Accepted Manuscript Review Non-equilibrium spatial dynamics of ecosystems Frederic Guichard, Tarik C. Gouhier PII: DOI: Reference:

S0025-5564(14)00119-9 http://dx.doi.org/10.1016/j.mbs.2014.06.013 MBS 7499

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Mathematical Biosciences

Please cite this article as: F. Guichard, T.C. Gouhier, Non-equilibrium spatial dynamics of ecosystems, Mathematical Biosciences (2014), doi: http://dx.doi.org/10.1016/j.mbs.2014.06.013

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Non-equilibrium spatial dynamics of ecosystems

Frederic Guicharda,∗, Tarik C. Gouhierb a Department

of Biology, McGill University, 1205 Docteur Peneld, Montreal, Quebec H3A 1B1, Canada b Marine Science Center, Northeastern University, 430 Nahant Road, Nahant, MA 01908, USA

Abstract

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 ∗

Corresponding author

Email addresses: [email protected] [email protected] (Tarik C. Gouhier)

Preprint submitted to Mathematical Biosciences

(Frederic Guichard),

June 25, 2014

spatial synchrony as a central phenomenon for its inference in natural systems. 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 signicant 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 signicant departures from spatial or temporal steady states in the absence of extrinsic environmental variability [3, 4]. The impact of those nonlinear feedbacks on (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 long2

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 dene 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]. 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 identied a set of signals such as traveling waves of predators and prey 3

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 patternsevenly spaced bands or spots, traveling wavesthat 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 uctuations. Regularity is here a matter of perception and its working denition 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, 18, 19, 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 rst 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 scaledependent decomposition of variability is fundamentally limited in its ability to show how processes can give rise to patterns across dierent scales (e.g., small scale processes giving rise to large scale patterns). This idea of crossscale interactions between ecological processes and dynamics (e.g., emergent behavior) is ingrained in complex system theory [23, 24] and self-organization, 4

a phenomenon commonly observed across physical and biological disciplines [25]. These theories suggest that specic 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 oer 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 simplication 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 non-equilibrium theory will thus require (1) metrics providing a one-to-one mapping between ecological processes and signature phenomena common to 5

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. Nonlinear dynamics in space and time

Historically, mathematical ecology has identied 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 dened in many dierent ways to t specic ecological questions [28]. The local stability of xed points characterizing the equilibrium state of a dynamical system has been one of the main denitions stemming from theoretical studies. Using this denition, 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 uctuations, 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 uctuations of abundance [27], spatial dynamics predicts that the limited movement of organisms can simultaneously increase uctuations and promote persistence and coexistence [31]. Non-equilibrium spatial dynamics can result from a combination of intrinsic (nonlinear ecological processes) and extrinsic (environment) forces 6

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 = rH(1 − dt   dP = dt

aHP b+H

H ) K

−

aHP b+H

(1)

− δP P

The interior equilibrium point {H ∗ ; P ∗ } of system (1) is stable for K
0) at ∂H low prey density. This positive feedback 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. 7

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 (diusion) of individuals:

   dHi = rHi (1 − dt   dPi = dt

aHi Pi b+Hi

Hi ) Ki

−

aHi Pi b+Hi

+ DH Hj − DH Hi

i, j ∈ {1, 2} : i 6= j

(2)

− δP Pi + DP Pj − DP Pi

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 uctuations [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 selfsustained oscillations, limited movement between locations in system (2) can disrupt this feedback and aect 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, 8

which can rescue extinct local populations. Such movement can also limit the destabilizing eect of nonlinear feedbacks by decoupling growth (e.g., density-dependent growth of the parasitoid) from density. These stabilizing eects can operate as long as locations connected by movement have unequal abundances. In other words, movement must be limited such that 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 uctuating ecological systems [37]. This prediction has been extended to 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 resoving intrinsic and extrinsic drivers of spaptial 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]. Spatiotemporal heterogeneity

The potential complexity of spatial dynamics can be distilled down to one property: the spatial (among location) asynchrony of local temporal uctuations. Asynchrony can be measured as the imperfect correlation between time series, and is important because it is a necessary and sucient condition for spatiotemporal heterogeneity. The synchrony of uctuations between ecological systems has received much attention and is typically ascribed to two mechanisms: movement of organisms and/or matter between locations, 9

including the movement of a mobile predator, and correlated environmental uctuations across locations (i.e. Moran eect) [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 uctuations that denes 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 dierence in the periodic uctuations 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:

   dθi = Ω + δG(θj − θi ) dt   dθj = Ω + δG(θi − θj ) dt

(3)

Where θi is the periodic phase of oscillator (location) i, Ω is the natural frequency of each oscillator, and the interaction function Gi (θj − θi ) gives the eect of each oscillator on the phase of the other through weak coupling

