Mathematical Biosciences 255 (2014) 1–10
Contents lists available at ScienceDirect
Mathematical Biosciences
journal homepage: www.elsevier.com/locate/mbs
Review
Non-equilibrium spatial dynamics of ecosystems
Frederic Guichard a,⇑, Tarik C. Gouhier b
a
b
Department of Biology, McGill University, 1205 Docteur Penfield, Montreal, Quebec H3A 1B1, Canada
Marine Science Center, Northeastern University, 430 Nahant Road, Nahant, MA 01908, USA
a r t i c l e
i n f o
Article history:
Received 11 October 2012
Received in revised form 16 June 2014
Accepted 19 June 2014
Available online 28 June 2014
Keywords:
Ecological dynamics
Non-equilibrium ecosystems
Spatial dynamics
Spatiotemporal heterogeneity
Coastal ecosystems
Nonlinear dynamics
a b s t r a c t
Ecological systems show tremendous variability across temporal and spatial scales. It is this variability
that ecologists try to predict and that managers attempt to harness in order to mitigate risk. However,
the foundations of ecological science and its mainstream agenda focus on equilibrium dynamics to
describe the balance of nature. Despite a rich body of literature on non-equilibrium ecological dynamics,
we lack a well-developed set of predictions that can relate the spatiotemporal heterogeneity of natural
systems to their underlying ecological processes. We argue that ecology needs to expand its current toolbox for the study of non-equilibrium ecosystems in order to both understand and manage their spatiotemporal variability. We review current approaches and outstanding questions related to the study of
spatial dynamics and its application to natural ecosystems, including the design of reserves networks.
We close by emphasizing the importance of ecosystem function as a key component of a non-equilibrium
ecological theory, and of spatial synchrony as a central phenomenon for its inference in natural systems.
Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction
Decomposing biological variability into its spatial and temporal
components is one of the main tools available for understanding
the mechanisms of life. In ecology, the occurrence of significant
temporal and spatial heterogeneity at multiple scales has led to
many unresolved methodological and conceptual challenges [1].
As we attempt to understand life through its patterns and phenomena across levels of ecological organization, we often accept the
idea of equilibrium, or steady state, as a baseline expectation to
resolve its underlying mechanisms. Indeed, it is often assumed that
a steady state results from biological regulation, which is a balance
between ecological functions such as growth and resource supply
(i.e., intrinsic processes), and that spatiotemporal variability represents a shift from a steady state driven by external environmental
conditions (i.e., extrinsic processes). Examples of the steady state
approach can be found in recent macroecological theories of abundance predicting community responses to latitudinal gradients and
climate change [2]. However, a large body of theoretical studies
has long challenged this view by showing how nonlinearities in
ecological processes can lead to significant departures from spatial
or temporal steady states in the absence of extrinsic environmental
variability [3,4]. The impact of those nonlinear feedbacks on
⇑ Corresponding author. Tel.: +1 514 398 6464.
E-mail addresses: [email protected] (F. Guichard), tarik.gouhier@gmail.
com (T.C. Gouhier).
http://dx.doi.org/10.1016/j.mbs.2014.06.013
0025-5564/Ó 2014 Elsevier Inc. All rights reserved.
(spatio)temporal heterogeneity is only one mechanism driving
ecosystems away from a steady state, but it has become a strong
focus of non-equilibrium ecological theory. It has reignited one of
ecology’s long-standing debates regarding the relative importance
of intrinsic and extrinsic processes for spatially-extended natural
systems [5,6].
The idea that ecological systems undergo non-equilibrium
dynamics goes back to the foundation of ecology as a science: From
Darwin’s integration of ecology as a driver of species replacement
over evolutionary time scales [7], to disturbance-succession
theories of species replacement over ecological time scales [8,9].
However, ecologists have struggled to define spatiotemporal
phenomena that can be both predicted from models and measured
in natural systems. To overcome the need for exceedingly rare
datasets comprised of long and spatially-explicit community time
series, some non-equilibrium models have focused on distilling a
subset of predictions based on summary statistics such as temporally- and spatially-unresolved patterns of species-abundance
distributions (SADs; e.g., neutral theory of ecology; [10]). Although
these non-equilibrium models are able to reproduce the SADs
observed in natural systems (e.g., tree communities in tropical
forests), tests of non-equilibrium theory using these types of
summary statistics are known to be weak [11] because both
equilibrium and non-equilibrium models based on either niche
or neutral processes are able to generate SADs that are virtually
indistinguishable from those observed in empirical datasets
[e.g., 12].
