance in a given environmental signal that cannot mental condition will be experienced by the population in a be attributed to trends or cycles is assumed to arise from given finite period of time and that condition may be either statistically independent deviations from some expected beneficial or detrimental to population growth. We can then value. Many types of environmental variation, however, are use this intuition to examine the impact of different levels positively autocorrelated (Steele 1985, Halley 1996, Király of autocorrelation on population growth over finite periods. et al. 2006), and autocorrelation has been demonstrated to If we relax the extreme assumption and instead assume just alter the predictions regarding variation and extinction risk moderate autocorrelation, it becomes clear that again over (Cuddington and Yodzis 1999, Petchey et al. 1997). short finite periods it is likely that the environment will be in Autocorrelation in environmental variation indicates only one state. Low autocorrelation makes the finite period that departures from the mean expectation are correlated during which conditions are similar very short. Therefore, in time. If environmental fluctuations were independent, a we expect that autocorrelation in environmental conditions cold winter would be equally likely to be followed by a cold will increase the length of windows of time where there is an winter or a mild winter; however, climate data suggest that increase (and a decrease) in the likelihood that a population the best predictors of conditions of this year are the condi- will reach a large size before going extinct. tions last year. A cold winter this year is more likely to be In the following, we use a simple density-independent followed by a cold winter next year. Such positive autocor- population model to investigate invasion success defined relation leads one to suspect that a set of conditions favorable as surpassing a given population threshold before extinc- to establishment and spread are likely to persist for several tion and compare it to persistence probability. We find that time periods, and of course, vice versa. This question is positively autocorrelated environmental variation that yields especially of interest in devising management strategies for similarly correlated population growth rates can increase this marine systems and coastal areas. Climate data may suggest risk even for populations we do not expect to persist for long that these systems have a higher degree of autocorrelation in periods. environmental signals than continental regions (Steele 1985, Vasseur and Yodzis 2004). There are few studies of the effects of positively autocorre- Model description lated variation on invasive risk. Haccou and Vatutin (2003), using a Markov process, found that probability of successful We describe the growth of an invasive population as simple colonization in environments with positive autocorrelation geometric population growth: was lower than in uncorrelated environments. This modeling N  N exp (r ) (1) effort suggests that predicting invasion success and spread t  1 t t rates using constant conditions would be conservative com- where the population density, N, at any time t  1, is pared to the reality of temporal environmental variation. determined by the population growth rate exp(rt) at time t. That is, positively autocorrelated variation would reduce the Effects of positively autocorrelated environmental probability of successful invasion from rates predicted under variation are incorporated into the population growth rate constant conditions. as a 1 / f process. Halley (1996) provides a detailed overview In all cases reviewed here, invasion success has been of the family of 1 / f noise processes as a way to model cor- defined as population persistence over a given time interval. relations with different time scales in ecological processes. At The impact and further spread of an invasive species is, how- every time step the growth rate, exp(rt), is determined by a ever, often related to the population size that it attains. Indeed mean value, r0, to which is added an error term, et. The error some invasive species have no negative impacts at low den- term has a desired degree of autocorrelation, generated by b sity. For example giant hogweed Heracleum mantegazzianum drawing et from a 1 / f signal created with a given degree has no negative effect on the surrounding plant community of autocorrelation as indicated by the spectral exponent, b. until greater than 50% cover is reached (Thiele et al. 2010). Increasing positive autocorrelation in our signal is indicated

