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Contents lists available at ScienceDirect
Ecological Complexity
journal homepage: www.elsevier.com/locate/ecocom
Original Research Article
Alternative (un)stable states in a stochastic predator–prey model
Karen C. Abbott *, Ben C. Nolting 1
Department of Biology, Case Western Reserve University, Cleveland, OH 44106, USA
A R T I C L E I N F O
A B S T R A C T
Article history:
Received 10 March 2016
Received in revised form 25 October 2016
Accepted 22 November 2016
Available online xxx
Stochastic models sometimes behave qualitatively differently from their deterministic analogues. We
explore the implications of this in ecosystems that shift suddenly from one state to another. This
phenomenon is usually studied through deterministic models with multiple stable equilibria under a
single set of conditions, with stability defined through linear stability analysis. However, in stochastic
systems, some unstable states can trap stochastic dynamics for long intervals, essentially masquerading
as additional stable states. Using a predator–prey model, we demonstrate that this effect is sufficient to
make a stochastic system with one stable state exhibit the same characteristics as an analogous system
with alternative stable states. Although this result is surprising with respect to how stability is defined by
standard analyses, we show that it is well-anticipated by an alternative approach based on the system’s
‘‘quasi-potential.’’ Broadly, understanding the risk of sudden state shifts will require a more holistic
understanding of stability in stochastic systems.
ß 2016 Elsevier B.V. All rights reserved.
Keywords:
Alternative stable states
Stochastic dynamics
Unstable equilibrium
Predator–prey dynamics
Regime shifts
Quasi-potential
1. Introduction
When an ecological system has alternative stable states –
multiple stable equilibria in its underlying deterministic dynamics
– very interesting behaviors can result (May, 1977). For instance,
switching between alternative stable states through time can
occur either by a stochastic perturbation to the state itself that
moves the system into the alternative basin of attraction, or by a
perturbation to the external conditions that moves the system out
of the multi-stable regime (Fig. 1a; Beisner et al., 2003; Ridolfi
et al., 2007; Scheffer, 2009). It is alarming that small perturbations
can produce large state changes, and that an equal size
perturbation in the reverse direction will not return the system
to the original state (Fig. 1a).
Traditionally in ecology, alternative stable states in models are
identified through linear stability analysis, wherein the dynamics
following a perturbation from an equilibrium state are characterized via linear approximation. Decay of perturbations means
convergence to the equilibrium, and thus stability of that
equilibrium state. These perturbations are assumed to be very
small, so that the linear approximation is valid, and isolated, so
* Corresponding author.
E-mail address: [email protected] (K.C. Abbott).
1
Present address: Department of Mathematics and Statistics, California State
University Chico, Chico, CA 95929, USA.
their growth or decay can be examined without considering
further perturbations.
Ecological dynamics are driven by both deterministic and
stochastic components (Ellner and Turchin, 1995; Bjørnstad and
Grenfell, 2001; Coulson et al., 2004; Denaro et al., 2013), but linear
stability analysis only examines a system’s deterministic behavior
following a single, small, isolated perturbation. In this way, this
ubiquitous technique is ill-equipped to address situations where
stochasticity has a meaningful, qualitative effect on dynamics (e.g.
Chesson and Warner, 1981; Vilar and Solé, 1998; Anderies and
Beisner, 2000; Hastings, 2001; Mankin et al., 2002; Greenman and
Benton, 2003; Spagnolo et al., 2003, 2004; Valenti et al., 2004a;
Abbott et al., 2009). Counter-intuitive qualitative effects arise from
the interaction between stochasticity and nonlinearities. These
effects include noise enhanced stability and stochastic resonance
(Gammaitoni et al., 1998; Valenti et al., 2004b). The transient
behaviors of stochastic systems are intruigingly complex, as they
can approach a stable state in a multitude of ways (Fiasconaro
et al., 2003; Fukami and Nakajima, 2011) or avoid convergence on a
stable state altogether (Ridolfi et al., 2007). As a result of these
complex behaviors, deterministic analyses can miss important
features of stochastic systems (Spagnolo et al., 2003; Provata et al.,
2008). In this paper, we explore the implications of this disconnect
specifically for the study of alternative stable states.
Classic theory has focused on using linear stability analysis to
identify stable equilibria, under the assumption that these are the
http://dx.doi.org/10.1016/j.ecocom.2016.11.004
1476-945X/ß 2016 Elsevier B.V. All rights reserved.
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
Complex. (2016), http://dx.doi.org/10.1016/j.ecocom.2016.11.004
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Fig. 1. Bifurcation diagrams for deterministic model (1), with alternative stable states in the ‘‘region 2SS’’ range. (a) Equilibrium predator density versus a. Solid lines
represent stable equilibria and dashed lines are unstable or infeasible (i.e. negative) equilibria. A population at A can shift to the upper equilibrium by direct perturbation to its
density (to arrive at B), or by a perturbation in a (to arrive at C). Populations at B or C subjected to an equal but opposite perturbation will recover to B, not A. (b–d) Diagrams
showing regions 1SS and 2SS in a–g space for 3 b values. Horizontal gray stripes show g values and a ranges used with each b in our simulations. (Note, d affects the
equilibrium population sizes but not the locations of these regions, so these diagrams hold for any value of d. Noise intensity, s, is by definition 0 in the deterministic model
depicted here.)
only states that matter in the long-term. However, many
fascinating and well-documented examples exist where there is
a clear long-term influence of unstable equilibria (Rohani et al.,
2002; Coulson et al., 2004; D’Odorico et al., 2005; Mankin et al.,
2007; Tél, 1990; Rand and Wilson, 1991; Aparicio and Solari, 2001;
Greenman and Benton, 2003; Dwyer et al., 2004). For example,
saddle nodes, equilibria that are unstable yet attracting from some
states along a stable manifold, can cause a stochastic trajectory to
remain nearby for long, albeit transient, intervals (Cushing et al.,
1998; Henson et al., 1999; Hastings, 2004; Parker et al., 2011).
Continual stochastic perturbations can allow frequent visits to
such a saddle. In this way, systems could appear as though they
have alternative stable states even when classic theory says they
do not. Currently, we lack theory on whether it is possible or even
informative to distinguish between true (classical) multi-stability
and stochastic look-alikes (Fukami and Nakajima, 2011), making
applications of these concepts to data particularly challenging.
