Timescale analyses of fluctuations in coexisting populations of a native and invasive tree squirrel

Abstract

Competition from invasive species is an increasing threat to biodiversity. In Southern California, the western gray squirrel (Sciurus griseus, WGS) is facing competition from the fox squirrel (Sciurus niger, FS), an invasive congener.
We used spectral methods to analyze 140 consecutive monthly censuses of WGS and FS within a 11.3 ha section of the California Botanic Garden. Variation in the numbers for both species and their synchrony was distributed across long timescales (>15 months).
After filtering out annual changes, concurrent mean monthly temperatures from nearby Ontario Airport yielded a spectrum with a large semi‐annual peak and significant spectral power at long timescales (>28 months). The cospectrum between WGS numbers and temperature revealed a significant negative correlation at long timescales (>35 months). Cospectra also revealed significant negative correlations with temperature at a six‐month timescale for both WGS and FS.
Simulations from a model of two competing species indicate that the risk of extinction for the weaker competitor increases quickly as environmental noise shifts from short to long timescales.
We analyzed the timescales of fluctuations in detrended mean annual temperatures for the time period 1915–2014 from 1218 locations across the continental USA. In the last two decades, significant shifts from short to long timescales have occurred, from <3 years to 4–6 years.
Our results indicate that (i) population fluctuations in co‐occurring native and invasive tree squirrels are synchronous, occur over long timescales, and may be driven by fluctuations in environmental conditions; (ii) long timescale population fluctuations increase the risk of extinction in competing species, especially for the inferior competitor; and (iii) the timescales of interannual environmental fluctuations may be increasing from recent historical values. These results have broad implications for the impact of climate change on the maintenance of biodiversity.

Full text

Ecology and Evolution. 2022;12:e8779.  
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https://doi.org/10.1002/ece3.8779
www.ecolevol.org
Received:12Novemb er2021
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Revised:4M arch202 2
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Accepted :15March20 22
DOI: 10.1002/ece3.8779
RESEARCH ARTICLE
Timescale analyses of fluctuations in coexisting populations of
a native and invasive tree squirrel
Robert A. Desharnais1 | Alan E. Muchlinski1 | Janel L. Ortiz2 | Ruby I. Alvidrez1 |
Brian P. Gatza1
Thisisanop enaccessarti cleundertheter msoftheCreativeCommonsAttributionL icense,whichpe rmitsuse,dis tribu tionandreprod uctioninanymed ium,
provide dtheoriginalwor kisproperlycited.
©2022TheAuthor s.Ecolog y and EvolutionpublishedbyJohnWiley&S onsLtd.
1Depar tmentofBiologi calSci ences,
Califo rniaStateUniver sityatLosAngeles,
LosAngeles,California,USA
2CenterforExcellenceinMathematics
andScienceTeaching ,CaliforniaSt ate
PolytechnicUniversityatPomona,
Pomona,California,USA
Correspondence
Rober tA.Desharnais,Dep artm entof
Biologi calSci ences,Califo rniaState
UniversityatLosAngeles,5151State
UniversityDr ive,LosA ngeles,California ,
90032 ,USA.
Email:rdeshar@calstatela.edu
Funding information
Nationa lScienceFoundation,Grant/
AwardNumber:DMS-1225529
Abstract
1. Competition from invasive species is an increasing threat to biodiversity. In
Souther n California, t he western gray squi rrel (Sciurus griseus, WGS) is facing
competitionfromthefoxsquirrel(Sciurus niger,FS),aninvasivecongener.
2. We used spect ral methods to analyze 140 consecutive monthly censuse s of
WGSandFSwithina11.3hasectionoftheCaliforniaBotanicGarden.Variation
inthenumbersforbothspeciesandtheirsynchronywasdistributedacrosslong
timescales(>15months).
3. Afterfilteringoutannualchanges,concurrentmeanmonthlytemperaturesfrom
nearby Ontario Airport yielded a spectrumwith a largesemi-annual peak and
significant spectral power at long timescales (>28 months). The cospectrum
between WG S numbers and temp erature revealed a sig nificant negati ve cor-
relation at lon g timescales (>35 months). Cosp ectra also revealed sig nificant
negativecorrelationswithtemperatureatasix-monthtimescalefor bothWGS
andFS.
4. Simulations f rom a model of two compe ting species indic ate that the risk of
extinctionfortheweakercompetitor increasesquickly asenvironmentalnoise
shiftsfromshorttolongtimescales.
5. Weanalyzedthetimescalesoffluctuationsindetrendedmeanannualtempera-
turesforthetimeperiod1915–2014from1218locationsacrossthecontinental
USA. In the last two decades, significant shifts from short tolong timescales
haveoccurred,from<3yearsto4–6years.
6. Ourresults indicate that (i) population fluctuations inco-occurringnativeand
invasivetreesquirrelsaresynchronous,occuroverlongtimescales,andmaybe
drivenby fluctuationsin environmental conditions;(ii)longtimescale popula-
tionfluctuationsincreasetheriskofextinctionincompetingspecies,especially
fortheinferiorcompetitor;and(iii) the timescalesof interannual environmen-
tal fluctuationsmay be increasing from recenthistorical values. These results

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1 | INTRODUCTIO N
Competitionfromnon-native,invasivespeciesisanincreasingthreat
tothe biodiversityofnative species in aglobalized world.Invasive
species areof ten considered one of the mostimportant threats to
ecologicalfunctionandatopdriverofspecies extinctions(Dueñas
etal.,2021;Flory&Lockwood,2020).Thepresenceofinvasivespe-
ciescanalteranimalcommunities,triggertrophiccascades,displace
nativespecies,andevenleadtohybridizationswithsimilarorrelated
species ( Doody et al., 2017; Huxel, 1999). The a bility to be mor e
competitiveoverlimitedresourcesisoneofthecharac teristicsthat
enablesinvasivespeciestobesuccessful.Inaddition,theyareoften
characterizedbyhavinglifehistorytraitswithcolonizercharacteris-
ticsasfollows:shortgenerationtimes,highreproductionrates,and
fast grow th rates (Sa kai et al., 20 01). With this comp etitive edge ,
theycaninvadeanddisplacenativespecies.
Anexamplewhere a nativespeciesisthreatened in somehab-
itats by co mpetition fro m an invasive species o ccurs in Southe rn
California, where the westerngray squirrel (Sciurus griseus, WGS,
Figure 1a) is facing increasing competition from the fox squirrel
(Sciurus niger,FS,Figure1b),anon-native,invasivecongener.WGSs
arenative to thewesterncoast of Nor th America withahistoric al
distrib ution exte nding from cent ral Washingto n to Baja Califor nia
(Carraway & Verts, 1994; Escobar-Flores et al., 2011). Populations
ofWGSshave beendeclining in areasof Washing ton, Oregon,and
Califor nia (Cooper, 2013; Cooper & M uchlinski, 2015; Muc hlinski
etal.,2009;Stuart,2012).InWashington,theyarelistedasastate-
threatenedspecies (Linders& Stinson,2007),while inOregonthey
areanOregon ConservationStrategySpecies(OregonDepartment
ofFish &Wildlife, 2016).Whiletherehavebeenonlyafewstudies
regardin g populatio ns of WGSs in Calif ornia, the re is a noticeabl e
trendinthedeclineofthesesquirrelsinareasbelowanelevationof
457m (Cooper,2013;Cooper & Muchlinski,2015).As of now,the
WGSdoesnothavespecialconservationstatusinCalifornia.
The FS has a historical native range inthe eastern and central
United Statesand the southern prairie provinces of Canada, south
ofapproximately 48ºN latitude (Koprowski,1994),where theyare
known to live in forests, woodlands, agricultural landscapes, and
urbanareas(Kleimanetal.,2004).Throughbothnaturalandhuman-
assistedrangeexpansion,theFSisnowcommoninmanyareaswest
of its hist orical range (i Naturalist acce ssed 24 July 2021, htt ps://
www.inaturalist.org/taxa/46020-Sciurus-niger). Fox squirrels
have been int roduced or have ex panded thei r range into Arizo na,
California,Colorado,Idaho, Montana, New Mexico,Oregon,Utah,
havebroadimplicationsfortheimpactofclimatechangeonthemaintenanceof
biodiversity.
KEYWORDS
climatechange,foxsquirrel,invasivespecies,populationtimescales,spectralanalysis,Western
graysquirrel
TAXONOMY CLASSIFICATION
Conservationecology;Globalchangeecology;Invasionecology;Populationecology;
Theorecticalecology
FIGURE 1 Photographsofthetwo
diurnallyactivetreesquirrelsthatare
presentinSouthernCalifornia:(a)the
nativeWesternGraySquirrel,Sciurus
griseusand(b)thenon-nativeFoxSquirrel,
Sciurus niger.TheFoxSquirrelhasreplaced
theWesternGraySquirrelinsome
habitat s,whilethetwospeciescoexist
inotherhabitats.PhotographsbyAlan
Muchlinski
(a) (b)

