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How ecosystems recover from pulse perturbations: A theory of short- to long-term responses
Jean-François Arnoldi, Azenor Bideault, Michel Loreau, Bart Haegeman
To cite this version:
Jean-François Arnoldi, Azenor Bideault, Michel Loreau, Bart Haegeman. How ecosystems recover
from pulse perturbations: A theory of short- to long-term responses. Journal of Theoretical Biology,
Elsevier, 2018, 436, pp.79-92. �10.1016/j.jtbi.2017.10.003�. �hal-02331185�
J.-F. Arnoldi , A. Bideault ,M. Loreau ,B. Haegeman
aCentre for Biodiversity Theory and Modelling, Theoretical and Experimental Ecology Station, CNRS and Paul Sabatier University, Moulis, France
bIntegrative Ecology Lab, Département de Biologie, Université de Sherbrooke, Sherbrooke, QC, Canada
a rt i c l e i n f o
Article history:
Received 2 March 2017 Revised 21 September 2017 Accepted 4 October 2017 Available online 4 October 2017 Keywords:
Ecosystem stability Return to equilibrium Asymptotic resilience Transient dynamics Reactivity Rare species
a b s t r a c t
Quantifyingstabilitypropertiesofecosystemsisanimportantprobleminecology.Acommonapproach isbased ontherecovery frompulseperturbations, and posits that thefaster anecosystemreturn to itspre-perturbationstate,themorestableitis.Theoreticalstudiesoftencollapsetherecoverydynamics intoasinglequantity:thelong-termrateofreturn,calledasymptoticresilience.However,empiricalstud- iestypicallymeasuretherecoverydynamicsatmuchshortertimescales.Inthispaperweexplainwhy asymptoticresilienceisrarelyrepresentativeoftheshort-termrecovery.First,weshowthat,incontrast toasymptoticresilience,short-termreturnratesdependonfeaturesoftheperturbation,inparticularon thewayitsintensityisdistributedoverspecies.Wearguethatempiricallyrelevantpredictionscanbe obtainedbyconsideringthemedianresponseoverasetofperturbations,forwhichweprovideexplicit formulas.Next,weshow thattherecoverydynamicsarecontrolledthroughtimeby differentspecies:
abundantspeciestend togovernthe short-termrecovery,whilerarespeciesoftendominate thelong- termrecovery.Thisshift fromabundantto rarespeciestypically causesshort-term returnrates tobe unrelatedtoasymptoticresilience. We illustratethat asymptoticresilience canbe determinedbyrare speciesthathavealmostnoeffectontheobservablepartoftherecoverydynamics.Finally,wediscuss howthesefindingscanhelptobetterconnectempiricalobservationsandtheoreticalpredictions.
© 2017ElsevierLtd.Allrightsreserved.
1. Introduction
Ecosystem stability,in particular,the wayecosystemsrespond to perturbations, is a longstanding topic of interest in ecology (May, 1973; Pimm, 1984; Tilman andDowning, 1994). Ecologists haveusedavarietyofprocedures toquantifythistype ofecosys- temstability,differinginthecharacteristicsofperturbationsandin thewaythesystemresponseismeasured.Aperturbationcancon- sistofachangeinanenvironmentalparameterlastingforshortor longtimes.Itcancorrespondtobiomass additionorremoval, ap- pliedonceorrepeatedly.Theecosystemresponse canbeassessed soon after the perturbation ormuch later, measuring the overall state ofthe ecosystem oran ecosystem variable ofspecific inter- est.Thismultitude ofprocedures hasledtoan overabundanceof stabilitymeasures,whoserelationshipsareoftenunclear(Donohue etal.,2013;GrimmandWissel,1997;IvesandCarpenter,2007).
∗ Corresponding author.
E-mail addresses: [email protected] (J.-F. Arnoldi), [email protected] (A. Bideault), [email protected] (M. Loreau), [email protected] (B. Haegeman).
Wefocushereonmeasures basedonan ecosystem’sresponse topulseperturbations, i.e., perturbationsof relativelyshortdura- tion(Benderetal.,1984).Weassumethatafterasufficientlylong time following a perturbation the ecosystem returns to the pre- perturbedstate,whichwecallequilibrium.Wepositthatthefaster thereturnthemorestabletheecosystemis.Severalstabilitymea- surescan thenbedefined,differinginthetimeatwhich,andthe ecosystemvariableofwhich,thereturntoequilibriumisassessed.