In this paper, we study a stochastic version of a predator–prey
model (Freedman and Wolkowicz, 1986; Kot, 2001) that allows us
to compare dynamics from highly analogous systems with and
without alternative stable states in their underlying deterministic
skeletons. We find that several key characteristics that we would
normally associate with alternative stable states appear even when
the model has only a single stable equilibrium. This occurs when a
saddle effectively poses as an additional attractor, as described
above. It may not be strictly impossible to distinguish between
systems with true alternative stable states and other stochastic
systems with similar dynamics, but our results demonstrate that
the distinction is likely to be significantly more subtle than is
generally recognized, and that the stable/unstable classification
dichotomy should be reconsidered.
Because we find that sudden state shifts may occur even in the
absence of ‘‘alternative stable states’’ (as defined through classical
linear stability analysis), our results challenge us to seek more
informative measures of stability for stochastic systems. We
recently proposed the quasi-potential as a useful means of
quantifying and visualizing stability in stochastic ecological
models (Nolting and Abbott, 2016). We therefore close this article
by exploring whether ‘‘stability’’ as measured by the quasipotential better aligns with the stochastic behavior of the model.
2. Methods
2.1. Model
One step toward a better understanding of alternative stable
states is to better appreciate whether and how they differ from
other types of stochastic dynamics. We explore this issue
beginning with a deterministic predator–prey model (Freedman
and Wolkowicz, 1986),
dN
N
NP
¼ N 1
(1a)
dt
g
ð1=aÞN2 þ N þ 1
dP
bdNP
¼
dP;
dt
ð1=aÞN 2 þ N þ 1
(1b)
where N and P are prey and predator population densities, g is a
rescaled prey carrying capacity, b governs the rate at which
consumed prey are converted to predator population growth, and d
is the predator’s rescaled death rate. We deliberately use the same
parameterization here as Kot (2001, Chapter 9), to which we refer
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
Complex. (2016), http://dx.doi.org/10.1016/j.ecocom.2016.11.004
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readers interested in a very accessible summary of this deterministic model’s behaviors. Predation follows a type-IV functional
response, which occurs when abundant prey benefit from group
N
defense, and is described by the term
in which a
2
ð1=aÞN þNþ1
controls the intermediate prey density where per-capita predation
rate is maximized. Although observed in nature (e.g van de Koppel
et al., 1996), type-IV predation is not commonly invoked in
ecological theory. We use it here simply as an example, as it
produces a model with desirable features that we describe below;
we have no reason to believe that our findings are specific to typeIV predation.
By changing a single parameter (a, Fig. 1a), model (1) can have
alternative stable states or a single stable state. We will refer to the
parameter range with alternative stable states as ‘‘region 2SS’’ (for
2 stable states) and an adjacent region with a single stable state as
‘‘region 1SS’’. In region 2SS, a stable interior equilibrium and a
stable boundary (prey-only) equilibrium coexist. In region 1SS,
only the interior equilibrium is stable while the boundary
equilibrium is a saddle. After adding stochasticity, we compared
simulations from these two regions in search of recognizable
signatures of alternative stable states in the deterministic skeleton.
We added Gaussian white noise to Eqs. (1) to produce a
stochastic version of the model,
" #
N
NP
dN ¼ N 1
(2a)
dt þ s dW N ; N; P 0
g
ð1=aÞN 2 þ N þ 1
"
dP ¼
bdNP
ð1=aÞN2 þ N þ 1
#
dP dt þ s dW P ;
N; P 0
(2b)
where WN and WP are independent Wiener processes, each with
infinitesimal mean 0 and infinitesimal variance 1. The noise
intensity, s, characterizes the strength of stochastic disturbances.
Readers interested in a broad introduction to the formulation and
study of stochastic biological models may refer to one of the many
good introductory texts on the subject (e.g. Nisbet and Gurney,
1982; Horsthemke and Lefever, 2006; Allen, 2010; Gardiner, 2010).
Biologically, the stochastic perturbations in Eqs. (2) represent
disturbances that occur independently of the current population
densities, such as random, density-independent immigration or
emigration. The boundary equilibrium is therefore not absorbing in
the stochastic model, reflecting the realistic situation where local
extinction is not permanent in viable populations (Hanski, 2003).
In other words, it is possible for positive perturbations to rescue
extinct populations.
The constraint N, P 0 in model (2) prevents the combination of
small population density and large downward perturbation from
causing negative population densities. There are a variety of ways
to ensure non-negative values for state variables that, biologically,
cannot be negative. A very common approach is to model
stochasticity such that the size of perturbations shrinks to zero
as population density approaches zero (e.g. Allen, 2010; Sharma
et al., 2015). This is appropriate for demographic stochasticity, but
not for density-independent perturbations like the ones we model
here. Our question requires that we hold noise intensity constant
across parameter sets (i.e. that we not allow the distribution of
perturbations size to change as we change parameter values and
thus mean population densities). As such, the noise formulation
shown in Eqs. (2) is ideal for comparing dynamics with and
without alternative stable states.
2.2. Simulations
We simulated stochastic model (2) to obtain time series that we
treated as though they were data. This allowed us to take an
3
empirical viewpoint and ask, given noisy data, do we recognize the
presence of alternative stable states in the underlying deterministic process? We began by selecting 3 values of the parameter b
and, for each b, 2–3 values of g that run through both regions 1SS
and 2SS. For each [b, g] combination, we identified a range of a
values that spanned the transcritical bifurcation between regions
1SS and 2SS; these ranges are shown by the horizontal gray bars in
Fig. 1b–d. We ran simulations for each of 50 evenly-spaced a
values within these ranges. Simulations were repeated for 2 values
of d (2.5 and 5) and 3 values of s (0.01, 0.05, and 0.1). Because d had
virtually no qualitative effect on our results (see Appendix 2), we
simulated most but not all d–s combinations with each of the [b, g]
pairs. In total, we ran simulations with 38 different [b, g, d, s]
combinations (listed in Appendix 2). We refer to these [b, g, d, s]
combinations as parameter sets hereafter, and we remind the
reader that within each parameter set, we repeated the simulations at 50 different a values spanning regions 1SS and 2SS.
With each parameter set and for every a value, we simulated
100 realizations of stochastic model (2) for 500 time units. This
number of realizations was sufficient to give precise estimates of
all candidate statistics described below (K. Abbott, unpublished
results). We initiated simulations near the interior equilibrium,
since this is the stable equilibrium shared by both regions (1SS and
2SS). We discarded the first 300 time units to eliminate transient
model behaviors due only to initial conditions, then calculated
several candidate statistics, described below. We compared the
average value of each statistic across a values and recorded the
percentage of the 38 parameter sets that conformed to our
expectation for how each candidate statistic should change as we
change a to gain or lose alternative stable states.
