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1 Sampling Four billabongs were chosen from two separate catchments in Kakadu National Park (NT, Australia); Jaja, Jabiluka, and Island in the East Alligator River catchment and Minggung in the South Alligator River catchment. Although Minggung is in a different catchment, it is less than 30 km from Island, and we have no evidence or suspicion that it differs ecologically or hydrologically from the three other billabongs. Data were collected at roughly monthly intervals over a 13-year period, from August 1991 to February At each site, salvinia cover, salvinia biomass, number of weevils, and percent ramets damaged by weevils were measured. First, salvinia cover was measured for each billabong. This was done by filling in areas with salvinia present on a paper map of each billabong and then calculating the area using a planimeter. Then 100 ramets were haphazardly selected from three locations in the billabong and the percent of plants with weevil damage was determined. Six samples of salvinia (30x30cm quadrat) were haphazardly collected from around the billabong, and these were placed into Berlese funnels to extract and count the adult weevils. The salvinia samples were then dried to constant weight at 65 C and weighed to the nearest 0.01g. Salvinia biomass was calculated by multiplying the percent cover of salvinia by the average dry weight per sample. The entire data set consists of 114, 134, 113, and 131 samples from Jaja, Jabiluka, Minggung, and Island billabongs. We removed the last 3 sample points from Jaja, and the last sample point from the other 3 billabongs, because they were separated by several months from the rest of the time series. We had data on both weevil damage (percentage of ramets damaged) and adult weevil density. Because weevil damage is most important for salvinia control, we selected it as our variable of weevil abundance for fitting the model. There were 44 missing values of weevil damage from the data set. We fit a logistic curve to the relationship between weevil damage and number of beetles, and from this estimated the 44 missing values using data on the number of beetles at the same samples (Fig. S1). Also, there were 5 missing salvinia biomass points from Minggung; because none of these were consecutive, we estimated each using log-linear interpolation from the two adjacent points. WWW NATURE.COM/NATURE 1

2 Preliminary data analyses We performed preliminary analyses to search for two patterns in the data that would suggest simple processes underlying the biological control of salvinia by the weevil. First, for successful control the intensity of weevil attack on salvinia should increase at high salvinia density to limit the growth of the weed. Second, the effect of weevil control should be seen as decreases in salvinia biomass when weevil attacks are high. Neither of these patterns is strong in the billabongs (Fig. S2a,b). Changes in log salvinia biomass between monthly samples are characterized by frequent small steps and occasional large jumps, leading to high kurtosis in the distribution of changes in log biomass (Fig. S2c). The values of kurtosis were 4.99 (p < 0.002, Anscombe-Glynn test), 8.00 (p < ), 5.69 (p < 0.001), and 8.02 (p < ) for Jabiluka, Minggung, Jaja, and Island Billabongs, respectively. These results indicate episodically large changes in the ecological character of the salvinia system. Both of these features of the data lack of clear dynamical coupling between salvinia and weevil dynamics, and high kurtosis in changes in salvinia log biomass suggest the possibility of alternative stable states underlying the dynamics. If alternative stable states are present, then control of salvinia will occur largely by weevils maintaining salvinia biomass within the domain of attraction to the lower state. Outside this domain, however, little control by weevils will be exerted. Therefore, weevils are not by themselves capable of consistently bringing salvinia biomass down from high levels, and therefore the anticipated pattern for effective biological control that high pest abundances cause increases in control agents and subsequent reductions in pests will be obscured. The main sources of population fluctuations are the large jumps between the relatively low and high states of salvinia biomass. As a simple, generic investigation of alternative stable states, we fit a time-delay inverse-logit model with measurement error to the log salvinia biomass data: [ [ ( )]] "1 + # t u t +1 = b 4 + b 3 1+ exp " b 0 + b 1 u t + b 2 u t"1 (S1) u * t = u t + $ t where u t * is the observed log salvinia biomass, ε t is the process error, α t is the measurement error, and u t is the true log salvinia biomass. Alternative stable states would be given if the fitted line in a plot of u t+1 vs. u t crossed the one-to-one line three times, giving 2 stable points and one unstable point. Over almost the entire range of log salvinia biomass (Fig. S3), the fitted line WWW NATURE.COM/NATURE 2