1028 by increasing values of the coefficient b. White noise (no this case, 64 timesteps), where extirpation is defined as falling autocorrelation) will have b  0, while highly autocorrelated below one half the initial population density (initial density, noise has b  1. N0 , is 10 in all cases). We compare this metric with a second To produce a time-series signal with a given spectral definition of success that we call establishment: the prob- exponent, we use the method previously described in ability of exceeding an upper bound of population density Cuddington and Yodzis (1999). A number, M, of random before falling below an extinction threshold. We use an upper phases, q1, …, qM, uniformly distributed on [0,2p] and M threshold of Nt  5000, and the same extinction threshold normally distributed random numbers, with zero mean and of Ne  5 as previously. We also calculated the probability b unit variance, r1, …, rM, are generated. For 1 / f noise, the of establishment and persistence over a given duration, that amplitudes of the individual elements of a time series are is, passing the establishment threshold remaining above the coupled to their frequencies via a power law; as the scale of extinction threshold for a period of time (20 timesteps) after the elements decreases, their amplitudes decrease at a rate establishment. All results are based on 100 000 replicate determined by the spectral exponent, b. Therefore, to gen- simulations for each parameter combination. erate the amplitudes for a given spectral exponent, each b/2 normally distributed number (rf) is multiplied by 1 / f to form the amplitudes af . The complex coefficients are Results calculated as: af exp(i qf )  a f cos (qf )  iaf sin(qf ), and the inverse fast Fourier transform of the coefficients is taken. The As expected (i.e. Lewontin and Cohen 1969), for uncorre- real part of the resulting sequence is used to form the tempo- lated (b  0), sequences of population growth rates, exp(r1)… ral noise series t1, …, tM (Saupe 1988, Voss 1988, Hastings exp (rt) with relatively low standard deviation (s  0.2), and Sugihara 1993). Since this method generates a periodic the probability of persistence (no extirpation) increases with sequence (Saupe 1988), where variance only increases to the geometric mean of the series exp(rt), lG (Fig. 1a). A value M / 2, we use only the first half of the series (64 timesteps). of lG greater than one is required for persistence. Where The generated series t1, …, t64 were standardized using a Z the geometric mean of the population growth rate is greater transform to have a given mean and variance. than one, the probability of persistence decreases with an In addition, the technique of spectral mimicry was increase in signal autocorrelation. Where lG is less than used to produce sets of noise signals with exactly the same one, there is no impact of increasing autocorrelation on elements, but differing autocorrelation (Cohen et al. 1999). probability of extirpation. In this technique, a white noise signal is generated, and its This expectation at low standard deviation (s  0.2), mean and variance standardized. The elements of the white is slightly different for establishment probability (i.e. the noise signal are rearranged so that they are in the same rela- probability of passing an upper population threshold tive order as the autocorrelated signal. Finally, the standard- before extinction) (Fig. 1b), and the probability of estab- ized mimic series e1, …, e64 was added to r0 producing a new lishment and persistence for 20 timesteps following estab- signal, with a desired spectral slope, b, generated as described lishment (Supplementary material Appendix 1 Fig. A1). 2 above. The elements with mean r0 and a given variance, s , Similar to persistence probability, the establishment prob- Since this series has a fixed arithmetic mean, the transformed ability increases with an increase in the geometric mean series exp(r1) , …., exp(r64) has a fixed geometric mean, lG. of the population growth rate for zero to low autocorrela- This technique ensured that our results are not driven by tion, although a growth rate greater than one is required minute difference in the geometric mean of the series, the for positive probability. Where lG is much greater than probability of occurrence of extreme values, or skew in the one, increasing autocorrelation decreases the establishment probability distribution of element values, that may occur probability (Fig. 1b). However, where lG is close to one, in the generation in highly autocorrelated signals. Signals increasing autocorrelation leads to increasing establish- with exactly the same elements, rearranged, are being com- ment probability. There is no effect on populations where pared (e.g. replicate simulation 1 with autocorrelation b  0, lG is less than one. would have the same series elements as replicate 1 with auto- When the variance of the signal is higher (s  0.8), correlation b  2). we find again that for large lG positive autocorrelation Throughout we report the degree of autocorrelation in can reduce persistence, but for lG greater than but close to terms of the measured spectral slope, b. The coefficient was one positive autocorrelation can slightly increase persistence estimated from the slope of a least squares line fit to power (Fig. 1c). There is no impact of positive autocorrelation on spectral density versus log frequency of the standardized the persistence of populations with lG much less than one series, where the first three data points where excluded. In (Fig. 1c). At this higher variance, however, the positive all cases, signals outside our error tolerance of  0.01 of the autocorrelation of the signal increases the establishment target value for b were discarded. This simulation technique probability for all but the largest values of lG (Fig. 1d). The successfully produced replicate series with identical elements, effect is quite striking for low lG where populations with fixed geometric mean, fixed variance, and autocorrelation 100% probability of going extinct also have a 40% prob- with the tolerance limits. All calculations were completed in ability of establishment with large positive autocorrelation. MATLAB (ver. R2013a). The predictions for probability of establishment and con- We compare the probability of a successful invasion using tinued persistence are very similar (Supplementary material two different definitions of invasion success. The first defini- Appendix 1 Fig. A1). In addition, similar results are obtained tion, persistence, is the probability that a population in the when positive autocorrelated series are generated using an new location does not go extinct over a finite time period (in AR(1) process (Supplementary material Appendix 1 Fig. A2),