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F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10
A more popular approach for testing non-equilibrium model
predictions using existing datasets has been to focus on the
regularity of ecological patterns to disentangle the relative importance of environmental forcing and of nonlinearities contained in
ecological interactions [13,14]. Although identifying regularity in
non-equilibrium spatial dynamics is challenging, theory has identified a set of signals such as traveling waves of predators and prey
or regular bands of vegetation that can be linked to their underlying mechanism [e.g., 15,16]. These studies have collectively laid out
a set of regular spatiotemporal patterns – evenly spaced bands or
spots, traveling waves – that can be associated with nonlinear
ecological processes (Fig. 1). They have more generally associated
a non-equilibrium view of ecosystems with regular patterns of variation, and built this association into a framework of inference:
regular patterns of variability can be explained by nonlinear
ecological processes, while irregular spatiotemporal variability is
driven by environmental fluctuations. Regularity is here a matter
of perception and its working definition evolves with the set of statistics used to represent ecological dynamics. Indeed, many studies
have derived novel methods for extracting regularity from noisy
signals and recover the regular or repeatable component of
variability that can be assigned to non-equilibrium dynamics
[17–20]. This framework has also been greatly extended to
integrate interactions between environmental or demographic
stochasticity and deterministic nonlinear ecological processes in
the production of regular patterns [21].
Detecting and understanding the causes of regularity in patterns of spatiotemporal variability is first and foremost a problem
of scale, and the decomposition of variability into its spatial and
temporal components can help identify the scales of the underlying processes [22]. However, a scale-dependent decomposition of
variability is fundamentally limited in its ability to show how processes can give rise to patterns across different scales (e.g., small
scale processes giving rise to large scale patterns). This idea of
cross-scale interactions between ecological processes and dynamics (e.g., emergent behavior) is ingrained in complex system theory
[23,24] and self-organization, a phenomenon commonly observed
across physical and biological disciplines [25]. These theories suggest that specific relationships between variability and scale [e.g.,
scale-free, see 26] can be used to link processes operating over very
short ranges to variability observed over very large scales. Their
use in ecology has been controversial, in part because ecological
systems often fail to offer the amount of control and data availability over broad ranges of scales required to validate predictions [23].
There is still much room for ecologists to discover natural phenomena that can reveal important ecological processes, and be detected
in existing ecological systems. Bridging the gap between theory
and reality will require the parallel development of novel model
predictions and statistical analyses of complex spatiotemporal data
to link ecological phenomena to their underlying mechanisms
across scales.
Here, we review novel insights and management strategies
that resulted from the creative integration and simplification of
spatiotemporal variability across scales. By doing away with summary statistics and embracing the full spatiotemporal variability
of species abundance and the environment, we show how temporally- and spatially-resolved methods can be used to extract simple phenomena such as (i) phase and amplitude synchrony from
complex datasets and (ii) scaling relationships near important
ecological transitions to better understand and manage natural
systems. These signatures of non-equilibrium spatial dynamics
can be generalized to whole-ecosystem dynamics, and thus have
direct implications for ecosystem-based management strategies
such as the design of reserve networks. Overall, validating nonequilibrium theory will thus require (1) metrics providing a
one-to-one mapping between ecological processes and signature
phenomena common to both model and natural systems and
(2) the use of modern statistical approaches to detect these key
phenomena in spatially- and temporally-resolved (but limited)
datasets.
2. Nonlinear dynamics in space and time
Fig. 1. Examples of regular spatial dynamics in natural populations. (A) Time series
of lynx density in 6 populations in northern Canada (from [16]). Populations show
stable phase-locked oscillations. (B) Illustration of traveling waves in larch
budmoth outbreaks in the European Alps (adapted from [112]). Colours show the
gradient of phase difference (measured as a phase angle) from south-east to northwest of study area, and provide evidence of a traveling wave. (For interpretation of
the references to color in this figure legend, the reader is referred to the web version
of this article.)
Historically, mathematical ecology has identified the stability of
equilibria as one of its main questions, and tied the concept of stability to steady states: constant abundance is the property of a stable population or community [27]. Stability can be defined in many
different ways to fit specific ecological questions [28]. The local stability of fixed points characterizing the equilibrium state of a
dynamical system has been one of the main definitions stemming
from theoretical studies. Using this definition, stability has been
analyzed through the ability and rate of return to equilibrium following a small perturbation. This analysis has pervaded our understanding of population regulation and persistence in heterogeneous
environments, of species coexistence, and of community assembly.
Almost in parallel, non-equilibrium theories have been developed to
show how strong fluctuations, beyond small perturbations of
steady-states [13] can lead communities to contrasting patterns of
coexistence and exclusion [29] and of temporal dynamics [30].
While early studies of stability posited that the persistence of communities is negatively correlated to fluctuations of abundance [27],
spatial dynamics predicts that the limited movement of organisms
can simultaneously increase fluctuations and promote persistence
and coexistence [31].
Non-equilibrium spatial dynamics can result from a combination of intrinsic (nonlinear ecological processes) and extrinsic
F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10
(environment) forces disrupting steady states in mathematical
models [4]. Although early approaches have attempted to partition
the relative importance of intrinsic and extrinsic disrupting forces
additively [see 13, for a review] more recent theories have focused
instead on their interaction to explain the maintenance of
non-equilibrium spatial dynamics [e.g. 32].