3 [ [ ( )]] "1. almost coincides with the one-to-one line in a plot of y = b 4 + b 3 1+ exp " b 0 + ( b 1 + b 2 )u Therefore, the fitted model is on the cusp of alternative stable states. Nonetheless, there are two clusters of points, a denser cluster around u = 4 and a broader cluster around u = 3, that are suggestive of alternative stable states even if these are not revealed by the fitted inverse-logit model. In summary, the tendency of the dynamics to jump up and down suggests that there are alternative stable states, and the fit of the inverse-logit model to log salvinia biomass suggests that the dynamics of salvinia alone are on the cusp of alternative stable states. Although suggestive of alternative stable states, a proper analysis requires fitting a non-linear model to the data that incorporates weevils, water flow, and biologically plausible interactions among these and salvinia. Model description We assume that salvinia biomass is divided into two categories. Category 1 consists of meristematic tissue that is fed upon by adult weevils, whereas category 2 is vegetative tissue and thus is less suitable for the weevil. In the model, the allocation of salvinia biomass to categories 1 and 2 depends on the total biomass of salvinia x; specifically, let g(x) be the proportion of salvinia biomass in category 2. We assume that g(x) = g min + ( g max " g min )logit -1 b 2 x " b 1 ( ( )) (S2) where logit -1 () is the inverse-logit function, g min and g max give the minimum and maximum proportion of salvinia biomass in category 2, b 1 gives the location of the inflection point at which g(b 1 ) = (g max + g min )/2, and b 2 determines the slope at the inflection point. The intrinsic rate of increase of salvinia in category 1 is r, while by definition the intrinsic rate of increase for salvinia in category 2 is zero. Category 1 salvinia experiences density dependent growth given by the function ( 1+ ( 1" g(x) )x) "v (S3) where (1 g(x))x is the biomass of salvinia in category 1, and v determines curvature of the density-dependent function; for higher values of v, the net reproduction rate becomes more humpshaped. We assume that adult weevils search for and attack individual meristems (in category 1) according to a negative binomial distribution with dispersion parameter k, so that the proportion of non-attacked meristematic tissue is WWW NATURE.COM/NATURE 3

4 $ a" 1+ t y ' & ) % k( 1# g(x) )x ( #k (S4) Here, the proportion of non-attacked tissue is assumed to depend on the density of weevils y relative to the category 1 biomass (1 g(x))x, scaled by aτ t where a is the attack rate per day and τ t is the time (in days) between times t and t+1. We also allow the possibility of the continuous inflow of salvinia from surrounding refuge areas, with the daily rate of immigration equal to m. Finally, the reproduction of weevils is assumed to be proportional to the total amount of category 1 tissue consumed with a scaling term c. From prior biological information about the salvinia-weevil system, we know that high flow rates depress weevil damage by flushing salvinia and weevils from billabongs. We included this effect as δ s (z t ) = d s z t and δ w (z t ) = d w z t where z t is the log-transformed flow rate at time t, and δ s (z t ) and δ w (z t ) are mortality rates for salvinia and weevils associated with flow rates z t. Thus, negative values of d s and d w lead to increasing loss of salvinia and weevils with increasing flow rates. 1 * 3 x t +1 = 2, x t g x t, 43 + The full dynamical equations are ( )e r# t 1+ ( 1" g( x t ))x t ( ) + x t 1" g( x t ) $ & % ( ) "v & 1+ a# t y ' t k( 1" g( x t ))x ) t ( "k - / / exp 0 s(z t ). ( ) + m# t exp( 8 s (z t )) 1 * 3 $ y t +1 = cx t ( 1" g( x t ))e r# a# t 1" 1+ t y ' 2, t &, % k( 1" g( x t ))x ) 43 + t ( "k - 5 / / exp 0 3 ( w(z t )) exp( 8 w (z t )) (S5) Here, process error (environmental stochasticity) is represented by the zero-mean random variables ε s (z t ) and ε w (z t ) whose variances may depend on flow rates. Specifically, we assume that the variances of ε s (z t ) and ε w (z t ) are (σ s1 exp(σ s2 z t )) 2 τ t and (σ w1 exp(σ w2 z t )) 2 τ t, so that the variances are proportional to the number of days between samples, τ t, and that ε s (z t ) and ε w (z t ) have correlation coefficient ρ. Positive values of σ s2 and σ w2 imply that the variability in loss/retention of salvinia and weevils increases with the flow rate. The parameters σ s1 and σ w1 govern the overall variability in salvinia and weevil loss/retention. Equations (S5) are nondimensional, in the sense that there are no units associated with salvinia biomass; the scaling of salvinia biomass is done in the model fitting, as described below (Fitting model to data). WWW NATURE.COM/NATURE 4

