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Coupled phase-amplitude dynamics in heterogeneous
metacommunities
Russell Milnea,∗, Frederic Guichardb
aDepartment of Applied Mathematics, University of Waterloo
bDepartment of Biology, McGill University
Abstract
Spatial synchrony of population fluctuations is an important tool for predict-
ing regional stability. Its application to natural systems is still limited by the
complexity of ecological time series displaying great variation in the frequency
and amplitude of their fluctuations, which are not fully resolved by current
ecological theories of spatial synchrony. In particular, while environmental fluc-
tuations and limited dispersal can each control the dynamics of frequency and
amplitude of population fluctuations, ecological theories of spatial synchrony
still need to resolve their role on synchrony and stability in heterogeneous meta-
communities. Here, we adopt a heterogeneous predator-prey metacommunity
model and study the response of dispersal-driven phase locking and frequency
modulation to among-patch heterogeneity in carrying capacity. We find that
frequency modulation occurs at intermediate values of dispersal and habitat
heterogeneity. We also show how frequency modulation can emerge in meta-
communities of autonomously oscillating populations as well as through the
forcing of local communities at equilibrium. Frequency modulation was further
found to produce temporal variation in population amplitudes, promoting local
and regional stability through cyclic patterns of local and regional variability.
Our results highlight the importance of approaching spatial synchrony as a non-
stationary phenomenon, with implications for the assessment and interpretation
of spatial synchrony observed in experimental and natural systems.
Keywords: Synchrony; dispersal; habitat heterogeneity; entrainment;
variability
Introduction
Spatial synchrony among populations has long been studied to understand
its biotic and abiotic drivers, such as dispersal and correlated environmental
∗Corresponding author
Email address: russell.milne@mail.mcgill.ca (Russell Milne)
Preprint submitted to Journal of Theoretical Biology March 22, 2021
fluctuations [1, 2]. It has also received much attention for its ecological implica-
tions, including the prediction of metapopulation and metacommunity stability5
across scales [3, 4, 5, 6, 7]. However, the complexity of ecological time series is
such that our current metrics of spatial synchrony can fail to detect important
properties of spatial dynamics and indicators of stability and persistence. In
particular, theory shows that disturbance events can temporarily force spatial
synchrony and promote population stability [8]. Synchrony between popula-10
tions can also be a dynamic property, with the potential for populations to cy-
cle between synchrony and asynchrony without any environmental disturbance
[9]. This treatment of synchrony predicts fluctuations in the timing of fluctu-
ations (phase) among populations, but also in their amplitude and frequency.
The full integration of these multiple facets of stability into metapopulation and15
metacommunity theories has yet to be achieved. Here, we study a predator-prey
metacommunity model with environmental heterogeneity and predict conditions
for the emergence of coupled fluctuations in spatial synchrony and amplitude
of population cycles, as well as its effects on population stability across spatial
and temporal scales.20
Metapopulation and metacommunity theories allow defining stability across
spatial scales, and have used spatial synchrony among local populations as a
measure of regional stability. For oscillatory dynamics, in-phase locking of pop-
ulation oscillations has been linked to higher regional variations and higher
extinction rates due to co-temporality of local population minima [10, 1, 2],25
while out-of-phase or asynchronous populations show less regional variability
due to statistical averaging and to higher persistence by allowing for rescue
effects [11, 12]. Drivers of spatial synchrony include dispersal and movement
of individuals between populations [13, 14], with time to synchrony being in-
versely proportional to coupling strength [15]. This result is robust to the type30
of model chosen, and has been observed in predator-prey [16] and discrete-time
matrix-based models [17], and in mass-action dynamics derived from agent-
based models [18]. Spatial synchrony can also be driven by correlated environ-
mental variability affecting local populations (the Moran effect, [19, 16]). The
relationship between dispersal and synchrony of oscillatory populations is not35
fully resolved by predictions of long-term phase synchrony. Intermediate levels
of dispersal can for example synchronize the phase but not the amplitudes of
local population fluctuations [13]. Broad ranges of dispersal strength can also
lead to long term fluctuations in the dynamics of both phases and amplitudes
[9, 20].40
Spatial synchrony theory in ecology has emphasized phase-locking as the
equilibrium response of phase difference to weak dispersal and habitat hetero-
geneity. In two-patch heterogeneous predator-prey systems, weak dispersal leads
to a phase-locked state between pulsed oscillations, and heterogeneity in local
parameters (carrying capacity, predator death rate) can prevent phase-locking45
and cause phase drifting [21]. Larger networks can display a broader range of
behaviours with three-patch predator-prey systems yielding cyclic solutions of
phase differences [9]. This frequency modulation suggests that phase dynamics
beyond drift and locking may have a much larger presence in natural systems
2
characterized by large networks. Non-equilibrium phase dynamics can also be50
caused by heterogeneity between habitats [21]. In sufficiently heterogeneous
systems where populations converge to either a stable point or limit cycle in the
absence of coupling, non-stationary fluctuations in population size can be forced
by dispersal [22]. Spatial synchrony of amplitudes is also associated with non-
stationarity caused by limited dispersal and habitat heterogeneity with chaotic55
amplitudes persisting despite long-term phase locking [13].
