Regime Shifts Under Forcing of Non-Stationary Attractors: Conceptual Model and Case Studies in...

Journal of Contaminant Hydrology xxx (2014) xxx–xxx

CONHYD-03042; No of Pages 11

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Journal of Contaminant Hydrology

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Regime shifts under forcing of non-stationary attractors:Conceptual model and case studies in hydrologic systems

Jeryang Park a,b,⁎, P. Suresh C. Rao b,c

a School of Urban and Civil Engineering, Hongik University, Seoul, 121–784, South Koreab School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USAc Agronomy Department, Purdue University, West Lafayette, IN 47907, USA

a r t i c l e i n f o

⁎ Corresponding author at: School of Urban anHongik University, Seoul, 121–784, South Korea. Tefax: +82 2 322 1244.

E-mail address: [email protected] (J. Park).

http://dx.doi.org/10.1016/j.jconhyd.2014.08.0050169-7722/© 2014 Elsevier B.V. All rights reserved.

Please cite this article as: Park, J., Rao, P.S.Cstudies in hydrologic systems, J. Contam. H

a b s t r a c t

Article history:Received 11 May 2014Received in revised form 16 August 2014Accepted 19 August 2014Available online xxxx

We present here a conceptual model and analysis of complex systems using hypothetical cases ofregime shifts resulting from temporal non-stationarity in attractor strengths, and then presentselected published cases to illustrate such regime shifts in hydrologic systems (shallow aquaticecosystems; water table shifts; soil salinization). Complex systems are dynamic and can exist intwo ormore stable states (or regimes). Temporal variations in state variables occur in response tofluctuations in external forcing, which are modulated by interactions among internal processes.Combined effects of external forcing and non-stationary strengths of alternative attractors canlead to shifts from original to alternate regimes. In systems with bi-stable states, when thestrengths of two competing attractors are constant in time, or are non-stationary but change in alinear fashion, regime shifts are found to be temporally stationary and only controlled by thecharacteristics of the external forcing. However, when attractor strengths change in time non-linearly or vary stochastically, regime shifts in complex systems are characterized by non-stationary probability density functions (pdfs). We briefly discuss implications and challenges toprediction and management of hydrologic complex systems.

© 2014 Elsevier B.V. All rights reserved.

Keywords:ResilienceTipping pointsComplex systemsMulti-stabilityNon-stationarity

1. Introduction

A regime is a region in state-space in which a complexsystem retains the same attributes such as structure, functions,and feedbacks. An attractor refers to a stable equilibrium stateand the basin of attraction, which is equivalent to a regime,defines the state-space region in which initial conditions willtend to move towards the attractor (Walker et al., 2004).Complex systems often possess more than one attractorbetween which repellers (or unstable equilibrium state) existas a boundary for the basins of attraction (Scheffer, 2009).Regime shift occurs when the system crosses such repellers.Dynamics of the system will be governed by completely

d Civil Engineering,l.: +82 2 320 1639;

., Regime shifts under forydrol. (2014), http://dx.

different processes in an alternative regime (Scheffer et al.,2001). Regime shifts are often driven by external stochasticevents or drivers. Stochastic events, characterized by fastvariables, cause regime shifts by pushing the complex systemtoward alternate regime across the repeller. External drivers,characterized by slow variables, also can cause regime shifts bygradually changing the size of the regimes (Walker et al., 2012).When an external driver reaches a critical value, which isknown as bifurcation point, the original regime disappears andonly the alternative regime remains (Lenton, 2013; May, 1977;Scheffer et al., 2001). Thus, the complex system ends up beingin an alternate regime because that is the only regime existingunder the new conditions as changed by the external driver(s).Here, both external drivers and stochastic events are referred toas external forcing(s).

Considering groundwater quality as state variable, clean(potable) and contaminated (non-potable) groundwater aretwo bi-stable states (regimes) of an aquifer. Successive loads of

cing of non-stationary attractors: Conceptual model and casedoi.org/10.1016/j.jconhyd.2014.08.005

http://dx.doi.org/10.1016/j.jconhyd.2014.08.005

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2 J. Park, P.S.C. Rao / Journal of Contaminant Hydrology xxx (2014) xxx–xxx

contaminants can lead to a regime shift, when the contaminantloading rates exceed the natural attenuation rate. Naturalattenuation capacity serves as the negative feedback tomaintain the groundwater quality within the original potableregime. Once in a contaminated non-potable regime, aquiferbiogeochemical conditions change (e.g., redox potential de-creases due to lack of oxygen), and different negative feedbacksnow dominate, and the aquifer persists in the contaminatedregime, even after some management practices are introducedto lower the contaminant loading rate below the level thatcontributed to the regime shift. The contaminated aquiferreturns to the potable regime only when the biogeochemicalconditions (negative feedbacks) are restored.

While external forcings are often direct causes for regimeshifts, changes in internal system dynamics, which forms thesize and shape of regimes, can also bring about regime shifts(Hastings and Wysham, 2010; Scheffer et al., 2001). A regimecan be defined by two geometric features of the attractionbasin: (1) width (or latitude), which characterizes how far asystem state is allowed tomove before it crosses a repeller; and(2) depth (or resistance), which characterizes how difficult isfor the system state to bemoved from its attractor, or how fast adisturbed system state can return to the attractor (Walker et al.,2004). In the following discussion, we integrate these twocharacteristics of a regime in a single term, attractor strength.