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DESHA RNAIS E t Al.
Washington,andWyoming(Bradyetal.,2017;Flyger&Gates,1982;
Jordan&Hammerson,1996;Koprowski,1994;Steele&Koprowski,
2001;Wolf&Roest,1971).
Foxsquirrelshavedispersedfromoriginalpointsofintroduction
through naturaldispersalandthrough intentionalmovement ofan-
imalsbyhumans(Frey &Campbell,1997;Geluso,2004;Kinget al.,
2010).SincetheoriginalintroductiontoLosAngelesCount y(Becker
&Kimball,1947),theFS hasexpandeditsrangeata rateof1.60to
3.00 km/year in heavilysuburbanized areasofSouthern California
(Garcia&Muchlinski,2017).AlthoughtheFShasgenerallyremained
restri cted to areas of h uman habit ation, with co ntinued ran ge ex-
pansion th e FS has become sym patric in some isol ated suburban
habitat fragmentsandincertainfoothillareaswiththenativeWGS
(Hoefler&Harris,1990).
FSsmaycompete with native WGSsforresources suchasnest-
ing sites an d food, and the FS ha s replaced the WGS w ithin cer-
tain habitats in Southern California (Cooper & Muchlinski, 2015;
Muchlinskietal., 2009).LosAngelesCounty canbeconsideredan
ideal location forinvasion by theFSgiven the mild Mediterranean
climateandyear-roundfoodsupplyof feredbyexoticplantspecies,
accompaniedbytheabsenceofthenativeWGSthroughoutmuchof
theLosAngelesBasin.TheFSisbothmorphologically,ecologically,
andbehaviorally similartothis native species; thus,these overlaps
inform, function, activity,andpresence provideasituation where
interactionsbetweenthetwospeciescanbestudied(Ortiz,2021).
Manyfactorscaninfluencepopulationpersistence,butonethat
has received comparatively less attention is the timescale of envi-
ronmentalandpopulationfluctuations. In the currentstudy,we in-
vestigatedtheeffectsoftimescalesusingthree approaches.First,
weapplied spectralanalysestoexaminethe timescale distribution
of the varia nce and covari ance of WGS, FS , and weather t ime se-
ries. Se cond, we cond ucted mod el simulati ons to examine t he im-
plicationsofchangesinthetimescaleofenvironmentalfluctuations
on the coexistenceof a native species facing competition from an
invasive spe cies. Third , we used spec tral and wavel et methods to
examine thelong-termchanges in thetimescale distributionofcli-
matedata.
By analog y with the s pectru m of visible lig ht, time ser ies fluc-
tuations t hat occur over lo ng timesca les are refer red to as having
ared spectrum and thoseoccurringovershort timescales as having
ablue spectrum(Lawton,1988). Thesearedistinguishedfromwhite
noiserandomfluctuations,whichhavenoserialautocorrelations.In
general, theoretical analysesfromsingle-species discrete-time un-
structured populationmodelssuggestthattheresponsetocolored
environmental noisedependson the type of population dynamics.
Deterministic models with stable equilibria can exhibit undercom-
pensatory dynamics, where thepopulation approaches equilibrium
monotonically,orovercompensatorydynamics,wheretheapproach
to equilibrium exhibits damped oscillations. Many studies have
shown that reddened environmental spectra increase extinction
risk for undercompensator y populations and blue spectraincrease
extinctionriskforovercompensatorypopulations(Danielian,2016;
García-Carreras&Reuman,2011;Mustinetal.,2013;Petcheyetal.,
1997; Ripa & Heino, 1999; Ripa & Lundberg, 1996; Ruokolainen
et al., 20 09; Schwager et al., 20 06). While some s tudies reached
differ ent conclusions ( Heino, 1998; Heino et al ., 2000), S chwager
et al. (20 06) showed that these contrasting results depend onthe
modeling details and a consideration of the likelihood of cata-
strophic events. In their simulations of three competing species,
RuokolainenandFowler(20 08)foundthatextinctionriskincreased
with reddened environmental noise when species responded in-
dependentlyto the environmentbutdecreasedwhen there was a
strongcorrelationbetweenspecies-specificresponses.Ontheem-
pirica l side, Pimm and Redfe arn (1988) looked at 100 time s eries
frominsects,birds,andmammalsandfoundthatthevarianceofthe
populationfluctuations increased withthewindow of timeusedin
thecalculation,suggestingthatthesepopulationshaveredspectra.
García-Carreras and Reuman (2011)analyzed the dynamics of 147
animalpopulationsandclimatedataforthepopulationlocationsand
found a pos itive correl ation bet ween the biot ic and climati c spec-
tralexponents(ameasureofspectralcolor),withmostspectrabeing
red-shifted. Inchaustiand Halley (2003) directly examined the re-
lationshipbetweenpopulationvariabilityandquasi-extinctiontime
(measuredasthetimerequiredtoobservea90%declineofpopula-
tionabundance)foralargesetofdatacomprisedof554populations
for123 animalspeciesthatwerecensusedfor more than 30 years.
The results showed that the quasi-extinction time was shorter for
populationshavinghighertemporalvariabilityandredderdynamics.
Inalaboratory microcosm experiment, Fey and Wieczynski (2017)
lookedathow the autocorrelationin thermalwarming affectedthe
abilityofanon-nativecladoceran,Daphnia lumholtzi,toestablishit-
selfinthepresence ofa nativecongener,D. pulex. The non-native
specieswasabletoattainsignificantlyhigherpopulationdensitiesin
thetreatmentwithanautocorrelatedwarmingregimerelativetothe
treatmentwithuncorrelatedwarmingbutthe same meantempera-
ture andthe unwarmed control. Although, in all three treatments,
D. lumholtziwentextinctbytheendoftheexperiment,theirresults
demonstratedthatthetimescaleofenvironmentalfluctuationscan
impact t he ability of an inv asive species to est ablish itsel f in the
presenceofanativecompetitor.
Spect ral methods ar e a powerful too l for character izing the
timescales of fluctuations in a time series (Brillinger, 2001). A
univariatetime seriescan be transformedintoapower spectrum,
whichdescribesthedistributionofthevarianceofthetimeseries
at diffe rent freque ncies. Th e sum of the spec tral powe rs across
freque ncies is prop ortiona l to the total va riance of the ti me se-
ries. If the time series is multivariate, in additionto the spectra,
therearealsocross-spectraforeachpairoftimeseriesvariables.
The cross-spect rum is a complex-valu ed function of fre quency.
Therealpartisthecospectrum,whichdescribesthedistributionof
thein-phasecovariancebetweenthetimeseriesatdif ferentfre-
quencies,andtheimaginar ypartisthequadrature spectrum,which
isaphase-shiftedcovariance.The sum of the cospectral powers
across frequencies is proportional tothe total covariance of the
two time series. The cospectrum can also be viewed as the dis-
tributionofthecorrelationcoefficient across frequencies. Since

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DESHARNAIS E t Al.
frequency,f,istheinverseoftheperiod,thespectralandcospec-
tral powe r provide inform ation on the vari ance and correl ation,
respectively,atthetimescale1/f.
Thecolorofapowerspectrumcanbecharacterizedusingaspec-
tral exponent (Gar cía-C arreras & Re uman, 2011; Vasseur & Yodzis,
2004).IfSfisthe powerofthe spectrumatfrequency f,thespec-
tral exponent canbecomputedasthe slopeofaleast-squares lin-
ear regression of log(Sf) versuslog(f). Negativespectralexponents
arecharacteristicofspectradominated by long timescale variation
(red spec tra), and posi tive values a re indicati ve of short ti mescale
variation(bluespectra).Whitenoisespectrawillhaveaspectralex-
ponentofzero.Whenappliedtoenvironmentalandpopulationtime
series,spectralcolor allowsone tobetterassess the riskofecolog-
icalextinction.
Wavelet analyses have been used in ecology to identify
changes in t he spect ral distri butions of po pulation an d environ-
mental fluctuations over time (Cazelles et al., 2008). Whereas
spec tra la na lysesassu me th at thest at is tic alprope rti es ofth et im e
series do not vary with time, wavelet analysis can be applied to
non-s tationar y time serie s. A filteri ng functio n is applied to th e
timeseriessignaltoallowalocalestimationofspectralcharac ter-
isticsofthesignalatapointintime.Thefilteringfunctioncanbe
adjustedto lookatdifferenttimes and frequencies.The result is
atwo-dimensionalpictureof thewaveletpower as afunction of
frequencyandtime.Waveletscanbeused,forexample,toinvesti-
gateth eim pactone cologicalp opula tionsofclimatereg imeshif ts ,
such as the North Atlantic Oscillation(Sheppardet al., 2016), or
changesinthetimescaleofenvironmentalfluctuationsduetocli-
matechange.
Global c limate is undergoi ng rapid change s (Masson-D elmotte
etal.,2021).Whilethethreats tobiodiversity havefocusedmostly
onincreasingtemperatures,itisfeasiblethatdisruptionstoclimate
patternsmayalsoaffectthetimescaleofenvironmentalfluctuations,
and,ifso,thismayhaveecologicalimplicationsforpopulationper-
sistence.Forexample,García-CarrerasandReuman(2011)analyzed
detrendedmeansummertemperaturetimeseriesfromweathersta-
tionsonsixcontinentsandfoundsignificantshiftstoshortertimes-
cales(blueshif ts)inthespectralexponentsfortheyear s1951–1990
comparedwith1911–1950.WangandDillon(2014)analyzedannual
global temperature cyclesfrom1975to2013and foundsignificant
increas es in the autoco rrelation of te mperature v alues (red shif ts)
intropicalandtemperateregions.DiCeccoand Gouhier(2018) ex-
amined air temperature values predicted by 21 global circulation
modelsunderthebusiness-as-usualscenarioandfoundthatspectral
expone nts were pre dicted to shi ft negati vely to longer t imescal es
fromtheyears1870through2090.Forconservationpurposes,it is
importanttogainabetterunderstandingofhowchangesinclimate
maybeassociated with changes in the timescaleofenvironmental
fluct uations and how this m ay impact ext inction risks f or natural
populations.
The objec tives of thepresentstudy were(1)to evaluate, using
spectralmethods,thetimescaleofpopulationfluctuationsinalong
time ser ies (14 0 months) where the WG S and FS have coexisted
together,(2)to determine the extenttowhichthetimescaleofthe
squirrel population fluctuationsare determined by environmental
factor s, (3) to infer, using model si mulations, how cha nges in the
timescaleofenvironmentalfluctuationscouldimpactthetimescale
ofpopulationfluctuations and theriskofextinction in a systemof
two comp eting specie s, and (4) to assess t he extent to whi ch the
timescale of year-to-year environmental fluctuations aroundt heir
trendsischanging,possiblyasaresultofhumanimpactsonclimate,
andto assesstheimplicationsof theseresultsonthepotentialloss
ofnativebiodiversit y.
2 | MATERIALS AND METHODS
2.1 | Collection of census data
Weestablishedthree transect lineswithina11.3 ha sectionofthe
CaliforniaBotanicGarden (CBG) inClaremont,CA,during October
of2009.Wedefinedsamplingpo in tsal on gt ra ns ec tl in es at40 -mi n-
tervalspr ov iding35view point sw ith in thest ud yarea .Twores ear ch -
ersconductedacensusalongthetransectlinesoncepermonthfrom
October2009throughMay2021.Theresearchersspent3minutes
ateach samplingpoint,witheachresearcherresponsible forcount-
inganimalswithinaseparate180-degreearcfromtheviewpoint.We
began each monthlycensus at080 0handendedatapproximately
1030h.Weswitchedthestartingtransectlineforthemonthlycen-
susbetweenLine1andLine3onalternatemonths.
Researchersconductingeachcensuswereconservativeincount-
ingthe numberofsquirrelsobser ved,therebygivinganestimateof
observablepopulationsizeatapointintime.Iftherewasanychance
thatasquirrel observedat a samplingpoint had beencountedata
previous samplingpoint, that individual wasnot countedasanew
observationunlessthe animalwas obviouslydifferentfromthe an-
imal previ ously obser ved (a juvenile in stead of an adul t or a male
insteadofafemale,whengendercouldbeassessed).Numbersmay
vary due to fac tors such as natality, mortality,dispersal,andactiv-
ity level s, which could cha nge due to seasona lity or repro ductive
activity.
The four corners of the 11.3 ha study area were defined by
the follow ing GPS coord inates: SE 34 .110262 & −117.714651, SW
34.110258 & −117.715921, NE 34.115883 & −117.714419, NW
34.115684&−117.715891.CBGisanativeCaliforn iagard en,mean-
ingall plant s are nativetoCalifornia, but notspecifically Southern
California.Atthebeginningofthestudyin20 09,thehabitatwithin
thestudyareaincluded1048treesalongwithnumerousshrubsand
bushes.Ofthetrees,31%ofthespeciesweredeciduous,with69%
beingnon-deciduous.Seventeenpercentofthetotaltreeswereco-
niferous (8 3% not conifero us); 42% of all trees wer e in the genus
Quercus;and 6%ofall treeswereinthe genusPinus. The composi-
tion of the study area did change over the timeperiodof thecen-
suses with the death and removalof several trees. Death of trees
in the stu dy area was due mainl y to a prolonged drought within
SouthernCaliforniafrom2011through2016.

