Alternative stable states and unpredictable biological control of salvinia in Kakadu
Shon S. Schooler1, Buck Salau2, Mic H. Julien1, and Anthony R. Ives3
1
CSIRO Ecosystem Sciences, Long Pocket Laboratories, Indooroopilly, QLD, 4068, Australia
2
Department of the Environment, Water, Heritage, and the Arts, Kakadu National Park, Jabiru,
NT, Australia
3
Department of Zoology, University of Wisconsin, Madison, WI, 53706, USA
Contact details: email: [email protected]; phone: 608-262-1519
Keywords: domains of attraction, alternative stable states, Salvinia molesta, regime shifts,
bistable systems
http://www.nature.com/nature/authors/gta/
1
Suppression of Salvinia molesta by the salvinia weevil is an iconic example of successful
biological control. However, in the billabongs of Kakadu National Park, Australia, control
is fitful and incomplete. By fitting a process-based non-linear model to 13-year datasets
from 4 billabongs, we show that this can be explained by alternative stable states1-4 – one
state in which salvinia is suppressed and the other in which salvinia escapes weevil control.
The shifts between states are associated with annual flooding events. In some years, high
water flow reduces weevil populations, allowing the shift from a controlled to an
uncontrolled state; in other years, benign conditions for weevils promote the return shift to
the controlled state. In most described ecological examples, transitions between alternative
stable states are relatively rare, facilitated by slow-moving environmental changes, such as
accumulated nutrient loading or climate change5(but see 6). The billabongs of Kakadu give
a different manifestation of alternative stable states which generate complex and seemingly
unpredictable dynamics. Because shifts between alternative stable states are stochastic,
they present a potential management strategy to maximize effective biological control:
when the domain of attraction to the stable state of salvinia control is approached,
augmentation of the weevil population may allow the lower stable state to trap the system.
Awareness and concern about the ecological consequences of alternative stable states is
growing as more examples have been identified1,5,7. In many of these examples, the alternative
states are very stable, so that in the absence of an extraordinary perturbation, the system remains
at its "natural" state. Concern arises because states can change abruptly even when the
environmental drivers responsible for the change occur gradually; if the ecological system is
perturbed by a slow-moving driver, the system may remain largely unchanged until it reaches a
threshold catastrophe and abruptly shifts to its other state1. A current theoretical enterprise is
identifying the early warning signs of these abrupt shifts8,9. Equally disturbing, once the shift
occurs, the system will exhibit hysteresis1; even if the environmental perturbation were reversed,
the system would linger at its new state, greatly inhibiting the ability of managers to repair the
system to its natural state10.
Ecological systems, however, are subjected to stochastic or cyclic perturbations, and if
alternative states are weakly stable and perturbations are large enough, then shifts between states
may be routine1,11. Alternative stable states may thus generate underlying forces that govern the
stochastic dynamics of the system, leading to complex and seemingly irregular behaviour. In
fact, it may be difficult to identify the alternative stable states, yet at the same time be difficult to
understand the dynamics of the system without first identifying that alternative stable states exist.
2
Salvinia molesta, a South American aquatic plant, is one of the most widespread and
economically, environmentally, and socially destructive invasive plant species. Since 1939, it
has invaded lake and river systems in tropical and subtropical habitats around the world12. Its
success owes to its ability to double in biomass every three to four days and to regenerate
vegetatively even after severe damage or drying13,14. It is capable of forming dense mats up to 1m
thick that make waterways unnavigable and displace aquatic organisms12. Nonetheless, highly
successful biological control is often provided by the salvinia weevil (Cyrtobagous salvineae:
Curculionidae) that since 1980 has been introduced into most regions where salvinia has
invaded15. The salvinia weevil is a strict specialist on salvinia; adults feed on growing
meristematic tissue, whereas larvae tunnel through underwater vascular tissues12. The damage
caused by the weevils often leads to dramatic reductions in salvinia cover and biomass,
successfully controlling the weed with no additional expenditure of resources12,15.
Salvinia became established in Kakadu National Park in 198314, and the salvinia weevil
was released later that year16. Although the weevils rapidly established and successfully
controlled salvinia for several years, in 1988-1990 salvinia resurged to form thick mats despite
the presence of the biological control agent. This led to an intensive research project conducted by
CSIRO on salvinia management in Kakadu16 and to the establishment of a long-term monthly
sampling program of salvinia biomass and proportion of meristems damaged by weevils in four
billabongs (Fig. 1). Unlike lake systems which experienced continuously successful biological
control of salvinia, the Kakadu billabongs are subjected to annual flooding that mixes salvinia
among billabongs within the same floodplain. Floods also translocate salvinia from terrestrial sites
into the billabongs; salvinia persists in these terrestrial sites during the dry season, sometimes at high
biomass, where it has a refuge from the strictly aquatic weevils. Thus, the billabong systems of
Kakadu are highly perturbed.
Fieldwork during 1991-1994 led to the following hypothesis for the failure of continuous
biological control. When salvinia is at low density, it has a growth form with an open architecture in
which weevils have ready access to submerged meristematic tissue. This biomass is highly
susceptible to weevils and control. Conversely, when salvinia is able to establish thick mats, much
of the plant biomass occurs as vegetative, non-meristematic tissue often piled high above the water
surface. This biomass is inaccessible to both adult and larval weevils, and is therefore able to
withstand biological control. Although not stated as such, this is a hypothesis of alternate stable
states determined by the growth form of salvinia: a low biomass state maintained by weevil attack,
and a high biomass state which cannot be penetrated by weevils. As found in other cases of
3
alternative stable states, the salvinia-weevil system involves a species that has distinct life forms1,17
and a herbivore that potentially can lose control of its food plant18,19.
Standard analyses of the times series from the 4 billabongs reveal none of the dynamical
hallmarks of successful biological control: there is little evidence that weevil damage is associated
with declines in salvinia biomass (Supplementary Material). Furthermore, changes in salvinia