1029 σ = 0.2 σ = 0.8

1 (a) 1 (c)

0.8 0.8

0.6 0.6

0.4 0.4 Probability of

persistence (n>5) 0.2 0.2

0 0 0 0.5 1 1.5 0 0.5 1 1.5

1 (b) 1 (d) λ 0.8 G 0. 8 0.75 0.6 0.85 0. 6 0.95 0.4 1.00 0. 4 1.05 0.2 1.15 0. 2 1.25

Probability of establishmen t 0 0

(n=>1000) before extinction (n<5 ) 0 0.5 1 1.5 0 0.5 1 1.5 Positive autocorrelation (β) Positive autocorrelation (β)

Figure 1. Probability that a population with geometric growth as in equation (1) will persist (a, c), or establish (b,d) over a range of per capita growth rates and autocorrelation of those rates. The standard deviation of the growth rate is either (a, b) 0.2, or (c, d) 0.8. All simulations are 64 timesteps long. Each data point used to form the curves is based on 100 000 replicate simulations.

or when a density-dependent model of population growth is (Readers should note the mechanism described here is not used (Supplementary material Appendix 1 Fig. A3). the same as the “inflationary” effect explored by Gonzalez In addition, at large standard deviation (s  0.8), and Holt 2002 for metapopulations, Supplementary increased positive autocorrelation increases the proportion of material Appendix 1 ). populations that achieve a large size (Fig. 2). This increased This increase in the probability of an early run of good probability of increased size is not restricted to those popula- conditions is driven by a change in the distribution of l tions that persist indefinitely: even populations with lG  1 values early in the invasion timeline. This distribution of may reach exceeding large sizes before extirpation (Fig. 2b). initial conditions ultimately determines the maximum pop- In addition, the magnitude of the maximum population size ulation size, without having much impact on the probability increases with increased autocorrelation for both lG greater of extirpation since the geometric mean of the entire series than and less than one. So not only do populations have remains the same. The distribution of the geometric mean a greater probability of passing some moderate threshold of the first 10 elements of the sequence is approximated by of population size before extirpation with increased posi- a normal curve for uncorrelated signals. However, for highly tive autocorrelation, they also have increased probability of autocorrelated signals the distribution is much flatter, with reaching extremely large outbreak densities. a fat tail (Fig. 4). As a result, populations are more likely The mechanism that drives these differences between to initially experience either strongly detrimental or strongly simple persistence and thresholded establishment is the beneficial conditions after introduction. Under highly auto- distribution of large values in the temporal series. A series correlated environmental conditions (b  1.6) many popu- with complete positive autocorrelation would correspond lations will experience initial conditions with lG less than to long continuous sequences with an upward or downward one, but a significant proportion will experience extremely trend. Therefore, when series have large positive autocorrela- beneficial conditions with the geometric mean of the first 10 tion, we expect longer sequences of beneficial (and detrimen- elements reaching 2–4. Consequently, there is a clear correla- tal) conditions. For example, the probability that the first tion between populations with such a beneficial early history 10 elements of a series has a geometric mean greater than and populations that pass an upper population threshold 1.2 is positively related to the autocorrelation of the series before extirpation. In contrast, there is much greater over- (Fig. 3). Although for series with large lG, this probability lap between the distribution of beneficial and detrimental does not change much with increasing autocorrelation, it conditions in series with b  0.2 and large lG, and therefore can increase rapidly for series with intermediate lG and low the extirpation risk, and the probability of passing an upper variance (Fig. 3a). For series with large variance and low lG, threshold before extirpation have more similar probabilities. positive autocorrelation can cause a dramatic increase in the Therefore, the most beneficial conditions for passing probability of an initial run of good conditions (Fig. 3b). an upper population threshold for an invasive species with

