Mathematical Biosciences 255 (2014) 1–10

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Mathematical Biosciences

journal homepage: www.elsevier .com/locate /mbs

Review

Non-equilibrium spatial dynamics of ecosystems

http://dx.doi.org/10.1016/j.mbs.2014.06.0130025-5564/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author. Tel.: +1 514 398 6464.E-mail addresses: fred.guichard@mcgill.ca (F. Guichard), tarik.gouhier@gmail.

com (T.C. Gouhier).

Frederic Guichard a,⇑, Tarik C. Gouhier b

a Department of Biology, McGill University, 1205 Docteur Penfield, Montreal, Quebec H3A 1B1, Canadab 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 2012Received in revised form 16 June 2014Accepted 19 June 2014Available online 28 June 2014

Keywords:Ecological dynamicsNon-equilibrium ecosystemsSpatial dynamicsSpatiotemporal heterogeneityCoastal ecosystemsNonlinear dynamics

a b s t r a c t

Ecological systems show tremendous variability across temporal and spatial scales. It is this variabilitythat 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 todescribe 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 naturalsystems to their underlying ecological processes. We argue that ecology needs to expand its current tool-box for the study of non-equilibrium ecosystems in order to both understand and manage their spatio-temporal variability. We review current approaches and outstanding questions related to the study ofspatial 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-equilibriumecological theory, and of spatial synchrony as a central phenomenon for its inference in natural systems.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Decomposing biological variability into its spatial and temporalcomponents is one of the main tools available for understandingthe mechanisms of life. In ecology, the occurrence of significanttemporal and spatial heterogeneity at multiple scales has led tomany unresolved methodological and conceptual challenges [1].As we attempt to understand life through its patterns and phenom-ena across levels of ecological organization, we often accept theidea of equilibrium, or steady state, as a baseline expectation toresolve its underlying mechanisms. Indeed, it is often assumed thata steady state results from biological regulation, which is a balancebetween ecological functions such as growth and resource supply(i.e., intrinsic processes), and that spatiotemporal variability repre-sents a shift from a steady state driven by external environmentalconditions (i.e., extrinsic processes). Examples of the steady stateapproach can be found in recent macroecological theories of abun-dance predicting community responses to latitudinal gradients andclimate change [2]. However, a large body of theoretical studieshas long challenged this view by showing how nonlinearities inecological processes can lead to significant departures from spatialor temporal steady states in the absence of extrinsic environmentalvariability [3,4]. The impact of those nonlinear feedbacks on

(spatio)temporal heterogeneity is only one mechanism drivingecosystems away from a steady state, but it has become a strongfocus of non-equilibrium ecological theory. It has reignited one ofecology’s long-standing debates regarding the relative importanceof intrinsic and extrinsic processes for spatially-extended naturalsystems [5,6].

The idea that ecological systems undergo non-equilibriumdynamics goes back to the foundation of ecology as a science: FromDarwin’s integration of ecology as a driver of species replacementover evolutionary time scales [7], to disturbance-successiontheories of species replacement over ecological time scales [8,9].However, ecologists have struggled to define spatiotemporalphenomena that can be both predicted from models and measuredin natural systems. To overcome the need for exceedingly raredatasets comprised of long and spatially-explicit community timeseries, some non-equilibrium models have focused on distilling asubset of predictions based on summary statistics such as tempo-rally- and spatially-unresolved patterns of species-abundancedistributions (SADs; e.g., neutral theory of ecology; [10]). Althoughthese non-equilibrium models are able to reproduce the SADsobserved in natural systems (e.g., tree communities in tropicalforests), tests of non-equilibrium theory using these types ofsummary statistics are known to be weak [11] because bothequilibrium and non-equilibrium models based on either nicheor neutral processes are able to generate SADs that are virtuallyindistinguishable from those observed in empirical datasets[e.g., 12].

2 F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10

A more popular approach for testing non-equilibrium modelpredictions using existing datasets has been to focus on theregularity of ecological patterns to disentangle the relative impor-tance of environmental forcing and of nonlinearities contained inecological interactions [13,14]. Although identifying regularity innon-equilibrium spatial dynamics is challenging, theory has iden-tified a set of signals such as traveling waves of predators and preyor regular bands of vegetation that can be linked to their underly-ing mechanism [e.g., 15,16]. These studies have collectively laid outa set of regular spatiotemporal patterns – evenly spaced bands orspots, traveling waves – that can be associated with nonlinearecological processes (Fig. 1). They have more generally associateda non-equilibrium view of ecosystems with regular patterns of var-iation, and built this association into a framework of inference:regular patterns of variability can be explained by nonlinearecological processes, while irregular spatiotemporal variability isdriven by environmental fluctuations. Regularity is here a matterof perception and its working definition evolves with the set of sta-tistics used to represent ecological dynamics. Indeed, many studieshave derived novel methods for extracting regularity from noisysignals and recover the regular or repeatable component ofvariability that can be assigned to non-equilibrium dynamics[17–20]. This framework has also been greatly extended tointegrate interactions between environmental or demographicstochasticity and deterministic nonlinear ecological processes inthe production of regular patterns [21].