biomass between monthly samples are characterized by frequent small steps and occasional large
jumps, suggesting non-linear processes driving salvinia dynamics (Supplementary Material).
Therefore, we built a non-linear model from what we know about the biology of the system (Box 1).
Briefly, the model assumes that salvinia biomass is divided into two categories, one (meristematic
tissue) that is vulnerable to weevils and the other that is not, with the proportion represented by these
two categories depending on total salvinia biomass16. The growth of salvinia biomass depends on
the amount of vulnerable tissue. Weevils attack this vulnerable tissue non-uniformly, so that attacks
can be aggregated among meristems. The population growth rate of the weevils depends on the
amount of salvinia biomass damaged. There is net migration of salvinia into billabongs from
external areas, and mortality/flushing of both salvinia and weevils that depends on the water flow
through the drainages. There is also stochastic variation that affects the per capita growth rates of
both salvinia and weevils, and this variation can increase with increasing water flow. The latter
property accounts for possible increases in unpredictable flushing or filling of billabongs with
salvinia or weevils during flooding events. The model fit the data well, with most parameters
showing statistically strong effects on the observed dynamics (Fig. 2, Table 1).
In the absence of stochastic variation, and with water flow fixed at its long-term average, the
parameterized model exhibits alternative stable states (Fig. 3, Supplemental Material). Key
biological insights from the fitted parameter values of the model include the following. The
proportion of salvinia biomass that is invulnerable to weevil attack, g, is generally high, ranging
from gmin = 0.91 to gmax = 0.94, which is consistent with weevils attacking meristematic tissues
(adults) and underwater vascular tissue (larvae). Flooding events reduce the abundance of weevils
(dw < 0) yet have little net effect on the mean abundance of salvinia (ds = 0). However, flooding
events increase the variability in sample-to-sample fluctuations in salvinia biomass (s2 > 0). These
patterns could be caused by flooding events moving salvinia among billabongs, and between
billabongs and surrounding land, which simultaneously increases the variance in salvinia biomass
and causes a net reduction of weevil abundance as salvinia colonizes billabongs from weevil-free
terrestrial refuges.
4
Existence of the alternative stable states requires the proportion of salvinia biomass in the
category vulnerable to weevil attack to change with salvinia biomass; if the model is constrained so
that this proportion does not change (gmin = gmax), then alternative stable states are impossible, and
the fit of the model is statistically significantly reduced (2 = 52.76, p << 0.001, likelihood ratio
test). This provides strong support for the existence of alternative stable states. In three of the
billabongs, the system spends time in the domains of attraction to both stable states, while Minggung
has generally high salvinia abundance and rarely occupies the domain of weevil control (Fig. 3). We
fit the model assuming that the underlying processes were identical across all billabongs. The
explanation for Minggung having poor salvinia control could be due simply to the stochastic nature
of the dynamics, by chance never staying long in the region of weevil control. There may be other
differences between Minggung and other billabongs that we cannot identify, although such
differences are not required to explain the observed dynamics.
Due to the high stochasticity in the system and the annual flooding events, the fit of the
model relies not only on the existence of alternative stable states, but also the transient dynamics of
the corresponding deterministic model. Although the deterministic model gives alternative stable
states when the logarithm of water flow is fixed at its mean value, when the log water flow is fixed
at one standard deviation below its mean, there is only a single stable point, and when fixed at one
standard deviation above its mean, the weevil is eliminated from the system (Supplemental
Material). Therefore, as the water flow regime fluctuates through its annual cycle, the alternative
stable points alternately disappear. Despite this disappearance, however, the "ghost" of the boundary
between stable states still affects the transient dynamics of the system6,20,21. An added complexity is
that the dynamical forces around the two alternative stable states differ. In the domain of attraction
to the state with high salvinia biomass, changes in biomass are slow compared to change in weevil
damage, as illustrated by the deterministic trajectories of the model (Fig. 3). This allows salvinia
biomass to dynamically wander between high and moderate values. The domain of attraction to the
lower stable state contains trajectories that tend to approach the stable point through increases in
both salvinia biomass and weevil damage, causing a positive correlation between these two variables
in the stochastic case (Fig. 3a). All of these dynamical patterns contribute to the strength of fit of the
model to the data.
These analyses point to possible opportunities to foster biological control of salvinia at
Kakadu. For many examples of ecological systems with alternative stable states, the stability of the
states makes transitions between them ecologically difficult and operationally challenging from a
management perspective 10. The Kakadu billabongs, however, are highly stochastic and experience
periodic flooding, giving a window opportunity to shift the system between states22,23. Even for
5
Minggung, where biological control has been ineffective, the system occasionally grazes the domain
of attraction to the lower stable state of salvinia control. This occurs towards the end of the dry
season after weevil populations recover from depression during flooding. If weevil control could be
augmented at this time by inoculating Minggung with infested salvinia from other billabongs, the
system could be captured in the lower domain of attraction. There is no guarantee that this would
work the first, the second, or even the third try. However, the demonstration that this lower state
likely exists for Minggung should give hope for repeated attempts. Even the other billabongs with
better salvinia control may benefit from this approach.
Technical Box
We fit a single model to the data simultaneously for all 4 billabongs, thereby assuming that
their dynamics are governed by the same processes. The model has a non-linear state-space form24,
with one set of equations describing the biological processes driving the dynamics and the other
describing the sampling used to generate the data. Both process and measurement equations contain
stochastic elements, with process error encapsulating environmental variation and measurement
error describing any deviations between the “true” state of the process variables and the data.
The process equations are