The Rosenzweig–MacArthur predator–prey model describes the
nonlinear interaction between predators and their prey and
provides a classic example of nonlinear dynamics and the resulting
positive feedbacks in non-equilibrium ecosystems:
(
dH
¼ rH
dt
dP
aHP
¼
dt
bþH
aHP
1 HK bþH
ð1Þ
dP P
3
local oscillations are not perfectly in phase. These studies have led
to the general prediction that spatiotemporal dynamics can stabilize
and increase persistence of locally fluctuating ecological systems
[37]. This prediction has been extended to species coexistence
[40,41] and to communities [42]. While system (2) is deterministic,
spatiotemporal patterns can result from the interaction between
noise and deterministic processes [21]. Such interactions have been
central to recent progress in resolving intrinsic and extrinsic drivers
of spatial dynamics in natural systems [43,44]. Spatial systems such
as (2) can be studied as stochastic systems to show how
demographic [45] and environmental [46,47] stochasticity can
interact with nonlinear ecological processes to produce and
maintain spatiotemporal heterogeneity through noise induced transitions or stochastic resonance [48].
The interior equilibrium point fH ; P g of system (1) is stable for
P rbÞ
, where K crit corresponds to a Hopf bifurcation.
K < K crit ¼ bð2bþrd
rðbdP Þ
A stable limit cycle emerges and the equilibrium point becomes
unstable for K P K crit . Because the carrying capacity K is a measure
of enrichment of the ecosystem and can be used as a control
parameter driving the destabilization of the equilibrium point, this
dynamical response has been formulated as the paradox of enrichment [33]. More generally, the nonlinear (saturating) per capita
response of predator growth to prey abundance, along with the
nonlinear (logistic) per capita response of prey growth to their
own abundance result in a positive feedback between prey density
and prey growth @H=dt
> 0 at low prey density. This positive feed@H
back destabilizes the equilibrium point and leads to a stable limit
cycle. Nonlinear dynamics associated with positive feedbacks is
an important driver of non-equilibrium ecological dynamics.
This model can be expanded into a minimal model of non-equilibrium spatial dynamics, with predator–prey dynamics within
two discrete habitats and passive movement (diffusion) of
individuals:
8
< dHi ¼ rHi 1 Hi aHi Pi þ DH Hj DH Hi
dt
Ki
bþHi
: dPi ¼ aHi Pi d P þ D P D P
P i
P j
P i
dt
bþHi
i; j 2 f1; 2g : i – j
ð2Þ
This system can display multiple equilibria, some of which are
spatially heterogeneous [34,35]. Non-equilibrium spatial dynamics
emerges when any spatially homogeneous limit cycle solution is
unstable and is replaced with spatially heterogeneous fluctuations
[36]. Non-equilibrium dynamics have important implications for
population persistence and species coexistence. Oscillatory dynamics is predicted to decrease persistence by bringing abundance away
from its positive steady state and leading to population extinction in
the presence of perturbations during phases of low abundance.
When nonlinear feedbacks, such as those found in system (1), lead
to self-sustained oscillations, limited movement between locations
in system (2) can disrupt this feedback and affect the amplitude or
onset (bifurcation point) of oscillations by decoupling local growth
from abundance [37]. This phenomenon was studied in experimental and model predator–prey [38], and host-parasitoid systems,
where time delays and nonlinear feedbacks predict no persistence
in the absence of limited movement [39]. These studies provided
one of the earliest motivations for studying spatiotemporal dynamics: asynchronous persistence mediated by limited movement. In
the presence of limited parasitoid movement (explicit or implicit),
increase in local host abundance can be achieved through passive
movement from other locations, which can rescue extinct local populations. Such movement can also limit the destabilizing effect of
nonlinear feedbacks by decoupling growth (e.g., density-dependent
growth of the parasitoid) from density. These stabilizing effects can
operate as long as locations connected by movement have unequal
abundances. In other words, movement must be limited such that
2.1. Spatiotemporal heterogeneity
The potential complexity of spatial dynamics can be distilled
down to one property: the spatial (among location) asynchrony
of local temporal fluctuations. Asynchrony can be measured as
the imperfect correlation between time series, and is important
because it is a necessary and sufficient condition for spatiotemporal heterogeneity. The synchrony of fluctuations between ecological systems has received much attention and is typically ascribed
to two mechanisms: movement of organisms and/or matter
between locations, including the movement of a mobile predator,
and correlated environmental fluctuations across locations (i.e.
Moran effect) [44]. When full synchrony is reached through these
mechanisms, the ensemble of locations is spatially homogeneous.
It is therefore the maintenance of asynchrony between population
fluctuations that defines spatiotemporal heterogeneity. The onset
of spatial dynamics can be studied through the stability analysis
of the spatially homogeneous solution [36]. This method predicts
the onset of spatial dynamics for arbitrary values of coupling
strength between communities, but it is based on small perturbations of the homogeneous solution, and provides little information
about the properties of the resulting spatiotemporal heterogeneity
[49]. It is also possible to simplify the dynamics of multiple communities to that of a phase difference in the periodic fluctuations
of predators and prey between communities. The solution to system (2) for each oscillator displays a closed orbit. This periodic
orbit can be described by the phase of the predator–prey system
oscillator in each community:
( dh
i
dt
dhj
dt
¼ X þ dGðhj hi Þ
¼ X þ dGðhi hj Þ
ð3Þ
where hi is the periodic phase of oscillator (location) i; X is the natural frequency of each oscillator, and the interaction function
Gi ðhj hi Þ gives the effect of each oscillator on the phase of the other
through weak coupling d. This method assumes periodic oscillations
and its analysis is based on weakly coupled communities. Under
those assumptions, the stability analysis of phase differences
ðhj hi Þ can predict a great diversity of spatiotemporal regimes
characterized by phase locking, including the existence of multiple
stable heterogeneous states [50]. This formalism has been applied
to system (2) where predator–prey interactions and their movement between two habitats provide a minimal set of coupled ecological oscillators [50,51].