Detecting and understanding the causes of regularity in pat-terns of spatiotemporal variability is first and foremost a problemof scale, and the decomposition of variability into its spatial andtemporal components can help identify the scales of the underly-ing processes [22]. However, a scale-dependent decomposition of

Fig. 1. Examples of regular spatial dynamics in natural populations. (A) Time seriesof lynx density in 6 populations in northern Canada (from [16]). Populations showstable phase-locked oscillations. (B) Illustration of traveling waves in larchbudmoth outbreaks in the European Alps (adapted from [112]). Colours show thegradient of phase difference (measured as a phase angle) from south-east to north-west of study area, and provide evidence of a traveling wave. (For interpretation ofthe references to color in this figure legend, the reader is referred to the web versionof this article.)

variability is fundamentally limited in its ability to show how pro-cesses can give rise to patterns across different scales (e.g., smallscale processes giving rise to large scale patterns). This idea ofcross-scale interactions between ecological processes and dynam-ics (e.g., emergent behavior) is ingrained in complex system theory[23,24] and self-organization, a phenomenon commonly observedacross physical and biological disciplines [25]. These theories sug-gest that specific relationships between variability and scale [e.g.,scale-free, see 26] can be used to link processes operating over veryshort ranges to variability observed over very large scales. Theiruse in ecology has been controversial, in part because ecologicalsystems often fail to offer the amount of control and data availabil-ity over broad ranges of scales required to validate predictions [23].There is still much room for ecologists to discover natural phenom-ena that can reveal important ecological processes, and be detectedin existing ecological systems. Bridging the gap between theoryand reality will require the parallel development of novel modelpredictions and statistical analyses of complex spatiotemporal datato link ecological phenomena to their underlying mechanismsacross scales.

Here, we review novel insights and management strategiesthat resulted from the creative integration and simplification ofspatiotemporal variability across scales. By doing away with sum-mary statistics and embracing the full spatiotemporal variabilityof species abundance and the environment, we show how tempo-rally- and spatially-resolved methods can be used to extract sim-ple phenomena such as (i) phase and amplitude synchrony fromcomplex datasets and (ii) scaling relationships near importantecological transitions to better understand and manage naturalsystems. These signatures of non-equilibrium spatial dynamicscan be generalized to whole-ecosystem dynamics, and thus havedirect implications for ecosystem-based management strategiessuch as the design of reserve networks. Overall, validating non-equilibrium theory will thus require (1) metrics providing aone-to-one mapping between ecological processes and signaturephenomena common to both model and natural systems and(2) the use of modern statistical approaches to detect these keyphenomena in spatially- and temporally-resolved (but limited)datasets.

2. Nonlinear dynamics in space and time

Historically, mathematical ecology has identified the stability ofequilibria as one of its main questions, and tied the concept of sta-bility to steady states: constant abundance is the property of a sta-ble population or community [27]. Stability can be defined in manydifferent ways to fit specific ecological questions [28]. The local sta-bility of fixed points characterizing the equilibrium state of adynamical system has been one of the main definitions stemmingfrom theoretical studies. Using this definition, stability has beenanalyzed through the ability and rate of return to equilibrium fol-lowing a small perturbation. This analysis has pervaded our under-standing of population regulation and persistence in heterogeneousenvironments, of species coexistence, and of community assembly.Almost in parallel, non-equilibrium theories have been developed toshow how strong fluctuations, beyond small perturbations ofsteady-states [13] can lead communities to contrasting patterns ofcoexistence and exclusion [29] and of temporal dynamics [30].While early studies of stability posited that the persistence of com-munities is negatively correlated to fluctuations of abundance [27],spatial dynamics predicts that the limited movement of organismscan simultaneously increase fluctuations and promote persistenceand coexistence [31].

Non-equilibrium spatial dynamics can result from a combina-tion of intrinsic (nonlinear ecological processes) and extrinsic

F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10 3

(environment) forces disrupting steady states in mathematicalmodels [4]. Although early approaches have attempted to partitionthe relative importance of intrinsic and extrinsic disrupting forcesadditively [see 13, for a review] more recent theories have focusedinstead on their interaction to explain the maintenance ofnon-equilibrium spatial dynamics [e.g. 32].

The Rosenzweig–MacArthur predator–prey model describes thenonlinear interaction between predators and their prey andprovides a classic example of nonlinear dynamics and the resultingpositive feedbacks in non-equilibrium ecosystems:

dHdt ¼ rH 1� H

K

� �� aHP

bþHdPdt ¼ aHP

bþH � dPP

(ð1Þ

The interior equilibrium point fH�; P�g of system (1) is stable for

K < Kcrit ¼ bð2bþrdP�rbÞrðb�dP Þ

, where Kcrit corresponds to a Hopf bifurcation.