1030 β = 0.2 β = 1.6 7 7 (a) (b) 6 6 ) 4 5 5 4 4 3 3 2 2 Populations (x 10 1 1 0 0 101102103104105106107108109 101102103104105106107108109

7 7 (c) (d) 6 6 ) 4 persisted 5 5 extirpated 4 4 3 3 2 2 Populations (x 10 1 1 0 0 101102103104105106107108109 101102103104105106107108109 Maximum population size Maximum population size

Figure 2. Histogram of the maximum population size (before extirpation) for small per capita growth rates (lG  0.85) (a, b), and large per capita growth rates (lG  1.15) (c, d), with either low autocorrelation (b  0.2, a, c) or large positive autocorrelation (b  1.6, b, d) for populations that remained above N  5 (black segments, ‘persisted’) and populations that did not (white segments ‘extirpated’).

lG less than or equal to one will be that of large variance (1988) reasons that the negative effect of runs of bad condi- and large positive autocorrelation in the population growth tions will always outweigh the beneficial impacts of runs of rates. good conditions. In support of this prediction, Haccou and Vatutin (2003) used an inhomogeneous branching process to prove that when a system exhibited a positive autocor- Discussion relation in the switching probability between beneficial and detrimental states, autocorrelation would increase extinc- We demonstrate that predictions regarding invasive threats tion probability for populations with expected growth rates may depend on whether the definition of a successful inva- greater than one. We are not aware of any experimental sion is simply the continued presence of the non-native work that tests the effects of autocorrelated environmental species (persistence), or exceeding some upper threshold conditions on extinction risk; however, Pike et al. (2004) population density prior to extirpation (establishment). We experimentally culled populations of springtails Folsomia therefore suggest that the risk of negative impact from inva- candida, and found decreased time to extinction (or inter- sive species cannot simply be read as the inverse of extinction preted another way, increased probability of extinction in a probabilities (Andersen et al. 2004). In addition, although shorter fixed time period) for autocorrelated sizes of popula- it has been previously predicted that positive autocorrelation removals. tion of environmental signals will negatively affect popula- As expected (Lewontin and Cohen 1969), increased tion persistence and therefore the establishment of invasive variance decreases the probability that a population will species (Lawton 1988), we show that large variance and persist for some given period of time (compare Fig. 1a and autocorrelation can benefit invaders with low population 1b). In agreement with these previous studies, we also find growth rates by increasing the probability that runs of good that when the geometric mean of a time-varying rate of conditions will occur early on in the invasion process. There- population growth is greater than one (lG  1) increasing fore, positively autocorrelated environmental signals have the autocorrelation of the signal decreases the probability the potential to allow those populations we would not ordi- that a population will persist. We note however, that when narily identify as invasive risks to successfully establish a signal variance is large, positive autocorrelation may slightly large population. increase the persistence probability of populations with All environmental signals such as temperature and geometric growth rates close to one (lG ≈ 1). rainfall are positively autocorrelated such that runs of good In contrast, if we instead define an invasive threat as the and bad conditions are likely to occur. Previous authors have probability of crossing an upper threshold of population suggested that positive autocorrelation of environmental size before extirpation (i.e. establishment) we find that the signals translates into increased extinction risk. Lawton effect of positive autocorrelation in population growth rates