Typically, two competing forcings are identified in human-nature coupled complex systems: (1) persistent natural forcing(e.g., hydro-climatic forcing) under which the system evolvesthrough self-reorganization; and (2) anthropogenic forcing(e.g., land-use changes) that involves utilization of naturalresources to maximize socio-economic services but with a lossof other ecosystem services. Depending on the relative strengthof forcing, state variables of a complex systemmay remain in thecurrent regime, or move toward alternate regimes. Complexsystems can also oscillate between alternate regimes in a cyclicway (van Nes et al., 2007) or flip between alternate regimes inhighly stochastic systems (Scheffer et al., 2012). Nonetheless, allof these phenomena occur because of multi-stability, which is aconsequence of multiple competing forcings that shape thestability landscape inherent to the complex system.

Human-impact gradients and ecosystem alterations overtime and space can be observed in many parts of the world atmultiple scales (Scheffer et al., 2001). Interesting cases ofecological regime shifts, often undesirable and sometimesirrevocable, occur along these impact gradients when anthropo-genic forcing, combined with natural forcing, eventually propelsthe ecosystem out of its current regime. Furthermore, for a givenecosystem, multiple natural (e.g., rainfall, temperature, nutrientavailability, etc.) and anthropogenic (e.g., irrigation; urbaniza-tion; climate change, etc.) forcings exhibit stochastic temporalfluctuations. Such stochastic drivers control the temporaltrajectories of the ecosystems, displacing it from an attractor.

Current literature on regime shifts of complex systems isfocused on a deterministic perspective; that is, the attractors areassumed to be stationary, with their “strength” changing slowlywith the changes in drivers (see Rockström et al. (2009), forexample). In such cases, regime shifts between the two attractorsoccur at known bifurcation points (Scheffer and Carpenter,2003). However, complex natural systems are often character-ized by non-stationary responses to external and internal forcing(i.e., contingent on initial conditions), and non-linear positive

Please cite this article as: Park, J., Rao, P.S.C., Regime shifts under forstudies in hydrologic systems, J. Contam. Hydrol. (2014), http://dx.

and negative feedbacks (Hastings and Wysham, 2010), all ofwhichmake the attractors to be non-stationary and the responseof the systems, including regimes shifts, to be non-stationary.

We examine here complex system dynamics in which theattractors are non-stationary, in that their strengths change intime, either in a deterministicmanner or evenmore interestinglychanges are defined by some stochastic processes. Many cases ofinterest are, in fact, represented by such non-stationarity, as willbe illustrated with some minimalistic hypothetical models, andalso with few case studies using hydrologic systems (e.g., vadosezone and aquifers) which (1) are subject to anthropogenicforcing (e.g., land use and land cover changes) and naturalforcing (e.g., precipitation; evapotranspiration); and (2) canhave at least two or more regimes and shifts under bothexternal forcing and internal changes.

2. Conceptual model

2.1. Regime shifts

Consider regime shifts in a complex system with twocompeting attractors, which are characterized by meanstrength μA and μB, and with respective variances of σA

2 and σB2.

Note that we assume the system condition is within the rangeof multiple regimes. For a bi-stable system, the two regimesA and B are separated by a transition zone, which is aheterogeneous hybrid space of A and B and includes therepeller (see Fig. 1). A system in the either regime A or B isassumed to have homogeneous system attributes, and main-tains its crucial forms and functions. We identify A as theoriginal regime (e.g., a pristine forest, a wetland or anuncontaminated aquifer), where ecologically healthy functionsexist with minimal or no human interventions, and let B as analternate regime, where structure and functions are dramati-cally altered as a result of human intentions. The system state,while in either regime A or B, fluctuates around themean valuewith some variance, but tends to return to themean because ofnegative feedbacks with a characteristic reverting rate. Thesystem state does have some probability of entering thetransition zone, and eventually shifts to an alternate regime. Asimilar process describes the transition from regime B to A;thus, in this analysis we assume that regime shifts between Aand B are revocable.

Once the system state is in the transition zone, the revertingrate that drives the system state back to the original regime Adiminisheswith the increasing counter-balancing force from thealternate attractor in regime B. This phenomenon is equivalentto a critical slowing down in dynamic systems theory (Schefferet al., 2009). In this transition zone, whether the net feedbackrate is negative or positive is determined by the vector sum oftwo attracting strengths. By using this analogy, we define atipping point aswhere the vector sum equals zero, and thresholdsas the boundaries within which two attractors co-exist.Generally, tipping points have been referred to as bifurcationpoints which have often been treated as a deterministic criticalvalue of a known external driver (or control parameter).

2.2. Non-stationary attractors

We argue here that regime shifts do not always happen in adeterministic fashion, at a fixed value of the control parameter



A B

Fig. 1. Schematic representation of change in stability landscape in hypothetical complex systems, responding to non-stationary forcing from two attractors. Sizes ofcircles (attractor strength) stochastically and independently change with some mean and variance. Black solid line indicates the mean position of repeller and dashedlines are 95% confidence interval. Transition zone is defined as a space where two attractor strengths overlap. State variable values are presented as normalized valuessuch that the distance (in state-space) between two attractors is unity.