Termsusedforthesemeasures includereturntime,recoveryrate, andresilience.1
Quantifyingecosystemstabilityusingthereturntoequilibrium isa common approach inboth empirical and theoretical studies.
Indeed, pulse perturbations are an appropriate model for many naturaldisturbances, such asfloods, forest fires anddisease out- breaks,andhavebeenwidelyappliedinexperimentalecosystems.
In the latter, it is typically the short-term return to equilibrium thatisstudied,duetopracticaldifficulties ofcollectinglongtime series(e.g.,Steineretal.,2006;DowningandLeibold,2010;Hoover et al., 2014; Wright et al., 2015). This stands in sharp contrast
1The term resilience might lead to confusion, because it is also used for a rather different set of stability measures ( Gunderson, 20 0 0; Holling, 1973 ).
https://doi.org/10.1016/j.jtbi.2017.10.003
0022-5193/© 2017 Elsevier Ltd. All rights reserved.
withtheoreticalwork,inwhichthereturntoequilibriumismainly studiedatlongtimescales(e.g.,Rooneyetal.,2006;Loeuille,2010;
Thébault andFontaine, 2010;Gellner and McCann,2016). This is dueto the fact that the long-termrate ofreturn to equilibrium, knownasasymptoticresilience,candirectlybecomputedfromthe dominanteigenvalueofthecommunitymatrix(werevisitthisthe- oryinthenextsection).
The problem that ecological theory and data do not neces- sarily address the same time scales has been emphasized be- fore (reviewed in Hastings, 2010). In particular, Neubert and Caswell (1997) argued that the initial response of an ecosystem toa pulseperturbationcan stronglydiffer fromits long-termre- sponse.Theydescribedecosystemsthateventuallyreturntoequi- librium for any perturbation but initially move away following some perturbations. Our work can be seen as an extension of NeubertandCaswell’s theory.Specifically, while their work dealt withtheperturbationthatcausesthestrongestresponse,weshall studytheecosystem average,ortypical,response,andextendthe analysisoveralltimescales.
We begin witha precise definition ofreturn ratesand return timescovering the range between initial to asymptotic response toaperturbation.Weshowthatshort-andlong-termreturnrates differin their dependence on theperturbation direction, i.e., the wayits intensityisdistributedoverspecies.Thisdependencecan be strong forshort times,but vanishes in the limit ofvery long times(i.e.,asymptoticresilience).Tocompareshort-andlong-term return rates on an equal footing, we propose to summarize the distributionofreturnratesfollowingdifferentperturbationsbyits median, forwhichwe presenta simpleandaccurate approxima- tion.Usingthisapproach,wefindthatspeciesabundancecanplay a predominant role in the recovery dynamics. In particular, rare species(that is,those with low abundance) often have a strong effectonthelong-termresponse,whiletheir effecton theshort- term response is typically very weak. We describe the underly- ingmechanism,andillustrateitsgeneralityusingarandommodel ofmany-speciescompetitive communities.2 Ourresultsshowthat asymptoticresilienceandshort-termreturnratesaretypicallydis- connected.Whileasymptoticresilienceprovidesonlyapartialview ontherecoverydynamics,empiricallyrelevantpredictionscanbe obtainedfrom short-term return rates,such as those introduced andstudiedinthispaper.
2. Definingreturnratesandreturntimes
The study of the recovery dynamics starts by specifying the statefrom whichthe ecosystemis perturbed andto whichit re- turnsafter the perturbation. Empirically,this referencestate is a dynamicequilibrium,characterizedbyrelativelysmallfluctuations around a fixed average. The pulse perturbation then induces a muchlargerdisplacement,such thattheecosystem leavesits ref- erencestate,thusinitiatingtherecoverydynamics.
It ispractically impossibletostudytherecovery oncethe dis- placementinduced bytheperturbationhasbecome indistinguish- able from the fluctuationsof the dynamicequilibrium. This is a commonproblemintheanalysisofempiricaltime series.Yet,al- mostalltheoretical work focusesonthe long-termreturn,which is,inprinciple,observable onlyifequilibriumfluctuationsareab- sent.Inother words,theory typicallyassumesthereferencestate tobeastaticequilibrium(May,1973,1974),afixedpointofade- terministicdynamicalsystem. Wealsomakethisassumption,em- phasizinghoweverthatourresultsontheshort-termrecoveryalso holdforafluctuatingreferencestate.
2Note, however, that our theory does not require any assumptions on interaction types.