1031 (a) σ = 0.2 at least 20 timesteps after passing the upper threshold can also be as high as 30% (Supplementary material Appendix 1 Fig. A1). Therefore, one would draw different conclusions regarding the effects of variance and positive autocorrelation on invasion depending on the definition of success that is 0.6 used and the expected population growth rate. We note that 0.4 our findings are unrelated to the method of generating autocorrelated signal variance (i.e. AR(1) versus 1/f noise), or the 0.2 underlying population dynamics (i.e. density independent versus density-dependent growth of small populations; see 0 supplementary figures) (Supplementary material Appendix 1 the first 10 elements > 1.2 1.2 2 Fig. A2, A3). Of course, it is true that invading populations Probability the geometric mean of Geometric mean of with extremely low initial carrying capacities may experience 1 1 entire series (λ density-dependent impacts not found here. Moreover, the 0.8 β) 0 Positive elements of the signals compared across autocorrelation values were identical, so the results can also not be attributed to G ) autocorrelation ( small differences in the mean, variance, or extreme values. Overall, we can summarize our results by saying that in (b) σ = 0.8 an environment with large variance, positive autocorrelation has a small negative impact on both the persistence or establishment probability of populations with geometric growth rates much greater than one. However, in such environments positive autocorrelation substantially increases the establish- 0.6 ment probability of populations with geometric growth rates 0.4 less than one, but has little or no effect on persistence probability. That is, positive autocorrelation increases establish- 0.2 ment risk precisely in those populations we may not expect to invade when using a definition of population persistence 0 for invasion success: those with low growth rates subject to the first 10 elements > 1.2 1.2 large variance. Probability the geometric mean of 2 Geometric mean1 of From an ecological or economic point of view, a entire series (λ 1 0.8 population which achieves high population density and then β) 0 Positive crashes can be nearly as devastating as one that reaches a high population and remains at this density (see Parker et al. G ) autocorrelation ( 1999, Thiele et al. 2010 regarding the impacts of high density invaders). In addition, populations which do reach high Figure 3. Probability that the geometric mean of the first 10 ele- density will inevitably establish refugia in protected areas. ments of a 64 element series is greater than 1.2 for a given overall geometric mean and degree of positive autocorrelation of the series, Therefore, when assessing invasion risk, it may be impor- for either (a) small standard deviation (s  0.2), or (b) large stan- tant to separate extirpation from a low initial density versus dard deviation (s  0.8). Each point on the surface is based on 100 extirpation from high density. 000 replicate simulations. Colour gradiations for ease of visual The mechanism that generates the difference in interpretation only. establishment between environments with low versus high autocorrelation is related to the probability of long sequences of beneficial or detrimental sequences in the early history depends on the variance in the signal. In low variance envi- of the invasion. Positive autocorrelation increases both the ronments, positive autocorrelation will increase establish- probability of beneficial and detrimental sequences at the ment probability for populations with lG close to one. For beginning of the invasion, relative to uncorrelated noise, and populations with lG greater than one, positive autocorrela- therefore produces some initial sequences with much larger tion will decrease the establishment probability, while there geometric means than the average expectation of the entire is no effect on populations with lG much less than one. series (compare Fig. 3a and 3b). For low population growth Strong autocorrelation and large variance may dramatically rates, the increased probability of a sequence of detrimen- increase establishment probability for all populations except tal conditions associated with positive autocorrelation adds those with very large growth rates (Fig. 1d). little to the probability of quick extirpation. However, the The predictions regarding persistence and establishment increased probability of a sequence of beneficial conditions diverge most strongly for populations whose growth rate has a disproportionately beneficial impact. Similarly, when is predicted to be at or below replacement (lG  1) Under population growth rates are large, positive autocorrelation the same conditions, populations with 100% probability in these rates can decrease establishment probability, espe- of being extirpated (0% persistence), can have as much as cially when variance is small, because of the disproportionate a 40% change of passing a large upper population thresh- effect of a detrimental sequence early in the invasion his- old before falling below the extinction threshold (Fig. 1c–d). tory. These findings support the idea that species may be Moreover, the probability that the population will persist for successful during an “invasion window” (Johnstone 1986,