Asynchrony between oscillators can be explained by heterogeneous environmental (external) forcing due to differences in the
parameters affecting the period of local limit cycles (natural frequency) [52,53], or by the weak mixing of biomass among communities. Phase dynamics greatly simplifies the description of coupled
systems by reducing the explicit dynamics of abundance of each
species at each location (4 variables in the case of system (2)) to
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F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10
the dynamics of their phase difference between locations. The
stability analysis of phase differences further limits their complexity by imposing weak spatial coupling. With such simplifications
come some limits to the applicability of phase analysis to natural
systems. Phase stability analysis has been mostly limited to two
or few coupled ecological oscillators, and it still has to be scaled
up to many locally-coupled systems [but see 54]. As we will
discuss below, applications to ecology also lack integration of
ecosystem-level dynamics involving the recycling and movement
of material, including nutrients and (in)organic matter [55].
Feedbacks between the recycling of nutrient from biomass and
its movement between ecosystems can cause both local fluctuations and their asynchrony [56]. Addressing these challenges
would contribute to a general non-equilibrium theory of largescale ecosystems and to a new ecosystem-based management
framework. Most importantly, approaches based on phase analysis
have been expanded to describe both phase and amplitude [16],
and a much broader range of spatiotemporal phenomena that are
not captured by phase locking. These studies can improve the
use of spatial dynamics as a signature of local ecological processes,
even for systems showing no apparent regularity in fluctuations.
2.2. The challenge of decomposing complex dynamics into their phases
and amplitudes
Phase dynamical approaches have emphasized phase locking
and studied dynamical systems by extracting the instantaneous
phase of population fluctuations from time series [57,50]. However, beyond phase locking and traveling waves, systems of asynchronous coupled oscillators can display a wide range of patterns
in time and space (Fig. 2A) that are not sufficiently described by
phase dynamics [49]. These patterns also involve strong changes
in the amplitude of fluctuations [16], which could be important
drivers of patterns [58]. By extracting both the phase and the
amplitude of oscillations, we can thus improve our ability to
discriminate between multiple plausible causal mechanisms. For
instance, local fluctuations in hare and lynx abundance in northern
Canada follow ten-year cycles that are phase-locked over large
scales (Fig. 1A; [16,59]). However, as discussed above, phaselocked oscillations alone can be associated with a number of
bottom-up (resource supply) and top-down (predator regulation)
mechanisms. Hence, simplifying the spatial dynamics of the
hare-lynx interaction to phase differences fails to resolve its
underlying cause. Only when the amplitude of oscillations was
integrated into a more complete phase-amplitude model of predator–prey-resource dynamics were Blasius and colleagues able to
show that phase-locking and chaotic amplitudes together provide
a signature of oscillations driven by trophic interactions and
coupled through the movement of predators [16].
Many spatiotemporal series from natural systems lack any
striking regularity that can be captured by simple stationary statistical properties such as their phase and amplitude. For example,
individual-based models can predict the maintenance of asynchrony and of complex spatial patterns through the limited movement of individuals [60–62]. This complexity requires that we
expand the range of statistics we use to characterize spatiotemporal series if we wish to infer their underlying drivers [44]. One
important feature of spatiotemporal dynamics is the cross-scale
feedback between local processes and regional heterogeneity that
eludes existing scale-dependent inference frameworks that associate variability with their underlying mechanisms over corresponding scales [e.g., 22]. For example, predator–prey communities
connected by dispersal can have a stable spatially homogeneous
periodic solution [36] associated with long-term synchrony
between locations [16,50]. However, transient dynamics can reveal
strong heterogeneity in amplitudes of fluctuations with important
Fig. 2. (A) Complex spatial dynamics of predator–prey interactions. (B) Comparing
time series of neighboring communities (see corresponding black and gray
transects in (A)) reveal transient ‘defects’ in the alignment of phases between
communities. (C) These defects are intermittent over time and results in regular
fluctuations in the frequency and amplitudes of individual time series.
consequences for local and regional persistence [54,63]. More
generally, any phase perturbation between two phase-locked
oscillators necessarily involves a transient return to one stable
phase-locked solution (Fig. 2B). Across multiple communities, such
transient dynamics means that one local perturbation can cause
additional phase perturbations to neighboring communities within
that transient period. This cascading effect leads to local gradients
of phase differences with characteristic spatial and temporal scales
that depend on the resilience of the phase locked solution. This
phenomenon has been referred to as the propagation of spatial
defects, or ‘kink breeding’ [64]. These spatial defects can be associated with transient changes in the amplitude and frequency of
individual communities (Fig. 2C) and give rise to spatial patchiness
with a characteristic spatial scale [65]. During the transient return
to in-phase synchrony equilibrium between two oscillators, their
amplitude can decreases while their frequency increases
(Fig. 2C). This transient pattern spreads across oscillators to form
transient patches of low-amplitude and high-frequency oscillations that are progressively replaced with synchronous oscillations
at proper frequency and amplitude. Kink breeding has been studied in physical systems [66], and patterns of spatial synchrony that
are compatible with kink-breeding have been detected in benthic
marine communities over continental scales [65]. The direct observation of such signatures in large ecological datasets is certainly a
great challenge [67] and could offer a set of novel statistical signatures of non-equilibrium spatial dynamics.