x t 1  x t gx t   x t 1 gx t e r t 1 1 gx t x t



v

exps (zt )
1 pt exps (zt )  m t 

y t 1  cx t 1 gx t e r t pt expw (z t )expw (z t )
where xt and yt are the salvinia biomass and weevil abundance at time t, and

k
a t y t
pt  1 
1 k 1 gx  x 
 is the proportion of accessible salvinia attacked by weevils.

t  t 

Because the sampling times were unevenly spaced, the time points at which the model was
iterated are also unequal. Between each pair of sample points, we iterated the model at intervals
<10 days apart, selecting the interval t to minimize the number of iterations under this constraint;
over the entire data set, the average time between iterations was 8.83 days. The function g(xt)
gives the proportion of salvinia biomass that is inaccessible to weevil damage. g(xt) follows a
logit function with minimum and maximum values gmin and gmax, inflection point where xt = b1
and slope at the inflection point given by b2. The input variable zt is the log water flow of the
drainage, and s(zt) and w(zt) give the response of salvinia and weevil survival rates dependent
6
on water flow. Model parameters are described in Table 1, and a detailed model description is
given in the Supplemental Material.
The measurement equations are


X t *  X t  log ge X t   C   s
W t *  logit pt    w
where Xt = log xt is the “true” log salvinia biomass from the process equations, Xt* is the
observed log salvinia biomass assuming that only inaccessible (vegetative) biomass is sufficiently
dense to occur in sampling, and Wt* is the logit of the observed weevil damage. The constant C
is an overall scaling term for salvinia biomass, because the process equations are nondimensional. The random variables s and w give measurement error, with s assumed to have
a Gaussian distributing with mean 0 and variance ms, and w assumed to have a quasi-binomial
1
1
2 
distribution with variance    mw
; if mw = 0, this would be the variance under a

n
 pt (1 pt )
binomial distribution, but to allow greater-than-binomial distribution, we also estimated mw.
The model was fit to the data using an extended Kalman filter to estimate the likelihood
function24. The parameter r, the intrinsic rate of increase of the salvinia, was set at 0.08 d–1 as
determined by extensive experiments at Kakadu (Julien, unpublished data). Backwards model
selection was performed to find the best-fitting model, and the best-fitting model was confirmed
with forward selection. Likelihood ratio tests were used to assess the statistical departure of key
variables from zero.
Acknowledgements
We thank S. Winderlich (Kakadu NP) for facilitating data processing, and S. Cruickshank and G.
Willis (NRETAS) for providing gauging station data. M. Storrs, C. Murakami, F. Hunter, M. Hatt
and F. Baird provided field assistance. A. Bourne assisted with data interpretation. S. S. Schooler
received funding from DAFF and A. R. Ives received funding from a CSIRO McMaster’s
Fellowship and US-NSF funds through the North Temperate Lakes LTER. S. R. Carpenter and X
anonymous reviewers provided constructive comments on earlier versions of this manuscript.
7
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8
Table 1: Estimates from the best-AIC fitting model of the biologically relevant parameters. The
model without alternative stable states (gmin = gmax, b1 = b2 = 0) has a AIC of 48.76 and provides a
statistically significantly inferior fit to the data (2 = 52.76, p << 0.001). For the model, R2 = 0.73
and 0.38 for log salvinia biomass and logit weevil damage, respectively. Values in bold are
statistically significant by a likelihood ratio test.
Parameter Value
Description
gmin 0.91
Minimum weevil-inaccessible tissue
gmax 0.94
Maximum weevil-inaccessible tissue
b1 5.41
Salvinia morphology infection point
b2 385.7
Slope of morphology change at b1
a 0.09
Weevil attack rate
k 3.38
Aggregation parameter for weevils
c 0.94
Weevil reproduction scale parameter
m 0.0085
Salvinia net immigration
v 0.791
Salvinia self-regulation form parameter
ds 02
Change in salvinia with water flow, s(zt) = dslog(zt)
dw –0.24
Change in weevils with water flow, w(zt) = dwlog(zt)
s1 0.21
Salvinia process error, SD{s(zt)} = s1+s2log(zt)
s2 0.35
Water flow effect on salvinia process error
w1 0.42
Weevil process error, SD{w(zt)} = w1+w2log(zt)