3J. Park, P.S.C. Rao / Journal of Contaminant Hydrology xxx (2014) xxx–xxx

(i.e., bifurcation); for example, Artic ice melting occurs at acertain CO2 level and temperature (as in Rockström et al., 2009).Instead, tipping points can be stochastic, represented by someprobability density function (pdf), because the outcomes of acomplex system state are forced by multiple non-stationaryattractors, whose strengths grow or shrink asymmetrically andstochastically in time. In the transition zone, heterogeneity in thestrengths of attractors change along the gradient, as a function ofthe system state causing nonlinearity in the reverting rate.

Conventionally, an equilibrium continuation is drawn upona control parameter or condition (see Figure 2 in Scheffer et al.(2001)). When the control parameter reaches a critical value,the system state abruptly shifts to an alternate regime, and it isknown as bifurcation-tipping (Ashwin et al., 2012). However,even if the condition is before the critical value, the system canstill shift to the alternative regime by external stochastic eventswhich push the system state toward a repeller. Then, thelikelihood of tipping to the alternative regime depends on bothhow far is the original attractor from the repeller in state spaceand the magnitude of the stochastic events. Other workers (seeBeisner et al. (2003) and Guttal and Jayaprakash (2007), forexample) have dealt with cases where the control variable isassumed to be a slow variable (especially compared to theexternal noises) and the system responses as being gradual.

Regime shifts are not only the product of external forcing butalso result from change in internal dynamics that producenegative feedbacks which retain the system to be in the originalregime after being perturbed from stochastic events. Parametersthat represent internal dynamics (e.g., growth rate of vegeta-tion) have long been treated as a constant (or stationary), thusproducing fixed or at least stationary size for basin of attractionat a fixed control parameter. We argue here that internaldynamics of a system are influenced by continuously changingenvironments, including those for control parameters, withinwhich the system exists. Thus, “non-stationary attractors” arealso the products of non-stationary and stochastic internaldynamics. A possible result from non-stationarity in internaldynamics is the shifting shape of an equilibrium continuationcurve. If the parameters for internal dynamics change stochas-tically, then the curve will also stochastically fluctuate resultingin the stochastic size for the basin of attraction or resilience.


Previous models typically illustrate how the size of basin(or regime) changes when its control parameter graduallychanges (Scheffer et al., 2001). However, we argue that relativestrengths of the attractors, thus size of the regime, can changeeven if control parameters donot change. This is due to changesin the internal dynamics of the system. For example, againusing groundwater quality as an illustration, input flux ofcontaminants to the vadose zone can be represented as controlparameter. However, contaminant fluxes leaving the vadosezone and reaching groundwater can have inter- and intra-annual stochastic fluctuations (e.g., Harman et al. (2011)).Thus, even if the control parameter value is fixed, strengths ofthe attractors that tend to maintain the quality of groundwaterfluctuate temporally.

Resilience of complex systems, under non-stationary forc-ing, may be lost without being recognized and unexpectedregime shiftsmay occur evenwith a small external disturbance.Current understanding of complex systems is based on thenotion of regime shifts being stationary (i.e., they occur at aspecific tipping point), and therefore predictable in a deter-ministic sense, albeit with uncertainties, and appropriatemitigation strategies being adopted from a range of choices.In our analysis, however, regime shifts are, in fact, non-stationary because the strengths of the two competingattractors are temporally variable.

2.3. Non-stationary regime shifts

Let us define, within the state-space, normalized (orscaled) location of the regime shift (i.e., tipping point) as: θ =(1 + μA/μB)−1, where the strengths of the attractors for the tworegimes, A and B, are μA and μB. Allowing μA and μB to betemporally non-stationary, and selecting two simple linearfunctions, μA(t)=αt and μB(t)=βt, whereα andβ are constantsand t is time,we have θ=(1+β/α)−1. This suggests that, in thiscase, regime shift will be stationary, even with time-varyingattractor strengths. This is the case because the rates atwhich thestrength of the two attractors change is fixed in time. Therefore,the relative size of the basin of attraction does not change, andthus the regime shift is stationary (deterministic).




Selecting some non-linear functions to represent the tempo-ral changes in attractor strengths (e.g., Fig. 2a for exponentialwith twodifferent characteristic rates, or Fig. 2c for sinusoidal forμA and μB with two different periods, with errors, εA and εB)would result in regime shifts occurring along a trajectory θ(t),defined by the pdf for regime shifts (Fig. 2b for exponential;Fig. 2d for sinusoidal). Growth of most anthropogenic stressorscan be approximated as exponential functions (e.g., see Figure 4in Zimmerman et al. (2008)), while intra- and inter-annual andinter-decadal changes can be represented by periodic functions(e.g., responses to climatic oscillations). We note that while the

Fig. 2. Simulations of various types of non-stationary regime shifts in hypothetical co(e) stochastic forcing, modeled as marked, Poisson processes. All forcings are normalocations of tipping points, θ = (1 + μB/μA)−1, associated with each forcing at the firsand k2 = 0.1. Error terms (ε1 and ε2) are scale varying white noises, such that CVs ofdotted lines are 95% prediction interval for μB and red dotted lines are for μA from theμB(t) = B0 + (20/3)B0 sin(f2t+ π) where f1 = 2π/365 and f2 = 2π/90. (e) Two parammean intensity of forcing, are (0.5, 0.1) and (0.2, 0.1) for μA and μB, respectively. Each estrealizations of Monte-Carlo simulation.


strength of one attractor increases, strength of the competingattractor might in fact decrease. Setting the strengths of thetwo competing attractors as a stochastic process (e.g., markedPoisson processes which may represent rainfall for the naturalregime and oil price shocks for anthropogenic regime) (Fig. 2e)also leads to regime shifts being non-stationary, even thoughthe two parameters (inter-arrival time and event intensity)defining the stochastic process are stationary.