F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10
As we attempt to validate theories of phase and amplitude
dynamics, we are greatly limited by the availability of ecological
data that provide extensive information across broad ranges of
spatial and temporal scales. One goal of non-equilibrium ecological
theory is then to scale-up current mathematical models to ensembles of ecosystems, and develop predictions that are compatible
with our ability to monitor natural ecosystems.
3. Inferring spatiotemporal dynamics from ecological data
As discussed above, there is a rich literature on spatial dynamics
that has identified and classified a number of template patterns for
linking local nonlinear dynamics and limited movement to largescale heterogeneity. However, this non-equilibrium theory has
yet to yield methods for linking these complex spatiotemporal
patterns to their underlying mechanisms.
3.1. Signatures of nonlinear spatial dynamics
Template spatiotemporal patterns are those that have been
adopted for the validation of spatial dynamical models because
their regularity can be inferred with relatively simple statistical
methods. They include traveling waves [68,69], scale invariant
patch dynamics [26], and stationary spatial periodicity [e.g. bands
and spots in arid vegetation; 15]. These templates have been used
to infer the nature of the processes governing ecosystems: Regular
patterns are linked to intrinsic nonlinear spatial dynamics and
irregular patterns to extrinsic environmental perturbations. For
example, ecologists are trying to resolve biotic and abiotic causes
of individual aggregation into discrete patches. The size distribution of those patches can be used as a signature of local biotic
antagonistic processes such as predator–prey, host-parasite or disturbance-recovery interactions [25]. One prediction borrowed
from critical phenomena theories is that local antagonistic interactions can lead to scale-invariant distribution FðsÞ of patch size s,
with FðsÞ / sb . This prediction was tested in tropical forests characterized by (antagonistic) interactions between tree colonization
and disturbance by wind or fire. In these systems, scale-invariance
in the size distribution of tree patches was used as a signature of
local and nonlinear biotic interactions driving the propagation of
fire and windthrows between neighboring trees, and the dispersal
of seeds from adult trees into nearby disturbed areas [70]. This
interpretation of scale invariant patch size distribution is ecologically important because it means that trees are dynamically connected across whole forests through purely localized ecological
processes [25]. Characteristic scales rather than scale invariance
can also be used as signature of the importance of local nonlinear
processes for spatiotemporal dynamics. Traveling waves are for
example observed in vole populations and characterized by their
frequency and amplitudes [69]. Ecologists have applied theories
of diffusive (Turing) instabilities that predict static or dynamic patterns with characteristic spatial and temporal scales. These theories predict the onset of spatial or spatiotemporal heterogeneity
when local ecological processes involve coupled positive and negative feedbacks that operate at different spatial scales [71,15]. Static and periodic aggregation of vegetation in arid systems results
from such feedbacks driven by short range positive effect of aggregation on water retention, and by long range negative effect of
aggregation through below-ground competition for nutrients
[15]. Here, the emergence of diffusive instabilities, periodic spatiotemporal variability, and other regular patterns are associated with
localized interactions, whereas irregular patterns are associated
with environmental stochasticity. In oscillatory systems where
patterns are dynamic, synchrony and phase locking of oscillations
across locations can be used to assess the regularity of spatial
5
dynamics. However, patterns that are perceived as regular (as
defined from the examples above) are relatively rare in ecological
systems.
3.2. Irregular spatiotemporal heterogeneity
Studies of complexity in ecology have greatly expanded the set
methods that can be used to detect statistical regularity emerging
from biological regulating mechanisms, from regular spatiotemporal periodicity of abundance to the regular scaling (i.e. power laws)
of spatiotemporal variance. However, ecologists are still facing the
challenge of detecting such regularity in natural systems, or with
our limited ability to ‘read the signs’ [72]. For example, phase locking – a clear manifestation of spatiotemporal patterns – has been
shown to emerge from 2-patch systems in relation to a separation
of temporal scales between predators and their prey [50], and to
time to extinction (Lyapunov exponent) relative to dispersal [58]
in trophic communities. These predictions can be tested in experimental systems where these parameters can be estimated, but the
phenomenon itself – stable phase locking – is rarely observed in
natural systems. When it is observed, a number of alternative
interpretations exist to explain its occurrence. In such cases, inference of the underlying mechanisms can be developed through
other properties of spatial dynamics (amplitude in addition to
phase) that can be measured from existing spatiotemporal series.