Changes in the magnitude and in the spatial distribution of phase and am-
plitude differences among local populations have strong ecological implications.
While amplitude is directly related to measures of variance and of local mea-
sures of ecological stability [23], variations in phase difference directly affect60
regional stability and persistence through rescue effects and spatial averaging.
Correlation and coupling between variations in phase difference and amplitudes
are also expected. The portfolio effect predicts that amplitude of regional time
series will be reduced by out-of-phase fluctuations among local populations [11].
This effect is observed because passive immigration into local populations is in-65
versely proportional to their relative population size, and has a further damping
effect on the amplitude of population fluctuations [24]. Because this effect on
amplitude is driven by the phase difference, fluctuations in phase and ampli-
tude differences between populations should be correlated. Correlation between
fluctuations of phase difference and amplitude has been predicted in model meta-70
populations displaying pulsed oscillations [20]. However, there is still no theory
of non-equilibrium dynamics of both phase-difference and amplitude integrating
the role of both dispersal and habitat heterogeneity.
Here we study the dynamics of phase difference and amplitude among small
ecological networks of predator-prey communities characterized by heterogene-75
ity in the carrying capacity of the prey. In this paper, the existence, robustness,
and implications for stability of modulation in phase difference and amplitude
are established. To do this, a model featuring one predator and one prey iter-
ated over two and three patches is investigated. A simple linear model of phase
[13, 25] is used to check pairs of oscillators for phase differences. This is fur-80
ther utilized to establish relationships between coupling strength and patchwise
parameter heterogeneity (i.e. migration rates and habitat heterogeneity), and
the type of phase dynamics expected. The wide parameter ranges considered
offer information about phase dynamics beyond the restrictions of weak cou-
pling and weak heterogeneity. Additionally, a common measure of ecological85
stability, namely the coefficient of variation, is used to determine how local and
regional stability respond to synchrony regimes. The establishment of phase
difference and amplitude modulation as distinct types of dynamics, separate
from locked systems and phase drift, provides new insight as to how oscillatory
natural populations can be evaluated.90
3
Methods
Metacommunity model
We adopt the Rosenzweig-MacArthur predator-prey model (1) to study net-
works of local communities that can display either stable equilibrium point or
limit cycle, to make our results directly comparable with theoretical and empir-95
ical studies of spatial synchrony in ecology [26, 14, 21, 16, 27]. The dynamics
of prey (Ni) and predators (Pi) in each local habitat iare described as:
dNi
dt =riNi1−Ni
Ki−siNiPi
1 + siτiNi
+X
j6=i
mN
ij Nj−X
j6=i
mN
ji Ni
(1)
dPi
dt =bisiNiPi
1 + siτiNi
−ciPi+X
j6=i
mP
ij Pj−X
j6=i
mP
ji Pi
where i= 1,2 or i= 1,2,3 and riis the intrinsic prey growth rate, Kithe
carrying capacity of the prey, sithe rate at which a given predator finds prey
(the search rate), τithe handling time, bithe predator’s conversion efficiency of100
prey biomass, and cithe predator mortality rate. Prey and predator populations
were assumed to move passively: mN
ij denotes the migration rate of prey (N)
and mP
ij denotes the migration rate of predators (P).