In the foregoing discussion, we have presented severalhypothetical cases of regime shifts occurring between twocompeting non-stationary attractors to illustrate the effects of

mplex systems: (a) exponentially decaying forcing (c) sinusoidal forcing, andlized by their maximum observed values. (b), (d), and (f) are the normalizedt column. (a) μA tð Þ ¼ μA;0e

−k1 t þ ε1 and μB tð Þ ¼ μB;0e−k2 t þ ε2 where k1 = 0.2

forcing are set to be stationary. Black solid line is the mean trajectory and bluemean. (c) The trajectory of sinusoidal forcing, μA(t) = A0 + 0.5A0 sin(f1t) andeters (λT, λD), where λT is mean inter-arrival frequency and λD is the inverse ofimated pdf shown in the second columnof plots (on the right) is based on10,000




non-stationarity types. The magnitude of pdfs representingstochastic regime shifts changes depending on the parameterschosen and the type of non-stationarity. Although, our hypo-thetical scenarios may represent competition between varioustypes of forcing, we acknowledge that other scenarios for non-stationary regime shifts are also possible.

Based on the foregoing scenarios, we note that: (1) non-stationarity of regime shifts can be characterized by anasymmetrical pdf, (2) the pdf itself is also non-stationary,meaning that the parameters defining of the pdf shifttemporally; (3) for the deterministic cases, the temporalshifts of pdfs are continuously moving in a certain directionor fluctuating; and (4) for random cases, the shift are non-continuous making the pdf to be anywhere regardless ofwhere the pdf was previously. Other types of forcing withmultiple attractors (e.g., forest vs. agricultural vs. urbanland-uses in a watershed serving as three anthropogenicattractors) may be involved in causing even more complexpatterns of regime shifts. Our approach can be extended ton-dimensional analysis to consider regime shifts whenmultiple competing attractors are at work in determiningthe system state trajectory (Lintz et al., 2011).

In the following section,we present selected case studies forregime shifts in hydrologic systems conceived as complexsystems, which exhibit multi-stability, with regime shiftsoccurring as a result of interactions between natural(e.g., hydro-climatic forcing) and anthropogenic (e.g., LULCchanges) drivers. Our goal is not to present a comprehensivereview; the reader is referred to an excellent book (Scheffer,2009) on natural and social systems as complex systems.Rather, we highlight some recent case studies that illustratehow over the past decade research interest has increased inhydrological applications to studies on surface and groundwa-ter systems as complex systems.

3. Case studies

Study of hydrologic systems as complex systems, exhibitingtwo or more stable states (or distinct regimes), has gainedincreasing attention (Dent et al., 2002; D'Odorico and Porporato,2004; Peterson et al., 2009), following a much longer explora-tion of these systems conceived as having only one stable state(a single regime). Several state variables define the dynamics ofcomplex hydrologic systems, including scalar (e.g., soil-waterstorage; contaminant concentration; water table position) andvector (e.g., water and solute fluxes) quantities used to assessthe systems of interest. Longitudinal studies for monitoringtemporal variations in these state variables, or their spatialpatterns in synoptic studies, are used to determine whether thehydrologic system is in an original or an alternate hydrologicregime. Such changes are driven by external forcing (e.g., hydro-climatic or anthropogenic forcing), and interactions amongsystem processes. Increasing evidence (e.g., see literature herein,and summarized in Table 1) suggests that multi-stable systemsare perhaps the norm in hydrologic complex systems.

3.1. Eco-hydrology applications

We examine, in the following section, selected recent casestudies which acknowledged the existence of multiple steadystates in various hydrological systems. Also, other systems not


comprehensively examined here include shallow lake waterquality (Scheffer et al., 1993), lake ecosystems (Carpenter et al.,1999b), wetland vegetation (D'Odorico et al., 2011; Foti et al.,2012, 2013), vegetation-soil moisture dynamics (D'Odoricoet al., 2005, 2008; Guttal and Jayaprakash, 2007), unconfinedaquifer-vegetation interactions (Ridolfi et al., 2006), drylandecosystems (D'Odorico et al., 2007) and river channel mor-phology (Schumm, 1973). We summarize these in Table 1.

3.2. Shallow aquatic ecosystems in antipodean landscapes

Shallow lakes or wetlands have often been used as theachetypical example ofmulti-stability in ecosystems (Carpenteret al., 1999b, 2003; Dent et al., 2002; Scheffer and Carpenter,2003; Scheffer and van Nes, 2007). As shown in the Table 1,nutrient (e.g., phosphorus) loadings have been identified as theprimary drivers that change resilience of such systems, andincreasing loads eventually push these systems toward analternate regime. Turbidity or algal biomass have beenmonitored as state variables to track the stability and internaldynamics of several biotic and abiotic factors, including theexistence of submerged vegetation and phosphorus accumu-lated in and released from sediments.