For instance, phase locking accompanied by chaotic amplitudes
can be used as a signature of coupled dynamics in tri-trophic food
chains [16]. Similarly, the rate at which synchrony decays with distance between ecosystems can help identify (i) the relative importance of biological regulating mechanisms and environmental
drivers and (ii) the scale of dispersal [65]. The relationship between
angular velocity, phase, amplitude and net dispersal also offers a
rich set of predictions to explain dispersal-mediated persistence
of communities with antagonistic interactions [73]. These examples illustrate how ecologists can develop new theory to improve
our ability to infer intrinsic and extrinsic causes of non-equilibrium
dynamics in natural communities.
Ultimately, distinguishing extrinsic and intrinsic causes of
population fluctuations in the real world is likely to require both
temporally-replicated and spatially-explicit data (Figs. 3 and 4).
For instance, only when spatial (e.g., autocorrelation or synchrony)
and temporal (e.g., wavelet) analyses are combined can the drivers
of population abundance be identified in a simple metapopulation
model characterized by equilibrium vs. non-equilibrium local
dynamics and constant vs. stochastic environmentally-mediated
dispersal (Figs. 3 and 4). In this case, wavelet analysis of the local
abundance of a representative subpopulation shows that the magnitude, duration and periodicity of temporal fluctuations in models
characterized by non-equilibrium dynamics and constant dispersal
are quite different from those observed in models characterized by
equilibrium local dynamics and stochastic dispersal (Fig. 4a and d
vs. b and e). However, the magnitude, duration and periodicity of
temporal fluctuations are largely identical in models characterized
by non-equilibrium local dynamics and either constant or stochastic
dispersal (Fig. 4a and d vs. c and f). Hence, although wavelet analyses
can distinguish between equilibrium and non-equilibrium local
dynamics, they cannot be used to infer the role of constant vs. stochastic dispersal. Conversely, analyses of spatial autocorrelation in
abundance can be used to identify models characterized by nonequilibrium local dynamics and constant vs. stochastic dispersal
kernels (Fig. 3b vs. d), but not models characterized by equilibrium
dynamics and constant vs. stochastic dispersal kernels (Fig. 3c and
d). Specifically, models characterized by non-equilibrium local
dynamics and constant dispersal exhibit nonlinear and non-stationary patterns of autocorrelation (Fig. 3b), whereas those characterized by stochastic dispersal exhibit largely linear and stationary
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F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10
patterns of autocorrelation (Fig. 3d). However, models characterized
by stochastic dispersal and either equilibrium or non-equilibrium
local dynamics exhibit similar linear and stationary patterns of
autocorrelation (Fig. 3c and d). Hence, only by employing both
spatial and temporal analyses can the source of population dynamics at local and regional scales be fully resolved. These examples
illustrate the potential power in combining modern spatial and temporal analyses (e.g., wavelet methods, spatial autocorrelation and
synchrony) to study how dispersal, nonlinear ecological processes
and environmental stochasticity interact to shape patterns of
population fluctuations in models [74,75,46,76] and nature [43,44].
4. The implications of non-equilibrium dynamics for reserve
design: a function for pattern formation
In marine systems, fisheries have long been managed as equilibrium and single-species systems. This view has led to the widespread adoption of maximum sustainable yield assuming
Stochastic dipersal
Spatial autocorrelation
Constant dipersal
Spatial autocorrelation
Equilibrium local dynamics
1
Non−equilibrium local dynamics
a
b
c
d
0.6
0.2
−0.2
−0.6
−1
1
0.6
0.2
−0.2
−0.6
−1
0
25
50
75 100 125 0
25
50
75 100 125
Lag distance (number of sites)
Lag distance (number of sites)
Wavelet analysis
Period
Local timeseries
Abundance
Fig. 3. Annual spatial autocorrelation of abundance for metapopulation models
undergoing local equilibrium (a and c) and non-equilibrium (b and d) dynamics
with either constant (a and b) or stochastic (c and d) dispersal for 2000 years. The
annual patterns of spatial autocorrelation are represented in gray and the mean is
represented in red. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
equilibrium and homogeneously distributed populations. Managers have now embraced ecosystem-based management, which
emphasizes the non-equilibrium and heterogeneous nature of
population dynamics and the interdependence of biotic and abiotic
processes across scales. This shift has lead to major developments
in management strategies such as marine protected areas, which
constitutes an additional anthropogenic driver of spatiotemporal
heterogeneity.
Ecological reserves, areas protected from all extractive and
destructive activities, are increasingly being heralded as important
components of a broader strategy for both conserving and managing natural systems [77–80]. There is growing recognition that networks of interconnected reserves can be even more effective than
individual reserves of the same size because they (1) distribute the
societal costs more evenly over space, (2) protect species over a
larger portion of their range, (3) provide spatial redundancy that
reduces the effects of local or spatially-autocorrelated catastrophes
(e.g., toxic spills, wildfires), and (4) promote the recovery of
individual reserves via regional subsidies [77,81,78,80]. Because
most of these network benefits require connectivity between individual reserves, theory based on equilibrium models has long
extolled the benefits of building strongly interconnected reserve
networks [82,83].