w2 –0.083
Water flow effect on weevil process error
1
not different from 1 (AIC = 1.08)
2
not different from 0.01 (AIC = 1.82)
3
not different from 0 (AIC = 1.86)
9
Fig. 1: Log biomass of salvinia in 4 Kakadu billabongs: Jabiluka (green), Minggung (blue), Jaba
(black), and Island (red). The vertical axis is scaled to have mean zero across all billabongs.
10
Fig. 2: Fitted model to log salvinia biomass (black) and logit weevil damage (red) for 4
billabongs; raw data are given by dots, and the fitted model by lines. Data and fitted model are
standardized to have mean zero across all billabongs. The water flow in the drainages are given
by the blue line that is standardized to have mean zero and variance one.
11
Fig. 3: Phase portraits of logit weevil damage against log salvinia biomass for the 4 billabongs;
values have been standardized to have mean zero across all billabongs. Plotted trajectories (black
dots and lines) are model values fitted to the data, thereby eliminating measurement error. The
green line demarks the domains of attraction of the two stable points, marked with red dots, and
example deterministic trajectories are included (red lines) when the drainage water flow is fixed
at its mean value, the time step between samples is assumed to be  = 8.8257 d, and there is no
stochasticity in the model. The blue x gives the equilibrium abundance of salvinia when there is
no stochasticity and the log water flow is fixed at one standard deviation below its mean value.
The blue arrow gives the deterministic abundance of salvinia in the absence of weevils.
12
Supplementary Methods
Alternative stable states and unpredictable biological control of salvinia in Kakadu
Shon S. Schooler, Buck Salau, Mic H. Julien, and Anthony R. Ives
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 2004. At each site, salvinia cover, salvinia biomass, number
of weevils, and percent ramets damaged by weevils was 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. Total salvinia biomass was calculated by multiplying the percent cover of salvinia by the area
of each billabong 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 meristems 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 used a simple linear interpolation from the two
adjacent points.
13
Preliminary time-series analyses
We performed preliminary analyses to determine whether there were obvious patterns in the
data that indicate biological control and relatively simple dynamics. For successful control, the
intensity of attack on the pest by the control agent must increase at high pest density to limit the
growth of the pest. Similarly, the effect of the control agent should be seen as decreases in the pest
abundance when attacks by the control agent are high. Neither of these patterns is strong in the
billabongs (Fig. S2a,b). Furthermore, 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). This distribution indicates episodically large
changes in the 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 – motivate the need for a nonlinear model.
Model description
We assume that salvinia biomass is divided into two categories. Category 1 consists of
meristematic and other tissue that is fed upon by weevils, whereas category 2 is tissue that is
inaccessible to weevils and, while photosynthetic, is not actively growing. Biologically, three
different growth forms have been described for salvinia, primary to tertiary, that decrease in the
proportion of meristematic tissue. Thus, the primary growth form has the highest proportion of
category 1 biomass, while the tertiary growth form has the highest non-growing tissue
unavailable to weevils (category 2). 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) follows a logit function
g(x) = gmin  gmax  gmin logit b2 x  b1
14
(1)
where logit() is the logit function, gmin and gmax give the minimum and maximum proportion of
salvinia biomass in category 2, b1 gives the location of the inflection point at which g(b1) = (gmax
+ gmin)/2, and b2 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
(2)
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 (contributing to
biomass in category 2) according to a negative binomial distribution with dispersion parameter k,
so that the proportion of non-attacked meristematic tissue is