All simulations were performed in Matlab. We discretized the
model for simulation using a time step of dt = 0.01, which is
sufficiently small to provide convergence of the Euler method for
the deterministic ODE model (Eqs. (1)) for every parameter
combination we considered (not shown). We included stochasticity by drawing an independent value of a Gaussian random
variable with mean 0 and variance s2 dt for use at each time step
and each species. To standardize the stochastic environment across
parameters, we used the same 100 sequences of this random
variate to generate our 100 realizations with each parameter
combination. If population sizes ever became negative, the density
was set immediately to zero instead. This correction had no effect
on the dynamics besides preventing negative population densities
(Appendix 1).
2.3. Candidate statistics and expected patterns
We compared simulations across regions 1SS and 2SS using 3
candidate statistics that quantify (i) the tendency of the populations to remain near the interior equilibrium, (ii) the temporal
variance in population densities, and (iii) the appearance of
multiple modes in the distributions of population densities. These
are not intended as diagnostics of bistability per se, but are rather
quantities that, on an intuitive level, we expect to change in
predicable ways as we lose bistability by increasing a. We treated
each simulated time series as a dataset and computed the 3
candidate statistics, then averaged each statistic across the 100
replicate simulations for each parameter set and a value.
The first of these 3 statistics was (i) the difference between the
average simulated population density and the population density
at the interior equilibrium for each species. If simulated populations from region 2SS split time between the interior and boundary
equilibria, and populations from 1SS remain mainly near the
interior equilibrium since the boundary is unstable, then we expect
the following (Fig. 2): in region 2SS, average prey density will be
higher than the interior equilibrium and mean predator density
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
Complex. (2016), http://dx.doi.org/10.1016/j.ecocom.2016.11.004
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(a) 1.6
region 1SS
(b)
1.4
region 2SS
1.4
1.2
Predator density
1.6
1.2
densities at interior equilibrium
1
1
0.8
0.8
expected mean densities
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.5
1
1.5
2
2.5
3
0
0
0.5
1
1.5
2
2.5
3
Prey density
Prey density
Fig. 2. Equilibria of deterministic model (1) with b = 2, g = 2.6, d = 2.5 and (a) a = 4.5 or (b) a = 4.1. (a) From region 1SS and (b) from region 2SS. Stable equilibria are marked by
black circles and unstable equilibria are marked by s. Gray shapes represent a hypothesized cloud of population densities that might be observed when stochasticity is
added to the model (Eqs. (2)) and the white star represents the approximate mean of this hypothesized cloud. In region 1SS, we might expect a cloud of densities roughly
centered on the only deterministically-stable equilibrium. In region 2SS, the stochastic system may split time between the two stable equilibria and have a mean somewhere
between them. This would give higher mean prey densities and lower mean predator densities than the interior equilibrium in 2SS.
will be lower. As we increase a and move into region 1SS (Fig. 1b–
d), both of these differences should shrink so that within 1SS, the
means of both populations are closer to the interior equilibrium.
Another effect of the populations staying near the interior
equilibrium in region 1SS, but splitting time between equilibria in
2SS, could be a lower temporal coefficient of variation (CV) in 1SS.
Thus, we used (ii) the CVs of the population sizes as our second
candidate statistic, and asked whether these decline as we increase
a and cross into 1SS.
Lastly, because stochasticity can cause a bistable system to
switch between alternative stable states (e.g. Guttal and Jayaprakash, 2007; Ridolfi et al., 2007; Zeng et al., 2015; Sharma et al.,
2015), it would be reasonable to expect bimodal distributions of
population sizes in region 2SS but not in 1SS. Indeed, multimodality has been used by others as evidence for alternative stable
states (e.g. Hirota et al., 2011). We therefore used an index of multimodality as our third statistic. Reliably testing for multi-modality is
difficult, and most approaches have some weaknesses, so we
initially considered a few complementary methods. For the critical
window size test (Silverman, 1981), we estimated kernel densities
from the distribution of prey population densities using different
smoothing windows. We estimated the kernel densities using
Matlab’s ‘ksdensity’ function, beginning with a fairly narrow
smoothing window (0.1 density units) and gradually widening it
(in increments of 0.01) until the estimate had only 1 peak. We refer
to the smallest smoothing window that results in a unimodal kernel
estimate as the critical window size. A larger critical window size
indicates stronger evidence of multi-modality, because it means
more smoothing was necessary to force the distribution of prey
population densities to appear unimodal. We therefore expect the
critical window size to decrease with increasing a.
The dip test, which assesses the significance of deviations
between the observed distribution of prey densities and the best-fit
unimodal distribution (Hartigan and Hartigan, 1985), is another
way to test for multi-modality. The trough between multiple modes
would cause a large deviation and lead to rejection of unimodality.
We performed the dip test on the distribution of prey densities and
computed its p-value using F. Mechler’s Matlab code, adapted from
P. Hartigan’s original Fortan algorithm and downloadable from
nicprice.net/diptest. A low p-value indicates strong evidence
against the null hypothesis of unimodality. Higher values of
1 p thus correspond to stronger evidence of multi-modality and
we expect 1 p to decrease as we transition from 2SS to 1SS.
The critical window size test and the dip test were not always in
agreement, suggesting unreliability of either or both tests. To
explore this issue, we generated nearly 4000 simulated time series
(using this and several other stochastic population models),
plotted histograms of the population densities, and visually scored
the number of modes without knowing which model was being
shown. The results from this exercise were in very close agreement
with the results of the critical window size test (K. Abbott,
unpublished results), leading us to trust that test over the dip test.
We therefore include (iii) the critical window size as a candidate
statistic in the main text. For dip test results, see Appendix 2.
3. Results
We found strikingly little difference between stochastic
dynamics simulated from region 2SS, with multiple stable states,
and those from region 1SS, with a single stable state. With low
noise intensity, simulations started near the interior equilibrium
stayed nearby in both cases, as one would expect (not shown).
However, when we increased noise intensity enough to see
switching between the interior and boundary equilibria, switching
occurred even when the boundary was unstable (Fig. 3). Thus, with
both low and high intensity noise, the dynamics appear quite
similar in the two regions.
Indeed, none of our candidate statistics reliably changed in the
expected ways as we changed a (Table 1). We had predicted that as
we increased a and transitioned from region 2SS to region 1SS, the
mean densities would move toward the interior equilibrium (from
a mean intermediate between the interior and boundary), that the
CVs of the densities would decrease, and that evidence of multimodality would become weaker (i.e. the critical window size
would decrease). Predictions about mean density held up for only
31.6% of the 38 parameter sets we examined and predictions about
CV were virtually never realized (Table 1). The critical window size
test showed stronger evidence of multi-modality in 2SS for only
39.5% of parameter sets (Table 1), and the dip test was worse
(15.8%, Appendix 2). Different statistics worked best in different
regions of parameter space (Appendix 2, Fig. A2.1), so we found no
sign that particular parameter values (certain values of s, for
example) caused our predictions to fail. Instead, successes and
failures occurred throughout the parameter range (Appendix 2),
with failures most common overall.