Spatial synchrony of phase and amplitude
Our study deals with the periods and amplitudes of population time series,105
and the changes in these properties over time. Phase synchrony corresponds to
a constant difference between the phases of populations associated with a ho-
mogeneous periods across populations. The phase difference between two given
periodic population time series takes values from 0-2πand can be expressed as
the length of time that passes between their peaks. Two populations are phase110
synchronous, or phase-locked, if their phase difference converges to a constant
value. They can either be phase-locked in phase if this value is 0, or phase-
locked out of phase for any other value. We take phase asynchrony to be any
deviation from phase-locking [9]. For example, we refer to phase drift when two
populations’ phase difference continually increases as a result of heterogeneous115
periods. This is also known as decoherence (see e.g. [16]). We use intrinsic
frequency and/or amplitude modulation to refer to periodic fluctuations in the
frequency and/or amplitude of population cycles that are not the result of direct
environmental forcing.
Spatially heterogenous habitats and dynamical regimes120
Our goal is to study the role of habitat heterogeneity on spatial synchrony
through its impact on the distribution of dynamical regimes (steady state vs
limit cycles) over the metacommunity. We thus set all parameters to spatially
homogeneous values except for the carrying capacity K, which is used to control
4
for local dynamical regime across the Hopf bifurcation value. Kwas chosen as125
the parameter of habitat heterogeneity because it was also shown to drive a
transition from phase drift to synchronization in a two-patch system [21].
Parameter values were chosen based on previous studies [14, 21] to produce
sinusoidal rather than pulse-relaxation oscillations. This was done in order to
test for the roles of habitat heterogeneity and spatial coupling on the mainte-130
nance of spatial heterogeneity while controlling for the known effect of pulse-
relaxation oscillations on the maintenance of phase-locked synchrony [15, 14]
and of phase asynchrony [frequency modulation, 9]. We used ri= 10, si= 0.5,
τi= 1, bi= 1 and ci= 0.4. We took mN
ij =mP
ij in all simulations to minimize
the confounding effects of asymmetric migration, using a single value mij for135
both.
We studied 2-patch and 3-patch networks with weak, intermediate and strong
levels of coupling. This was done to control for the role of weak coupling in
driving phase dynamics in 2-patch systems [14, 28] and in the emergence of
frequency modulation in 3-patch systems [9]. In our 3-patch system, we set one140
patch (the “intermediate-K” patch) to K= 5, while the other two ( “high-
K” and “low-K” patches) had K-values of 5 + ∆Kand 5 −∆Krespectively,
with ∆Kvarying from 0-0.5. For the two patch system, each patch was set to
K= 5 ±∆K. These values were chosen so that one patch crosses the Hopf
bifurcation with increasing ∆K, thus driving spatial heterogeneity in dynamical145
regimes among spatially coupled habitats (equilibrium or limit cycle). The
system was integrated using MATLAB (MathWorks inc.), using the ODE45
function. In each simulation, the system was integrated until t= 3000 and
the results up until t= 1000 were thereafter discarded in order to eliminate
transient dynamics.150
Analysis of phase and amplitude
The phase of each time series was generated over oscillating regimes following
[13, 25], and based on the method of marker events [29] by extracting local
maxima of oscillations and assuming linear phase growth from 0 to 2πbetween
local maxima. This method is less computationally intensive and produces155
similar results to other methods such as the Hilbert transform [30, 31, 29], and
is not built on assumptions of weak coupling and weak heterogeneity required by
phase equations used by weakly-coupled oscillator theories [32, 21]. Amplitudes
were measured from time series as the difference in abundance between local
minima and maxima of each oscillation.160
We assessed spatial synchrony between time series, as the pairwise phase dif-
ferences (see Figure 1b) were analyzed for phase synchrony. Equilibrium phase
differences corresponded to phase synchrony while non-equilibrium phase differ-
ences, measured as long-term variations in their values, corresponded to phase
asynchrony. Non-equilibrium cases characterized by continuous growth of the165
phase difference corresponded to phase drift (Figure 1b, upper panel). Equilib-
rium values of phase difference corresponded to phase-locked oscillators (Figure
1b, lower panel), with in-phase locking corresponding to the specific case of a 0
equilibrium phase difference. Non-equilibrium (phase asynchronous) cases with
5
bounded (0 −2π) phase difference values corresponded to frequency modulation170
(Figure 1b, middle panel). Amplitude was similarly assessed, with time series
distinguished between those that showed temporal variation in amplitude (am-
plitude modulation) and those where the amplitude converged to an equilibrium
value (no amplitude modulation).