A stable limit cycle emerges and the equilibrium point becomesunstable for K P Kcrit . Because the carrying capacity K is a measureof enrichment of the ecosystem and can be used as a controlparameter driving the destabilization of the equilibrium point, thisdynamical response has been formulated as the paradox of enrich-ment [33]. More generally, the nonlinear (saturating) per capitaresponse of predator growth to prey abundance, along with thenonlinear (logistic) per capita response of prey growth to theirown abundance result in a positive feedback between prey density

and prey growth @H=dt@H > 0

� �at low prey density. This positive feed-

back destabilizes the equilibrium point and leads to a stable limitcycle. Nonlinear dynamics associated with positive feedbacks isan important driver of non-equilibrium ecological dynamics.

This model can be expanded into a minimal model of non-equi-librium spatial dynamics, with predator–prey dynamics withintwo discrete habitats and passive movement (diffusion) ofindividuals:

dHidt ¼ rHi 1� Hi

Ki

� �� aHiPi

bþHiþ DHHj � DHHi

dPidt ¼

aHiPibþHi� dPPi þ DPPj � DPPi

8<: i; j 2 f1;2g : i – j ð2Þ

This system can display multiple equilibria, some of which arespatially heterogeneous [34,35]. Non-equilibrium spatial dynamicsemerges when any spatially homogeneous limit cycle solution isunstable and is replaced with spatially heterogeneous fluctuations[36]. Non-equilibrium dynamics have important implications forpopulation persistence and species coexistence. Oscillatory dynam-ics is predicted to decrease persistence by bringing abundance awayfrom its positive steady state and leading to population extinction inthe presence of perturbations during phases of low abundance.When nonlinear feedbacks, such as those found in system (1), leadto self-sustained oscillations, limited movement between locationsin system (2) can disrupt this feedback and affect the amplitude oronset (bifurcation point) of oscillations by decoupling local growthfrom abundance [37]. This phenomenon was studied in experimen-tal and model predator–prey [38], and host-parasitoid systems,where time delays and nonlinear feedbacks predict no persistencein the absence of limited movement [39]. These studies providedone of the earliest motivations for studying spatiotemporal dynam-ics: asynchronous persistence mediated by limited movement. Inthe presence of limited parasitoid movement (explicit or implicit),increase in local host abundance can be achieved through passivemovement from other locations, which can rescue extinct local pop-ulations. Such movement can also limit the destabilizing effect ofnonlinear feedbacks by decoupling growth (e.g., density-dependentgrowth of the parasitoid) from density. These stabilizing effects canoperate as long as locations connected by movement have unequalabundances. In other words, movement must be limited such that

local oscillations are not perfectly in phase. These studies have ledto the general prediction that spatiotemporal dynamics can stabilizeand 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 betweennoise and deterministic processes [21]. Such interactions have beencentral to recent progress in resolving intrinsic and extrinsic driversof spatial dynamics in natural systems [43,44]. Spatial systems suchas (2) can be studied as stochastic systems to show howdemographic [45] and environmental [46,47] stochasticity caninteract with nonlinear ecological processes to produce andmaintain spatiotemporal heterogeneity through noise induced tran-sitions or stochastic resonance [48].

2.1. Spatiotemporal heterogeneity

The potential complexity of spatial dynamics can be distilleddown to one property: the spatial (among location) asynchronyof local temporal fluctuations. Asynchrony can be measured asthe imperfect correlation between time series, and is importantbecause it is a necessary and sufficient condition for spatiotempo-ral heterogeneity. The synchrony of fluctuations between ecologi-cal systems has received much attention and is typically ascribedto two mechanisms: movement of organisms and/or matterbetween 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 thesemechanisms, the ensemble of locations is spatially homogeneous.It is therefore the maintenance of asynchrony between populationfluctuations that defines spatiotemporal heterogeneity. The onsetof spatial dynamics can be studied through the stability analysisof the spatially homogeneous solution [36]. This method predictsthe onset of spatial dynamics for arbitrary values of couplingstrength between communities, but it is based on small perturba-tions of the homogeneous solution, and provides little informationabout the properties of the resulting spatiotemporal heterogeneity[49]. It is also possible to simplify the dynamics of multiple com-munities to that of a phase difference in the periodic fluctuationsof predators and prey between communities. The solution to sys-tem (2) for each oscillator displays a closed orbit. This periodicorbit can be described by the phase of the predator–prey systemoscillator in each community:

dhidt ¼ Xþ dGðhj � hiÞdhj

dt ¼ Xþ dGðhi � hjÞ

(ð3Þ

where hi is the periodic phase of oscillator (location) i;X is the nat-ural frequency of each oscillator, and the interaction functionGiðhj � hiÞ gives the effect of each oscillator on the phase of the otherthrough weak coupling d. This method assumes periodic oscillationsand its analysis is based on weakly coupled communities. Underthose assumptions, the stability analysis of phase differencesðhj � hiÞ can predict a great diversity of spatiotemporal regimescharacterized by phase locking, including the existence of multiplestable heterogeneous states [50]. This formalism has been appliedto system (2) where predator–prey interactions and their move-ment between two habitats provide a minimal set of coupled eco-logical oscillators [50,51].