Denotingthevectorofdynamicalvariables(e.g.,thebiomassof the species in the ecosystem) by N(t) and the equilibrium point by N∗, we focus on the dynamics for the displacement vector x(t)=N(t)−N∗.Apulseperturbationappliedattimet=0tothe ecosystempreviouslyatequilibrium(i.e.,x(t)=0fort<0)ischar- acterized by a vector u and describes the ecosystem’s state im- mediately after the perturbation (i.e., x(0+)=u). For pulse per- turbationsthatarenot toostrong,alinearizationofthedynamics around theequilibriumyields aqualitatively accurate, yetanalyt- ically tractable, picture of therecovery dynamics(we come back tothisassumptionanditslimitationsinthediscussion).Theselin- earized dynamics aregoverned by the communitymatrix A, that is,theJacobianofthenon-lineardynamicalequationsevaluatedat N∗,
dx
dt =Ax. (1)
Eq.(1)yieldsthe recoverytrajectory x(t)following thepulseper- turbation,
x(t)=eAtu fort>0, (2)
where eA denotes the matrix exponential of A. We assume the equilibriumto be stableinthe senseof thestabilitycriterion, so that the system returns to equilibrium following any sufficiently smalldisplacement,sothatlimt→∞x(t)=0.
We are interested in quantifying how stable the system is, basedontheideathatamorestablesystemreturnsfastertoequi- librium. This general idea can be implemented in several ways.
Here we introduce one classic measure that will serve as a ref- erencethroughout.Itisbasedontheasymptoticreturntoequilib- rium,
R∞=lim
t→∞−lnx(t)
t , (3)
where the Euclidean norm x(t)=
ix2i(t) measures the phase-spacedistancetoequilibrium. Eq.(3)statesthat x(t)de-
caysasymptoticallyase−R∞t.Inprinciple,R∞coulddependonthe perturbationvectoru.However,R∞isinfactthesameforvirtually anyperturbation u (see Appendix A). This commonvalue, called asymptoticresilience, isequalto−e(λdom(A)),whereλdom(A)is theeigenvalueofAwiththelargestrealpart.3
Return rates. While the asymptotic returnyields a stabilitymea- surewithelegantmathematicalproperties,onlythefinite-timere- covery is of practical interest. We define two finite-time return rates:theinstantaneousreturnrateattimet,
Rinst =− 1
x(t)
dx(t)
dt =−d
dtlnx(t), (4)
andtheaveragereturnrateovertheinterval[0,t], Ravgt =−lnx(t)−lnx(0+)
t . (5)
Definitions(4) and(5) are illustrated in Fig.1, where we ap- ply a pulse perturbation to a two-species community atequilib- rium.FromtherecoverydynamicsofvariablesN1(t)andN2(t),we deduce the distance to equilibrium x(t) asa function of time
(panelA).ToconstructthereturnratesRinst andRtavg,weplotthis distanceonalogarithmicscale(panelB).Theinstantaneousreturn rateRinst attime t istheslope (withopposite sign) ofthiscurve attimet.TheaveragereturnrateRavgt attimetistheslope(with oppositesign)ofthesegmentconnectingthedistancestoequilib- rium at times 0 andt. Those rates can substantially differ; they
3The stability criterion is equivalent to e (λdom(A ))< 0 , so that R ∞is positive for stable systems.
Fig. 1. Definition of return rates. The response of an ecological system to a pulse perturbation contains information about the system’s stability, as illustrated here for a system of two interacting species. Panel A: We apply a pulse perturbation after which the species biomass N 1( t ) (blue) and N 2( t ) (green) return to their equilibrium values N ∗1and N 2∗. We monitor the recovery dynamics by the distance to equilibrium (red), x (t)=
x 21(t)+ x 22(t)with x i(t)= N i(t)−N ∗i. Panel B: The relative rate at which the distance to equilibrium diminishes is a commonly used stability measure (note the logarithmic scale on the y -axis). Here we distinguish between the average rate of return R avgt over the period [0, t ], and the instantaneous rate of return R inst at time t . These two measures can largely differ, and can even have opposite sign. Parameter values:
N ∗= (1 . 8 , 1 . 2), A = −1 −4
0 −2
and u = (0 . 9 , 0 . 4). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
canevenhaveoppositesign.Forexample,inFig.1,attimet≈1.2, wehaveRinst <0andRavgt >0meaningthatthetrajectorymoves away fromequilibriumatthat time, whilehaving comecloserto equilibriumsincetheendoftheperturbation.