k
a t y
1

 k 1 g(x)x 
(3)
Here, the proportion of non-attacked tissue is assumed to depend on the density of weevils y
relative to the category 2 biomass (1–g(x))x, scaled by at where a is the attack rate per day and t
is the time (in days) between samples 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
2 tissue consumed with a scaling term c.
From prior biological information about the salvinia-weevil system, we suspected that
high flow rates would depress weevil damage and presumably flush salvinia from billabongs.
High flow rates might also increase the environmental variance impacting both salvinia and
weevils. We included these effects of flow rates as s(zt) = dslog(zt) and w(zt) = dwlog(zt) where
zt is the log-transformed flow rate at sample t, and s(zt) and w(zt) are mortality rates for salvinia
and weevils associated with flow rates zt. Thus, negative values of ds and dw lead to increasing
loss of salvinia and weevils with increasing flow rates.
15
The full dynamical equations are


x t 1  x t gx t   x t 1 gx t e r t 1 1 gx t x t





v


k 

a t y t


1 k 1 gx x 
 exps (zt )  m t exps (z t )
t
t 






 
k 

a t y t
r t 
exp (z )

y t 1  cx t 1 gx t e 1 
1
w
t expw (z t )
 

k 1 gx t x t 








(4)
Here, process errors (environmental stochasticity) is represented by the zero-mean random
variables s(zt) and w(zt) whose variances may depend on flow rates. Specifically, we assume
that the standard deviations of s(zt) and w(zt) are s1+s2log(zt) and w1+w2log(zt), and that s(zt)
and w(zt) have correlation coefficient . Positive values of s2 and w2 imply that the variability
in loss/retention of salvinia and weevils increase with the flow rate. The parameters s1 and w1
govern the overall variability in salvinia and weevil loss/retention. Equations (4) 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 (Model fitting).
Fitting model to data
We fit the model given by equations (4) using a non-linear state-space model24. The
process equations used to describe the dynamics of the salvinia-weevil system
are
k