Asynchrony between oscillators can be explained by heteroge-neous environmental (external) forcing due to differences in theparameters affecting the period of local limit cycles (natural fre-quency) [52,53], or by the weak mixing of biomass among commu-nities. Phase dynamics greatly simplifies the description of coupledsystems by reducing the explicit dynamics of abundance of eachspecies at each location (4 variables in the case of system (2)) to

Fig. 2. (A) Complex spatial dynamics of predator–prey interactions. (B) Comparingtime series of neighboring communities (see corresponding black and graytransects in (A)) reveal transient ‘defects’ in the alignment of phases betweencommunities. (C) These defects are intermittent over time and results in regularfluctuations in the frequency and amplitudes of individual time series.

4 F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10

the dynamics of their phase difference between locations. Thestability analysis of phase differences further limits their complex-ity by imposing weak spatial coupling. With such simplificationscome some limits to the applicability of phase analysis to naturalsystems. Phase stability analysis has been mostly limited to twoor few coupled ecological oscillators, and it still has to be scaledup to many locally-coupled systems [but see 54]. As we willdiscuss below, applications to ecology also lack integration ofecosystem-level dynamics involving the recycling and movementof material, including nutrients and (in)organic matter [55].Feedbacks between the recycling of nutrient from biomass andits movement between ecosystems can cause both local fluctua-tions and their asynchrony [56]. Addressing these challengeswould contribute to a general non-equilibrium theory of large-scale ecosystems and to a new ecosystem-based managementframework. Most importantly, approaches based on phase analysishave been expanded to describe both phase and amplitude [16],and a much broader range of spatiotemporal phenomena that arenot captured by phase locking. These studies can improve theuse 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 phasesand amplitudes

Phase dynamical approaches have emphasized phase lockingand studied dynamical systems by extracting the instantaneousphase of population fluctuations from time series [57,50]. How-ever, beyond phase locking and traveling waves, systems of asyn-chronous coupled oscillators can display a wide range of patternsin time and space (Fig. 2A) that are not sufficiently described byphase dynamics [49]. These patterns also involve strong changesin the amplitude of fluctuations [16], which could be importantdrivers of patterns [58]. By extracting both the phase and theamplitude of oscillations, we can thus improve our ability todiscriminate between multiple plausible causal mechanisms. Forinstance, local fluctuations in hare and lynx abundance in northernCanada follow ten-year cycles that are phase-locked over largescales (Fig. 1A; [16,59]). However, as discussed above, phase-locked oscillations alone can be associated with a number ofbottom-up (resource supply) and top-down (predator regulation)mechanisms. Hence, simplifying the spatial dynamics of thehare-lynx interaction to phase differences fails to resolve itsunderlying cause. Only when the amplitude of oscillations wasintegrated into a more complete phase-amplitude model of preda-tor–prey-resource dynamics were Blasius and colleagues able toshow that phase-locking and chaotic amplitudes together providea signature of oscillations driven by trophic interactions andcoupled through the movement of predators [16].

Many spatiotemporal series from natural systems lack anystriking regularity that can be captured by simple stationary statis-tical properties such as their phase and amplitude. For example,individual-based models can predict the maintenance of asyn-chrony and of complex spatial patterns through the limited move-ment of individuals [60–62]. This complexity requires that weexpand the range of statistics we use to characterize spatiotempo-ral series if we wish to infer their underlying drivers [44]. Oneimportant feature of spatiotemporal dynamics is the cross-scalefeedback between local processes and regional heterogeneity thateludes existing scale-dependent inference frameworks that associ-ate variability with their underlying mechanisms over correspond-ing scales [e.g., 22]. For example, predator–prey communitiesconnected by dispersal can have a stable spatially homogeneousperiodic solution [36] associated with long-term synchronybetween locations [16,50]. However, transient dynamics can revealstrong heterogeneity in amplitudes of fluctuations with important

consequences for local and regional persistence [54,63]. Moregenerally, any phase perturbation between two phase-lockedoscillators necessarily involves a transient return to one stablephase-locked solution (Fig. 2B). Across multiple communities, suchtransient dynamics means that one local perturbation can causeadditional phase perturbations to neighboring communities withinthat transient period. This cascading effect leads to local gradientsof phase differences with characteristic spatial and temporal scalesthat depend on the resilience of the phase locked solution. Thisphenomenon has been referred to as the propagation of spatialdefects, or ‘kink breeding’ [64]. These spatial defects can be associ-ated with transient changes in the amplitude and frequency ofindividual communities (Fig. 2C) and give rise to spatial patchinesswith a characteristic spatial scale [65]. During the transient returnto in-phase synchrony equilibrium between two oscillators, theiramplitude can decreases while their frequency increases(Fig. 2C). This transient pattern spreads across oscillators to formtransient patches of low-amplitude and high-frequency oscilla-tions that are progressively replaced with synchronous oscillationsat proper frequency and amplitude. Kink breeding has been stud-ied in physical systems [66], and patterns of spatial synchrony thatare compatible with kink-breeding have been detected in benthicmarine communities over continental scales [65]. The direct obser-vation of such signatures in large ecological datasets is certainly agreat challenge [67] and could offer a set of novel statistical signa-tures of non-equilibrium spatial dynamics.