4. Discussion of simulation results
We failed to find reliable signatures of bistability on the
stochastic dynamics from our simulations because stochasticity
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
Complex. (2016), http://dx.doi.org/10.1016/j.ecocom.2016.11.004
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region 1SS
(b) 2.5
2.5
2
1.5
1
1.5
1
0.5
0.5
0
0
region 1SS
2
Predator density
Predator (black) and prey (gray) density
(a)
100
200
300
400
0
0
500
0.5
1
Time
3
1.5
2
2.5
Prey density
region 2SS
(d) 1.8
region 2SS
1.6
2.5
1.4
2
Predator density
Predator (black) and prey (gray) density
(c)
5
1.5
1
1.2
1
0.8
0.6
0.4
0.5
0.2
0
0
100
200
300
400
500
Time
0
0
0.5
1
1.5
2
2.5
3
Prey density
Fig. 3. Example realizations of stochastic model (2) in (a, b) region 1SS and (c, d) 2SS with high-intensity stochasticity (s = 0.1). In all panels, b = 2, g = 2.6, d = 2.5; in (a, b)
a = 4.5 and in (c, d) a = 4.1. (a, c) Time series from one stochastic realization of the model for each value of a (predator densities in black, prey densities in gray). Equilibria are
marked by dotted lines (the middle two are predator and prey densities are the interior equilibrium; the highest dashed line is the prey density at the boundary equilibrium,
where equilibrium predator density is 0). (b, d) The same realizations as (a, c) in phase space plotted as a continuous thin gray line. Deterministic trajectories are also shown
from two initial conditions (black lines), emphasizing the absence of bistability in region 1SS. Stable equilibria are marked by white-outlined black circles and unstable
equilibria are marked by s.
affected the dynamics in non-intiutive ways. If we had only time
series data, we would not be able to determine if they came from
region 1SS or 2SS. Dynamics arising from stochasticity acting on a
deterministic system with a single stable equilibrium may very
well appear to have alternative stable states (e.g. Fig. 3a). In best
case scenarios in ecology, we have not only time series data but
also enough knowledge of the system to build a realistic model for
its deterministic skeleton. Even with a detailed understanding of
the deterministic skeleton, though, we would not be able to tell if
we were observing alternative stable states without very precise
knowledge of the parameter values. In short, whether or not there
are alternative stable states appears rather inconsequential, at
least in our model; their presence did not qualitatively change the
observed dynamics in consistent ways.
Concrete examples of alternative stable states in nature are
scarce (Walker and Meyers, 2004; Schröder et al., 2005), due at
least in part to the difficulty of identifying them (Petraitis and
Dudgeon, 2004). The question of whether it is even possible to
distinguish systems with alternative stable states from those
without is an interesting one. Others have presumed the answer is
‘‘yes’’ and proceeded to ask how (Andersen et al., 2009). Our
results, and the known importance of unstable states in stochastic
systems (e.g. Cushing et al., 1998; Henson et al., 1999; Hastings,
2004; Parker et al., 2011), make us suspicious that the answer is
‘‘no.’’ We have obviously not done an exhaustive search for better
candidate statistics and our intent here is not to claim that it is
impossible to diagnose multi-stability. Rather, we have demonstrated that the issue is not trivial and it at least might behoove us
not to assume that such a distinction is possible.
More importantly, our results challenge us to consider whether
and when we actually care about true stability, in the classical
(linear stability) sense. If stochastic dynamics spend significant
time near a state (like the boundary equilibrium in region 1SS), is it
really helpful to categorize that state as ‘‘unstable’’? Our results are
a deep reminder that the stability concepts on which we rely most
heavily in ecology are grounded in deterministic systems theory,
and are thus bound to have limits in their application to ecological
systems subjected to stochasticity (Provata et al., 2008). Rigorous
mathematical definitions of stochastic stability exist (Arnold,
2010), but there has been little guidance on how to apply them to
ecological problems. Furthermore, there has been very little effort
to establish the relationships between these stochastic stability
concepts and the deterministic theory that most ecologists are
comfortable with and upon which decades of good ecological
insights rest. As a small step toward stronger theory for stochastic
ecological systems, we devote the final part of this article to an
approach that extends a concept already familiar to many
ecologists, and that successfully captures the stable-like behavior
of the unstable boundary equilibrium in this predator–prey model.
5. Potentials and quasi-potentials
For a restricted class of deterministic models (so-called
‘‘gradient systems’’) we can write down a function U, known as
the potential, that has a very convenient physical interpretation:
the state of the system can be envisioned as the position of a ball
rolling on a surface specified by U (e.g. Livina et al., 2010).
Mathematically, the potential is defined as the function U that
dX
satisfies dt¼rU
ðXÞ, where X is a vector of population densities giving
N
. The ball will always
the current state of the system, say X ¼
P
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
Complex. (2016), http://dx.doi.org/10.1016/j.ecocom.2016.11.004
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Table 1
Summary of results from analyzing simulated time series. For each of the candidate statistics, we show examples of the different qualitative patterns we observed and report
the percent of the 38 parameter sets considered that showed each pattern. The examples show average values of each statistic for the predator (gray) and prey (black) across
the 50 values of a considered under one of the parameter sets (set of fixed b, g, d, and s values). In each figure, the x-axis is a, with the location of the transcritical bifurcation
marked with a dashed vertical line. The bistable region 2SS is always on the left. Candidate statistics are shown on the y-axes, as labeled. For the critical window size test,
higher values on the y-axis indicate stronger evidence of multi-modality. See Appendix 2 for details.
Outcome
% of parameter sets
with this outcome
Example
(i) Difference between mean population density and density at the interior equilibrium – Predictions: (1) Mean prey density greater than density at interior
equilibrium in 2SS; (2) Mean predator density less than density at interior equilibrium in 2SS; (3) Distance from mean prey density and interior equilibrium decreases as
bistability is lost (as we move from 2SS into 1SS); (4) Distance from mean predator density and interior equilibrium increases as bistability is lost.
Conforms with all predictions and shows clear change near bifurcation
18.4%
Conforms with all predictions but no clear change near bifurcation
13.2%
Prediction 2 violated for all a
21.1%
Prediction 2 violated for some a
42.1%
Prediction 4 violated
5.3%
(ii) Temporal coefficient of variation in population densities – Predictions: (1) CV in prey density is greater in 2SS and decrease as we move into 1SS; (2) CV in predator
density is greater in 2SS than 1SS.