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DESHA RNAIS E t Al.
2.2 | Spectral analyses of census data
We used spectral methods to analyze the monthly census data.
We used fast Fo urier tra nsforms to com pute the raw sp ectra an d
cross-spectrum of the bivariate time series. Computations were
conducted using thespec.pgram algorithm fromRmodified to run
inMATLAB.Notrendswereremovedfromthedataprior toanaly-
sis.Since raw spectraandcross-spectra are usuallyjagged,weap-
pliedtwoiterationsofawindow-averagingsmoothingDaniellkernel
with spansof5and 7 datapoints,modified with clipped windows
attheendpointstopreser vethenumberofdatavalues.Wedivided
thespectralpowersby their sumacrossfrequencies. Thisyieldeda
normalized spectral power plot for each species,which shows the
distributionofvariationacrosstimescales.Weusedtherealpartof
thecross-spectrumtoobtainasmoothedcospectralpowerplotfor
thecovariancebetweenthetwospecies.Wenormalizedthecospec-
trumsothatitssumequalsthecorrelationcoefficient.
Weco nd uc tedc om pu tationstodete ctsign if ic ant( p <.05)peaks
orvalleysintheobservedspectraforthenullhypothesisthatthere
isnofrequency dependencein the variance andcovarianceofthe
timeseriesfluctuations(i.e.,independent“whitenoise”timeseries).
Weshuffledthetemporalorderofthebivariatetimeseriesbygen-
erating a r andom per mutation of th e integers 1 thro ugh n, where
n =140 isthe numberof monthly obser vations.Wethenusedthe
permutation to reorder the bivariate monthly censuses of the two
species.Next,we computedtwosmoothednormalizedspec traand
asmoothednormalizedcospectruminthesamewayasthecorrectly
ordered d ata. We repeate d this random re shufflin g process 20 00
times.Forthespectra,whichmustbenonzero,weusedthe95thper-
centileateachfrequencytodefineone-sidedupper95%confidence
limitsforthenullhypothesisthattherearenotimescalecomponents
tothevariance.Forthecospectra,whichcanbepositiveand/orneg-
ative,we usedthe2.5thand97.5thpercentileat eachfrequencyto
definetwo-sided95%confidencelimitsforthenullhypothesisthat
therearenotimescale components tothecorrelation.Thismethod
ofgeneratingthespectra preserves thetime-independentstatisti-
calpropertiesofthetwotimeseries(means,variances,distribution,
total corr elation, et c.), while var ying only t he time-depend ence of
thebivariatedatavalues.
2.3 | Analyses of weather data
We obtained weather data for Ontario Airport (ONT) from the
Climate Data Online website of NOA A’s National Centers for
EnvironmentalInformation(https://www.ncdc.noaa.gov/cdo-web/).
ONTislocatedabout12kmfromtheCBGandshouldbeanaccurate
representationof the temperature profileofthestudy site. Wefo-
cusedonthereported“averagemonthlytemperature,”whichiscom-
putedbyaveraging the dailymaximum and minimumtemperatures
for each mo nth. We avoided rainf all totals be cause many mont hs
havezeros,whichisaproblemforspectralanalyses,andmuchofthe
vegetationintheCBGisirrigated. Weobtaineda temperaturetime
seriesforthesamemonthsasthecensusdataandappliedthesame
spectralmethodstoobtainasmoothednormalizedpowerspectrum.
Sinceannualseasonalchangesdominatedthetemperaturetime
series, we used the MATLAB “band-stop” function to attenuate
cyclic component s with periodicities inthe range of 9–15months.
This prod uced a filtere d time serie s with annual ef fects remove d.
Wethenproducedasmoothednormalizedspec trumforthefiltered
temperaturetime series. Wealso generated smoothednormalized
cospectra betweenthe filtered temperaturetime series and both
the WGS andFS census time series. Usingthe methods described
above,weobtained95%confidenceinter valsforthesespectraand
cospectra.
2.4 | Model simulations
Weconducted modelsimulationstoobtain a betterunderstanding
of the implications of timescale-specific environmental variation
onthedynamic softwocompetingspecies. Weused the following
discrete-timeversionoftheLotka–Volterracompetitionequation:
where r1 and r2are the intrinsic rates ofpopulationincrease,K1and
K2arethecarryingcapacities,andαandβarethecompetitioncoeffi-
cients for thet wospecies.The variables ε1(t)and ε2(t)representran-
domenvironmentalnoise with ameanofzero and varianceof .5. We
usedthecoefficient sσ1andσ2tosc al ethemagnit udeofthenoise .Fo r
the purposes ofdiscussion, species 1 will represent a native species
andspecies2willrepresentaninvasivespecies.
Weintroducedfrequency-specific biases into thenoise variables
us ing ana lgo r ith mde v ise dby Cha m ber s (19 95) .T h ism etho dge n era tes
amultivariate random time series based on any specifiedtheoretical
spectralmatrixthatisafunctionoffrequency.Thediagonalelements
ofthatmatrixare thetheoreticalspectra(frequencydecompositions
ofthe variances),andthe off-diagonalelementsaretheoreticalcross-
spectra(complexnumbers).Therealpartsofthecross-spectraarethe
theoreticalcospectra(frequencydecompositionsofthecovariances),
andthecomplexpartsarethequadraturespectra(frequency-specific
phase shifts). For the model (1), we used identical spectra that were
linear fu nctions of freque ncy for the two species. High-frequenc y-
biased blue noisewas representedwitha linearspectrumthatvaried
froma power of0.0for afrequenc yoff =0.0toapowerof1.0fora
frequencyof f =0.5 (maximumpossible frequency).Low-frequency-
biased rednoise wasrepresented with a linear spectrum that varied
froma power of1.0forafrequencyoff =0.0toapowerof0.0fora
frequencyoff =0.5.Unbiasedwhitenoisehadaconstantpowerof
0.5 acro ss all freque ncies. A gra dual shif t from blue to whi te to red
noisewasaccomplishedbyvaryingtheslopeofthenoisespectrumin
101incrementswhilekeepingtheaverageofthespectrumconstantat
0.5. This produced aconstanttotal varianceofε1(t)and ε2(t)equal to
0.5 while changing only its frequency-specificity. For the covariance
(1)
N
1(t+1) =N1(t)exp
(
r1
(
K1−N1(t)−𝛼N2(t)
)
∕K1+𝜎1𝜀1(t)
),
N2
(t+1) =N
2
(t)exp
(
r
2(
K
2
−N
2
(t)−𝛽N
1
(t)
)
∕K
2
+𝜎
2
𝜀
2
(t)
),

6 of 23
|
DESHARNAIS E t Al.
betweenthe random variables ε1(t) and ε2(t),w e used a cospec trum
function that was equal toa constant fraction, 0.9,of the spec trum.
Thisresulted in afrequency-specificcorrelationof 0.9acrossall fre-
quencies. A high correlation was usedsince it wasassumedthat the
native and invasive species are ecologically similar and occupy the
samehabitat.Thequa draturesp ec trumwassettozero(nofrequen cy- 
specificphaseshifts).Tosummarize,thetimescalesoftherandomen-
vironmentalnoise were variedfromshort (blue)to uniform(white)to
long (red) witha frequenc y-independent correlation in the ef fect sof
thenoiseonthegrowthofthetwospecies.
Inadditiontothespectralfrequencyoftheenvironmentalnoise,
thesimulationprotocolalsoinvolvedvaryingthecompetition coef-
ficient α, which represent sthe intensityofthecompetitive effects
ofthe invasive species on the native species. Weset the value of
the αto.25(weakcompetition),.50(moderatecompetition),and.75
(strong competition).Wekeptthecompetitiveeffectsofthenative
speciesontheinvasivespeciesatavalueofβ =.25whileincreasing
αbecauseweareinterestedinsituationsofconcerntoconserva-
tionist swhere an invasive species outcompetes the nativespecies.
Wechosevaluesfortheintrinsicrateofincreaser1 = r2 =.3t hatare
appropriatefortreesquirrelsofthegenusSciurus(AppendixA).The
remainingmodel parameters had constant values of K1 = K2 = 50,
andσ1 = σ2 =0.75.Fortheassessmentofextinctionrisk,whenthe
populationdensityofaspeciesfellbelow5%ofitscarryingcapacity,
weset it to zero. For simulations not involving extinctionrisk, the
th res h old wa sse tto ze ro. Weran eac hs i mul at i onf or 10 0 tim est eps .
Forevery set ofparametervalues andenvironmentalnoise color,
weconducted2000replicatesimulations.Forblue,white,andreden-
vironmentalnoise,wecomputedsmoothednormalizedpowerspectra
andcospectrum ofthespeciesandaveraged theseoverreplicatesto
see how the timescale for population fluctuations is affected by dif-
ferent col ors of noise. To investi gate gradua l shifts i n the effec ts of
frequency-biasedenvironmental noiseon the population spectraand
probabilityofextinction,wechoseaslopefortheenvironmentalspec-
tra, var ying the s lopes in 101 gradu al incremen ts, beginn ing at blue
noise(slope=2)andendingatrednoise(slope=−2).Foreachchoice
oftheenvironmentalspectra,wesimulatedthepopulationtrajectories
ofthetwospecies,estimatedtheunsmoothednormalizedpopulation
spect ra, compute d the two sp ectral ex ponents , and averaged t hem.
In the sam e way,we co mputed average sp ectral exp onents for the 
environmental noise realizations. We repeated these computations
for each of th e 2000 re plicate simu lations and co mputed an overa ll
average forthe spectral exponents ofthe populations and noise. To
investig ate the effect s of frequency-biase d environmenta l noise on
ec ol ogi c al per sis ten ce,th enu mbe ro finsta nce sw h er eea chp opu la tio n
wentextinctwasdividedby200 0toyieldestimatesoftheextinction
risksforboththenativeandtheinvasivespecies.
2.5 | Analyses of climate data
We obtained climate data from the U.S. Historical Climatology
Network (USHCN), which is freely available online (https://ww w.
ncei.noaa.gov/products/land-based-station/us-historical-clima
tology-net work).We usedversion 2.5ofthemonthlytemperature
records ,w hich contai ns long-term da ta from 1218 stat ions across
thecontinentalUnitedStates.Menne etal.(20 09)describethe ad-
justments used to removebiases duetofactorssuch as relocation
of recording stations, changes in instrumentation, and urbaniza-
tion.USHCNmonthly average temperatureswerecomputedasthe
average overthe month of the dailymaximum and daily minimum
temper atures. The mea n annual temper ature for each yea r is the
average ofthe12meanmonthly temperatures.Weusedthemean
temperatures for the 100-yearrange from 1915through 2014,the
latterbeingthelatestyearavailable.
Welookedat changesinthedistribution of spectral exponents
forthefluctuationsinthemeanannualtemperatures.First,webroke
the100-yearrangeintofour25-yearspans.Next,wedetrendedthe
temperaturetimeseriesforeach25-yearspanbyfittingaquadratic
polynomialusing least-squaresregressionand computedthestan-
dardized re siduals. Then , we computed an unsm oothed spect rum
foreachresidualtimeseriesandestimatedthespectralexponentas
theslopeofalinearregressionoflog(spectralpower)versuslog(fre-
quency).Histogramswerecreatedwiththe1218spectralexponents
(oneperstation)foreachofthe25-yeartimespans.
Althou gh it would be temptin g to analyze the cha nges in the
spect ral exponent s using a repeate d measures ANOVA, w ith sta-
tions as thesubjects, spatial autocorrelations existamong stations
thatareinthesamegeographicalproximity,inflatingtheTypeIerror
rates. A so lution to this pro blem was sugges ted by Cliffor d et al.
(1989)and modified by Dutilleul (1993), which yields an “effective
samplesize”basedonthespatialstruc tureofthedata.Itisappropri-
ateforpaired observationsdistributed in space. Weused the soft-
warepackageSAM(SpatialAnalysisinMacroecology;Rangeletal.,
2006)tocomputeeffectivesamplesizesforthefollowingthreeset s
ofpaireddata:[1915–1939]vs.[1940–1964],[1940–1964]vs.[1965–
1989], and [1965–1989] vs. [1990–2014]. We conducted paired
samplet-testsforthespectralexponents from thesethreepaired
datasetsandadjustedthestandarderrorsfortheteststatisticsand
degrees of freedom for thestatistical significancevalues usingthe
effective samplesizes. Wethen appliedaBonferronicorrection to
accountforthemultiplecomparisons.
Wealsoconductedameanfieldwaveletanalysisonthe100-year
timeseriesof meanannual temperatures.Foreach station, we de-
trendedthetimeseriesusingaquadraticpolynomialandcomputed
thestandardizedresiduals.Next, weusedtheMATLABcontinuous
wavelet tr ansform fun ction “cwt ” to compute wavel et powers for
the resid ual time serie s using the analy tic Morse f ilter (Olhede &
Walden, 20 02) with the d efault valu es of 3 for the sy mmetry pa -
rameter a nd 60 for the time -bandwidt h product. L astly, we aver-
agedthewaveletpowersacrossallstationsforeachtime–frequency
combination. Wechose theMorse wavelet because it is useful for
analyz ing signals with tim e-varyin g amplitude and fr equency. We
investigatedvaryingthesymmetr yandtime-bandwidthproductpa-
ra met er s ,b u tt her esu lt s we ren ot muc hd iffe ren tf rom wh atw as ob-
tainedusingthedefaultvalues.WealsousedaMorletwaveletwhich