1032 β = 0.2 β = 1.6 4 4 ) (a) (b) 4 3 3

2 2

1 1 Populations (x 10 0 0 0 2 4 0 2 4

4 4 ) (c) (d) 4 established 3 3 failed

2 2

1 1 Populations (x 10 0 0 0 2 4 0 2 4 Geometric mean of Geometric mean of first 10 elements first 10 elements

Figure 4. Histogram of the geometric mean of the first 10 elements a 64 element series for small per capita growth rates (lG  0.85 for whole series) (a, b), and large per capita growth rates (lG  1.15 for whole series) (c, d), with either low autocorrelation (b  0.2, a, c) or large positive autocorrelation (b  1.6, b, d) for populations that increase beyond 5000 individuals before extirpation (black segments, ‘established’) and populations that did not (white segments ‘failed’).

Davis et al. 2000, Gabler and Siemann 2012) even when growth rates. Moreover, in no case does increasing vari- mean conditions do not indicate a significant risk. ance predict increased establishment for positive population The role played by autocorrelation in stochastic influences growth, as found for large autocorrelation. on population dynamics has previously been studied in other Environmental signals in marine and coastal envi- contexts by others, notably Gonzalez and Holt (2002), Holt ronments are more likely to be positively autocorrelated et al. (2003) and Roy et al. (2005) who focused on long term (Pelletier 2002, Vasseur and Yodzis 2004) and generally persistence rather than the transient issues that are the focus of have moderate variation in environmental signals such as our study. Roy et al. (2005) describe how positive covariance temperature. Therefore, one might predict that the intro- in population growth rates and population size can “inflate” duction of a species is more likely to result in a successful the average population abundance and consequently the invasion in these regions. However, it also seems probable long term persistence of coupled sink populations. They find that species introductions are more frequent in coastal increased average abundance of persistent sink metapopula- and marine locations than in isolated terrestrial regions, tion patches with increased variance and positive autocor- and so this effect may be of little importance (see Haccou relation in growth rates, but note that these effects disappear and Vatutin 2003 regarding the effects of repeated intro- without dispersal between patches. This finding is not incon- ductions). This point may still be of interest, however, sistent with our results, where the change in the distribution when estimating risk for areas with similar rates of species of covariance between population growth and population introductions, but dissimilar degrees of autocorrelation in size can lead to a larger number of populations with both environmental conditions. large and small maximum population size in finite time (i.e. Our findings suggest that studies which attempt to the distribution has heavy tails with large autocorrelation, explain the potential of a given species to pose a significant Supplementary material Appendix 1 Fig. A4). invasive threat are doomed to failure when such assignments We note that the effect of beneficial or detrimen- are made independently of considerations regarding tempo- tal sequences cannot be mimicked through the use of an ral environmental variation that impacts population growth uncorrelated noise signal with greater variance as has been rates. Invasive risk is clearly related to both the life history suggested (Lande et al. 2003). In particular, the effect of characteristics of the invader, features of the invaded commu- increasing autocorrelation is dependent on signal variance nity, and last, but not least, the nature of the environmental and the geometric mean of the sequence of population variation in the region. Management policies in areas with

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Supplementary material (available online as Appendix oik- 02859 at  www.oikosjournal.org/appendix/oik-02859 ). Appendix 1.

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