Conforms with both predictions
2.6%
Prediction 1 violated
60.5%
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7
Table 1 (Continued )
Outcome
% of parameter sets
with this outcome
Both predictions violated; both CVs increase
26.3%
Both predictions violated; both CVs U-shaped
10.5%
Example
(iii) Evidence of multi-modality from the critical window size test – Prediction: Coarser smoothing is needed to force a multi-modal distribution to look unimodal. The
critical smoothing window size should thus decrease as we move from 2SS to 1SS.
Conforms with prediction
39.5%
Prediction violated; window sizes increase
15.8%
Prediction violated; no systematic change in window sizes
44.7%
roll downhill on the surface defined by U and eventually settle at a
local minimum; these valleys correspond to stable equilibria of the
system.
The potential function is particularly useful for describing
gradient systems with alternative stable states because it provides a
way to compare the stability of different equilibria, via differences
in the shapes and depths of the valleys. Potential functions are more
than just a metaphor. Once the potential function is known,
properties like the stationary probability distribution and mean first
passage times can be calculated. Potentials have even been used to
study the effect of stochasticity on unstable states (Fiasconaro et al.,
2003; Dubkov et al., 2004; Guttal and Jayaprakash, 2007; Agudov
et al., 2010). In most cases, however, only one-dimensional
potential functions have been examined (even when interacting
populations are included in the model). The use of potential
functions in higher dimensional systems has been limited, because
gradient systems are a special case.
For non-gradient systems (such as model (1)), we recently
suggested the quasi-potential (Cameron, 2012; Freidlin and
Wentzell, 2012; Zhou et al., 2012; Xu et al., 2014) as an analogous
way to gain additional insight into a model’s behavior (Nolting and
Abbott, 2016). On an intuitive level, the quasi-potential, like the
traditional potential, quantifies how difficult it is to move from one
state to another. Stochastic perturbations allow trajectories to
deviate from the vector field that describes the deterministic
skeleton of the system. Hence the stochastic perturbations can be
thought of as performing ‘‘work’’ against the deterministic vector
field. Following the notation of Cameron (2012), the amount of
work required to move the system from state a to b is described by
the functional,
ST ðfÞ ¼
1
2
Z
T
˙
2
jfðt Þf ðfðt ÞÞj dt;
(3)
0
where f(t) is a path between a and b and f ðXÞ is the deterministic
skeleton of the model. Minimizing this functional over all paths
between the points yields a function Wa(b). The lower the value of
Wa(b), the less work needs to be done by stochastic perturbations
to move from a to b, and hence the more likely it is for such a
transition to occur. Using this construction, it is possible to define a
global quasi-potential, W(X), that generalizes the potential
function to non-gradient systems. In practice, the quasi-potential
is computed numerically (Cameron, 2012; Nolting and Abbott,
2016; Moore et al., 2016).
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
Complex. (2016), http://dx.doi.org/10.1016/j.ecocom.2016.11.004
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The quasi-potential is closely related to the stationary
probability distribution of the system and the expected times it
will take trajectories to exit each basin of attraction. These can be
approximated very accurately using the quasi-potential when the
noise intensity, s, is small relative to the radii and steepness of the
basins of attraction. For relatively high noise intensities, the
relationship between the quasi-potential and the stationary
probability distribution and the mean first passage times is no
longer guaranteed to be highly accurate (although in many cases,
e.g. Nolting and Abbott, 2016, we find that the approximations are
still accurate with relatively high noise intensity). For a more
complete and technical description of how approximation
accuracy changes with noise intensity, we refer readers to Nolting
and Abbott (2016). That paper also provides information on
applications of the quasi-potential to models with different noise
formulations, such as demographic stochasticity or colored noise,
and the relationship between the quasi-potential and other
familiar concepts like the effective potential and stationary
distribution.
The deterministic part of the model, f ðXÞ, can be thought of as a
vector field that describes, from each point X in the state space,
how fast and in what direction the system is expected to change in
the near-term. An extremely useful feature of the quasi-potential is
that the vector field f ðXÞ can be decomposed as the gradient of the
quasi-potential function and an orthogonal remainder term,
f ðXÞ ¼ rW ðXÞ þ f r ðXÞ;
hf r ; rWi ¼ 0:
(4)
This decomposition provides important insights about the dynamics of the system. Like the potential in a gradient system, the hills
and valleys of the surface specified by W determine how difficult it
is for a system to transition between different states. Unlike the
potential in gradient systems, the surface specified by W is not the
sole driver of the dynamics: the f r ðXÞ term causes circulatory
motion in directions perpendicular to the gradient of W.
Quasi-potentials for our predator–prey model are shown in
Figs. 4a and b and 5a and b. We see that in both regions 1SS and 2SS,
the quasi-potential shows a deep basin around the interior
equilibrium and is shallow at the boundary. The qualitative shapes
Fig. 4. Quasi-potential of stochastic model (2) in region 1SS (b = 2, g = 2.6, d = 2.5, and a = 4.5 as in Fig. 3a and b). (a) 3-D plot and (b) contour plot of the quasi-potential
function. Combinations of predator and prey density associated with low quasi-potential values (wells in (a), bluer shades in (b)) are more ‘‘stable’’ in the sense that the
populations are very likely to spend time in their vicinity. The horizontal axis in (a) is truncated; the quasi-potential continues steeply upward above the range plotted. (c)
Gradient vector field and (d) remainder vector field of the quasi-potential. Arrows show the direction and rate (proportional to arrow size) of the population trajectories.
Broadly, the gradient vector field shows the influence of the deterministic equilibrium and the remainder vector field shows the response of the system to stochastic
perturbations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
Complex. (2016), http://dx.doi.org/10.1016/j.ecocom.2016.11.004
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9
Fig. 5. Same as Fig. 4 except showing region 2SS (b = 2, g = 2.6, d = 2.5, and a = 4.1 as in Fig. 3c and d).
of the quasi-potentials are quite similar, thus anticipating that their
stochastic dynamics should be similar. In Fig. 5a and b we see
another interesting feature: the basin depth at the boundary
equilibrium is very similar to the depth at the unstable equilibrium
intermediate between the boundary and interior. Using standard
linear stability analysis, we would categorize the boundary and
interior equilibria as stable in this case, and the intermediate
equilibrium as unstable. Using the quasi-potential, though, we can
see that the boundary and intermediate points have very similar
stabilities, suggesting that those two points should be classified
together, with the interior equilibrium being put in a different
category. In other words, ‘‘stability’’ as measured by the quasipotential (basin depth) and ‘‘stability’’ as measured in linear
stability analysis (the dominant eigenvalue; essentially local basin
steepness) can lead to different conclusions. In light of our
simulation results, we suggest that the quasi-potential is better
capturing the pertinent information, at least for this model.