5 Fitting model to data We fit the model using a non-linear state-space model 1. In addition to the process equations S5, the state-space model includes measurement equations for the sampling of salvinia and weevil damage, specifically X t * = X t + log g e X t W t * = logit p t ( ( )) + C + " ms ( ) + " mw (S6) Here, X t * is the observed log biomass (where zero values have been replaced in the data set with 0.5 times the lowest observed value of 0.25); this structure of the measurement error equation for salvinia assumes that measurement error is lognormally distributed. We assume that α ms is a Gaussian random variable with mean zero and variance σ 2 ms. The parameter C is a scaling constant that sets the overall magnitude of the observed log salvinia biomass. W t * is the observed logit-transformed proportion weevil damage. This structure of the weevil measurement equation approximates the situation in which weevil damage is binomially distributed with the probability $ a# that a sampled ramet is damaged given by p t =1" 1+ t y ' t & % k( 1" g( x t ) ) )x t ( "k. Measurement error in this approximation is a Gaussian random variable α mw with mean zero. For a purely binomial process, the variance of α mw is 1 n 1 p t (1" p t ) where n is the sample size. To account for possible inflated measurement error variance that could occur in taking only 3 samples of 100 ramets from # 1 an entire billabong, we set the variance of α mw equal to n + " 2 & 1 % mw( $ ' p t (1) p t ), where σ2 mw is estimated from the data. The time intervals between samples in the data set were generally a month, although some intervals were longer due to missing data. We wanted to iterate the process model on time steps shorter than the interval between samples, because the extended Kalman filter approximates the non-linear process equations using stepwise linear approximations. These approximations will be more accurate over shorter time steps. Therefore, for each pair of consecutive sample points, we added the minimum number of evenly spaced iteration points to make the interval between iteration points <10. Note that the state-space model does not require that the time points of iteration of the process equations be the same as the time points of the samples, although for each sample point there needs to coincide an iteration point. To mesh the data on flow rates with the iteration time points, we averaged the daily flow rates over the 10-day period before the iteration point. WWW NATURE.COM/NATURE 5

6 From extensive experiments, we know that r = 0.08 day -1 which we did not estimate from the time-series data (Julien, unpublished data). Therefore, there are 16 parameters remaining in the process equations that must be estimated: g min, g max, b 1, b 2, a, k, m, v, c, d s, d w, σ s1, σ s2, σ w1, σ w2, and ρ. There are 3 parameters in the measurement error: C, σ 2 ms, and σ 2 mw. This gives a total of 19 parameters. Note that all billabongs were fit using the same parameter values, so the model assumes that there are no differences among billabongs in salvinia-weevil dynamics. To fit the model with an extended Kalman filter, initial points for each billabong were taken as the first observed values, and the initial covariance matrix of process errors was taken as the covariance matrix of ε s (z t ) and ε w (z t ). The extended Kalman filter is applied iteratively: i. From the current estimate of the true population sizes of salvinia and weevils, and a current estimate of the uncertainty (variance) in the estimates, the population sizes at the next time point are predicted using the processes equations. The covariance matrix for the population sizes are also predicted using both the process equations and the process (environmental) error (which is a covariance matrix whose elements are estimated parameters). Because the process equations are non-linear, the projection of the covariance matrix for population size uncertainty is approximated by linearising the model. ii. When the iteration of the model coincides with an observed data point, the estimates of population sizes and the corresponding covariance matrix are updated. This updating depends on the measurement error variances (which are estimated parameters). If the measurement error variances are low, then the estimates of the "true" population sizes are brought closer to the observed values. There is a similar updating of the covariance matrix for the estimates of the true population sizes. During this iterated process, the extended Kalman filter produces an estimate of the likelihood function that we minimized to obtain the Maximum Likelihood parameter estimates. We compared models generated by removing parameters that successively caused the smallest increase in AIC values, that is, backwards stepwise selection (Table S1), and confirmed the model selection with forward stepwise selection. The values displayed as fitted trajectories in figures 2 and 3 are the updated values from the extended Kalman filter the model-predicted values that are updated using the observed values WWW NATURE.COM/NATURE 6