However, non-equilibrium theory has established that connectivity is not always conducive to positive outcomes such as
increased resilience and persistence. Instead, connectivity is more
of a double-edged sword that can promote persistence by allowing
regional subsidies to rescue local populations, but also increase
global extinction risk by spatially synchronizing the dynamics of
interconnected populations [84,85]. Hence, in addition (and sometimes contrary) to current guidelines, reserve networks established
for populations undergoing non-equilibrium dynamics will have to
abide by design principles that seek to avoid the synchronizing
effect of connectivity and its negative impact on persistence
[86,87].
Limiting connectivity between reserve networks not only
reduces the risk of global extinction, but it can also facilitate the
management of trophically- and spatially-coupled systems undergoing non-equilibrium dynamics [87]. Indeed, in equilibrium systems, reserve network design typically involves trade-offs with
respect to the size and spacing of reserves in order to achieve conservation vs. commercial objectives [88]. For instance, achieving
Non−equilibrium local dynamics Equilibrium local dynamics Non−equilibrium local dynamics
and constant dispersal
and stochastic dispersal
and stochastic dispersal
1
0.8
a
b
c
0.6
0.4
0.2
0
4
8
16
32
64
128
256
512
0
d
e
500
1000
Time
1500
2000 0
f
500
1000
Time
1500
2000 0
500
1000
1500
2000
Time
Fig. 4. Applying wavelet analysis to local model abundance time series (a–c) to test predictions about the source of population fluctuations in metapopulations. The
magnitude, duration and periodicity of local population fluctuations revealed by wavelet analysis (d–f) can help differentiate between intrinsic (e.g., nonlinear species
interactions) and extrinsic (e.g., environmentally-mediated stochastic dispersal) drivers of population fluctuations. (d–f) Regions depicted by warm (cold) colors correspond
to high (low) variability in the time series. Black contours indicate statistically-significant variability and the black dashed line represents the cone of influence (COI). Values
that lie outside the COI are subject to edge effects. These results are based on the same simulations that were analyzed in Fig. 3. (For interpretation of the references to color in
this figure legend, the reader is referred to the web version of this article.)
7
F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10
(A) Closed meta-ecosystem
dP
P
Pi
Pj
fP
fP
H
P
Hi
dH
H
Hj
fH
fH
Ni
dN
Nj
(B) Low dispersal, dN=2
Ni
Hi
Pi
Nj
Hj
Pj
Nutrient stock
conservation goals such as maximizing abundance typically
requires large and fully connected reserves, whereas achieving
commercial goals such as maximizing yield typically requires
smaller and partially connected reserves in order to promote the
‘spillover’ of resources into unprotected areas where they can be
harvested [88]. Additionally, there are community-level trade-offs
with respect to the optimal size and spacing of reserves: dominant
competitors and predators benefit from small and aggregated
reserves that maximize connectivity and its positive effects on
single species growth. In contrast, weaker competitors and prey
species benefit from large and isolated reserves that minimize connectivity and release those species from competition and predation
[89,87]. Altering connectivity by varying the size and spacing of
reserve networks can only dampen this community-level tradeoff or tilt the balance towards one set of species (dominant
competitors and predators vs. weaker competitors and prey) [87].
In non-equilibrium systems, these trade-offs disappear because
cross-scale ecological feedbacks between local population fluctuations and regional dispersal result in a separation of scales between
ecological processes and patterns (distribution of abundance),
which decouples the intraspecific benefits of connectivity from
the interspecific costs of trophic cascades. Networks that exploit
this separation of scales by using the extent of patterns (i.e., the
extent of patchiness or spatial autocorrelation) as the size and
spacing of reserves are thus able to maximize the abundance, yield,
and persistence of trophically-coupled species [87]. Overall, this
suggests a function for pattern formation: the optimal management of trophically- and spatially-coupled communities with
respect to both conservation and commercial objectives.
Time
(C) High dispersal, dN =4
A non-equilibrium theory of spatial dynamics certainly needs to
consider the recycling and movement of organic and inorganic
matter. The theoretical framework of meta-ecosystems predicts
ecological consequences of limited flows of organic and inorganic
matter across space (Fig. 5A). This emerging ecological theory
[90] and its application to natural systems will provide a good case
study to build and validate a theory of non-equilibrium spatial
dynamics. A simple meta-ecosystem model can be formulated as
an extension of the predator–prey metacommunity (system (2)),
where biomass is recycled into nutrients, and where primary
producers (the prey) take-up nutrients for growth:
8
dN i
Hi N i
>
¼ dH Hi þ dP Pi abþN
þ DN N j DN Ni
>
>
i
< dt
dHi
aHi Ni
aHi P i
¼ bþNi bþHi dH Hi þ DH Hj DH Hi
dt
>
>
> dPi aHi Pi
:
¼ bþHi dP Pi þ DP Pj DP Pi
dt
i; j 2 f1; 2g : i – j
ð4Þ
System (4) describes a closed meta-ecosystem with no external
input or output of nutrients. Recent theory based on similar models suggests that flows of organic and inorganic matter can deeply
transform the structure and stability of communities [91], generate
nonlinear feedbacks and explain spatiotemporal heterogeneity
[92]. Fluctuations in the concentration of (in)organic matter can
similarly lead to patterns of synchrony among ecosystem compartments (e.g. between nutrient concentration and primary producer
biomass) and among locations. Such patterns have been predicted
from simple 2-patch meta-ecosystem models [Fig. 5B–C; 92], and
recently extended to larger irregular networks [93]. These predictions need to be extended to large meta-ecosystems and matched
with long-term monitoring data of biogeochemical cycles. These
theoretical studies also echo recent calls for whole-ecosystem
management of natural resources that are embedded in complex
interaction networks and flows of (in)organic matter.