Because of its simplicity for conveying the idea of regimeshifts, themechanisms and primary factors identified initially forthe shallow lakes model have been widely accepted. However,factors that constitute shallow lake environments can be bothtemporally stochastic and non-stationary and spatially hetero-geneous. Scheffer and van Nes (2007) revisited these ideas byaccounting for the effects of climate and hydrologic regimes thatdefine lake depth and size. Theirmain argumentwas to illustratethe effects of stochasticity and heterogeneity of environmentalfactors on critical nutrient levels where regime shift may occur.This re-examination of regime shifts did not, however,consider how the environmental factors induced regimeshifts by changing internal dynamics, thus basin of attrac-tion, even without the change in nutrient levels.

Davis et al. (2010) examined how regime shifts can occur inshallow inland aquatic ecosystems by accounting for pathways,i.e., contingency, ofmultiple drivers and stressors resulting fromanthropogenic forcing (e.g., urbanization; agricultural intensi-fication). These changes can be interpreted as primary sourcesfor non-stationary change in attractor strengths, even if thecontrol parameters, such as nutrient level or salinity, are fixed.Using our conceptual model in the Section 2, non-stationarityin those anthropogenic forcings are likely to be exponential orlogistic growth and overwhelming the natural forcings thatmaintained the wetland hydrologic regimes.

Davis et al. (2010) started by identifying four differentregimes (persistent states) observed in Australian inland wet-lands, shallow lakes and river pools: (1) clear, aquatic plant-dominated, (2) clear, benthic microbial community-dominated,(3) turbid, phytoplankton-dominated, and (4) turbid sediment-dominated regimes (Fig. 3). They recognized that changes inhydrologic regimes were an additional primary driver thatchanges the type of alternate wetland regime as well as themechanism underlying regime change. Changes in hydrologicregime, resulting from anthropogenic forcing, have not beengenerally considered in modeling regime shift in shallow lakes.Davis et al. (2010), however, illustrated that shift from atemporary, surface water-dominated regime to a persistent,



Table 1Examples of hydrologic complex systems showing system state variable, multiple regimes, drivers, and possible causes for non-stationary attractors.

Hydrologic complexsystem

System state variable Stable states drivers Possible causes fornon-stationary attractors

Refs.

Shallow lake waterquality

Turbidity “Clear”;“turbid”

Nutrient load(e.g., Phosphorous)

Other nutrients (e.g., Nitrogen);Nutrients stored in sediments;Spatial heterogeneity (lake size,depth); Climate

Scheffer et al.(1993); Schefferand van Nes (2007)

Lake ecosystems Algal biomass Oligotrophic;Eutrophic

Nutrient load Other nutrients (e.g., Nitrogen);Nutrients stored in sediments;Spatial heterogeneity (lake size,depth); Climate

Carpenter et al.(1999a)

Wetland vegetationpatterns

Vegetation Type(abundance of tallsawgrass and pinesavanna)

Dominance of eitherof the two species

Hydrologic Regimes Climate change; anthropicactivity (e.g., irrigation,fertilizer uses)

Foti et al. (2012);Foti et al. (2013);D'Odorico et al.(2011)

Groundwaterdependent vegetation

Vegetation density Vegetated;Un-vegetated

Water table depth Anthropic activity (e.g., logging,groundwater pumping); Forestfire regimes

Ridolfi et al. (2006)

River channel Vegetation Patterns inRiver Channels

Wide, vegetatedvalleys; vs. Barren,entrenched arroyos

Plant coverage Climate change; stream incision Schumm (1973)

Dryland ecosystems Woody VegetationDynamics

Vegetated state;No canopy cover

Soil –water Content Anthropic activity (e.g., logging,groundwater pumping); Forestfire regimes; Climate change

D'Odorico et al.(2007)

Land–atmosphereexchange system

Soil–water content Dry;Wet

Rainfall frequency Land use land cover change D'Odorico andPorporato (2004)

Fig. 3.Multiple pathways that lead to different types of alternative regimes and regime shift mechanisms in shallow lentic ecosystems (Source: Davis et al., 2010).


Please cite this article as: Park, J., Rao, P.S.C., Regime shifts under forcing of non-stationary attractors: Conceptual model and casestudies in hydrologic systems, J. Contam. Hydrol. (2014), http://dx.doi.org/10.1016/j.jconhyd.2014.08.005



groundwater-dominated regime occurs with large-scale land-use changes (when shallow-rooted cropping species replacedeep-rooted native vegetation). The result is salinization ofwetlands by the increased salt inputs from saline groundwater.When surface runoff increases with urbanization (withincreases in impermeable paved areas), the hydrologic regimein urban wetlands shifts from seasonal to permanent. Morefrequent and flashy runoff events carry more nutrients intowetlands, resulting in their eutrophication. Increased nutrientinputs induce regime shift. That is, nutrient and sedimentinputs into the urban wetlands serve as stochastic drivers; thetemporal variability in runoff events is determined by the dailyand seasonal rainfall patterns, and how impermeable urbanareas influence translation of rainfall events into runoff inputs.However, had the wetland hydrologic regime remainedseasonal, the consequence would have been different regard-less of nutrient inputs. With a seasonal hydrologic regime,consolidated sediments formed during drying season couldhave supported extensive stands of submerged macrophytes,which are known to strengthen internal feedbacks for clearingwater during the wet seasons.