We show the gradient (rW ðXÞ) and remainder (f r ðXÞ) vector
fields of this model in Figs. 4c and d and 5c and d. The gradient
component of the vector field makes trajectories move ‘‘downhill’’
and represents the forces that move trajectories toward or away
from equilibria. Note that this vector field has a similar configuration near the boundary equilibrium, regardless of its stability
(Figs. 4c and 5c). The remainder component of the vector field is
orthogonal to the gradient of the quasi-potential and represents
forces that lead to circulation around equilibria, without moving
trajectories uphill or downhill on the quasi-potential surface. This
captures the oscillatory nature of the approach to equilibrium seen,
for example, in the black trajectories in Fig. 3b and d.
Notice that when perturbations add density-independent noise,
as in this example, the quasi-potential does not depend on the
noise intensity, s (Zhou et al., 2012). This coincides with the
physical interpretation of a randomly perturbed ball rolling on a
surface. The perturbations affect the ball’s trajectory, but not the
surface it rolls on. The surface is completely defined by the
deterministic skeleton of the system. Nonetheless, the quasipotential provides different information than traditional linear
stability metrics. Because eigenvalues only provide information
about infinitesimal neighborhoods of equilibria, and stochastic
perturbations move trajectories outside of these neighborhoods,
stability information from the quasi-potential is much more
relevant for perturbed systems.
The quasi-potential is just one of many possible methods for
summarizing the behavior of a model, but we highlight it here
because we feel it is especially useful for understanding stochastic
dynamics and because it demystifies our counter-intuitive
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
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simulation results presented above. Classification of equilibria as
either ‘‘stable’’ or ‘‘unstable’’ is standard practice in ecology but it
can be highly misleading for stochastic systems. According to
linear stability analysis, the distinguishing feature between
regions 1SS and 2SS is that the prey-only boundary equilibrium
is unstable in 1SS and stable in 2SS. From this, we proposed that the
dynamics in region 2SS should appear more bistable, but of course
we failed to see this in our simulations. While linear stability
analysis emphasizes that a transcritical bifurcation separates these
two scenarios, the quasi-potential analysis shows that they are not
so different, and the boundary equilibrium is shallow in both
situations. Thus, the quasi-potential is able to capture an essential
feature of the stochastic dynamics: that the boundary equilibrium
has a very similar influence on dynamics in both regions 1SS and
2SS.
to know more than the location and stability classification of the
equilibrium solutions if we are to understand the dynamics of
perturbed systems. Figuring out how much more we need to know,
and developing methods for how to know it, represents an exciting
frontier in ecological research.
Acknowledgements
This work was supported by a Complex Systems Scholar grant
to K.C.A. from the James S. McDonnell Foundation. Special thanks
to Maria K. Cameron for assistance with implementing the quasipotential analysis, to Lee Altenberg for very thoughtful feedback on
the project, and to Frithjof Lutscher for suggesting the analysis in
Appendix 1. We thank Sam Catella, Katie Dixon, Chris Moore, and
Chris Stieha for a helpful discussion on an earlier version of the
manuscript.
6. Concluding remarks
Our work has two main take-home messages. First, we showed
that a system with alternative stable states may not differ
functionally and meaningfully from a similar system with a single
stable state when we account for stochasticity. This is an important
insight: much of our present concern over the possibility of sudden
state shifts is focused on systems that are believed to be multistable in the traditional sense. Understanding when stochastic
systems with a single stable state are expected to show the same
shifts in qualitatively the same ways will allow ecologists to
refocus research and management efforts. Meanwhile, it may be
helpful to think about ‘‘alternative persistent states’’ rather than
‘‘alternative stable states,’’ focusing on identifying which states a
particular system is likely to be found in and removing the
emphasis on stability per se (while also gaining a more usable
acronym).
Second, our results point more generally to a need for clear
ways to think about stability in stochastic systems (Bjørnstad and
Grenfell, 2001; Spagnolo et al., 2004; Provata et al., 2008). Given
theoretical ecology’s firm roots in deterministic dynamical
systems theory, it would be difficult for many ecologists to make
the leap to stochastic stability theory that is conceptually quite
different. Approaches like the quasi-potential method outlined
above, that have clear connections to familiar analyses yet improve
our ability to anticipate and interpret stochastic behavior, will be
especially valuable. Interestingly, and somewhat reassuringly, we
have shown that we can improve our understanding of stochastic
dynamics by more carefully studying the deterministic skeletons:
the quasi-potential of our model did quite a good job of capturing
the stochastic model’s behavior, despite being defined only by the
deterministic part. We are therefore not suggesting that the
practice of analyzing the deterministic skeleton of a stochastic
system necessarily be set aside, but rather that we should consider
more carefully what information we wish to extract about a model
and how best to extract it. For anticipating which states stochastic
trajectories are likely to visit, classifying equilibria as stable or
unstable by standard stability analysis may not always be
particularly helpful. When the formalism is used to understand
observed dynamics in time series data, it could actually be quite
misleading.
More than a decade has passed since Alan Hastings began his
appeal to ecologists to consider transient dynamics (Hastings,
2001, 2004, 2010), and significant progress is being made to better
understand what ecological systems do when they are away from a
deterministic equilibrium (e.g. Spagnolo et al., 2004; Briggs and
Borer, 2005; Koons et al., 2005; Tenhumberg et al., 2009; Fukami
and Nakajima, 2011; Stott et al., 2011). We agree fully with
Hastings’ message, and emphasize that non-equilibrial dynamics
are not always ‘‘transient’’ per se. Even at long timescales, we need
Appendix 1
To avoid negative population sizes in our simulations, if a
stochastic perturbation ever caused either N or P to become negative,
that negative density was immediately set to 0; a subsequent positive
perturbation could then restore the population. Due to this
correction, the average perturbation actually experienced by the
populations was slightly positive, even though we drew perturbations from a distribution with mean zero. We did not save information
on the average perturbation sizes when we ran the simulations
presented in the main text, so to estimate these averages, we repeated
10 realizations of each parameter combinations used in the paper. In
these new simulations, the average perturbation to the prey across all
of our parameter combinations was 1.5 T 10S5 and the maximum
perturbation (on average, for any one parameter combination) was
3.8 T 10S5. For the predator, the average perturbation was 1.3 T 10S5
and the maximum was 3.3 T 10S4.