1
a t eQt 
Xt
Xt
Xt
Xt
r t
Xt
Xt

X t 1  loge ge  e 1 ge  e 1 1 ge  e
1

m



t  ds (z t )  1 (z t )
k






k
  a eQt  
Yt 1  Yt  Qt  logc   r t  log1 1 t   dw (zt )  2 (zt )
k  


 

  
 
(5)
These are the same as equations (4), with Xt = log(xt), Yt = log(yt), and
Qt  log


yt
 Yt  log 1 ge X t   X t .
1
g
x
x
  t  t
The measurement error equations are
16


X t *  X t  log ge X t   C   ms
W t *  logit pt    mw
Here, Xt* is the observed log of 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 measure error equation for
salvinia assumes that measurement error is lognormally distributed. We assume that ms is a
Gaussian random variable with mean zero and variance 2ms. The parameter C is a scaling
constant that sets the overall magnitude of the observed log salvinia biomass.
Wt* is the observed logit-transformed proportion weevil damage. This structure of the
measurement equation approximates the situation in which weevil damage is binomially
distributed with the probability that a sampled ramet is damaged given by
k
 a expQt 
pt  1 1 t
 . Measurement error in this approximation is a Gaussian random
k


variable mw with mean zero. For a purely binomial process, the variance of mw is
1
1
n pt (1 pt )
where n is the sample size (n = 100). To account for possible inflated measurement error
variance that could occur in taking only 3 samples of 100 ramet samples from an entire billabong,
1
1
2 
, where 2mw is estimated from the data.
we set the variance of mw equal to    mw

n
 pt (1 pt )
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 iteration points to make the interval between iteration points <10.
Note that the state-space model does not require the time points of iteration of the process
equations to 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 flow rate over the 10 day period before the iteration point.
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: gmin, gmax, b1, b2, a, k, m, v, c, ds, dw, ss1, ss2, sw1,
sw2, and . There are 3 parameters in the measurement error: C, 2ms and 2mw. This gives a total
17
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.
We fit the model to the data with an extended Kalman filter. Initial points for each
billabong were taken as the first observed values, and the initial variance was taken as the process
error variance (i.e., the variances of ms and mw). 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).
Theoretical Exploration of the Model
The fitted model, when removing stochasticity (ss1 = ss2 = sw1 = sw2 = 0), assuming a
constant time between model iterations ( = 8.8257 d), and setting the water flow rate to its mean
value (zt = 0), has two stable points separated by an unstable point (Fig. S3a); note that this phase
portrait is graphed in terms of salvinia biomass x and weevil abundance y, rather than log biomass
of visible salvinia and logit weevil damage as in figure 3. Trajectories approach both stable
points horizontally via changes primarily in salvinia biomass, rather that weevil abundance.
Thus, there is a broad range of salvinia biomass over which trajectories move relatively slowly.
We can use the water flow rate, zt, as a convenient and biologically relevant bifurcation
parameter. Decreasing zt causes the stable point at high salvinia biomass to collide with the
unstable point through a fold (saddle-node) bifurcation25, annihilating both points and leaving
only the lower stable point (Fig. S2b); this occurs at zt = –0.29486. The "ghost" of the annihilated
unstable point still remains20,21, causing a slowdown in the trajectories and a sudden change in
direction. This corresponds to the salvinia biomass x = b1 at which the proportion of biomass in
the inaccessible category, g(x), switches from gmax to gmin as x decreases. In the fitted model, this
switch from gmax to gmin is rapid, since b2 = 385. In the statistical fitting, the value of b2 is poorly
estimated; reducing the value of b2 changes the fit very little and leads to smoother deterministic
trajectories. Inability to estimate b2 precisely is to be expected, because stochasticity makes
trajectories stay only briefly in the vicinity of the switch point b1.
In a similar way, increasing zt causes the stable point at low salvinia biomass to collide
with the unstable point through a fold (saddle-node) bifurcation, annihilating both points and
leaving only the higher stable point (Fig. S2c); this occurs at zt = 0.22876. As before, the ghost
18
of the annihilated unstable point still remains, causing a slowdown in the trajectories and a
sudden change in direction.
19
Table S1: Parameter values of models compared to the best-fitting AIC model
AIC
npar
lnlike
gmin
gmax
b1
b2
a
k
c
m
v
0.79
d2
ss1
ss2
sw1
sw2