δ . This method assumes periodic oscillations and its analysis is based on weakly coupled communities. Under those assumptions, the stability analysis of phase dierences (θj −θi ) can predict a great diversity of spatiotemporal regimes characterized by phase locking, including the existence of multiple 10

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 dierences in the parameters aecting the period of local limit cycles (natural frequency)[52, 53], or by the weak mixing of biomass among communities. Phase dynamics greatly simplies 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 the dynamics of their phase dierence between locations. The stability analysis of phase dierences further limits their complexity by imposing weak spatial coupling. With such simplications 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 uctuations and their asynchrony [56]. Addressing these challenges would contribute to a general non-equilibrium theory of large-scale 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 11

spatial dynamics as a signature of local ecological processes, even for systems showing no apparent regularity in uctuations. 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 uctuations 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 suciently described by phase dynamics [49]. These patterns also involve strong changes in the amplitude of uctuations [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 uctuations 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, phase-locked 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 dierences 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 in-

teractions and coupled through the movement of predators [16]. Many spatiotemporal series from natural systems lack any striking regu12

larity 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, 61, 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 uctuations with important 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 neighbouring communities within that transient period. This cascading eect leads to local gradients of phase dierences 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 13

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 oer a set of novel statistical signatures of non-equilibrium spatial dynamics. 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. Inferring spatiotemporal dynamics from ecological data

As discussed above, there is a rich literature on spatial dynamics that has identied and classied a number of template patterns for linking local nonlinear dynamics and limited movement to large-scale heterogeneity. However, this non-equilibrium theory has yet to yield methods for linking these complex spatiotemporal patterns to their underlying mechanisms.

14

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, hostparasite or disturbance-recovery interactions [25]. One prediction borrowed from critical phenomena theories is that local antagonistic interactions can lead to scale-invariant distribution F (s) of patch size s, with F (s) ∝ s−β . This prediction was tested in tropical forests characterized by (antagonistic) interactions between tree colonization and disturbance by wind or re. 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 re 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 15

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 diusive (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 dierent spatial scales [71, 15]. Static and periodic aggregation of vegetation in arid systems results from such feedbacks driven by short range positive eect of aggregation on water retention, and by long range negative eect of aggregation through below-ground competition for nutrients [15]. Here, the emergence of diusive 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 dynamics. However, patterns that are perceived as regular (as dened from the examples above) are relatively rare in ecological systems. 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 ex16

ample, phase lockinga clear manifestation of spatiotemporal patternshas 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 itselfstable phase lockingis 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 oers 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 uctuations in the real world is likely to require both temporally-replicated and spatially-explicit data (Figs. 3, 4). For instance, only when spatial (e.g., autocorrelation or synchrony) and temporal (e.g., wavelet) analyses 17

are combined can the drivers of population abundance be identied in a simple metapopulation model characterized by equilibrium vs. non-equilibrium local dynamics and constant vs. stochastic environmentally-mediated dispersal (Figs. 3, 4). In this case, wavelet analysis of the local abundance of a representative subpopulation shows that the magnitude, duration and periodicity of temporal uctuations in models characterized by non-equilibrium dynamics and constant dispersal are quite dierent from those observed in models characterized by equilibrium local dynamics and stochastic dispersal (Fig. 4a,d vs. b,e). However, the magnitude, duration and periodicity of temporal uctuations are largely identical in models characterized by nonequilibrium local dynamics and either constant or stochastic dispersal (Fig. 4a,d vs. c,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 non-equilibrium 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, d). Specically, 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 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,d). Hence, only by employing both spatial and temporal analyses 18

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 uctuations in models [74, 75, 46, 76] and nature [43, 44]. The implications of non-equilibrium dynamics for reserve design: a function for pattern formation

In marine systems, sheries have long been managed as equilibrium and single-species systems. This view has led to the widespread adoption of maximum sustainable yield assuming 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, 78, 79, 80]. There is growing recognition that networks of interconnected reserves can be even more eective 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 19

the eects of local or spatially-autocorrelated catastrophes (e.g., toxic spills, wildres), and (4) promote the recovery of individual reserves via regional subsidies [77, 81, 78, 80]. Because most of these network benets require connectivity between individual reserves, theory based on equilibrium models has long extolled the benets 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 eect 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 trophicallyand spatially-coupled systems undergoing non-equilibrium dynamics [87]. Indeed, in equilibrium systems, reserve network design typically involves tradeos with respect to the size and spacing of reserves in order to achieve conservation versus commercial objectives [88]. For instance, achieving 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 20

promote the `spillover' of resources into unprotected areas where they can be harvested [88]. Additionally, there are community-level trade-os with respect to the optimal size and spacing of reserves: dominant competitors and predators benet from small and aggregated reserves that maximize connectivity and its positive eects on single species growth. In contrast, weaker competitors and prey species benet 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 trade-o 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-os disappear because cross-scale ecological feedbacks between local population uctuations and regional dispersal result in a separation of scales between ecological processes and patterns (distribution of abundance), which decouples the intraspecic benets of connectivity from the interspecic 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.