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hasequalvarianceintimeandfrequency,but,again,theresultswere
liketheMorse waveletwith defaultparameters. We experimented
with cubic and quarticpolynomials for detrending, but these gave
mean fie ld wavelets t hat were much like t he one obtain ed with a
quadraticfunction.Ourmeanfieldwaveletdiffersfromthe“wavelet
meanfield”definedbySheppardetal.(2016),whichwasdesignedas
anaverage measureofthetime-andtimescalesynchronyfortime
seriesfromdifferentlocations.
To identify wavelet powers that were statistically significant,
weused the surrogate timeseries approach (Schreiber & Schmitz,
2000).Wetookara ndompermut ationofthemeanannualtem pera-
turetimeseriesforallstationsintandemandcomputedameanfield
wavelet as described above.Werepeated thisprocess 2000 times
andcomputedtheupper95thpercentile ofthewaveletpowersfor
eachcombinatio noftimeandf req uen cy.Thisp rovidedasetofcriti-
calvaluesforidentifying“hotspots”onthemeanfieldwaveletunder
thenullhypothesisofnotimescaledependenceinthefluctuations
ofthemeanannualtemperatureresidualsaroundthetrends.
3 | RESULTS
3.1 | Census data
Figure2showsthetimeseriesofmonthlycensusvaluesfortheWGS
andtheFS. The large increase in census numbers during2013and
2014corresponded with theproduction of a large acorn crop dur-
ingthefallof2013(mean±SEof608.3± 120.1 g/m 2ina1m2 plot
under each of six treesused toassess acornproduc tion, Appendix
B).Meanacornproductionmeasuredin thesame1m2plotsduring
otheryearsrangedfromalowof5.7± 2.7 g/m 2in2014toahighof
67. 5 +35.5g/m2in2012. Availabilityofacorns appearstohave a
majorimpactonthenumberofWGSsandFSsintheCBG.
The fluc tuations in census n umbers show signs of s ynchrony.
TheestimatedPearson correlation coefficientin animalnumbers is
R =.581whichisst at istica ll ysign if ic an tfromzero(p =5. 20× 10−1 4).
The total variation in the numbers for each species andtheir syn-
chrony see ms to be distribute d across differe nt timescales . Long
intervals can be seenwhere the numbers of both species areele-
vatedanddepressed(Figure2).Superimposedonthislongtimescale
variationarerandomshorttimescalefluctuations.Wequantifythis
timescalecomponentofvariationwiththe spectral analyses in the
nextsection.
3.2 | Population spectra and cospectrum
Figure 3a ,b show the smoot hed normalized sp ectra for the WG S
andtheFS. ForboththeWGS andtheFS,thespectra suggestthat
thelargestvariationinnumbersoccursatfrequenciesbelow0.0833
whichcorrespondstoatimescaleofmorethan12months.TheWGS
spect rum crosse st he upper sig nificance t hreshold a t timescal e of
around15months.TheFSspectrumcrossestheuppersignificance
thresholdattimescale ofaround 18 months.The spectrum forthe
FSshowsasmallpeakat6months,butthatpeakisnot statistically
significant. Since the total variation remains constant across fre-
quenciesfortheconfidencebandsfromtherandomlyordereddata,
thelargervariationinWGSandFSatlongtimescalesiscompensated
forbysmallervariationattimescalesofabout4monthsorless.
The smoot hed normalized cos pectrum (Figu re 3c) shows how
the totalcorrelation in populationnumbers between the two spe-
cies is distributed across timescales. Covariance between WGS
andFSissignificantlybiasedtowardlongtimescales,withasmaller
nonsignificant peak at a timescale of around 6 months. The co-
spect rum crosse st he upper sig nificance t hreshold a t timescal e of
about18months.Thetot alcorrelationbetweenthenumbersofthe
WGSandFSisR =.581.Usingtheunsmoothedcospec trum,wecan
partition this totalcorrelation by timescale intervals:R1 =.409 for
>12months,R2 =.162fo r 4–12m o nth s ,a n dR3 =.010f o r≤ 4mon ths , 
where R = R1 + R2 + R3.Thus,70%ofthetotalcorrelationoccursat
timesc ales exceedi ng one year. We can infer tha t population s yn-
chronyforthesetwospeciesoccursmostlyatlongtimescales.
FIGURE 2 MonthlytimeseriesfornumbersofWGSandFSatourstudysitefromOctober2009throughMay2021
SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDM
0
10
20
30
40
50
WGS
FS
Number of Animals
Month
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

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3.3 | Spectral analyses of weather data
Figure 4ashowsthetimeseries ofmeanmonthly temperaturesfor
OntarioAirport(ONT), whichis 12 km fromthestudy site.Asone
wouldexpect,thereisastrongseasonalcomponenttothesetem-
peratures.Figure 4bshows thesmoothednormalizedspectrumfor
the mean m onthly temper atures, which is d ominated by a stro ng
peak for theannual cycle. Since thesquirrelspectra show no indi-
cation of an a nnual cycle (F igure 3), we appli ed a band-s top filter
to remove the a nnual cycle and plo tted the resulti ng time series
(Figure4a,dashedline).Thesmoothednormalizedspectrumforthe
filteredmeanmonthlytemperaturesappearsinFigure4c.Thereisa
peakatlow frequencies whichcrossestheupperthresholdforsta-
tisticalsignificance at a timescale of approximately28 months and
reachesaminimumatatimescaleof12months.Thereisalsoalarge
spectralpeakat6months.
Thereisanegativecorrelationbetweenthefilteredtemperature
timeseries and the squirrelcensus data.For the WGS, the correla-
tionis statisticallysignificant(R = −.194 , p =.022)and,asindicated
bythe smoothed normalized cospectrum(Figure 5a),isdistributed
atlongtimescales(>35months)andatatimescaleof6months.The
correlationbetweenthefilteredtemperaturetimeseriesandtheFS
census dataisalso negative, but not statistically significant overall
(R =−.146,p =.085) .Thec ospe ct rumb et we enth efilteredtemper a-
ture time series andFScensus data shows a large significant peak
ata 6-month timescale(Figure 5b).These resultssuggest that the
distributionofvariationinthesquirrels’populationfluctuationsmay
bedriven,inpart,byfluctuationsinweatherandclimateoutsideof
theannualseasonalcycle.
3.4 | Simulation results
Our analyses of thesimulations of the Lotka–Volterra competition
model(1)aresummarizedinFigure6.Ourfocuswasontheef fects
ofthetimescaleofenvironmentalfluctuationsonthespectralprop-
erties of population numbers and the probability of extinction for
thenativespecies.
Figure 6a shows the protocolweusedforthe randomenviron-
mentalnoise.Weassumedalinearspectrumwhichvariedfromshort
timescale fluctuations (slope = 2, blu e noise), to fluc tuations wit h
noautocorrelation (slope=0,whitenoise),tolongtimescalefluc-
tuations(slope= −2,rednoise).Therandomtimeseriesgenerated
by these sp ectra have t he same mea n of zero and same v ariance,
thelatterbeingproportionaltothetotal areaunder the spectrum;
theydifferonlyintheirtimescaleproperties.Forthesimulationsin-
volvingthecomputationofspectr alexponent sandextinctionprob-
abilitie s, we varied the sp ectral slop e of the environmen tal noise
in101small increments from+2to −2, as indic ated by the curved
arrowinFigure6a.
Figure 6b s hows the popu lation spec trum and cos pectru m for
the simulations involving blue noise, white noise, and red noise
(Figure 6a).Sincetheparametervaluesforthetwocompeting spe-
cies are id entical, an d the prope rties of the ir environme ntal noise
inputsarethesame,themeancurvesshownapplytobothpopula-
tions. As described in section2.4, we used a cospectrumfunction
that was equal to a constant fraction, 0.9,of thespectrum, so, for
eachcolorofenvironmentalnoise,thepopulationspectrumandco-
spect rum are simil ar.For b lue environm ental noise , the smooth ed
normalizedspectrumandcospectrumhavelowpoweratlongtimes-
caleswhichincreasesandlevelsoffatfrequenciesexceeding.1.This
reflects the factthat, for theintrinsic ratesofincrease usedinthe
FIGURE 3 Smoothednormalizedspectrafor(a)theWGSand(b)
theFSandthe(c)smoothednormalizedcospectrumbetweenthe
twospecies.Dashedlinesare95%significancethresholdsforthe
nullhypothesisofnotimescaledependenceoftheWGS-FSpaired
observations
120 12 64
32
Timescale (months)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Normalized spectral power
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Normalized spectral power
0.0 0.1 0.2 0.30.4 0.5
Frequency (cycles/month)
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Normalized cospectral power
(a)
(b)
(c)