Nutrient stock
5. Towards a non-equilibrium meta-ecosystem theory of coastal
management
Time
Fig. 5. Meta-ecosystem dynamics illustrating emerging patterns of spatial synchrony. (A) Minimal meta-ecosystem model corresponding to system (4). (B–C)
Time series of nutrient stock contained in each ecosystem compartment (nutrients
N, Primary producer H and Consumer P), in each location (i in red and j in blue).
Patterns of spatial synchrony (compare red and blue lines) emerging as the rate of
nutrient movement is increased from dN ¼ 2 (B) to dN ¼ 4 (C). High nutrient
movement in (C) leads to in-phase synchrony of nutrients while the producer and
consumer become phase-locked with a period corresponding twice the period of
nutrient fluctuations. Adapted from [92]. (For interpretation of the references to
color in this figure legend, the reader is referred to the web version of this article.)
Coastal ecosystems are among the most productive and threatened by human activities and climate change [94], and thus prime
targets for ecosystem-based management approaches [95]. They
show strong fluctuations in the distribution of species and in ecosystem functions (e.g. nutrient recycling and primary production).
Temperate coastal ecosystems support large populations of benthic
invertebrates whose local abundances fluctuate by more than 60%
of their mean abundance between years in the North-East Pacific
[96] and North West Atlantic [97]. Over longer temporal scales,
Dungeness crab population abundance follow a 10–11 year cycles
in the North-East Pacific [98]. At the ecosystem level, fluctuations
in (in)organic matter has also been documented in nearshore
waters [99,100], and related to species interactions [101–103].
8
F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10
Despite empirical evidence, current theories of coastal (and other)
ecosystems assume that nutrients are well mixed across large
spatial scales. These simplifying assumptions contained within
ecological theories are in sharp contrast with our detailed
understanding of transport and cycles of nutrient and organic matters in oceans. Coastal systems strongly depend on coupling
between benthic and pelagic compartments [104,99,105], and
spatial fluxes are therefore controlled across a very broad range
of spatial and temporal scales [106], from fast advective mixing
and stratification [107] to storage over geological time scales.
Oceanographers have a long history of spatially resolved trophic
models even including cycling of matter through recycling or
excretion [108]. While many coupled bio-physical models have
been used to make quantitative predictions [109], many studies
have adopted a more heuristic approach and contributed to a
meta-ecosystem theory for marine systems [108]. For example,
spatially explicit (1D vertical) Nutrient–Phytoplankton–Zooplankton (NPZ) models [110] where diffusion couples the dynamics of
nearby nutrient stock, can induce stable profiles as well as oscillatory dynamical trajectories that become vertically phase-locked for
large mixing levels [110]. Given the accumulating evidence of
strong meta-ecosystem dynamics in coastal ecosystems, one task
is to test for the existence and implications of nonlinear feedbacks
over regional scales that are targeted by managers and policy
makers. The sudden shifts in the state of regional fisheries
documented over the last decades [111] certainly suggest the
relevance and urgency of this task.
6. Conclusion
Natural systems are constantly changing as they face external
perturbations, but also through the processes that are responsible
for their very persistence. Yet, the balance of nature is still a strong
paradigm in ecology, and the steady state is still a dominant target
for conservation biologists and managers. Great progress has been
achieved towards a general non-equilibrium theories of ecological
systems that can be applied to natural systems across scales.
However, our ability to understand dynamical and interconnected
ecosystems remains limited by our reliance on inferential tools
that associate regular patterns with intrinsic processes and
irregular patterns with extrinsic processes. We highlighted recent
development in non-equilibrium theories that contributed to
expanding the range of phenomena we can measure in natural systems and use as signatures of underlying mechanisms. Spatial synchrony has been central to recent development and application of
non-equilibrium spatial dynamics. However, understanding the
causes of synchrony, or lack thereof, between populations is still
challenging. We suggest that this task can be facilitated by testing
for temporal and correlated shifts in phase and amplitude differences between time series, which assesses the transient and
cross-scale response of local populations to limited dispersal as a
cause of spatiotemporal heterogeneity. In order to apply to natural
systems, ecological theory should certainly be able to describe
ecological systems of increasing complexity. But most importantly,
it must develop predictive frameworks built around metrics that
can reduce this complexity and provide simple signatures of
underlying causes of spatial dynamics. These metrics should in
turn inform and optimize the design and implementation of
long-term ecosystem monitoring programs. This is important
because non-equilibrium metacommunity and meta-ecosystem
theories have the potential to directly affect management and
policies such as reserve networks that are still largely based on
equilibrium models. The complexity of predictions rather than that
of models themselves is currently limiting the application of these
theories to natural and managed systems.
Acknowledgement
F. Guichard was supported by the Natural Science and Engineering Research Council of Canada through the Discovery Grant
program.
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