Habitat heterogeneity and coupling strength175
To evaluate the response of phase dynamics to habitat heterogeneity and
coupling strength, the pairwise spread in phase difference between oscillators was
plotted as a function of coupling strength (mij ) and interpatch heterogeneity
in carrying capacity (∆K). Parameter spaces were created for each pair of
oscillators. Since the assumptions of weak coupling and weak heterogeneity180
were relaxed, wide ranges of values for these parameters could be looked at.
Specifically, in each parameter space, ∆Ktook values in the range [0,0.5] in
order to capture areas on both sides of the Hopf bifurcation, which occurs at
K=14
3when the other system parameters are as specified above [33]. Likewise,
the migration rate mij took values in the range [0,0.045] in order to represent185
weak, intermediate and strong coupling scenarios. Additionally, the three-patch
results were compared with a two-patch control over a similar parameter space
(see Figures 2 and 3).
Effects on local and regional stability
In order to test for the effect of regimes of spatial synchrony on community190
stability both within and among patches, the regional and average local coeffi-
cients of variation (CV) were calculated for simulated time series from parameter
sets leading to phase drift, frequency modulation (FM) and phase-locking. The
oscillators in a system exhibiting FM should have cyclical periods by definition,
and thus FM was predicted to show non-stationary signals. To capture these,195
the CV needed to be taken over a window that would be long enough to include
multiple periods of the oscillators, but too short to contain an entire period of
the oscillators’ periods. Specifically, the CV was taken over windows of 20 and
200 time steps. Regional CV (CVr eg) for a given species was calculated by sum-
ming the populations of that species across patches, then taking the CV of the200
resulting time series within the aforementioned window, an approach consistent
with past studies [34, 35]. Average local CV (CVloc) for a given species was
obtained by averaging the CV from each patch in that species over the same
window [36].
Results205
Emergence of frequency modulation
Increasing heterogeneity in the two-patch system caused phase-locked oscil-
lators to start drifting as their proper frequencies diverged, but it also caused
the low-Koscillator to approach the Hopf bifurcation and thus have lower am-
plitudes. This meant that the low-Koscillator could be forced by the high-K210
6
oscillator, leading to phase-locking, which was further facilitated by increas-
ing coupling strength (Figure 2). As expected, increasing coupling strength
promoted phase-locked synchrony (Figures 1a and 3). In the three-patch sys-
tem, holding coupling strength constant while increasing heterogeneity caused a
transition from phase-locked synchrony to phase asynchrony and to phase drift215
(Figures 1a and 3).
Frequency modulation characterized by bounded fluctuations of phase differ-
ences was observed in the three-patch system in regions of our parameter space
between areas of phase drift and phase-locking (Figures 3 and 4). Frequency
modulation was present regardless of the pair of oscillators being compared220
(Figures 3a and 3b). The two-patch system also exhibited frequency modula-
tion (Figure 2), but over more restricted regions of parameter space compared
to the three-patch one (Figures 2 and 3). Specifically, frequency modulation oc-
curred where the low-Koscillator was just close to the Hopf bifurcation at the
transition between drifting and locking relative to the high-Koscillator (Figure225
2).
Frequency modulation of forced oscillations
Within the three-patch system, frequency modulation emerged in the pres-
ence of forced oscillations as well as in systems of three autonomous oscillators.
Large areas of parameter space gave rise to frequency modulation where the230
low-Kpatch did not oscillate on its own and was instead forced to oscillate
through dispersal from other patches with self-sustained oscillations (Figure
3a). This contrasted with the two-patch system, in which systems where one
patch was forced by the other had the forced oscillator systematically matching
the phase dynamics of the self-sustained oscillation (i.e. it was entrained into235
phase-locking; Figure 2).
Forced systems showing frequency modulation had stronger variation in
phase difference than systems with three autonomous oscillators (Figures 3a
and 3b). The phase difference fluctuations in these forced systems could have
amplitudes close to π(i.e. the high-Kand low-Koscillators could cycle from240
being almost in phase to almost anti-synchronous).