Davis et al. (2010) study illustrates the possibility of highercomplexity in the stability landscape of wetlands becauseseveral controlling factors, which were typically regarded asconstants (stationary), may play more important roles than itwas previously recognized. In case of shallow lake model,variability in hydrologic regimes have not been considered as adriver for examining regime shifts. However, Davis et al. (2010)study illustrated that hydrologic regime is indeed a primarydriver and determined the type of alternate regime, and in-fluenced the shape of the equilibrium continuation (e.g., simplethreshold model versus hysteresis model). Although, Davidet al. (2010) only presented the end points of four alternateregimes in wetlands arising from interaction of distincthydrologic regimes and land uses, both of those drivers aretemporally non-stationary and stochastic anthropogenic forc-ings. As such, the impact of these non-stationary parameters onequilibrium continuation may also be non-stationary andstochastic. Thus, an equilibrium continuation predicted bysolely considering control parameter may not well representthe state of resilience of such systems. A careful examination ofunderlying drivers that have not been well recognized or byexplicitly considering non-stationary and stochastic nature ofstability landscape is required to better understand resilienceand regime shifts in hydrologic complex systems.

3.3. Water-table fluctuations

Peterson et al. (2012) analyzed a model for the bi-stabilityof groundwater table in an unconfined aquifer. They definedthe dynamics of the water table elevation (h) as:

ASdhdt

¼ −ksatwb∂h∂x þ Rnet ð1Þ

where A [L2] is catchment area; S is aquifer specific yield; ksat[LT−1] is aquifer saturated lateral hydraulic conductivity; w [L]is catchment width; x [L] is catchment length; and b [L] issaturated thickness of the aquifer; and Rnet [LT−1] is netrecharge (see Figure 1 in Peterson et al., 2012). After


substituting with appropriate parameters, they finally derivedthe governing equation for the model as:

ASdDdt

¼ Caq D2 þ h0−2Eð ÞDþ E2−h0E� �h i

−A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimax P−r3;0ð Þ

r1e−r2D þ PETe−c1D

s ! ð2Þ

where D [L] is a reference depth below ground surface(e.g., confining layer); Caq [LT−1] is aquifer conductance,calculated as (2ksatw/x); h0 [L] is thewater table head elevationat the catchment outlet; E [L] is elevation of the land surface;P [LT−1] is the annual precipitation; r3 [LT−1] is a cutoff value,beyondwhich recharge starts; r1 is a parameter controlling therate at which the maximum recharge increases with precipi-tation; r2 is the decay rate of recharge with increasing watertable; PET [LT−1] is the annual potential evaporation; and c1[L−1] is the decay rate of water table evaporation withincreasing water table depth. This model assumes that thegross recharge increases as thewater table rises such that it canrepresent reduced transpiration due to saline water table rise.But, this type of phenomenon occurs onlywhenno transition tosalt-tolerant vegetation develops. This assumption assures thatthe model produces a positive feedback and ensures theexistence of multiple attractors.

The control parameter in this model is the aquifer conduc-tance, Caq. Given appropriate values for other parameters, adeterministic equilibrium continuation is drawn to have eithersingle or multiple steady states along the Caq, with two-foldbifurcations. Most studies on multi-stability of complex systemsexamine the existence of bifurcation points, and quantifyresilience as the distance between the current stable equilibrium(attractor) and the unstable equilibrium (repellor) at a givenvalue of the control parameter assuming only one equilibriumcontinuation exists. The system undergoes regime shift bycrossing a bifurcation point, due to a change in controlparameter; this is a deterministic perspective of systemdynamics, and represents system resilience as a static property.Regime shift can also occur when the strength of the external“shocks” exceeds resilience under antecedent conditions (e.g., ata certain value for control parameter); such shocks might resultfrom a stochastic process. However, these are rather limitedperspectives for examining regime shifts in complex systems.Resilience is not deterministic even when control parameter isfixed because of non-stationarity in attractor strengths caused byother driving factors.

Peterson et al. (2012) examined the stochastic nature ofregime shifts in complex systems subjected to also stochasticexternal disturbances, such as groundwater pumping orrecharge from flooding events. That is, parameters other thanthe control parameter (e.g., rate of decay of recharge, r2), canalso temporally change, causing a shift in the shape ofequilibrium continuation. Thus, even if the system state doesnot change by external forcing, the resilience (e.g., the distancefrom either repeller or bifurcation point) can change or eventhe system state can be in the alternate regime because of theshift in the equilibrium continuation.

This type of stochastic regime shift is illustrated in Fig. 4b inwhich the hydrologic system may stay in upper stableequilibrium when Caq = 750 m year−1. However, without




changing Caq or having external shocks that displaces thesystem state from the equilibrium, the system still may moveinto deeper groundwater (Fig. 4c), because there is only asingle equilibrium left due to the change of r2. This implies thatthe control parameter or external shocks are not the onlysources of regime shifts, and additional parameters that mayalter the resilience of given system should be carefully chosenand managed. Otherwise, the system may unexpectedly shiftinto an alternative regime. If the resilience at a given controlparameter value did not change, the system state would havesimply recovered to the original equilibrium or shifted to analternate stable equilibrium only after having shocks thatexceed system resilience.