21

Towards a non-equilibrium meta-ecosystem theory of coastal management

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 ows 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:

  dNi  i Ni  = δH Hi + δP Pi − αH + DN Nj − DN Ni  dt β+Ni   dHi i Ni i Pi = αH − aH − δH Hi + DH Hj − DH Hi dt β+Ni b+Hi      dPi i Pi  = aH − δP Pi + DP Pj − DP Pi dt b+Hi

i, j ∈ {1, 2} : i 6= j (4)

System 4 describes a closed meta-ecosystem with no external input or output of nutrients. Recent theory based on similar models suggests that ows 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 22

meta-ecosystem models [Fig 5B-C; 92], and recently extended to larger irregular networks [93]. These predictions need to be extended to large metaecosystems 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 ows of (in)organic matter. Coastal ecosystems are among the most productive and threatened by human activities and climate change [94], and thus prime targets for ecosystembased management approaches [95]. They show strong uctuations 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 uctuate by more than 60% of their mean abundance between years in the North-East Pacic [96] and North West Atlantic [97]. Over longer temporal scales, Dungeness crab population abundance follow a 10-11 year cycles in the North-East Pacic [98]. At the ecosystem level, uctuations in (in)organic matter has also been documented in nearshore waters [99, 100], and related to species interactions [101, 102, 103]. 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 uxes are therefore controlled across a very broad range of spatial and temporal scales [106], from fast advective mixing and stratication 23

[107] to storage over geological time scales. Oceanographers have a long history of spatially resolved trophic models (Steele and Henderson 1992b), 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 diusion couples the dynamics of nearby nutrient stock, can induce stable proles 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 sheries documented over the last decades [111] certainly suggest the relevance and urgency of this task. 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 inferen24

tial 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 dierences 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 aect 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.

25

Acknowledgements

F. Guichard was supported by the Natural Science and Engineering Research Council of Canada through the Discovery Grant program. References

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40

List of Figures

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 dierence (measured as a phase angle) from south-east to north-west of study area, and provide evidence of a traveling wave. . . . . . . . . . . . . . . . . . . . 44

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 uctuations in the frequency and amplitudes of individual time series. (A) Adapted from ; (B) and (C) unpublished . . . . . . 45

3

Annual spatial autocorrelation of abundance for metapopulation models undergoing local equilibrium (a, c) and nonequilibrium (b, d) dynamics with either constant (a, b) or stochastic (c, d) dispersal for 2000 years. The annual patterns of spatial autocorrelation are represented in gray and the mean is represented in red . . . . . . . . . . . . . . . . . . . . . . . 46

41

4

Applying wavelet analysis to local model abundance time series (a-c) to test predictions about the source of population uctuations in metapopulations. The magnitude, duration and periodicity of local population uctuations revealed by wavelet analysis (d-f) can help dierentiate between intrinsic (e.g., nonlinear species interactions) and extrinsic (e.g., environmentally-mediated stochastic dispersal) drivers of population uctuations. (d-f) Regions depicted by warm (cold) colors correspond to high (low) variability in the time series. Black contours indicate statistically-signicant variability and the black dashed line represents the cone of inuence (COI). Values that lie outside the COI are subject to edge eects. These results are based on the same simulations that were analyzed in Fig. 3 . . . . . . . . . . . . . . . . . . . . . . . . . 47

42

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 = 2 (B) to dN = 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 uctuations. Adapted from [92]. . . . . . . . . . . . . . . . . . . . . . . . . 48

43

Figure 1:

44

Figure 2:

45

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) Figure 3:

46

Local timeseries Abundance Wavelet analysis Period

Non−equilibrium local dynamics Equilibrium local dynamics Non−equilibrium local dynamics and constant dispersal and stochastic dispersal and stochastic dispersal 1 b c a 0.8 0.6 0.4 0.2 0 4 d 8 16 32 64 128 256 512 0 500

e

1000 Time

1500

2000 0

f

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Figure 4:

47

1000 Time

1500

2000 0

500

1000 Time

1500

2000

(A) Closed meta-ecosystem δP

Pi

Pj

δP

fP

fP δH

dP

Hi

dH

Hj

δH

fH

fH

Ni

dN

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(B) Low dispersal, dN=2 Hi

Pi

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Ni

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Time (C) High dispersal, dN=4

Time

48

Figure 5:

Nj

Hj

Pj

• • • •  

Ecological  variability  across  scales  eludes  equilibrium  theories   We  lack  predictions  of  spatial  dynamics  that  can  be  validated  in  natural   systems   Spatial  synchrony  can  simplify  the  description  of  complex  spatial  dynamics   Recycling  and  transport  of  matter  are  key  to  non-­‐equilibrium  ecological   theory