Relations between phase and amplitude
Dynamical regimes displayed by amplitudes of oscillations across our param-
eter space corresponded to regimes of phase dynamics (Figures 5a; 4). Fluctu-
ations of amplitudes were observed in phase-drifting systems, with maximum245
amplitude matching the intrinsic amplitude of each patch, but with minimum
amplitude decreasing with increases in coupling. For systems showing frequency
modulation, the difference between maximum and minimum amplitude was un-
affected by coupling strength. Under strong coupling leading to phase-locked
synchrony in the system, amplitude showed no variability (Figures 4 and 5).250
In the three-patch system, frequency modulation was strongly correlated
with temporal variation in amplitude (Figure 7b), leading to frequency-amplitude
modulation. Each habitat in a drifting system stayed at its own intrinsic am-
plitude and period (Figure 7a). In phase-locked systems, all habitats converged
7
to a common period and showed no temporal variation in period or amplitude255
(Figure 7c). However, systems showing frequency modulation also showed cyclic
and positively correlated patterns of period and amplitude (Figure 7b).
Effect of phase difference and amplitude modulation on stability
Local habitats involved in frequency-amplitude modulation showed lower
temporal variability in population size (local CV) than either phase drifting or260
phase-locked systems (Figure 6a). This is despite frequency-amplitude modu-
lation typically being intermediate between drift and locking in terms of cou-
pling strength and heterogeneity. Frequency-amplitude modulation also showed
lower maximum regional CV values compared to drift and phase-locked sys-
tems. These results were robust to the length of time series used to calculate265
CV (Figure 6b). Related to this is the result that under frequency-amplitude
modulation, all habitats had lower amplitudes than in the other two phase-
amplitude regimes (Figure 7).
Systems with frequency-amplitude modulation also had the distinct sig-
nature of oscillating in both CVloc and CVreg (Figure 6a), in keeping with270
frequency-amplitude modulation being a non-stationary phenomenon. In con-
trast, the parts of parameter space governed by phase drift had highly variable
CVr eg but comparatively little change in CVloc, while phase-locked systems had
no temporal variation in CVloc or C Vreg beyond that from artifacts related to
the sampling window (Figure 6a). Greatly lengthening the window to minimize275
these artifacts yielded no CV variation at all in phase-locked systems (Figure
6b), as would be expected from a system of completely synchronized oscillators.
Discussion
Spatial phase synchrony in heterogeneous metacommunities: from drift to fre-
quency modulation280
Field and theoretical studies have shown how connecting populations through
dispersal can lead to in-phase synchrony [37], with populations with greater abil-
ity to disperse displaying similarly stronger phase synchrony [38, 39]. However,
environmental heterogeneity is expected to result in corresponding spatial vari-
ation in the natural frequency of population cycles, leading to phase asynchrony285
where populations fail to lock into a constant phase difference and frequency
[9]. Our study builds on theoretical and field studies showing how dispersal
can synchronize populations to a single regional frequency when natural fre-
quencies (in the absence of dispersal) are heterogeneous. In both two-patch
and three-patch models, increasing coupling led phase-drifting populations to290
lock into a single frequency. Prior studies have shown this pattern both under
assumptions of weak coupling [14] and of higher rates of dispersal [40]. Our
results further illustrate how the ability of dispersal to synchronize heteroge-
neous and phase-drifting population cycles is strongly affected by the amount
of heterogeneity [21]. They more importantly reveal how phase asynchrony can295
be maintained under high habitat heterogeneity and dispersal rate in the form
8
of frequency modulation, even when habitat heterogeneity and coupling result
in forced oscillations of local equilibrium points.
Different patterns of phase (a)synchrony were found even in the two-patch
system. One possible outcome of strong habitat heterogeneity is the presence300
of local populations that lack self-sustained oscillations as is the case for local
habitat conditions able to drive local dynamics across a Hopf bifurcation. Our
two-patch results illustrate how self-sustained oscillations in high-Khabitats
can entrain and lock the frequency of the low-Koscillator that has passed the
Hopf bifurcation to stable equilibrium. The fact that coupling can alter the305
dynamics of oscillators near a Hopf bifurcation is well-known [41]. This is also
similar to the concept of master-slave oscillators where forced oscillations and
entrainment are caused by unidirectional coupling [42, 43].
In our two-patch metacommunity, altering heterogeneity and dispersal re-
sulted in fast transitions from phase drift to phase-locked synchrony. However,310
in a three-patch system, forced oscillations of one community leads to complex
interactions between forced and self-sustained oscillators. These interactions
result in frequency modulation over a broad range of heterogeneity and disper-
sal values, where phase difference displays bounded and oscillatory dynamics.