F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10 5

As we attempt to validate theories of phase and amplitudedynamics, we are greatly limited by the availability of ecologicaldata that provide extensive information across broad ranges ofspatial and temporal scales. One goal of non-equilibrium ecologicaltheory is then to scale-up current mathematical models to ensem-bles of ecosystems, and develop predictions that are compatiblewith our ability to monitor natural ecosystems.

3. Inferring spatiotemporal dynamics from ecological data

As discussed above, there is a rich literature on spatial dynamicsthat has identified and classified a number of template patterns forlinking local nonlinear dynamics and limited movement to large-scale heterogeneity. However, this non-equilibrium theory hasyet to yield methods for linking these complex spatiotemporalpatterns to their underlying mechanisms.

3.1. Signatures of nonlinear spatial dynamics

Template spatiotemporal patterns are those that have beenadopted for the validation of spatial dynamical models becausetheir regularity can be inferred with relatively simple statisticalmethods. They include traveling waves [68,69], scale invariantpatch dynamics [26], and stationary spatial periodicity [e.g. bandsand spots in arid vegetation; 15]. These templates have been usedto infer the nature of the processes governing ecosystems: Regularpatterns are linked to intrinsic nonlinear spatial dynamics andirregular patterns to extrinsic environmental perturbations. Forexample, ecologists are trying to resolve biotic and abiotic causesof individual aggregation into discrete patches. The size distribu-tion of those patches can be used as a signature of local bioticantagonistic processes such as predator–prey, host-parasite or dis-turbance-recovery interactions [25]. One prediction borrowedfrom critical phenomena theories is that local antagonistic interac-tions can lead to scale-invariant distribution FðsÞ of patch size s,with FðsÞ / s�b. This prediction was tested in tropical forests char-acterized by (antagonistic) interactions between tree colonizationand disturbance by wind or fire. In these systems, scale-invariancein the size distribution of tree patches was used as a signature oflocal and nonlinear biotic interactions driving the propagation offire and windthrows between neighboring trees, and the dispersalof seeds from adult trees into nearby disturbed areas [70]. Thisinterpretation of scale invariant patch size distribution is ecologi-cally important because it means that trees are dynamically con-nected across whole forests through purely localized ecologicalprocesses [25]. Characteristic scales rather than scale invariancecan also be used as signature of the importance of local nonlinearprocesses for spatiotemporal dynamics. Traveling waves are forexample observed in vole populations and characterized by theirfrequency and amplitudes [69]. Ecologists have applied theoriesof diffusive (Turing) instabilities that predict static or dynamic pat-terns with characteristic spatial and temporal scales. These theo-ries predict the onset of spatial or spatiotemporal heterogeneitywhen local ecological processes involve coupled positive and neg-ative feedbacks that operate at different spatial scales [71,15]. Sta-tic and periodic aggregation of vegetation in arid systems resultsfrom such feedbacks driven by short range positive effect of aggre-gation on water retention, and by long range negative effect ofaggregation through below-ground competition for nutrients[15]. Here, the emergence of diffusive instabilities, periodic spatio-temporal variability, and other regular patterns are associated withlocalized interactions, whereas irregular patterns are associatedwith environmental stochasticity. In oscillatory systems wherepatterns are dynamic, synchrony and phase locking of oscillationsacross locations can be used to assess the regularity of spatial

dynamics. However, patterns that are perceived as regular (asdefined from the examples above) are relatively rare in ecologicalsystems.

3.2. Irregular spatiotemporal heterogeneity

Studies of complexity in ecology have greatly expanded the setmethods that can be used to detect statistical regularity emergingfrom biological regulating mechanisms, from regular spatiotempo-ral periodicity of abundance to the regular scaling (i.e. power laws)of spatiotemporal variance. However, ecologists are still facing thechallenge of detecting such regularity in natural systems, or withour limited ability to ‘read the signs’ [72]. For example, phase lock-ing – a clear manifestation of spatiotemporal patterns – has beenshown to emerge from 2-patch systems in relation to a separationof temporal scales between predators and their prey [50], and totime to extinction (Lyapunov exponent) relative to dispersal [58]in trophic communities. These predictions can be tested in exper-imental systems where these parameters can be estimated, but thephenomenon itself – stable phase locking – is rarely observed innatural systems. When it is observed, a number of alternativeinterpretations exist to explain its occurrence. In such cases, infer-ence of the underlying mechanisms can be developed throughother properties of spatial dynamics (amplitude in addition tophase) that can be measured from existing spatiotemporal series.For instance, phase locking accompanied by chaotic amplitudescan be used as a signature of coupled dynamics in tri-trophic foodchains [16]. Similarly, the rate at which synchrony decays with dis-tance between ecosystems can help identify (i) the relative impor-tance of biological regulating mechanisms and environmentaldrivers and (ii) the scale of dispersal [65]. The relationship betweenangular velocity, phase, amplitude and net dispersal also offers arich set of predictions to explain dispersal-mediated persistenceof communities with antagonistic interactions [73]. These exam-ples illustrate how ecologists can develop new theory to improveour ability to infer intrinsic and extrinsic causes of non-equilibriumdynamics in natural communities.