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simulations(r1 = r2 =.3),populationgrowth is undercompensating,
thatis,per turbationsfromastableequilibriumdonotshowdamped
oscillationsinthedeterministicversionofthemodel.Previouswork
forsingle-speciespopulationmodelshasshownthatundercompen-
sating populations are sensitive to long timescale environmental
noise,whereasovercompensatingpopulationsaresensitivetoshort
timescalenoise(e.g.,Danielian,2016).Ineffect,theslowerresponse
times of populations with small intrinsic rates of increase “filter
out” the short timescale components of the environment alnoise
(Desharnaisetal.,2018). Thisphenomenoncanalsobeseenin the
smoothednormalizedspectrumandcospectrumforthepopulations
subjectedtowhiteenvironmentalnoise.Thepopulationfluctuations
are le sssensitivetotheshortertimesc al ecomponent softh ef laten-
vironmentalspectrumproducingapopulationspectrumandcospec-
trumthatisbiasedtowardlongtimescales(Figure6b).Lastly,when
thepopulationsaresubjectedtoenvironmentalnoisebiasedtoward
long timescales,the longertimescale componentsofthe noiseare
enhanced and the shorter timescale components are suppressed,
producingasmoothednormalizedspec trumandcospectrumthatis
morestronglybiasedtowardlongtimescalesthantheenvironmental
noise(Figure6b).
Figure6cshowshowthespectralexponentsfortherealizations
of the popu lation fluc tuations an d environment al noise chan ge as
theenvironmental noise isshiftedgraduallyfromblue,towhite,to
red(arrowinFigure6a).Positivespectralexponentsindicatespectra
whicharebiasedtowardshorttimescales,andnegativespectralex-
po ne ntsare in dic ati ve of lon gt ime sca lef lu c tua ti ons .B oth thepo pu-
lationandenvironmentalspectralexponentsdecreasemonotonic ally
as the spectra for theenvironmental noise redden. However, the
populationspectralexponentstartsoutnegativewhiletheenviron-
mentalspectrumisstillstronglyblue.Asmentionedabove,withthe
modelparametervaluesusedinoursimulations,thedynamicsofthe
twocompetingspeciesact sasa“reddeningfilter,”producingpopu-
lationspectrathataremorebiasedtowardlongtimescales.
Of interest for conservation purposes is how the timescale
of the fluctuations in the environmental noise influences the
FIGURE 4 (a)MeanmonthlytemperaturefortheOntarioAirport(ONT )timeseriesfromOctober2009throughMay2021.Thesolidline
isfortherecordedtemperaturesandthedashedlineisthetimeseriesobtainedafterfilteringouttheannualcycle.Smoothednormalized
spectraareshownforthe(b)unfilteredand(c)filteredONTtemperaturetimeseries.Dashedlinesin(b)and(c)are95%significance
thresholdsforthenullhypothesisofnotimescaledependenceintheorderingofthefilteredandunfiltereddata
SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDM
10
15
20
25
30
Mean monthly temperature (°C)
Unfiltered
Filtered
Month
120126 43 2
Timescale (months)
0 0.1 0.2 0.3 0.4 0.5
Frequency (cycles/month)
0.00
0.03
0.06
0.09
0.12
0.15
Normalized spectral power
12012643 2
TImescale (months)
00.1 0.20.3 0.
40
.5
Frequency (cycles/month)
0.00
0.02
0.04
0.06
0.08
Normalized spectral power
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
(a)
(b) (c)

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persistenceofthen ativesp ecies.Fig ure6disbas edonsim ulations
whereanex tinctionthresholdhasbeensetarbitrarilyto5%ofthe
carr yingc ap acit y.Allothermodelp ar amete rvaluesareidentic alto
theonesusedforthesimulationsinFigure6b,c.Whenthecompe-
titioncoefficient sareequal(α = β =. 25),thee xtinctionpro bability
forbothspeciesremainssmalluntilthecoloroftheenvironmental
noisebeginstoredden(Figure6d).For thereddestenvironmental
spectrum,bothspecieshaveabouta55%probabilityofextinc tion.
If the non-native sp ecies has a comp etitive adv antage, the i nflu-
enceofreddenedenvironmentalspectraonthepersistenceofthe
native spe cies becom es more pron ounced. Fi gure 6d shows h ow
increasingthe competition coefficient for the invading species to
α =.50 and α =.75increasesthelikelihoodthatthenativespe-
cieswill be lost,whileslightlylowering the extinctionrisk for the
invasive spe cies. For, α = .75, a reddening of the environmental
spectrumquicklyelevatestheprobabilityofextinctionforthena-
tivespeciesfromavalueofabout3%forthebluestenvironmental
noise to avalue which asymptotesat about 94% for the reddest
environm ental noise (Fig ure 6d). This sugge sts the possi bility of
asynergybetween theeffects ofreddening environmentalnoise
andcompetitionfrominvasivespeciesfortheriskofextinctionfor
nativepopulations.
3.5 | Climate data
Weknowthathumanimpactontheclimatesystemhasresultedinan
increasingtrendofwarmingtemperatures(Masson-Delmotteetal.,
2021).Giventheobservationsandresults oftheprevious sections,
animport antrelated question is whetherthere havebeen changes
inthetimescaleofrandomenvironmentalfluctuationsaroundthese
trends.Ouranalysesmakeuseofa100 -yearrecord(1915–2014)of
mean annu al temperatures f rom 1218 weather statio ns obtained
fromtheU.S.HistoricalClimatology Network(Menne etal.,2009).
Figure7showsthelocationsoftheweatherstations.Althoughnot
uniformintheirdistribution,theycovereverystateandregioninthe
continentalUnitedStates.
Toinvestigateevidenceforchangeinthecolorofthemeanan-
nualtemperaturespectraovertime,wedividedthe100-yearrecord
from each station into four 25-year inter vals and computed the
spectralexponentsforeachtimeinterval(seesection2.5).Figure8
shows the histograms of spectral exponentsforthe1218stations.
Thedashedlinerepresentsthezerovalue(whitenoiseenvironmen-
talfluctuations); spectral exponentsto the leftindicate arednoise
bias and those to the right represent a blue noise bias. The arrow
atthe topof each histogramshowsthemean.Themeanvaluesare
0.500,0.313,0.432,and−0.160fortherangeof years1915–1939,
1940–1964, 1965–1989, and 1990–2015, respectively. It appears
thattherewasashiftfrom1990to2014fromblue-shiftedspectra
tored-shifted spectra. The significancevalues forthechangesbe-
tweenadjacenttimeintervalsarep =.046for1915–1939vs.1940–
1964 ,p =.654for1940–1964vs.1965–1989,andp =1. 676 × 10 −10
for1965–1989vs.1990–2015.
The spec tral anal yses conduc ted for Figu re 8 assume that t he
residualdeviationsfromthefittedquadratictrendsforeach25-year
time period are stationary,that is,the probability distribution and
timescaleproper tiesoftheresidualtimeseriesareinvariant .Amean
field wave let analysis w hich relaxe s the stati onarity as sumption is
presentedin Figure9for the entire 100 -yeartime period. There-
gionsofstatisticallysignificantwaveletpowerareoutlinedinblack.
Theyindicate that thetimescaleofthe fluc tuationsin mean annual
temperature, whenaveragedoverall weather stations, has shifted
tolongtimescalevaluesofapproximately3.5–7yearsfortheperiod
after1980,againsuggestingthattherehasbeenarecentreddening
ofthe timescale forrandomfluctuations inmean annual tempera-
turesaroundtheirchangingtrends.
FIGURE 5 Smoothednormalizedcospectrabet weenthefilteredOntarioAirporttemperaturetimeseriesandthecensusnumbersfor(a)
theWGSand(b)theFS.Dashedlinesin(a)and(b)are95%significancethresholdsforthenullhypothesisofnotimescaledependenceinthe
orderingofthetemperature,WGS,andFSdatatriplets
120 12 643 2
Timescale (months)
0 0.1 0.2 0.3 0.4 0.5
Frequency (cycles/month)
-0.04
-0.03
-0.02
-0.01
-0.01
0.00
0.01
Normalized cospectral power
120 12 64
32
Timescale (months)
00.1 0.20.3 0.
40
.5
Frequency (cycles/month)
-0.04
-0.03
-0.02
-0.01
-0.01
0.00
0.01
Normalized cospectral power
(a) (b)

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4 | DISCUSSION
Ourcospectralcorrelationanalysisdoesnotimplyadirectcausal
linkbet wee npopulationnu mbersofthetwospe cie softreesquir-
relsandambienttemperature.Therearemanyenvironmentaland
biotic fa ctors which inte ract to impac t the dynamics of n atural
populations. Although we used temperature asaproxy forenvi-
ronmentalfluctuations,thismetricisoftenassociatedwithother
environmentalvariables.Forexample,ZhaoandKhalil(1993)have
shownthatmeanmonthlytemperatureandtotalmonthlyprecipi-
tation are negatively correlated insummer months over most of
thecontiguous UnitedStates.Excellent timeseriesontempera-
tures are av ailable and the d ata lend thems elves to analysis by
spectralmethods.Timeseriesoftotalmonthlyprecipitationdata,
whileavailableforthesouthwestUniteStates,arenotwell-suited
foranalysisusingspectralmethods,astheycontainmanyconsec-
utivevaluesofzero.
Our spectralanalyses of the WGS and FS census datasuggest
that most ofthe variation in animal numbers occurson timescales
thatexceed15months.InthecaseoftheFS,thereisalsoevidence
forvariationon asix-monthtimescale.Thistimesc ale-specificvari-
ationmaybeduetochangesinresourceabundance,thetimingand
frequencyofreproduction,andreproductiveoutput.
Changesinpopulationnumbersonalongtimescalecouldbedue
tova ria tio ni nth esu ppl yoffo odr eso urc eso nm ult i-ye a r,h igh lyv ari -
able time scales. Fo r example, a corns provide a v aluable sou rce of
foodfortreesquirrels(Steele&Yi,2020),butaverylarge(>600g/
m2)mastcropwas onlyproducedinoneof the nine yearsin which
wemeasuredrelativeacornproduction(AppendixB).Weobserved
the prod uction of a ver y large mas t crop within o ur study ar ea in
the fall of 2013 (TableA2). Census counts for both species began
to increas e in the late sprin g and summer of 2013 an d continued
toincreasethrough thespringof2014(Figure2).Aprecipitous de-
creaseinabundancewasobservedthroughoutthesummerof2014
FIGURE 6 (a)Linearenvironmentalspectrausedforthemodelsimulations.Eachspectrumhasthesamevariance,butadifferent
distributionofvariationovertimescales.Thearrowindicateshowthespectraweregraduallychangedfrombluenoise,towhitenoise,to
rednoiseforthesimulationsinpanels(c)and(d).(b)Meansmoothednormalizedspectraandcospectraforthepopulationswiththeblue,
red,andwhiteenvironmentalnoiseshowninpanel(a).Bothcompetingpopulationshadthesameparametervalues,sotheirspectrawere
identical.(c)Meanspectralexponentsforthepopulations(solidline)andenvironmentalnoise(dashedline)withenvironmentalnoisecolor
variedcontinuouslyfrombluetowhitetoredasshowninpanel(a).Morenegativeslopesindicatelongertimescalefluctuationsinpopulation
numbers.(d)Probabilitiesofextinctionforthenativespecies(solidlines)andinvadingspecies(dashedlines)for2000simulationsofthe
modelwithenvironmentalnoisecolorvariedcontinuouslyfrombluetowhitetoredasshowninpanel(a).Largervaluesofαrepresent
higherintensitiesofcompetitionfromtheinvadingspecies
(a)
(c)
(b)
(d)