This suggests that local equilibrium dynamics may play an important role in315
driving complex non-equilibirum spatial dynamics in metacommunities. The
convergence of phase-drifting populations to a phase-locked state is well docu-
mented [13, 14, 44, 28], but temporal changes in synchrony have received much
less attention [45]. The presence of frequency modulation reported here is no-
table, as it implies that spatial synchrony can be a non-equilibrium, oscillatory320
phenomenon in the absence of environmental fluctuations. It has been shown
to result from weak coupling of at least 3 homogeneous oscillating predator-
prey communities characterized by pulse-relaxation oscillations [9]. Here we
show how spatial heterogeneity can drive frequency modulation under a broad
range of coupling strength and in the absence of pulse-relaxation cycles. The325
robustness of frequency-modulation to increasing coupling strength is however
associated with fluctuations in the amplitude of population cycles and its dy-
namic coupling to frequency.
Coupled phase-amplitude dynamics in strongly coupled metacommunities
Strong spatial heterogeneity and strong dispersal were responsible for the330
maintenance of coupled fluctuations in both the frequency and amplitude of
oscillations. Out-of-phase populations have been found to temporarily dampen
each other’s amplitudes. This phenomenon is referred to as immigration-local
abundance decoupling [24, 46]. It entails immigration rates into patches becom-
ing negatively correlated with population levels, leading to high immigration335
to less populated patches and high emigration from more populated ones, and
hence reductions in variability. However, the dampening of oscillations driven
by dispersal depends on the maintenance of out-of-phase fluctuations and thus
on processes that can maintain out-of-phase fluctuations in the face of increasing
coupling strength. Within the regime of phase drift, greater coupling increased340
9
the strength of this dampening effect on amplitudes. Further increase in cou-
pling strength leads to non-equilibrium entrainment of local communities in the
form of coupled frequency-amplitude modulation, explaining the robustness of
this dampening effect to strong coupling. Frequency-amplitude modulation re-
sults from variations in amplitude that are negatively coupled and negatively345
correlated with fluctuations in frequency.
Studies of weakly-coupled oscillators have reported uniform phase and chaotic
amplitude in both bitrophic [21] and tritrophic models [13]. However, strong
coupling (>0.001) is typically assumed to lead to full (in-phase) synchrony [47].
Our study suggests that environmental heterogeneity can maintain both spa-350
tial heterogeneity and the dampening of local fluctuations in strongly coupled
metacommunities through coupled fluctuations in their frequency and ampli-
tude. The spectral analysis of long time series on a frequency domain has led to
many insights on the importance of environmental change to explain fluctuating
frequencies leading to non-stationary spectral properties [48, 9]. The coupled355
frequency-amplitude dynamics reported here can provide further signatures of
local dispersal and spatial heterogeneity as drivers of non-stationarity in both
local and regional time series of abundance. Such non-stationarity has impor-
tant implications for our understanding of synchrony as a predictor of local and
regional ecological stability.360
A non-equilibrium theory of coupled local-regional stability
Frequency-amplitude modulation has the potential to reduce variability by
damping the amplitudes of both local and regional time series. In coupled
oscillatory systems, the amplitudes of local time series can be damped via cou-
pling of out-of-phase oscillators [24], and variability in regional averaged time365
series can be damped by statistical averaging [11, 12]. Our results show that
frequency-amplitude modulation provides a mechanism leading to both local
damping and regional averaging. This occurs at intermediate levels of coupling,
consistent with previous results that have also shown intermediate coupling to
maximize stability [17, 49, 6]. In systems exhibiting frequency-amplitude mod-370
ulation, these local (damping) and regional (averaging) effects are combined
by alternating between exhibiting small local amplitudes and exhibiting out-of-
phase synchrony. At any given time, these local and regional properties become
mutually exclusive due to the negative correlation between amplitude and phase
difference. This result has important implications for the assessment of ecolog-375
ical stability, and stresses its dependence on both spatial and temporal scales.
Different metrics of ecological stability are often applied depending on the
spatial scale of the populations or metapopulations being assessed. Over re-
gional scales, out-of-phase dynamics are expected to maximize stability, while
in-phase dynamics such as coherent oscillations lead to greater regional variabil-380
ity and extinction risk [50, 17]. In contrast, single (local) time series analysis
often infers stability from the amplitude and amount of local variations (see e.g.