The effect of parameter other than the original controlparameter, Caq, is emphasized again by choosing annualprecipitation patterns as the stochastic driver for causingregime shifts (Fig. 5). Specifically, Peterson et al. (2012)examined the fixed condition in which only a single stableequilibrium existed (e.g., Caq= 1,100m yr−1) and showed thatanother equilibrium continuation, which causes multiplestability, can occasionally emerge by a change in precipitationrate. Thus, even if a system may be perceived as being able torecover to its attractor regardless of perturbation size, in reality,it may experience a regime shift caused by unrecognizedbifurcation in other dimensions.

Peterson et al. (2012) model describes the rise of salinewater table, which reduces transpiration, and they distinguish“deep” and “shallow”water tables as the original and alternateregimes, respectively. Attractor strength in the alternate regimewould result from the clearance of vegetation due to land usechange, which facilitates water table rise and further causeswilting of remnant vegetation cover. Thus, the level of LULCchange, which typically gradually increases under anthropicforcing (as in Section 2, with the exponential growth), mayrepresent the alternate attractor strength. In the absence ofhuman alterations, evapotranspiration losses from the naturalvegetation will act as the strong attractor to maintain a “deep”water-table regime, but typically with periodic seasonalfluctuations, and intra-annual stochastic fluctuations.

The conclusions drawn by Peterson et al. (2012) areimportant as they imply that even if a groundwater table ismanaged according to conventional methods based on aquiferconductance, especially by considering the resilience of the

Fig. 4. The change in equilibrium continuation for water table depth due


current regime, groundwater regime can still shift unexpectedlyto an alternate regime or shift earlier than the resourcemanagers would have perceived because: (1) stochasticity ofthe identified controls (e.g., precipitation patterns), or (2) un-identified factors, which are often ignored or considered asbeing constant. Non-stationarity then introduces uncertaintiesin predictingwater-table regime shifts. Such effects explainwhysome hydrologic systems experience regime shifts as surprises,even if the size of external shockwas smaller than the perceivedresilience.

3.4. Soil salinization

Soil salinization is a global problem, with about nine millionsquare kilometers of croplands affected by salt accumulation,either due to inputs fromexternal sources such as anthropogenicsources (wet/dry deposition) or from irrigation/drainage prac-tices causing redistribution of geogenic sources. Suweis et al.(2010) explored the shifts between two distinct salinity regimesin soils, as driven by stochastic rainfall patterns which controlthe balance between salt inputs from atmospheric deposition(wet and dry), and export from leaching. Drivers controllingshifts between the two stable states (i.e., salinity regimes) are ofinterest: “saline” regime,wheremean salt concentration exceeds2 dS/m causing phyto-toxicity to non-halophytic vegetation, and“non-saline” regime below that threshold salt concentration.Using a minimalistic model of water and salt balance in the rootzone, they derived the following stochastic differential equationfor the rate of change of salt mass (m):

∂m∂t ¼ π−mLt ð3Þ

At long timescales (e.g., Ndecades), assumption of aconstant salt mass input rate, π (M/T) at the soil surface maybe justified. Such input rate is given by the sum of dry and wetdeposition rates:

π ¼ Md þλpCR

γpð4Þ

where Md is dry deposition rate, CR is the salt concentration inrainfall. Stochasticity of the daily rainfall patterns is represented

to change in recharge decay rate r2 (source: Peterson et al., 2012).



Fig. 5. The occurrence of unperceived equilibrium continuation caused by non-control parameter of annual precipitation rate (adopted from Peterson et al., 2012).


as a marked Poisson process with λp and γp as the mean valuesfor rainfall frequency and precipitation depths, respectively,which are exponentially distributed. Note that the mean dailyrainfall amount is simply the product: λpγp. The resulting rate ofsalt loss (Lt), driven by temporally variable infiltration anddrainage events, is also represented as a stochastic process (amarked Poisson process) with mean frequency of λ, and mean,scaled intensity, μ = b/(wγp), where 0 ≤ b ≤ 1 is thedimensionless salt removal efficiency, w is the threshold soil-water storage above which salt leaching events are triggered.Note that themean salt concentration,bCN, is the ratio of the totalmean salt mass and the total mean volume of soil water in theroot zone: [bmN/(bs N Zr n)]. Themean values, bs N and bm N arecalculated from the exact solutions of the stochastic differentialequation (Eqs. 1 and 6 in Suweis et al. (2010)).

In this minimalist modeling approach, the emergent soilsalinity regime is controlled by linkage between two stochasticdrivers (rainfall patterns; net rate of salt accumulation) andtwo deterministic drivers (mean dry deposition rate; mean saltconcentration in rainfall). Suweis et al. (2010) reported good

Fig. 6. (a) Contour plot of the mean salt concentration b C N as a function of yearly(solid line) and numerical simulation (dotted line) of p(C) (Source: Suweis et al. (2010


agreement (Fig. 6b) between the outputs of numerical modelsimulations and theminimalist analytical model for time seriesand for long-term (steady state) pdfs of salt concentrations insoil water. They examined the impact of rainfall stochasticity(amounts; frequency) on salt accumulation in coastal andcontinental regions (Fig. 6a). As expected, with an order ofmagnitude larger salt input in coastal regions compared toinland regions, soil salinization is more likely even for humidclimates.