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which mayhavebeenbroughtaboutbydispersalofanimalsout of
our studysite. A ver y small acorn crop (<6 g/m2) wasproducedin
thefallof2014.Amodest-sizedcropofacorns(~35g/m2)produced
inthefallof2015wasfollowedbyanincreasein censuscountsfor
both spe cies throug h the summer of 2016. A corn produ ction was
verylowinthefallof2016,2017,2018,and2019,andthislongtime
FIGURE 7 Geographiclocationsof
the1218weatherstationsfromwhich
a100-yearrecord(1915–2014)ofmean
annualtemperatureswereobtained.Data
arefromtheU.S.HistoricalClimatology
Network
FIGURE 8 Spectralexponentsforthe1218timeseriesofmeanannualtemperatures.Each100-yearrecordwasbrokenintofour25-
yearinter vals(a–d).Thearrowsindicatethelocationsofthemeanvalues.Thedashedlinesrepresentaspectralexponentofzero.Positive
spectralexponentsindicatespectrabiasedtowardsshorttimescales(blue-tingednoise)andnegativevaluesindicateabiastowardslong
timescales(red-tingednoise)
-2 02
0
50
100
150
200
250
Frequency
-2
02
0
50
100
150
200
250
-2 02
Spectral exponents
0
50
100
150
200
250
Frequency
-2
02
Spectral exponents
0
50
100
150
200
250
1915-1939 1940-1964
1965-1989 1990-2014
(a)
(c)
(b)
(d)

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periodwithoutamodest tolarge-sizedacorncropcorrespondedto
relativelylowcensuscountsforbothspecies(≤20animals).Amod-
est acorn cropproducedinthefall of 2020 againcorrespondedto
anincreaseincensuscountsforbothspeciesduringthesummerand
fallof2020.Acornsarepresentinthe treesfor aprolongedperiod
before they appear in significant quantities on the ground, so this
foodresource isalsoavailabletotheanimalspriortothe fall ofthe
yearwhichmayaccount forthehighcensuscount sinthesummers
priortoouracorncropsamplingperiods.
Acornproductionbycoastalliveoaks(Quercus agrifolia),acom-
montreespecieswithinourstudyarea,isinfluencedbytheamount
of rain in th e one or two years p rior to the year in w hich acorns
areproduced (Koeniget al., 1996).MannandGleick(2015),aswell
as Diffe nbaugh et al. (2015 ), documented t hat an increase i n am-
bienttemperatures has beenaccompaniedby adecreasein rainfall
within C alifornia. A pl ot of temperatu re and precipit ation anoma-
lies over the period of 1895 through November 2014showed the
3-year period ending in 2014 was by far the hot test anddrieston
recordinCalifornia(Mann&Gleick,2015).Diffenbaughetal.(2015)
documentedthatalthoughtherehasnotbeenalargechangeinthe
probabilityofeithernegative ormoderatelynegativeprecipitation
anomalies in recent decades, the occurrence of drought years has
beengreaterinthetwodecadespriortotheirstudythaninthepre-
cedingcentur y.Inaddition,theprobabilitythatprecipitationdeficits
co-oc cur with warm co nditions an d the probabil ity that pre cipita-
tion defi cits prod uce drought have b oth increase d. Climate mo del
experiments by Diffenbaugh et al. (2015) revealed an increased
probability thatdry precipitation years are alsowarm years. Many
regions ofCaliforniawereinmoderate,severe,orextremedrought
conditionsformuchof2015through2021,exceptformostof2019
(NCEI website,accessed2Februar y 2022). So,the droughtislong
termandpersistedformuchofourstudy.
Theyearlyrecordofobservationsofjuvenileandsubadultindi-
viduals for bothspecies shown in Appendix C illustrates the effect
thatlong-termvariabilit yoffoodresources may haveonreproduc-
tionbythe WGSandtheFSoverlongtimescales.As statedabove,
productionofacornsvaried widelybetweenyearsandtheproduc-
tionofotherfoodsupplyitemscouldcertainlyvarywidelybetween
years. Vari ability in the av ailability of fo od items each year a long
withchangesin the numberofjuvenile and subadult animalscould
leadtopopulationvariabilityonlongtimescales,asobservedinthe
spectralanalysisofourdat a(Figure3).
The availa bility of food in ou r study site also va ried on a six-
month timescale.Itemssuchascatkinsfromoak and walnuttrees,
flowers on Fremontodendronspp.andArctostaphylosspp., andmale
cones on pin e trees bec ame availab le in the sprin g. Items such as
acorns, walnuts, and fruit bodies from the C alifornia Bay Laurel
(Umbellularia californica)andCaliforniaBuckeye( Aesculus californica)
becameavailable in thefall ofthe year (Or tiz & Muchlinski, 2015).
The timin gs (spring and fa ll) of the first av ailability of t hese food
itemsonayearlybasisfitwellwiththepotentialtimingofreproduc-
tiononayearlybasisbyboththeFSandWGS.
FIGURE 9 Meanfieldwaveletfor100-yeartimeseriesofmeanannualtemperatures.Thewaveletisavisualizationofhowthe
timescaledistributionofvariationinmeanannualtemperatureshasvariedfrom1915to2014.Each100-yeartimeserieswasquadratically
detrendedandaMorsewaveletwasobtainedusingtheMATLABdefaultvaluesof3and60forthesymmetr yandtime-bandwidthproduct,
respectively.Thewaveletsforthe1218stationswerethenaveraged.Thewhitedashedcurveistheconeofinfluencewhereedgeef fect s
canaffectthewaveletpower.Thedarkcurvesindicateareaswherethewaveletpowerisstatisticallysignificantatthe5%level,basedon
theupper95thpercentilesof2000surrogatedatasetswherethetimeseriesforallstationswerereorderedrandomlyintandem.Achange
tolongertimescalefluctuationsisindicatedbythesignificantshiftinwaveletpowertolowerfrequencies
20
10
6
5
4
3
2.5
Timescale (yrs)
1915 1939 1964 1989 2014
Year
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Frequency (cycles/year)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Wavelet power

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DESHARNAIS E t Al.
Two distinct periods of potential reproduction for the FS in
Southern California were documented by King’s (2004) study of
135Lsubmittedtothreewildliferehabilitationcentersduring20 02.
Approximately,60%oflitterproductiondocumentedbyKing(2004)
wasassociatedwiththemonthsofFebruary,March,andApril,with
the largest number of littersborn inMarch.A second pulse of lit-
ter production occurred during the months of August, September,
and October with thelargestnumber of litters borninSeptember,
sixmonths after the largest pulseoflittersborn duringthe spring.
Although production of litters by the FS on a semi-annual basis is
possible,thusleadingtoanincreaseinobservedpopulationsizeon
a semi-annual basis,t henumber of juvenile/subadult animals ob-
servedduringcensuscount sinthisstudyvariedwidelyamongyears
(FigureA1).
TheWGSappearstoe xhibitayearlypatternofr eproduc tiondif-
ferent th an the FS. Mos t research d ocument s breeding ac tivit y in
latefallandearlywinter monthswithbirth ofmostlittersin spring
and earl y summer months (C arraway & Verts , 1994; K ing, 2004).
A few pregna nt females wer e observe d in June, July, Augu st, and
Septemb er (Fletcher, 1963), and lactating females have been o b-
servedas late asOctoberinCalifornian(Swift, 1977). However,no
definite records of multiple pregnancies not attributable to intra-
uterine l oss of the first lit ter are availabl e (Bailey, 1936; Fletcher,
1963;Jameson& Peeters,1988; Swift,1977).The difference inre-
productivepatternsbetweentheFSandtheWGScouldbringabout
the pres ence of a 6-mon th cycle in abun dance of the FS and t he
absence of a si milar 6-month cycl e in the WGS. Th e differe nce in
reprodu ctive patte rns could also give a co mpetitive adv antage to
the FS in certain habitats through higher natalityin yearsof good
resourceproduction.
Muchlins ki et al. (2012) produce d a Habitat Suita bility Model
(HSM)fortheWGSandtheFSwhichallowedshort-termandlonger-
term coex istence habi tats to be id entified usi ng a linear com bina-
tion of three habitat variables: percent canopy cover, percent of
deciduoustrees,andaverageheightofground cover.Habitatswith
alowpercentageofcanopy cover,a high percentage of deciduous
trees,andalowheightofgroundcoverwereclassifiedasshort-term
coexiste nce habitat s. Location s with a high perce ntage of canopy
coverage, alowpercentage ofdeciduoustrees,andalow height of
groundcoverwereclassifiedaslonger-termcoexistencesites.(Sites
withahighheightofgroundcover,ahighpercentagecanopycover,
andalowpercentagedeciduous treeswere identifiedas“exclusion
habitat s” where on ly the WGS is foun d, but the FS e xists in ad ja-
cent habit ats.) For example, Muchlinski etal. (20 09) reported that
theFSreplaced the WGS in four yearsat a short-termcoexistence
habitat, California State Polytechnic University, Pomona, which
contain ed manicured an d more natural a reas on the cam pus with
paved pathwaysand buildings surroundedby a mixture of Juglans,
Eucalyptus,Washingtonia,Pinus,andothertreespecies.Incontrast,
thetwospecieshavecoexistedwithinlonger-termcoexistencehab-
itats of G riffith P ark in Los Ange les, CA , for more than 6 0 years,
whichweremorenaturalinappearanceconsistingofPinus,Quercus,
Umbellularia,Sequoia,andUlmusspecies,butwithhuman-influenced
aspectssuchaspicnictables,aplayground,andrestrooms(DeMarco
etal.,2020;King,200 4;Kingetal.,2010).ThestudyareaatCBGhas
been classified asa longer-term coexistence habitat by Muchlinski
etal. (2012).Howlongcoexistence cancontinueinlonger-termco-
existencehabitatsis unknown.Manylonger-term coexistence sites
arefragments of habitat wherethe FS, butnotthe WGS, exists in
surroundinghabitats.TheWGS isalsosubjecttolossofgenetic di-
versit y in these hab itat frag ments as des cribed by De Marco et al.
(2020).
The predictions of the competition model presented in sec-
tion 3.4 can be interpreted in terms of the HSM developed by
Muchlinskietal.(2012).TheHSMimpliesthat thecompetitiveef-
fe ctsofthe FS ont he WG Sar eh ighin as hor t-t erm co e xi s te ncesi te 
suchasC aliforniaStatePolytechnicUniversity,Pomona,andother
former lowland coexistence sites(Cooper & Muchlinski,2015).In
terms of the c ompetitio n model pre sented in Fig ure 6, the val ue
of the competition coefficient αwouldbelargerelativetothe
coefficient β, an d extinc tion of the WGS c ould occur un der con-
ditions of b lue and red env ironment al noise. Conve rsely, a lower
level of competitionin a longer-term coexistence site implies the
valuesofαandβaremoresimilarandahigherlevelofreddened
environmentalnoisewouldbeneededtobringaboutextinctionof
theWGS(Figure6d).Ourresultsfromsection3.5suggestthatcli-
matechangesareincreasingthetimescaleofyearlyenvironmental
fluctuations.Ourspectrala nalysesofmonthlycensusdatasu gge st
that mostofthe variation in numbers of the WGS and FSoccurs
over times cales of more than 15 m onths (Figure 2). Thu s, aside
fromt heef fec tsof awar mi ngclimate,a nychangesinthet im esc al e
oftemperaturefluctuationsaroundtheincreasingtrendcouldrep-
resent an additional riskfactor for thepersistenceoftheWGSin
someofitsnativerange.
After the annualchanges in meanmonthly temperaturewere
removedfromtheONTdat ausingaband-stopfilter,theremain-
ing variation in temperature fluctuations was composed of a
strongsix-monthcycleandsignificantvariationontimescalesthat
exceeded 28 months(Figure 4c). Meteorologists andclimatesci-
entistshaveusedharmonicanalysistodocumentsemi-annualcy-
clesinrainfallandtemperatureswhoseamplitudeandphaseshift
vary by geographical location, withmoderateamplitudes for the
southwestUnitedStates(Hsu&Wallace,1976;White&Wallace,
1978).AnalyzingNorthAmericantemperaturedatafrom1979to
2018, Nor th et al. (2021) used Baye sian analysis to f it a model
withannualandsemi-annualharmonicsthatvaryoverspaceand
time.Theyidentifygeographicalregionswithsignificantchanges
in the contributions of the two harmonics to seasonal cycles.In
Appendix D, we used le ast squares to fit a model with annual
andsemi-annual harmonics to the unfiltered meanmonthlytem-
peratu re data in Figu re 4a and show th at a model that i ncludes
both annual and semi-annual c ycles provides a significantly bet-
terfittothedata than a modelbasedontheannual cyclealone.
Figure4cshowsthat thefilteredtemperature timeseries hasthe
sameperiod,phase,andapproximateamplitudeasthetheoretical
semi-annualcycle.Onecanalsoseetheeffectsoflongtimescale