[11, 51]). Our results suggest that scaling up regional stability from local time
series may require the integration of long-term and non-stationary fluctuations
in both local and regional metrics. They more generally emphasize dynamical385
10
constraints limiting the simultaneous contributions of local and regional mech-
anisms promoting stability. Through this balance between local and regional
stability, frequency-amplitude modulation can maximize both local and regional
persistence by minimizing long-term local and regional variations, and by pro-
viding a mechanism for the maintenance of heterogeneous fluctuations under390
strong dispersal.
Conclusion
In ecology, spatial synchrony between populations has typically been treated
as a static or equilibrium phenomenon, with theoretical studies often predicting
the in-phase locking of populations at moderate levels of dispersal. Our study395
shows that synchrony of phase and amplitude can be viewed as non-equilibrium
properties, hence extending the understanding and applicability of spatial syn-
chrony in ecology. We established coupled frequency-amplitude modulation,
where both the phase and amplitude differences between populations follow
regular cycles. We also show the robustness of coupled phase-amplitude asyn-400
chrony to high levels of dispersal and habitat heterogeneity, thus contributing to
the maintenance of spatial heterogeneity over strongly coupled landscapes. This
includes cases where one population naturally converges to equilibrium but is
forced into oscillations by dispersal from adjacent populations. The emergence
of coupled phase-amplitude dynamics in the presence of forced oscillations sug-405
gests the importance of these forced systems when assessing stability on regional
scales. Frequency-amplitude modulation induces spatially heterogeneous and lo-
cally damped oscillations in population size under strong dispersal. Importantly,
the coupled fluctuations of amplitudes and phase differences produce greater
local stability and long-term regional persistence compared to spatially syn-410
chronous (either in- or out-of-phase) populations or to populations that exhibit
phase drift. Coupled phase-amplitude asynchrony in heterogeneous landscapes
offers distinct statistical signatures of ecological processes underlying long-term
ecological fluctuations, which can assist in evaluating the long-term stability of
natural systems. Our study stresses the importance of integrating environmen-415
tal heterogeneity and strong dispersal for studying the stability of communities
across spatial and temporal scales.
Acknowledgements
FG is grateful for funding from the Natural Sciences and Engineering Re-
search Council of Canada for funding through the Discovery Grant program420
(RGPIN-2017-04266) and for a Discovery Accelerator Supplement (RGPAS-
2017-507832).
11
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17
Tables
Parameter Value Units Description
ri10 time−1Prey intrinsic growth rate
Kivaried organisms Prey carrying capacity
si0.5 time−1·organisms−1Rate at which a given preda-
tor finds prey
τi1 time Time required for a predator
to handle prey before resum-
ing hunting
bi1 unitless Predators produced (through
reproduction) per prey organ-
ism consumed
ci0.4 time−1Predator death rate
mN
ij varied time−1Prey migration rate between
patches iand j
mP
ij varied time−1Predator migration rate be-
tween patches iand j
Table 1: Parameters, their units, and their associated values in this paper
18
Figure legends665
Figure 1: Example time series of systems of oscillating prey populations, and
phase difference between pairs of such oscillators, showing the three different
kinds of dynamics. Parameters are chosen for illustrative purposes. In Figure
1a, from top to bottom, parameters used are ∆K= 0.125 and mij = 0.005
∀i, j (low migration, low heterogeneity); ∆K= 0.25 and mij = 0.005 ∀i, j670
(low migration, high heterogeneity); and ∆K= 0.25 and mij = 0.03 (high
migration, high heterogeneity). In Figure 1b, from top to bottom, parameters
used are ∆K= 0.1 and mij = 0 (phase drift); ∆K= 0.1 and mij = 0.012
(frequency modulation); and ∆K= 0.1 and mij = 0.02 (phase-locking).
Figure 2: Spread in phase difference (maximum minus minimum) between the675
patches in the two-patch system as a function of migration (mij ) and hetero-
geneity (∆K). The green line represents the value of ∆Kat which the low-K
patch crosses the Hopf bifurcation and hence approaches equilibrium absent
coupling. Areas in white (with a spread of 2π) are where one oscillator has
overtaken the other at least once. Grey areas are where the phase difference680
between the two oscillators never changes. Other colours represent areas where
the phase difference changes over time but neither oscillator overtakes the other.
Figure 3: Spread in phase difference between two given patches in the three-
patch system as a function of migration and heterogeneity. All visual elements
are the same as in Figure 2. Figure 3a represents the phase differences between685
the middle-Kpatch and the high-Kpatch, while Figure 3b represents the phase
differences between the high-Kpatch and the low-Kpatch.