Ultimately, distinguishing extrinsic and intrinsic causes ofpopulation fluctuations in the real world is likely to require bothtemporally-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 driversof population abundance be identified in a simple metapopulationmodel characterized by equilibrium vs. non-equilibrium localdynamics and constant vs. stochastic environmentally-mediateddispersal (Figs. 3 and 4). In this case, wavelet analysis of the localabundance of a representative subpopulation shows that the magni-tude, duration and periodicity of temporal fluctuations in modelscharacterized by non-equilibrium dynamics and constant dispersalare quite different from those observed in models characterized byequilibrium local dynamics and stochastic dispersal (Fig. 4a and dvs. b and e). However, the magnitude, duration and periodicity oftemporal fluctuations are largely identical in models characterizedby non-equilibrium local dynamics and either constant or stochasticdispersal (Fig. 4a and d vs. c and f). Hence, although wavelet analysescan distinguish between equilibrium and non-equilibrium localdynamics, they cannot be used to infer the role of constant vs. sto-chastic dispersal. Conversely, analyses of spatial autocorrelation inabundance can be used to identify models characterized by non-equilibrium local dynamics and constant vs. stochastic dispersalkernels (Fig. 3b vs. d), but not models characterized by equilibriumdynamics and constant vs. stochastic dispersal kernels (Fig. 3c andd). Specifically, models characterized by non-equilibrium localdynamics and constant dispersal exhibit nonlinear and non-station-ary patterns of autocorrelation (Fig. 3b), whereas those character-ized by stochastic dispersal exhibit largely linear and stationary

6 F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10

patterns of autocorrelation (Fig. 3d). However, models characterizedby stochastic dispersal and either equilibrium or non-equilibriumlocal dynamics exhibit similar linear and stationary patterns ofautocorrelation (Fig. 3c and d). Hence, only by employing bothspatial and temporal analyses can the source of population dynam-ics at local and regional scales be fully resolved. These examplesillustrate the potential power in combining modern spatial and tem-poral analyses (e.g., wavelet methods, spatial autocorrelation andsynchrony) to study how dispersal, nonlinear ecological processesand environmental stochasticity interact to shape patterns ofpopulation fluctuations in models [74,75,46,76] and nature [43,44].

4. The implications of non-equilibrium dynamics for reservedesign: a function for pattern formation

In marine systems, fisheries have long been managed as equilib-rium and single-species systems. This view has led to the wide-spread adoption of maximum sustainable yield assuming

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nst

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dip

ersa

lS

patia

l aut

ocor

rela

tion

a

Non−equilibrium local dynamics

b

0 25 50 75 100 125−1

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chas

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dip

ersa

lS

patia

l aut

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c

0 25 50 75 100 125Lag distance (number of sites)

d

Fig. 3. Annual spatial autocorrelation of abundance for metapopulation modelsundergoing local equilibrium (a and c) and non-equilibrium (b and d) dynamicswith either constant (a and b) or stochastic (c and d) dispersal for 2000 years. Theannual patterns of spatial autocorrelation are represented in gray and the mean isrepresented in red. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

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Non−equilibrium local dynamicsand constant dispersal

Lo

cal t

imes

erie

sA

bund

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b

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Wav

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an

alys

isP

erio

d

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0 500 1000 1500 2000

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163264

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Fig. 4. Applying wavelet analysis to local model abundance time series (a–c) to testmagnitude, duration and periodicity of local population fluctuations revealed by waveinteractions) and extrinsic (e.g., environmentally-mediated stochastic dispersal) driversto high (low) variability in the time series. Black contours indicate statistically-significanthat lie outside the COI are subject to edge effects. These results are based on the same simthis figure legend, the reader is referred to the web version of this article.)

equilibrium and homogeneously distributed populations. Manag-ers have now embraced ecosystem-based management, whichemphasizes the non-equilibrium and heterogeneous nature ofpopulation dynamics and the interdependence of biotic and abioticprocesses across scales. This shift has lead to major developmentsin management strategies such as marine protected areas, whichconstitutes an additional anthropogenic driver of spatiotemporalheterogeneity.