7 according to the strength of measurement error. The reason for using these values is that, to the ability of the model, they "smooth out" the trajectories by accounting for measurement error. For example, in figure 2 there is considerable variability in the proportion of salvinia damaged by weevils (grey points and lines). This level of variability, however, is no greater than expected under binomial sampling; the parameter σ 2 mw that accounts for greater-than-binomial variability is estimated to be zero (Table S1). Simulations of the fitted model An informal manner of assessing a fitted model is to simulate population trajectories and compare them to the original data to which the model was fit. Figure S4 depicts the raw data from each of the four billabongs, and figure S5 gives a representative simulation from the fitted model. The simulated dynamics qualitatively match those of the observed data. However, in the simulations the log biomass of salvinia occasionally exceeded 6, as shown in figure S5d. When this occurred, the log salvinia biomass could wander to yet higher values. For figure S5d we set a maximum value of 6 for log salvinia biomass. This minor modification to the model is biologically plausible, because there are limits to salvinia biomass as plant mats cannot become too thick. Furthermore, modifying the model to include a maximum salvinia biomass does not change the fitting, because the fitted log salvinia biomass never exceeded 6. Thus, we could have used the modified model with a biological maximum salvinia biomass, and its fit to the data would be indistinguishable from the model we used. Biological insights from the fitted model The fitted parameters for the suite of well-fitting models are given in Table S1. From these, biological processes can be inferred about the salvinia-weevil system: i. Flooding events reduce the abundance of weevils (d w < 0) yet have little net effect on the mean abundance of salvinia (d s = 0). ii. Flooding events increase the variability in sample-to-sample fluctuations in salvinia biomass (σ s2 > 0), and this effect is very strong (ΔAIC = 81.97). Flooding also decreases the variability in weevil abundance (σ w2 < 0), although this effect is weak (ΔAIC = 1.86). iii. Most of the biomass of salvinia (g min = 0.91) is in category 2 and invulnerable to weevils. WWW NATURE.COM/NATURE 7

8 iv. Attacks on salvinia by weevils is only slightly aggregated (k = 3.38). v. The immigration of salvinia is small in magnitude (m = ), but important statistically (ΔAIC = 14.13). vi. There was a fairly strong positive correlation between process errors for log salvinia biomass and weevils (ρ = 0.51). This means that environmental fluctuations (excluding water flow) tend to have positive or negative effects simultaneously on fluctuations in salvinia biomass and weevil abundance. Theoretical exploration of the model To explore the model, we first removed stochasticity (σ s1 = σ s2 = σ w1 = σ w2 = 0), set the log water flow rate at a constant value z, and assumed a constant time between model iterations (τ = 8.83 d). The water flow rate z can be used as a bifurcation parameter in a bifurcation diagram depicting salvinia biomass (Fig. S6). As z increases from 1, a single stable point gives way to alternative stable states at z = through a fold (saddle-node) bifurcation that generates an upper stable and intermediate unstable point. When z reaches , the lower stable point collides with the unstable point, annihilating both. Even though outside the region < z < there is only a single stable point, the "ghost" of the unstable point remains (Fig. S6, dashed green line). At this point, the trajectories slow, although they do pass through to approach the stable point. More information about salvinia-weevil dynamics can be obtained from phase portraits showing trajectories for specific values of the log water flow rate. When z = 0, the fitted model has two stable points separated by an unstable point (Fig. S7a). (Note that this phase portrait is graphed in terms of salvinia biomass x and weevil abundance y, rather than log biomass of category 2 salvinia and logit weevil damage as in Fig. 3.) Trajectories approach the upper stable points mainly horizontally via changes in salvinia biomass, rather than weevil abundance. Thus, there is a broad range of relatively high salvinia biomass over which trajectories move relatively slowly. Decreasing the water flow rate to z = 1, the stable point at high salvinia biomass disappears (Fig. S7b); this occurs because low flow rates favour weevils that can then suppress salvinia. Nonetheless, the "ghost" of the annihilated unstable point still remains 3,4, causing a slowdown in the trajectories and a sudden change in direction at x = This corresponds to the salvinia biomass x = b 1 at which the proportion of biomass in the invulnerable category 2, WWW NATURE.COM/NATURE 8