Figure 4: Plot of maximum and minimum phase difference between the middle-
Kand high-Kpatches as a function of coupling strength, with ∆K= 0.1. This
graph is equivalent to a horizontal slice through Figure 3a.690
Figure 5: Plots of maximum and minimum amplitude (Figure 5a) and period
(Figure 5b) for the middle-Kpatch as functions of coupling strength. Cases
where ∆K= 0.1 (in yellow and purple) and ∆K= 0 (in grey) are included.
Figure 6: Regional CV (C Vreg ) and average local CV (CVloc) over time in
the three-patch system for a variety of parameter values, representing different695
types of phase dynamics. ∆K= 0.125 in all scenarios unless otherwise noted. In
Figure 6a, CVreg and CVloc were taken over a rolling window 20 units of time
long, with results plotted for systems with mij = 0.005 (representing phase
drift), mij = 0.015 (frequency modulation) and mij = 0.025 (phase-locking).
In Figure 6b, CVr eg and CVloc were taken over a 200 time unit rolling window,700
with the same three values for mij as in 6a. In Figure 6c, CVreg and CVloc were
taken over a 20 time unit rolling window for two systems featuring frequency
modulation. These were a system which featured three autonomous oscillators,
which had ∆K= 0.125 and mij = 0.015 (as seen in Figures 6a and 6b), and
one in which the low-Koscillator was entrained by the other two, which had705
∆K= 0.45 and mij = 0.03.
Figure 7: Amplitudes and periods of populations in the three-patch system
over time. ∆K= 0.1 in each graph. mij = 0, 0.012 and 0.02 in Figures 7a, 7b
and 7c, respectively.
19
Figures710
(a)
0
0.5
1
1.5
2
2.5
3
Prey density
K=5
K=5.125
K=4.875
0
0.5
1
1.5
2
2.5
3
Prey density
1000 1020 1040 1060 1080 1100
Time
0
0.5
1
1.5
2
2.5
3
Prey density
(b)
0
/2
3 /2
2
Phase difference
0
/2
3 /2
2
Phase difference
1000 1200 1400 1600 1800 2000
Time
0
/2
3 /2
2
Phase difference
Figure 1
20
0 0.01 0.02 0.03 0.04
System migration
0
0.1
0.2
0.3
0.4
0.5
System heterogeneity
0
Spread in phase difference
Figure 2
21
(a)
0 0.01 0.02 0.03 0.04
System migration
0
0.1
0.2
0.3
0.4
0.5
System heterogeneity
0
Spread in phase difference
(b)
0 0.01 0.02 0.03 0.04
System migration
0
0.1
0.2
0.3
0.4
0.5
System heterogeneity
0
Spread in phase difference
Figure 3
22
0 0.01 0.02 0.03 0.04 0.05
Migration
0
/2
3 /2
2
Phase difference
Max phase difference
Min phase difference
Figure 4
23
(a)
0 0.01 0.02 0.03 0.04 0.05
Migration
0
2
4
6
8
Amplitude
(b)
0 0.01 0.02 0.03 0.04 0.05
Migration
4
4.5
5
5.5
Period
Figure 5
24
(a)
0 0.1 0.2 0.3 0.4 0.5
Regional CV
0
0.1
0.2
0.3
0.4
0.5
Average local CV
mij=0.005 (drift)
mij=0.015 (FM)
mij=0.025 (locking)
(b)
0 0.1 0.2 0.3 0.4 0.5
Regional CV
0
0.1
0.2
0.3
0.4
0.5
Average local CV
mij=0.005 (drift)
mij=0.015 (FM)
mij=0.025 (locking)
(c)
0 0.1 0.2 0.3 0.4
Regional CV
0
0.1
0.2
0.3
0.4
0.5
Average local CV
K=0.45, mij=0.03 (entrained)
K=0.125, mij=0.015 (autonomous)
Figure 6
25
(a)
02468
3
3.5
4
4.5
5
5.5
6
Middle-K patch
High-K patch
Low-K patch
Amplitude
Period
(b)
02468
3
3.5
4
4.5
5
5.5
6
Middle-K patch
High-K patch
Low-K patch
Amplitude
Period
(c)
02468
3
3.5
4
4.5
5
5.5
6
Middle-K patch
High-K patch
Low-K patch
Amplitude
Period
Figure 7
26