Ecological reserves, areas protected from all extractive anddestructive activities, are increasingly being heralded as importantcomponents of a broader strategy for both conserving and manag-ing natural systems [77–80]. There is growing recognition that net-works of interconnected reserves can be even more effective thanindividual reserves of the same size because they (1) distribute thesocietal costs more evenly over space, (2) protect species over alarger portion of their range, (3) provide spatial redundancy thatreduces the effects of local or spatially-autocorrelated catastrophes(e.g., toxic spills, wildfires), and (4) promote the recovery ofindividual reserves via regional subsidies [77,81,78,80]. Becausemost of these network benefits require connectivity between indi-vidual reserves, theory based on equilibrium models has longextolled the benefits of building strongly interconnected reservenetworks [82,83].

However, non-equilibrium theory has established that connec-tivity is not always conducive to positive outcomes such asincreased resilience and persistence. Instead, connectivity is moreof a double-edged sword that can promote persistence by allowingregional subsidies to rescue local populations, but also increaseglobal extinction risk by spatially synchronizing the dynamics ofinterconnected populations [84,85]. Hence, in addition (and some-times contrary) to current guidelines, reserve networks establishedfor populations undergoing non-equilibrium dynamics will have toabide by design principles that seek to avoid the synchronizingeffect of connectivity and its negative impact on persistence[86,87].

Limiting connectivity between reserve networks not onlyreduces the risk of global extinction, but it can also facilitate themanagement of trophically- and spatially-coupled systems under-going non-equilibrium dynamics [87]. Indeed, in equilibrium sys-tems, reserve network design typically involves trade-offs withrespect to the size and spacing of reserves in order to achieve con-servation vs. commercial objectives [88]. For instance, achieving

local dynamicsstic dispersal

Non−equilibrium local dynamicsand stochastic dispersal

c

ime000 1500 2000

Time

f

0 500 1000 1500 2000

predictions about the source of population fluctuations in metapopulations. Thelet analysis (d–f) can help differentiate between intrinsic (e.g., nonlinear species

of population fluctuations. (d–f) Regions depicted by warm (cold) colors correspondt variability and the black dashed line represents the cone of influence (COI). Valuesulations that were analyzed in Fig. 3. (For interpretation of the references to color in

Time

Nutr

ientsto

ck

(A) Closed meta-ecosystem

(B) Low dispersal, dN=2

(C) High dispersal, dN=4

Pi

Hi

Ni

Pj

Hj

Nj

dP

dH

dN

P

H

P

H

fH

fP

fH

fP

Ni PiHi Nj PjHj

F. Guichard, T.C. Gouhier / Mathematical Biosciences 255 (2014) 1–10 7

conservation goals such as maximizing abundance typicallyrequires large and fully connected reserves, whereas achievingcommercial goals such as maximizing yield typically requiressmaller and partially connected reserves in order to promote the‘spillover’ of resources into unprotected areas where they can beharvested [88]. Additionally, there are community-level trade-offswith respect to the optimal size and spacing of reserves: dominantcompetitors and predators benefit from small and aggregatedreserves that maximize connectivity and its positive effects onsingle species growth. In contrast, weaker competitors and preyspecies benefit from large and isolated reserves that minimize con-nectivity and release those species from competition and predation[89,87]. Altering connectivity by varying the size and spacing ofreserve networks can only dampen this community-level trade-off or tilt the balance towards one set of species (dominantcompetitors and predators vs. weaker competitors and prey) [87].

In non-equilibrium systems, these trade-offs disappear becausecross-scale ecological feedbacks between local population fluctua-tions and regional dispersal result in a separation of scales betweenecological processes and patterns (distribution of abundance),which decouples the intraspecific benefits of connectivity fromthe interspecific costs of trophic cascades. Networks that exploitthis separation of scales by using the extent of patterns (i.e., theextent of patchiness or spatial autocorrelation) as the size andspacing of reserves are thus able to maximize the abundance, yield,and persistence of trophically-coupled species [87]. Overall, thissuggests a function for pattern formation: the optimal manage-ment of trophically- and spatially-coupled communities withrespect to both conservation and commercial objectives.

Time

Nutr

ientsto

ck

Fig. 5. Meta-ecosystem dynamics illustrating emerging patterns of spatial syn-chrony. (A) Minimal meta-ecosystem model corresponding to system (4). (B–C)Time series of nutrient stock contained in each ecosystem compartment (nutrientsN, 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 ofnutrient movement is increased from dN ¼ 2 (B) to dN ¼ 4 (C). High nutrientmovement in (C) leads to in-phase synchrony of nutrients while the producer andconsumer become phase-locked with a period corresponding twice the period ofnutrient fluctuations. Adapted from [92]. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