It is instructive to compare the behavior of return rates Rinst
andRavgt forvery smallandverylargetimes t.Itholds generally that Rins0 =limt→0Rtavg andthat limt→∞Ravgt =R∞.However,the analogous relationship limt→∞Rinst =R∞ does not always hold.
It does for the example of Fig. 1, but does not for the one of Fig. A.1 (Appendix A). In thelatter, return rateRinst continuesto oscillate betweenpositive andnegativevaluesforlargetime t,so that Rinst doesnot tend to a steady value. This is avoided when considering atime-average ofRinst ,such asRtavg.Thisis onerea- son why we shall focus on average return rates Ravgt . Finally, it should be noted that while thetheory in thispaper is basedon thedistancetoequilibrium,itcanbeextendedtootherecosystem variables(seeAppendixB).
Return times. Whilereturn ratesmeasure the speed atwhich an ecosystem approaches equilibrium, it might be more interesting to consider thetime it takesfor an ecosystem torecover froma perturbation, i.e., its return time. Return rates and return times are clearlyrelated. Returntime is definedastheamount oftime between the perturbation andthe instant at which the distance to equilibrium becomes smaller than a prespecified bound. In Appendix C we show that this yields a family of return times parameterized by this bound, andwe describe how thesereturn times are related to average returnrates Ravgt . This provides an- otherreasonwhyweshallmainlyfocusonthelatter.Ifthebound is chosen asthe typical extentof thefluctuationsin theequilib- rium state, then the return time corresponds to the time during whichtheecosystemresponseisdistinguishablefromequilibrium fluctuations.
Intheoreticalstudies thereturntime isoftenapproximatedas thereciprocalofasymptoticresilience.Thisapproach,initiatedby Pimm and Lawton (1977,1978), isnot self-evident as ituses the asymptoticregimetodescribetheentirerecoverydynamics.Itim- plicitlyassumesthattheasymptoticreturnrateisagoodproxyfor the returnratesatshorter times.As weargue extensivelybelow, thisneednotbethecase.Itisinfactmoreappropriatetoquantify the returntime asthe reciprocalof afinite-time return rate. For thismattertheaveragereturnrateRtavgisparticularlywellsuited,
asitisbasedonthesamepartoftherecoverythatcontrolsreturn times.
3. Returnratesdependonperturbationdirection
As mentionedabove, virtually anypulse perturbationleads to thesameasymptoticrateofreturntoequilibrium.Duetothisre- markableproperty,asymptoticresiliencehasbeencalledanintrin- sicstabilitymeasure (Arnoldi etal.,2016).In contrast,finite-time returnrates dodepend on features of theperturbation; they are notfullydeterminedbythesystemdynamics.Restrictingtolinear systems,wenowinvestigatethisqualitativedifference.
Apulseperturbationalongaperturbationvectorucausesadis- placementx(0+)=u.Bylinearity,theperturbationintensity,quan- tifiedbythenormu,hasatrivialeffect:whentheperturbation ismultipliedbyaconstantfactor,theresponseismultipliedbythe samefactor,whichthereforedoesnotaffectreturnrates.Wemay thusrestrictourattentiontonormalizedvectorsu=1,i.e.,per- turbationdirections.Inecologicalterms,thedirectionudefinesthe waytheperturbation intensityisdistributedover theconstituent speciesoftheecosystem.
Wefocus onthe averagereturn ratesRavgt buttheresults are similarfortheotherstabilitymeasuresintroducedintheprevious section.Recallthatlimt→∞Rtavg=R∞,andletusdenotetheinitial returnratebyR0=limt→0Rtavg.
Westartwithasimpleexampleoftwonon-interactingspecies (Fig. 2). The community matrix A=−4 0
0 −1
indicates that the first species responds four times faster to a displacement than the second. The species with the slowest recovery determines asymptoticresilienceR∞=1, thus followingan arbitrary pertur- bationthesystemeventuallyreturnstoequilibriumwithunitrate (Fig. 2B). This asymptoticrateis, however,not informative about the short-term recovery. In particular, the systemabsorbs a per- turbationthat mainly affects the firstspecies (perturbation‘a’ in Fig.2)muchfasterthanaperturbationthatmainlyaffectsthesec- ondspecies(perturbation‘b’inFig.2).
Asa result,atsmallt,the distributionofpossiblereturnrates Ravgt (associated to all possible perturbation directions) is quite broad, but becomes increasingly narrow at longer times t (see Fig.2D).Asymptoticresilience,whichisthelowerlimitofeachof thesedistributions,isnotagoodpredictoroftheshort-termreturn rateforanarbitraryperturbation.