9 g(x), switches from g max to g min as x decreases. In the fitted model, this switch from g max to g min is rapid, since b 2 = 385. In the statistical fitting, the value of b 2 is poorly estimated; reducing the value of b 2 changes the fit very little and leads to smoother deterministic trajectories. Inability to estimate b 2 precisely is to be expected, because stochasticity makes trajectories stay only briefly in the vicinity of the switch point b 1. In a similar way, at z = 0.5 the stable point at low salvinia biomass has been annihilated with the unstable point (Fig. S7c). As before, the ghost of the annihilated unstable point still remains, causing a slowdown in the trajectories and a sudden change in direction. Finally, to illustrate the case in which alternative stable states do not occur, we eliminated the transition between vulnerable and invulnerable salvinia categories by setting g(x) = (g min + g max )/2 = ; thus, a fixed 92.6% of the salvinia biomass is in category 2 invulnerable to weevils (Fig. S7d). Not only does this eliminate the alternative stable states and the ghost of the unstable state, compared to the upper alternative stable point (Fig. S7a) the trajectories now approach the stable point in a more diagonal fashion from either the top-right or the bottom-left of the phase portrait. Thus, in the top-right portion of the phase portrait, increases in salvinia biomass lead to increases in weevil abundance that then return salvinia biomass to the stable point. Similarly, in the bottom-left portion of the phase portrait, decreases in weevil abundance allow the salvinia biomass to recover to the stable point. These patterns are those that would be anticipated for "simple" biological control in which the control agent exerts continuous suppression of the pest (see Preliminary data analyses). These analyses have assumed that the water flow z is fixed. To examine the effects of cyclic environmental forcing (annual flooding), we allowed z to vary between two values with a period of a year, with a high value of z corresponding to annual high water flow and a low value for the rest of the year (Fig. S8). Periodically forcing the system generates stable environmentally forced cycles, and it is possible to have alternative stable cycles coexisting (Fig. S8a). The consequences of a periodic flow rate depend on the magnitudes of the different flow values and their duration. The higher and longer the high flow rate, the greater the detriment to weevils and the more likely it is for only a single cycle to exist at high salvinia biomass (Fig. S8b). Similarly, lower and longer low flow rates favour weevils and lead to a cycle at low salvinia biomass (not shown). However, when the model is parameterized to disallow alternative stable states (setting g(x(t)) = (g min + g max )/2 = ), there is a single simple cycle (Fig. S8c). Thus, the existence of the alternative stable states in the absence of environmental forcing has clear effects on the system when cyclic forcing is introduced. WWW NATURE.COM/NATURE 9

10 The cycles generated by environmental forcing can be explained as follows. The characteristic rates of return of the trajectories to the stable points can be described by the halflives of perturbations from the stable points; as trajectories asymptotically approach the lower and upper stable points, they half the distance to the stable points every 0.42 and 0.72 years, respectively. These time scales are roughly the same as the yearly periodicity of the environmental forcing. To illustrate the consequences of this, consider the trajectory leading to the upper stable cycle in figure S8a. Initially the flow rates are low, so the populations follow trajectories towards the lower stable point (Fig. S7b). Long before the lower stable point is approached, however, the flow rate switches to its high value (at roughly 10 o clock in the cycle), and the trajectory moves towards the higher stable point (Fig. S8c). Again the flow rate switches (at roughly 4 o clock), and the trajectory moves towards the lower stable point. The stable cycle arises when the two trajectories from the two flow rate regimes meet end-to-end, so that switches in flow rates return trajectories to their original locations. Our basis for describing the model as having alternative stable states is the fact that stripped of environmental forcing and stochasticity, the best-fitting statistical model exhibits deterministic alternative stable points. The deterministic, environmentally forced model can show alternative stable cycles (Fig. S8a), although this depends on the magnitude and frequency of forcing (Fig. S8b). Strictly speaking, the fully stochastic model does not have alternative stable states, because stochasticity causes trajectories to jump between domains of attraction to alternative stable points. Although the fully stochastic model does not have alternative stable states, it nonetheless shows the dynamical characteristics of the underlying deterministic model with alternative stable states. Furthermore, even when the underlying deterministic model is parameterized so that it does not show alternative stable states (Fig. S7b,c), the ghost of the system with alternative stable states nonetheless produces distinctly different dynamics from the model in which there is no possibility of alternative stable states (Fig. S7d). Therefore, we refer to the system as having alternative stable states because the deterministic, unforced model has alternative stable states, and the resulting dynamical characteristics dictate the behaviour of the full stochastic model. Inference from models and model alternatives There is a fundamental epistemological problem associated with fitting a statistical model to data and using this to infer the true dynamics of the system. Any inference is necessarily dependent on the model that is fit. This problem is not necessarily solved by investigating multiple models, because the number of possible alternative models is infinite. Therefore, one WWW NATURE.COM/NATURE 10