Our stochastic model thus effectively included a small amount of
net positive immigration. To ensure that our findings were due to the
added stochasticity and not the added immigration, we analyzed the
deterministic model (Eqs. (1)) with density-independent immigration at rate I. This allowed us to see how the dynamics are expected to
change due to immigration per se, in the absence of stochasticity. The
model is,
dN
N
NP
¼ N 1
þI
(1.1a)
dt
g
ð1=aÞN2 þ N þ 1
dP
bdNP
¼
dP þ I:
dt
ð1Þ=ðaÞN2 þ N þ 1
(1.1b)
We computed equilibria and eigenvalues for this model numerically for the parameter sets used in the main text and with a range
of I values. From the eigenvalues, we found the boundaries of
region 2SS; how these boundaries shift with I tells us the effects of
immigration on the model’s deterministic behaviors with respect
to bistability. We show these shifts, as well as example population
trajectories, in Fig. A1.1.
Fig. A1.1 shows that although our model included very small
amounts of net positive immigration, the rates of immigration were
so low as to have virtually no effect on the dynamics. Furthermore,
even if our net immigration rates had been high enough to affect the
dynamics, Fig. A1.1 shows that immigration cannot be responsible for
our main results. The striking finding from our simulations was that
dynamics can appear bistable even when they are not (i.e. even in
region 1SS). For immigration to explain this result, it would have to
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
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11
Fig. A1.1. Behavior of the model with immigration (Eq. (1.1)). (a) Example trajectories from region 1SS, comparable to black lines in main text Fig. 3b. We plotted trajectories
from 2 different initial conditions each for I = 0 (no immigration: black), I = 105 (about average for our simulations: green), I = 104 (around the highest net immigration we
ever saw: red), and I = 0.01 (much higher than we observed in our simulations, and shown here simply to demonstrate a more pronounced effect of immigration: blue). The
black and green lines are not visible because they coincide completely with the red lines; thus, unless immigration is much higher than what we observed (e.g. I = 0.01, blue
lines), immigration has a negligible impact on the dynamics. (b–d) Bifurcation diagrams as in main text Fig. 1b–d. Black lines repeat the information from Fig. 1b–d, showing
the bifurcation points for the deterministic model without immigration. Colored lines show how the bifurcations delineating region 2SS shift with immigration, for I = 104
(red) or I = 0.01 (blue). With I = 105 (about average for our simulations), the bifurcation does not perceptibly change from the no immigration case (black lines) and so it is not
shown. As in Fig. 1, gray horizontal lines in (b–d) show the a range for each g that was considered in the simulations. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
shift region 2SS to the right (to higher values of a) in Fig. A1.1b–d. If
this had occurred, then immigration could cause us to remain in
region 2SS even after we thought we had increased a far enough to
cross the bifurcation into region 1SS; this could then explain why we
retained the appearance of bistable dynamics beyond this critical a.
However, Fig. A1.1 shows exactly the opposite effect: immigration
shifts region 2SS to the left. Thus, even if our simulations had included
high enough immigration to affect the dynamics, this effect would
still not explain our key result (unexpected bistable-like behavior).
For these reasons, we attribute our findings to stochasticity, rather
than to the very small amount of net immigration that necessarily
arose in our stochastic simulations.
Appendix 2
Detailed simulation results (Table A2.1).
Table A2.1
Behavior of all 38 parameter sets summarized in main text Table 1. ‘‘Example shown’’ gives the parameter values used to create each example figure in main text Table 1.
b
d
Difference between mean population density and density at the interior
Conforms with all predictions and shows clear change near bifurcation
Example shown
4
5
Others
4
2.5
4
2.5
4
2.5
4
2.5
4
2.5
4
5
Conforms with all predictions but no clear change near bifurcation
Example shown
4
2.5
Others
4
2.5
4
2.5
4
5
4
5
g
s
a range
Bifurcation a
1
0.9
1
1.1
1
1.1
0.9
0.05
0.05
0.05
0.05
0.1
0.1
0.05
[0.45,
[0.45,
[0.45,
[0.45,
[0.45,
[0.45,
[0.45,
0.75]
0.75]
0.75]
0.75]
0.75]
0.75]
0.75]
0.50
0.48
0.50
0.53
0.50
0.53
0.48
1
0.9
0.9
0.9
1
0.01
0.01
0.1
0.01
0.01
[0.45,
[0.45,
[0.45,
[0.45,
[0.45,
0.75]
0.75]
0.75]
0.75]
0.75]
0.50
0.48
0.48
0.48
0.50
equilibrium
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
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12
Table A2.1 (Continued )
Prediction 2 violated for all a
Example shown
Others
Prediction 2 violated for some a
Example shown
Others
Prediction 4 violated
Example shown
Other
Temporal coefficient of variation
Conforms with both predictions
Example shown
Prediction 1 violated
Example shown
Others
b
d
g
s
a range
Bifurcation a
1.5
1.5
1.5
2
1.5
1.5
2
2
5
2.5
2.5
2.5
5
5
5
5
5
5.5
6
3.1
5.5
6
2.6
3.1
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
16.67
17.29
18.00