C
2ms
2mw
-0.24
0.21
0.35
0.42
-0.08
0.51
-3.57
0.24
0.00
-0.24
0.21
0.36
0.41
-0.08
0.51
-3.54
0.25
0.00
-0.25
0.21
0.36
0.44
-0.11
0.50
-3.54
0.23
0.00
-0.23
0.21
0.36
0.41
0.51
-3.57
0.23
0.00
0.00
-0.23
0.21
0.34
0.42
-0.08
0.51
-3.57
0.22
0.00
0.00
-0.25
0.21
0.32
0.42
-0.06
0.51
-3.49
0.25
0.00
0.00
-0.23
0.21
0.34
0.42
0.51
-3.56
0.22
0.00
-0.22
0.19
0.36
0.42
-0.12
0.50
-3.67
0.29
0.00
d1
0.00
18
-1360.39
0.91
0.94
5.41
385.70
0.09
3.38
0.94
0.0085
1.08
17
-1361.93
0.92
0.95
5.24
385.64
0.09
3.38
0.95
0.0084
1.82
18
-1361.29
0.92
0.96
5.13
295.94
0.10
2.41
0.85
0.0085
1.86
17
-1362.32
0.91
0.95
5.41
390.46
0.09
4.74
0.96
0.0088
0.82
3.53
19
-1361.15
0.89
0.93
5.46
459.45
0.10
3.13
0.87
0.0086
0.67
3.75
18
-1362.26
0.91
0.95
4.97
399.38
0.10
3.13
0.88
0.0089
6.51
18
-1363.64
0.90
0.93
5.36
491.91
0.10
3.38
0.87
0.0082
14.13
17
-1368.45
0.92
0.95
5.75
800.64
0.06
3.31
1.32
0.83
16.59
18
-1368.68
0.90
0.93
5.44
490.44
0.10
2.52
0.87
0.66
0.00
-0.23
0.21
0.34
0.43
-0.09
0.51
-3.59
0.22
0.00
48.76
16
-1386.77
0.90
0.00
1.32
22.04
0.6230
0.07
0.01
-0.28
0.16
0.36
0.46
-0.23
0.37
-8.22
0.70
0.00
52.58
15
-1389.67
0.86
0.06
1.32
1.46
0.0570
0.07
-0.27
0.16
0.40
0.48
-0.22
0.52
-6.22
0.68
0.00
81.97
17
-1402.37
0.92
0.94
6.06
270.94
0.09
2.31
1.02
0.0083
0.62
-0.25
0.23
0.44
-0.11
0.51
-3.63
0.31
0.00
155.44
18
-1438.10
0.90
0.94
4.72
427.91
0.10
3.07
0.86
0.0088
0.66
0.00
-0.22
0.24
0.43
-0.08
0.51
-3.45
0.24
0.00
160.43
18
-1440.60
0.89
0.89
6.07
470.19
0.10
3.46
0.75
0.0076
0.56
0.00
161.21
17
-1441.99
0.90
0.92
6.29
342.59
0.10
2.53
0.83
0.0100
0.71
20
0.01
0.68
0.20
0.31
0.46
-0.09
0.51
-3.70
0.39
0.00
0.22
0.28
0.48
-0.08
0.51
-3.90
0.34
0.00
Fig. S1: Logistic regression of proportion beetle damage against adult beetle density used to
interpolate missing values of beetle damage.
21
Fig. S2: Dynamical characteristics of salvinia biological control by weevils in Jabiluka billabong.
(a) The change in log biomass of salvinia between samples was weakly negatively correlated (cor
= –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 0.001 for Minggung, Jaja, and Island,
respectively. (b) The change in the intensity of weevil damage between samples was weakly
positively correlated (cor = 0.049) with log salvinia biomass in the first of the samples;
correlations for the other 3 billabongs are 0.041, 0.081, and –0.028 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 < 0.0001), 5.69 (p < 0.001), and 8.02 (p < 0.0001) for Minggung, Jaja, and
Island, respectively.
22
Fig. S3: Phase portraits of the salvinia-weevil model graphed in terms of salvinia biomass and
weevil abundance (both non-dimensional); the bifurcation parameter zt = 0, –1, and 0.5 for (a)(c), respectively. Stationary points are shown by black crosses, with the arms of the crosses
corresponding to the eigenvectors of the linearized system at the stationary points. Red linear 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. 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.
23