11 never knows whether the true model is found. For salvinia and weevils in Kakadu, the biological model we used was derived from a priori knowledge of the system. Therefore, in a sense it has primacy over other models that might be derived without biological information. Nonetheless, we cannot claim that our model is the true model. There is no obvious alternative model that we could suggest to compete against the model we have used. Nonetheless, we did investigate a generic model, x t +1 = x t e r" t {[ ( 1+ x t ) #v ( 1# p t )]exp( $ s (z t )) + m" t }exp % s (z t ) y t +1 = cx t e r" t { p t exp ( $ w (z t ))}exp(% w (z t )) ( ) (S7) where $ a# p t =1" 1+ t y ' t & % k( 1" g( x t ) ) )x t ( "k. This model is related to the biological model that we used for the main analyses but eliminates the two categories of salvinia; all salvinia biomass is vulnerable to weevil attack. This model is generic, in the sense that the resource density dependence and consumer-resource interactions are standard; salvinia density dependence is given by a generalized Beverton-Holt equation, and the herbivory rate by weevils involves a negative-binomial attack rate. This generic model did not fit the data well; the ΔAIC value is 220 compared to the bestfitting biological model. This means that it will be difficult to come up with a generic model that fits the data well and is simpler than the biological model that we analyzed in detail. The only thing non-standard about the biological model is the separation of salvinia into vulnerable and invulnerable categories, and this seems to be necessary to fit the data. There is no simple solution to the problem of making inferences from models when there is a potentially infinite number of models that could be used to fit a data set. Our approach is to build a model from biological understanding of the system and then fit this model to data, using it to explore dynamical properties. Descriptive analyses (see Preliminary data analyses) indicate that the salvinia-weevil dynamics are complex and suggest alternative stable states. In the fitted biological model, there is strong statistical support for the switching of salvinia biomass between categories that are vulnerable and invulnerable to weevils, which is known to be the case biologically. The model further shows the dynamical consequences of this switching: (i) alternative stable states in the deterministic unforced system; (ii) dynamical patterns that reflect the ghost of alternative stable states even when alternative stable states do not exist; and (iii) for the WWW NATURE.COM/NATURE 11

12 environmentally forced case, similar alternative stable cycles and dynamics reflecting the ghost of alternative stable cycles. Every model is a caricature of a real ecological system and therefore cannot be treated as the true model. Nonetheless, models that capture key biological and dynamical characteristics of a system can be used to understand it. References 1. Harvey, A. C. Forecasting, structural time series models and the Kalman filter. (Cambridge University Press, Cambridge, U.K., 1989). 2. Guckenheimer, J. and Holmes, P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. (Springer-Verlag, New York, 1983). 3. Hastings, A. Transients: the key to long-term ecological understanding? Trends in Ecology & Evolution 19 (1), 39 (2004). 4. Strogatz, S. H. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering. (Perseus Books, Cambridge MA, 1994). WWW NATURE.COM/NATURE 12

13 Table S1: Parameter values of models compared to the best-fitting AIC model. The number of parameters in the model is n par, and LL is the log likelihood computed from the extended Kalman filter. Blanks are given when a parameter is excluded from the model. The base values of all parameters (i.e., their values when excluded from the model) are zero except for v = 1 and d s = The possibility of alternative stable states is removed by setting g(x) = g min, in which case values of g max, b 1, and b 2 do not enter into the model. ΔAIC n par LL g min g max b 1 b 2 a k c m v d 1 d 2 σ s1 σ s2 σ w1 σ w2 ρ C σ 2 ms σ 2 mw WWW NATURE.COM/NATURE 13

14 Fig. S1: Logistic regression of proportion beetle damage against adult beetle density used to interpolate missing values of beetle damage. WWW NATURE.COM/NATURE 14