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DESHA RNAIS E t Al.
variabilityinthewaythefiltereddatameandersaboveandbelow
thesemi-annualcycle.ThecospectraofFigure5indicateasignif-
icantnegative correlation between thesquirrel census data and
theONTfilteredweatherdataatatimescaleofsixmonths.While
weca nno td emo nst r at ea dir ect cau sal me c ha nis mf ort hi scorr ela -
tion,thisobservationcouldmotivatefurtherresearch.
It was not pos sible to speci fy an est imated peak va lue for the
timescale of low-frequency variation in the squirrel numbers or
mean mont hly temperature s. The 140 months of the time series
representlessthan12years.Inspectralanalyses,estimatesoflong-
period,low-frequencyc ycles areless precise since theycannot be
asreadilyobserved asshort-period, high-frequencyoscillations. In
the estimated spectraofFigures3and 4, thespectral powercon-
tinuestoincreaseasthefrequenc ydecreases.However,therecord
ofmean annual temperatures forOntario Airport extendsback to
1999,providinga 22-year timeseries.InAppendixD,weshow that
asignificantpeakinthespectrumofannualtemperaturesoccurson
atimescale of about 7 years, which isconsistentwith thewavelet
analysis inFigure 9. If annual changesin environmentalconditions
aredrivingthelong-termvariationinsquirrelnumbers,whichseems
tobethec asefortheWGS(Figure5a),thisestimatecouldalsorep-
resentthetimescaleofthosefluctuations.
We presented s imulation resu lts in sect ion 3.4 that were de -
signedtoexploretheeffectsofchangesinthetimescaleofenviron-
mental noise ontheoutcomeofcompetitionbet weenecologically
similarnativeandnon-nativespecies.Weshowed thatanincrease
inthe timescale of environmental noisereddensthespectrumof
population fluctuations and decreases the likelihood of coexis-
tence, especially whenthe non-native is a better competitor.This
result dif fers fromone of the findingsofRuokolainenand Fowler
(2008), who concludedthat extinction risk decreased with a red-
deningofenvironmentalnoisewhen,likeinour model,therewas
astrong correlationin the species response to theenvironmental
fl u ctua tio n s.H owe ver,t hei r si m ula t ion pro t oco lsd iffe r edf r omo urs
inseveralways.First,theylookedatacommunit yofthreecompet-
ingspecies.Second,theirenvironmentalnoisewasgeneratedusing
an autoreg ressive proces s and was added to t he carryi ng capac-
ity for each species. Third, and most importantly, in their models
the intrinsic rate of increase for eachspecies was set to ri = 1 .8,
whereasinourmodelwechoser1 = r2 =.3,whichismoreconsis-
tentwiththereproductivecapabilitiesoftreesquirrels(Appendix
A).Indeterministic modelsofthetypeusedinoursimulationsand
those of Ruokolainen andFowler (20 08), values of 1 > r > 2lead
toovercompensatingdynamics,wheretheapproachtoequilibrium
exhibits damped oscillations. Although Ruokolainen and Fowler
(2008)claimedthattheirresultsarequalitativelysimilarforvalues
ofr <1,previousworkwithsingle-speciesmodels(e.g.,Danielian,
2016)hasshownthatextinctionriskincreaseswithareddeningof
environmentalnoise when thedeterministicmodel, like ours, has
undercompensating dynamics (ri <1),butdecreaseswithared-
deningof environmental noisewhenthedeterministicmodel has
overcompensatingdynamics(1< ri <2).Sincetheapparentcontra-
dictionbetweenourresultsandthose of Ruokolainen and Fowler
(200 8) occurred when th ere was a strong sy nchrony among th e
threespeciesduetothehighcorrelationoftheeffectsofenviron-
mentalnoise,theirresultisconsistentwithwhatonewouldexpect
from a singl e-species m odel. When we rep eated our simulat ions
usingvalues ofr1 = r2 =1.8,weobservedaresultconsistentwith
RuokolainenandFowler(2008).
Our simulation results were limited in their scope. They were
motivatedbyouranalysesoftimescaledifferencesinthevariability
ofpopulation fluctuations for two sympatric species of tree squir-
rels. Simulation analysesofthetypeconductedinthispaper could
beexpanded toincludea broaderexaminationof parameterspace,
other ecological interactions suchaspredatorand prey, and larger
communities of interacting species. Our approach could also be
adaptedtoappliedconservationmodelswhereenvironmentalvari-
abilityisapartofthesimulationprotocols.Theinferenceinsection
3.5 that timescale shifts in environmental fluctuations may be oc-
curring duetoclimatechange wouldbeamotivationtoexplorethe
impact s for populations of interest to conservationists and natural
resourcemanagers.
Our anal yses of spec tral expo nents for me an annual te mpera-
tures suggest that there has been a reddeningof the timescale of
climate fluctuations for the continental United States during the
time per iod 1990–2014. García-Carrer as and Reuman (2011) con-
ducted aglobal analysis ofspectral exponentsfor the time period
1911–1990.Theysplitthetimeseriesintotwohalvesandconcluded
that,whilemostofthespectralexponentswerered-shifted,thered
shiftwassmallerin1951–1990comparedwith1911–1950.Thiswas
trueforallcontinent sexceptA sia, which was redderin 1951–1990
than it was i n 1911–195 0. Thus, in gene ral, they obs erved a shif t
toshortertimescales in more recenttimes, whileweobservedthe
opposite.Thisinconsistencymaybeduetodif ferencesinour anal-
yses. García-Carrerasand Reuman(2011)useda linear func tion to
detrendtheirdata,whileweusedaquadraticfunction.Theydivided
their timeseries into twosegment sof40 yearseach, whilewedi-
vided oursintofoursegments of25years each.Most importantly,
our last timeseries segment covered 1990–2014 whichwent be-
yondthelatest yearthattheyexamined. Itwasinthislast 25-year
periodthatwesawastronglysignificantshiftfromblue-tingedfluc-
tuations tored-tinged fluctuations (Figure 8). Thisagrees with the
resultsofWangandDillon(2014)whoalsofoundanincreaseinthe
autocorrelation (timescale) ofannual temperaturesfromtheperiod
of1975through 2013.It may bethe case that the lengthening of
thetimescaleofmeanannualtemperaturefluctuationsisarelatively
recentphenomenon.
Our analyses of changes in climate fluctuations could be ex-
tendedinseveralways.ThedistributionsinFigure8showthatthere
isvariation inthevalues ofthespectral exponents amongweather
st ation s.The sam pl inglo cat io ns, wh ichar es c at ter edacr os sth ec on -
tinentalUnitedStates(Figure7),couldbesubdividedbyregion(e.g.,
northeastandsouthwest)toseewhethertherearesignificantdiffer-
encesinthevaluesofthespectralexponentsduetogeographiclo-
cation.FollowingGarcía-Carreras and Reuman (2011),ouranalyses
couldbeexpandedtoincludesamplingstationsonothercontinents