Regime shifts in soil salinity – from non-saline to saline(based on phytotoxicity) – are sensitive to annual meanrainfall, and especially to stochasticity of daily rain events. Forrainfall below 80 cm/y, soil salinization in coastal and conti-nental regions increases with increasing frequency of rainfallevents (larger λp), and associated decrease in mean rainfalldepth (αp), because salt leaching losses are smaller (see Fig. 6)relative to inputs. Interestingly, both mean rainfall amountsand frequencies are highly variable in arid regions (Park et al.,2013), and as a consequence stochastic regime shifts in soilsalinity are more likely in coastal regions. These findings have

rainfall depth and frequency. (b) Comparison between the analytical solution)).




even greater implications to soil management in coastalregions under climate change scenarios, with projected shiftsin spatial and temporal patterns of hydro-climatic forcing.

Suweis et al. (2010) distinguished two different regimes(saline and non-saline) based on phytotoxicity to non-halophytic vegetation, but did not examine equilibriumcontinuation of vegetation state along a control parameter(either rainfall frequency or annual rainfall). Primary purposeof Suweis et al. (2010) analysis was to derive an exact solutionfor the pdf for description of soil salinity analysis, and does notprovide information on the possibility of multi-stability ofvegetation state along a control parameter. However, based onother literature on feedbacks between vegetation and salinegroundwater (Runyan and D’Odorico et al., 2010), multi-stability for vegetation is likely. Given the possibility of havingmulti-stability, and thus resilience of vegetation in saline soils,Suweis et al. (2010) provided a hint of how multiple controlparameters can be explored to examine the effect of non-stationary attractors.

As the temporal scale for changes in annual rainfall(e.g., decadal oscillations) is much larger than rainfall frequency(e.g., at daily or seasonal scales), annual rainfall which is theslow variable, is typically chosen as a control parameter if multi-stability is to be examined. Then, an equilibrium continuationfor vegetation state along the annual rainfall can be drawn.However, rainfall frequency, as a fast variable, is not only actingas the external forcing for reducing vegetation density by addingsalt mass to the soil, but may also shifts internal dynamics ofvegetation via changing evapotranspiration rate and frequencyof root zone inundation, for example. Thus, rainfall frequencycannot be treated as a constant while observing temporaltrajectory of vegetation state at a given annual precipitation, andit should be treated as non-stationary stochastic variable whichmay change the shape of equilibrium continuation frequently.By taking at least 2-dimensional analysis by considering bothannual rainfall and rainfall frequency, resilience landscape canbe developed to depict how much of non-stationary attractorcan have effects on perceived resilience at a given condition.

4. Concluding remarks

A dominant feature of complex systems is the possibleexistence of tippingpoints atwhich the system state shifts to analternate regime, as governed by different feedback mecha-nisms. An external driver (e.g., climate change or LULC changes,which are generally treated as “slow” variable) is a majorcomponent that induces such regime shifts. If a tipping pointserves as bifurcation point, then the systemmay have multipleregimes under the same driver conditions, and returning to theoriginal regime exhibits hysteresis because strong positivefeedbacks pull the system to remain in the alternate regime.Other drivers serving as the “fast” variables,with randomnoise,can also cause regime shifts even if the slow driver is notchanging or changing in a desired direction. In both cases,resilience of a complex system is managed such that thecurrent system state is either far from bifurcation point orunstable steady state (repeller). These are well-known causalmechanisms for regime shifts in complex systems.

Less well known are the cases when the internaldynamics of the complex system are stochastically changingsuch that resilience of the system cannot be characterized


deterministically. We argue here that many abruptly changingphenomena we observe in complex systems are often unex-pected and surprising because of these stochastic characteristicsin systems resilience and regime shifts. We presented apreliminary analysis of the stochasticity in internal dynamicsand thus of resilience which are the results of the competitionbetween multiple attractors (e.g., natural forcing versusanthropogenic forcing). Because these forcings are temporallychanging, the regime shifts are fundamentally non-stationarysuch that resilience or the location of tipping points cannot bedetermined with a single steady-state probabilistic densityfunction. We reviewed here selected case studies in hydrologywhich show the existence of multiple regimes, for example,in shallow aquatic ecosystems, groundwater table, and soilsalinity. Then, for each case study, we discussed mechanismsfor stochastic regime shifts, and possible implications for waterresource management.

Acknowledgements

This researchwas initiatedwhile the senior author (JP)was aPh.D. student at PurdueUniversitywith the support provided bythe Lee A. Reith Endowment in the School of Civil Engineering atPurdue University. Significant portions of this work werecompleted after JP joined the faculty of Hongik University,Seoul, South Korea, and his work supported by 2013 HongikUniversity Research Fund. Themanuscriptwas completedwhenPSCR was on sabbatical leave at the Helmholtz Centre forEnvironmental Research (UFZ) at Leipzig, Germany. The authorsalso acknowledge the contribution of two anonymous re-viewers who provided substantive input, which helped inrevising the manuscript.

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