4.58
17.29
18.00
4.23
4.58
1.5
2
1.5
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
1.5
2
2
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
5
5
5
5
5
5.5
2.6
5
5.5
6
2.6
3.1
5
6
2.6
3.1
5
5.5
6
2.6
3.1
0.1
0.01
0.05
0.05
0.05
0.05
0.05
0.1
0.1
0.1
0.1
0.05
0.05
0.05
0.05
0.05
[16.06, 19.00]
[4.02, 5.00]
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
17.29
4.23
16.67
17.29
18.00
4.23
4.58
16.67
18.00
4.23
4.58
16.67
17.29
18.00
4.23
4.58
4
4
2.5
5
1.1
1.1
0.01
0.01
[0.45, 0.75]
[0.45, 0.75]
0.53
0.53
in population densities
1.5
5
5
0.01
[16.06, 19.00]
16.67
4
4
4
4
4
4
4
4
1.5
1.5
1.5
2
2
1.5
1.5
1.5
2
2
1.5
1.5
1.5
2
2
2.5
2.5
2.5
2.5
2.5
5
5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
5
5
5
5
5
1
0.9
1.1
1
1.1
0.9
1
0.9
5
5.5
6
2.6
3.1
5
5.5
6
2.6
3.1
5
5.5
6
2.6
3.1
0.05
0.05
0.05
0.1
0.1
0.05
0.05
0.1
0.05
0.05
0.05
0.05
0.05
0.1
0.1
0.1
0.1
0.1
0.05
0.05
0.05
0.05
0.05
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
0.50
0.48
0.53
0.50
0.53
0.48
0.50
0.48
16.67
17.29
18.00
4.23
4.58
16.67
17.29
18.00
4.23
4.58
16.67
17.29
18.00
4.23
4.58
Both predictions violated; both CVs increase
Example shown
2
Others
4
4
4
4
4
4
1.5
2
1.5
5
2.5
2.5
5
5
2.5
5
2.5
2.5
5
3.1
0.9
1
0.9
1
1.1
1.1
6
3.1
6
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
[4.02, 5.00]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[16.06, 19.00]
[4.02, 5.00]
[16.06, 19.00]
4.58
0.48
0.50
0.48
0.50
0.53
0.53
18.00
4.58
18.00
Both predictions violated; both CVs U-shaped
Example shown
1.5
Others
1.5
2
2
5
2.5
5
2.5
5.5
5.5
2.6
2.6
0.01
0.01
0.01
0.01
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
17.29
17.29
4.23
4.23
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
Complex. (2016), http://dx.doi.org/10.1016/j.ecocom.2016.11.004
G Model
ECOCOM-613; No. of Pages 15
K.C. Abbott, B.C. Nolting / Ecological Complexity xxx (2016) xxx–xxx
13
Table A2.1 (Continued )
b
d
g
s
a range
Bifurcation a
Evidence of multi-modality from the dip test (not shown in main text)
Conforms with prediction
1.5
5
1.5
5
1.5
2.5
1.5
5
2
5
2
2.5
5
5
5.5
5.5
2.6
2.6
0.01
0.05
0.01
0.01
0.01
0.01
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[0.45, 0.75]
[0.45, 0.75]
16.67
16.67
17.29
17.29
4.23
4.23
Prediction violated; evidence for multi-modality increases
2
5
4
2.5
4
2.5
4
2.5
4
2.5
2.5
4
4
5
4
5
4
2.5
1.5
2.5
1.5
2.5
2
2.5
1.5
2.5
1.5
2.5
2
2.5
1.5
5
1.5
5
2
5
2
5
4
2.5
4
2.5
4
5
4
5
3.1
0.9
1
1.1
1
1.1
0.9
1
0.9
5.5
6
3.1
5.5
6
3.1
5.5
6
2.6
3.1
0.9
1
0.9
1
0.01
0.05
0.05
0.05
0.1
0.1
0.05
0.05
0.1
0.05
0.05
0.05
0.1
0.1
0.1
0.05
0.05
0.05
0.05
0.01
0.01
0.01
0.01
[4.02, 5.00]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
4.58
0.48
0.50
0.53
0.50
0.53
0.48
0.50
0.48
17.29
18.00
4.58
17.29
18.00
4.58
17.29
18.00
4.23
4.58
0.48
0.50
0.48
0.50
Prediction violated; no systematic change in evidence for multi-modality
4
2.5
1.5
2.5
2
2.5
1.5
2.5
2
2.5
4
5
1.5
2.5
2
2.5
1.5
5
1.1
5
2.6
5
2.6
1.1
6
3.1
6
0.01
0.05
0.05
0.1
0.1
0.01
0.01
0.01
0.01
[0.45, 0.75]
[16.06, 19.00]
[4.02, 5.00]
[16.06, 19.00]
[4.02, 5.00]
[0.45, 0.75]
[16.06, 19.00]
[4.02, 5.00]
[16.06, 19.00]
0.53
16.67
4.23
16.67
4.23
0.53
18.00
4.58
18.00
Evidence of multi-modality from the critical window size test
Conforms with prediction
Example shown
1.5
2.5
Others
1.5
5
1.5
5
1.5
5
2
5
2
2.5
4
2.5
1.5
2.5
1.5
2.5
1.5
5
2
5
1.5
2.5
2
2.5
1.5
2.5
2
2.5
5.5
5
5
5.5
2.6
2.6
0.9
5.5
5.5
5.5
2.6
5
2.6
5
2.6
0.01
0.01
0.05
0.01
0.01
0.01
0.05
0.05
0.1
0.05
0.05
0.05
0.05
0.1
0.1
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[4.02, 5.00]
[0.45, 0.75]
[16.06, 19.00]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[16.06, 19.00]
[4.02, 5.00]
[16.06, 19.00]
[4.02, 5.00]
17.29
16.67
16.67
17.29
4.23
4.23
0.48
17.29
17.29
17.29
4.23
16.67
4.23
16.67
4.23
Prediction violated; window sizes increase
Example shown
2
Others
2
1.5
2
4
2
5
2.5
5
5
5
2.5
3.1
3.1
6
3.1
1.1
3.1
0.01
0.05
0.05
0.05
0.01
0.01
[4.02, 5.00]
[4.02, 5.00]
[16.06, 19.00]
[4.02, 5.00]
[0.45, 0.75]
[4.02, 5.00]
4.58
4.58
18.00
4.58
0.53
4.58
Prediction violated; no systematic change in window sizes
Example shown
4
2.5
Others
4
2.5
4
2.5
4
2.5
4
2.5
4
5
4
5
1
1
1.1
1
1.1
0.9
1
0.01
0.05
0.05
0.1
0.1
0.05
0.05
[0.45,
[0.45,
[0.45,
[0.45,
[0.45,
[0.45,
[0.45,
0.50
0.50
0.53
0.50
0.53
0.48
0.50
0.75]
0.75]
0.75]
0.75]
0.75]
0.75]
0.75]
Please cite this article in press as: Abbott, K.C., Nolting, B.C., Alternative (un)stable states in a stochastic predator–prey model. Ecol.
Complex. (2016), http://dx.doi.org/10.1016/j.ecocom.2016.11.004
G Model
ECOCOM-613; No. of Pages 15
K.C. Abbott, B.C. Nolting / Ecological Complexity xxx (2016) xxx–xxx
14
Table A2.1 (Continued )
b
d
g
s
a range
Bifurcation a
4
1.5
1.5
2
4
4
4
4
1.5
1.5
2.5
2.5
2.5
2.5
2.5
5
5
2.5
2.5
5
0.9
6
6
3.1
0.9
0.9
1
1.1
6
6
0.1
0.05
0.1
0.1
0.01
0.01
0.01
0.01
0.01
0.01
[0.45, 0.75]
[16.06, 19.00]
[16.06, 19.00]
[4.02, 5.00]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[0.45, 0.75]
[16.06, 19.00]
[16.06, 19.00]
0.48
18.00
18.00
4.58
0.48
0.48
0.50
0.53
18.00
18.00
Fig. A2.1. Graphical display showing which parameter combinations conformed
with our predictions for each of the candidate statistics.
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