15 Fig. S2: Dynamical characteristics of salvinia biological control by weevils in Jabiluka billabong. (a) The change in log biomass of salvinia between consecutive samples was weakly negatively correlated ( 0.036) with the observed intensity of weevil damage in the first of the samples; correlations for the other 3 billabongs are 0.029, 0.052, and for Minggung, Jaja, and Island, respectively. (b) The change in the intensity of weevil damage between consecutive samples was weakly positively correlated (0.049) with log salvinia biomass in the first of the samples; correlations for the other 3 billabongs are 0.041, 0.081, and for Minggung, Jaja, and Island, respectively. (c) The distribution of changes in log biomass of salvinia between samples was highly leptokurtic (kurtosis = 4.99, p < 0.002, Anscombe-Glynn test); kurtosis for the other 3 billabongs are 8.00 (p < ), 5.69 (p < 0.001), and 8.02 (p < ) for Minggung, Jaja, and Island, respectively. WWW NATURE.COM/NATURE 15

16 Fig. S3: Fit of a time-delay inverse-logit model (eq. S1) to the dynamics of log salvinia biomass. Data are plotted as consecutive samples [u t+1 vs. u t ], and the blue line gives the fitted function y = b 4 + b 3 [ 1+ exp ["( b 0 + ( b 1 + b 2 )u)]] "1. The red line gives the values at which u t+1 = u t. Data include all four billabongs, and all changes between samples are treated equally regardless of the true time between samples. WWW NATURE.COM/NATURE 16

17 Fig. S4: Raw data for log salvinia biomass (black) and logit weevil damage (red) for each of the four billabongs. Log water flow, standardize to have mean zero and variance 1, is given by the blue line on a different scale. WWW NATURE.COM/NATURE 17

18 Fig. S5: Representative stochastic simulation of log salvinia biomass (black) and logit weevil damage (red); the duration of the time series and the water flow rates are the same as the corresponding panels in figure S4 for each of the four billabongs. Maximum salvinia log biomass is set to 6, and this ceiling is hit only in panel d. WWW NATURE.COM/NATURE 18

19 Fig. S6: Bifurcation diagram for the salvinia-weevil model (eqs. S5) displayed for salvinia biomass x. Black lines give stable points, and the red line gives an unstable point. The unstable point and one stable point annihilate each other at fold (saddle-node) bifurcations at z = and z = Even when there is only a single stable point (z < or < z), there is the ghost of the collision between stable and unstable points (dashed green line); even though trajectories pass through this ghost, they slow down substantially as they pass through (see Fig. S7b,c). WWW NATURE.COM/NATURE 19

20 Fig. S7: Phase portraits of the salvinia-weevil model graphed in terms of salvinia biomass x and weevil abundance y (both non-dimensional); the bifurcation parameter z = 0, 1, and 0.5 for (a)- (c), respectively. In (d) the alternative stable states are eliminated by setting g(x) = (g min + g max )/2. Stationary points are shown by black crosses, with the arms of the crosses corresponding to the eigenvectors of the Jacobian matrix of the linearised system at the stationary points. Red lines are deterministic trajectories, with dots giving the initial values; in (b) and (c), dots are also placed at every fifth iteration to give an indication of the speed of the trajectories. Note that in (b) and (c), even though there is no stable point where log salvinia biomass x = 5.4, the ghost of the collision between stable and unstable points is seen in the slowing and shift in direction of the trajectories. Blue arrows give the flow rates in a grid across the phase portrait. In (a) the green line gives the boundary between domains of attraction to the two stable points. WWW NATURE.COM/NATURE 20

21 Fig. S8: Phase portraits of the salvinia-weevil model including cyclic forcing (flooding) graphed in terms of salvinia biomass x and weevil abundance y (both non-dimensional). In (a) the values of z are selected to correspond to figure S7b and S7c; z alternates between 1 and 0.5 with an annual period (15 and 26 consecutive values, respectively). In (b) the values of z alternate between the observed minimum and maximum values of 0.9 and 2.0 with a period of one year and a mean value of z = 0. In (c) parameter values in the model were selected to give no alternative states, g(x(t)) = (g min + g max )/2 = , and environmental forcing is identical to that in (a). Stationary points calculated with z = 0 are shown by black crosses, with the arms of the crosses corresponding to the eigenvectors of the Jacobian matrix of the linearised system at the stationary points. Red lines are deterministic, environmentally forced trajectories, with dots giving the initial values. In (a) and (b) the green line gives the boundary between domains of attraction to the two stable points. WWW NATURE.COM/NATURE 21