5. Towards a non-equilibrium meta-ecosystem theory of coastalmanagement

A non-equilibrium theory of spatial dynamics certainly needs toconsider the recycling and movement of organic and inorganicmatter. The theoretical framework of meta-ecosystems predictsecological consequences of limited flows of organic and inorganicmatter across space (Fig. 5A). This emerging ecological theory[90] and its application to natural systems will provide a good casestudy to build and validate a theory of non-equilibrium spatialdynamics. A simple meta-ecosystem model can be formulated asan extension of the predator–prey metacommunity (system (2)),where biomass is recycled into nutrients, and where primaryproducers (the prey) take-up nutrients for growth:

dNidt ¼ dHHi þ dPPi � aHiNi

bþNiþ DNNj � DNNi

dHidt ¼

aHiNibþNi� aHiPi

bþHi� dHHi þ DHHj � DHHi

dPidt ¼

aHiPibþHi� dPPi þ DPPj � DPPi

8>>><>>>:

i; j 2 f1;2g : i – j ð4Þ

System (4) describes a closed meta-ecosystem with no externalinput or output of nutrients. Recent theory based on similar mod-els suggests that flows of organic and inorganic matter can deeplytransform the structure and stability of communities [91], generatenonlinear feedbacks and explain spatiotemporal heterogeneity[92]. Fluctuations in the concentration of (in)organic matter cansimilarly lead to patterns of synchrony among ecosystem compart-ments (e.g. between nutrient concentration and primary producerbiomass) and among locations. Such patterns have been predictedfrom simple 2-patch meta-ecosystem models [Fig. 5B–C; 92], andrecently extended to larger irregular networks [93]. These predic-tions need to be extended to large meta-ecosystems and matchedwith long-term monitoring data of biogeochemical cycles. Thesetheoretical studies also echo recent calls for whole-ecosystemmanagement of natural resources that are embedded in complexinteraction networks and flows of (in)organic matter.

Coastal ecosystems are among the most productive and threa-tened by human activities and climate change [94], and thus primetargets for ecosystem-based management approaches [95]. Theyshow strong fluctuations in the distribution of species and in eco-system functions (e.g. nutrient recycling and primary production).Temperate coastal ecosystems support large populations of benthicinvertebrates 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 cyclesin the North-East Pacific [98]. At the ecosystem level, fluctuationsin (in)organic matter has also been documented in nearshorewaters [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 largespatial scales. These simplifying assumptions contained withinecological theories are in sharp contrast with our detailedunderstanding of transport and cycles of nutrient and organic mat-ters in oceans. Coastal systems strongly depend on couplingbetween benthic and pelagic compartments [104,99,105], andspatial fluxes are therefore controlled across a very broad rangeof spatial and temporal scales [106], from fast advective mixingand stratification [107] to storage over geological time scales.Oceanographers have a long history of spatially resolved trophicmodels even including cycling of matter through recycling orexcretion [108]. While many coupled bio-physical models havebeen used to make quantitative predictions [109], many studieshave adopted a more heuristic approach and contributed to ameta-ecosystem theory for marine systems [108]. For example,spatially explicit (1D vertical) Nutrient–Phytoplankton–Zooplank-ton (NPZ) models [110] where diffusion couples the dynamics ofnearby nutrient stock, can induce stable profiles as well as oscilla-tory dynamical trajectories that become vertically phase-locked forlarge mixing levels [110]. Given the accumulating evidence ofstrong meta-ecosystem dynamics in coastal ecosystems, one taskis to test for the existence and implications of nonlinear feedbacksover regional scales that are targeted by managers and policymakers. The sudden shifts in the state of regional fisheriesdocumented over the last decades [111] certainly suggest therelevance and urgency of this task.

6. Conclusion

Natural systems are constantly changing as they face externalperturbations, but also through the processes that are responsiblefor their very persistence. Yet, the balance of nature is still a strongparadigm in ecology, and the steady state is still a dominant targetfor conservation biologists and managers. Great progress has beenachieved towards a general non-equilibrium theories of ecologicalsystems that can be applied to natural systems across scales.However, our ability to understand dynamical and interconnectedecosystems remains limited by our reliance on inferential toolsthat associate regular patterns with intrinsic processes andirregular patterns with extrinsic processes. We highlighted recentdevelopment in non-equilibrium theories that contributed toexpanding the range of phenomena we can measure in natural sys-tems and use as signatures of underlying mechanisms. Spatial syn-chrony has been central to recent development and application ofnon-equilibrium spatial dynamics. However, understanding thecauses of synchrony, or lack thereof, between populations is stillchallenging. We suggest that this task can be facilitated by testingfor temporal and correlated shifts in phase and amplitude differ-ences between time series, which assesses the transient andcross-scale response of local populations to limited dispersal as acause of spatiotemporal heterogeneity. In order to apply to naturalsystems, ecological theory should certainly be able to describeecological systems of increasing complexity. But most importantly,it must develop predictive frameworks built around metrics thatcan reduce this complexity and provide simple signatures ofunderlying causes of spatial dynamics. These metrics should inturn inform and optimize the design and implementation oflong-term ecosystem monitoring programs. This is importantbecause non-equilibrium metacommunity and meta-ecosystemtheories have the potential to directly affect management andpolicies such as reserve networks that are still largely based onequilibrium models. The complexity of predictions rather than thatof models themselves is currently limiting the application of thesetheories to natural and managed systems.

Acknowledgement

F. Guichard was supported by the Natural Science and Engineer-ing Research Council of Canada through the Discovery Grantprogram.

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