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DESHARNAIS E t Al.
andothermeasures,suchasmeanseasonaltemperatures,couldbe
analyze d. To tal annual pre cipitation coul d also be include d in the
analyses.Furtherresearchisneededtosee whetherourinference
thatclimatechangesarelengtheningthetimescalesforenvironmen-
talfluctuations is robust.This could have implications,notonlyfor
conservationbiologyandresourcemanagement,butalsoforother
areassuchasforestfiremanagementandagriculture.
5 | CONCLUSIONS
Using spec tral analyses, we have shown that the variations in
monthly fluctuations of population numbers for native and non-
native tree squirrels coexistingin the same habitat aredistributed
mostly ove r long timescal es (>15 mont hs) and their numbe rs are
synchronous over long timescales. After annual cycles are filtered
from the time series for mean monthly temperatures from nearby
OntarioAirport,thereremainsastrong six-monthcycleandsignifi-
cantfluctuationsattimescalesthatexceed15months.Therewasa
significant negativecorrelation between the temperaturedata and
squirrelnumbersforbothWGSandFSatasix-monthtimescaleand
a signific ant amount of long t imescale corr elation betwee n mean
monthly temperatures and WGSnumbers.Weused modelsimula-
tionstoshowthatenvironmentallyinducedlongtimescalevariation
in popula tion numbers fo r two competing s pecies with mod erate
rates of reproduction greatly increases the probability of extinc-
tion of the inferior competitor. Finally,we conducted spectraland
wavelet analyses for 100 yearsofmeanannualtemperatures from
1218weatherstationsacrossthecontinentalUnitedStates.Ourre-
sults suggest that thetimescale of fluctuations around the chang-
ingclimatetrendshas increased inthelast fewdecades,providing
anotherenvironmentalaspectofclimatechangethatcouldthreaten
themaintenanceofbiodiversit y.
Thisstudy documentslongtimescale variationinnatural popu-
lationsofconservationinterest,showsthatlongtimescalevariation
can accelerate theloss of species diversity, and provides evidence
thatthetimescalesofenvironmentalfluctuationshaveincreasedin
recenttimes.Wehopethisservesasacautionarymessageforcon-
servationist sandnaturalresourcemanagersthatanexaminationof
timescales for environmental andpop ulationfluctuations are fac-
torsworthyofconsideration.
ACKNOWLEDGMENTS
WethankadministrationandstaffattheCaliforniaBotanicGarden
forallowing our studytobeconductedattheirlocation.Wethank
Ka t y aEr k abae v aa n dAar o nOc a r izf o rth e ira s sis t a nce with t hef i eld-
work.R.A.D.thanksRichardM.MurrayattheCaliforniaInstituteof
Technologyforhisho spi talit ywh ilepor ti onsofth isworkwerecom-
pleted there. This research was supported in partbyU. S. National
ScienceFoundationgrantDMS-1225529toR.A.D.
CONFLICT OF INTEREST
Nonedeclared.
AUTHOR CONTRIBUTIONS
Robert A. Desharnais: Conceptualization (equal); Formal analysis
(lead); Methodology (equal); Software (lead); Visualization (lead);
Writing – original d raft (lead); Wri ting – review & e diting (equal).
Alan E. Muchlinski:Conceptualization(equal); Datacuration (lead);
Investigation (equa l); Methodolog y (equal); Project administration
(lead); Supervision (lead);Writing – original draf t(equal); Writing–
review&editing(equal).Janel L. Or tiz:Investigation(equal);Writing
– orig inal draf t (equal); Writin g – review & e diting (equal ). Ruby I.
Alvidrez: Investigation (equal); Writing – original draft (equal);
Writing – review & editing (equal). Brian P. Gatza: Investigation
(equal);Writing–review&editing(equal).
DATA AVA ILAB ILITY STATE MEN T
Squirrelcensusdat aarearchivedonDr yad(https://doi.org/10.5 061/
dryad.w6m905qqv). Weather and climate data are accessible at
the Climat e Data Online web site of NOAA’s National Ce nters for
EnvironmentalInformation(https://www.ncdc.noaa.gov/cdo-web/).
MATLAB code usedforcomputingand smoothing the spectraand
cospectraandgeneratingunivariateormultivariatetimeserieswith
the asymptotic spectral properties of a supplied spectral matrix is
availableonlineat:https://doi.org/10.5281/zenodo.5753478 .
ORCID
Robert A. Desharnais https://orcid.org/0000-0003-2874-6835
Janel L. Ortiz https://orcid.org/0000-0001-5912-1366
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APPENDIX A
INTRINSIC RATES OF INCREASE FOR SQUIRREL S
WeusedtheEuler-Lotkaequationtocomputetheintrinsicrateofincrease,r,fortheSciurusspeciesintable1ofWoodetal.(20 07).Forthis
equation,risthesolutionto
where lxisthesurvivalprobabilitytoagexandbxisthenumberofof fspringborntoanindividualof agex.Intheirpopulationviabilityanalysisof
treesquirrels,Woodetal.(2007)givevaluesforthepercentageoffemalesbreedingperseason,P,thenumberofbreedingsperyear,β,thelitter
sizeperfemale,L,andthemor talit yratesforagezero,m0,andindividualsoneyearandolder,m(TableA1).Foreachspecie s,bre edingbeginsatage
one.Giventhisinformation,wecomputedannualsurvivalratesasl1 =1–m0andlx =(1–m0) ( 1–m)x– 1 forx≥2.(Bydefinition,l0 =1.)Thenumberof
offspringperindividualwascomputedasb0 =0andbx = PβL/2forx≥1.Wedivideby2becauseonlyfemalesproduceoffspringandonlyhalfthe
litterarefemale.TheEuler-Lotkaequationbecomes
Takinglogarithms,weget.
Aroot-findingalgorithmwasusedtonumeric allysolveequationA2forr.
Woodetal.(2007)providedthreesetsofthelifehistor yvaluesdescribedforthefollowingspecies:Sciurus aberti,S. carolinensis,S. granat-
ensis,S. niger(FS),andS. vulgaris.Thethreesetsofparametersarelabeledas“optimistic,”“average,”and“pessimistic.”Usingthevalueslabeled
asaverage,wecomputedthevaluesofr,respectively,forthefivespecies(TableA1).Theaverageofthesefivevaluesis0.32.Therefore,we
usedavalueofr1 = r2 =.3inoursimulations.Thisisclosetothevalueof0.29obtainedforFS(TableA1).
APPENDIX B
NUMBERS OF ACORNS
Estimatesofacornproductionweremadeinyears2012through2020bymeasuringduringthemiddleofOctoberthemassofacornsinaone
squaremeterplotundereachofsixoaktreeswithinthestudyarea.Thesametreesandthesameplot swereusedeachyearforestimationof
aco rnpr od uctiontoas se ss va ri ab ilit ya mo ng year sandrelationshi ptochangesi nabu nd an ceoft he tr ee sq ui rr el s. Th edat aapp ea rinTab leA2 .
(A1)
∞
∑
x=0
e−rxlxbx=
1
∞
∑
x
=1
(P𝛽L∕2)(1−m0)(1−m)x−1e−rx =(P𝛽L∕2)(1−m0)e−r
∞
∑
x=0[(1−m)e−r]
x
=(P𝛽L∕2)
(
1−m
0)
e−r
(
1−(1−m)e−r
)
−1=1.
(A2)
ln [
(P𝛽L∕2)
(
1−m
0)]
−r−ln
[
1−(1−m)e
−r]
=
0
TABLE A1 ParameterestimatesfromWoodetal.(2007,table1,“average”)forfivespeciesoftreesquirrelsfromthegenusSciurus
Parameter Description Sciurus aberti Sciurus carolinensis Sciurus granatensis Sciurus niger Sciurus vulgaris
P%femalesbreeding 62 70 70 55 90
βbreedingsperyear 2 2 2 2 2
Llittersize(perfemale) 3.4 2.8 2.2 2.2 2.2
m0mortality(1styear) 0.60 0.60 0.60 0.60 0.62
mmortality(≥1year) 0.28 0.338 0.48 0. 28 0.32
rrateofincrease(year–1 )0.45 0. 37 0.13 0.29 0.36

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DESHARNAIS E t Al.
APPENDIX C
NUMBERS OF JUVENILE AND SUBADULT SQUIRRELS
StartinginJanuary2015,observersatourfieldsitestarteddistinguishingamongadults,subadults,andjuveniles.Thelattertwocategoriescan
serveasaproxyforreproduction.FigureA1showsthenumbersofjuvenilesandsubadultsfortheWGSandtheFS.Forbothspecies,juveniles
andsubadultsareseenmoreofteninspringandfall(FigureA1,shadedareas),especiallyinthecaseoftheFS.However,thereisagreatdeal
ofyear-to-yearvariation(longtimescales)inthenumbers.
APPENDIX D
ANNUAL AND SEMI- ANNUAL TEMPERATURE CYCLES
Meteorologists and climatologists haveused harmonicanalysis to identify seasonalcyclesin atmospherictemperature ( White &Wallace,
1978).Inadditionto astrongannualcycle,asemi-annualcycle, whichcanvarybyyearandlocation,hasbeen identified(Northetal.,2021;
White&Wallace,1978).Intheirequation(1),Northetal.(2021)usedthefirsttwotermsofaFourierrepresentationofanannualtemperature
timeseries.Theirequation,usingmonthsasthetimeunit,isgivenby
where xtisthetemperature(in°C),tisthetime(inmonths),a0isthecentervalueforthetemperatureoscillations(in°C),Aiistheamplitude(in°C)
andϕiisthephaseshift(inmonths)oftheannual(i =1)andsemi-annual(i =2)componentcycles.
Wefit equation(A3)tothemeanmonthlytemperaturedatafromOntarioAirport(ONT)(Figure4a) using themethodofnonlinearleast
squares.TheparameterestimatesandtheirstandarderrorsappearinTableA3.Thephaseshift sarerelativetothemonthofSeptember.The
temperature data andthe fittedfunction appearinpanel A of FigureA2. Thefitted functionprovidesa gooddescription of the observed
temperaturetimeseries.
ThetwocomponentcyclesareshownseparatelyinpanelBofFigureA2.Theannualcycleisthelargerofthetwoandisobtainedbysetting
A2 =0inequation (A3). Itpeaksin July–Augustandhasitstrough inJanuar y–February.Thesmaller semi-annualcycle,obtainedby setting
A1 =0 in equation (A3),peaks twice per year in February–March andAugust–September and has it stroughsinNovember–December and
May–June.Theamplitudeofthesemi-annualcycleis21%thesizeoftheamplitudeoftheannualcycle.
Inpanel(c)ofFigureA2,weplottedthetemperaturetimeseriesobtainedafterfilteringouttheannualcycle(Figure4a,dashedline)withthe
semi-annualcyclefrompanel(b)ofFigureA2.Thetheoreticalandfilteredcycleshavethesameperiodicity,phase,andapproximateamplitude.
Moreover,onecanseetheeffectsoflongtimescalevariabilit yinthewaythefiltereddatameandersaboveandbelowthesemi-annualcycle.
Thisgivesusconfidencethattheband-stopfilterusedtoattenuatetheannualcycleworkedwell.
Wefitequation(A3)to theobservedtemperaturetime serieswithA2 =0,sothatonlytheannualcyclewaspresent.Theparameteresti-
matesandtheirstandarderrorsarealsopresentedinTableA3.WecomputedthefollowingAkaikeInformationCriterion(AIC)forbothfitted
models:
(A3)
xt=a0+A1cos

2𝜋

t+𝜙1

12

+A2cos

4𝜋

t+𝜙2

12
,
AIC =nln
(
SSE
n
)
+2k
,
TAB LE A 2 Massofacorns(g)ina1m2plotundereachtreeinthestudyarea
Tre e
Year
2012 2013 2014 2015 2016 2 017 2018 2 019 2020
Blueoak 0250 683 0 7 12 098
Canyonoak 230 960 053 54 19 27 84 68
Coastliveoak 18 290 18 29 353 0 5
Coastliveoak 75 880 60 2 40 1 16
Coastliveoak 69 590 234 39 40 0 24
Coastliveoak 2680 210 14 0 0 0 136
Mean±SE 65.7±35.5 608.3± 120.1 5.7± 2.7 34.8± 12.3 18 .7 ± 9. 2 6.5± 2.7 7. 0 ±4.4 14. 2± 14.0 57. 8  ± 21.2

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where n =140ist hesam plesize,SSEistheresid ualsumofs quare sfo rth ere gression,andkisthenumbe roffit te dpa ra meter s(k =3forth ean nual
modelandk =5forthemodelwithbothannualandsemi-annualcycles).TheannualmodelhadanAICof165.02andthemodelwithbothannual
andsemi-annualcycleshadanAICof114.82.ThesmallerAICsuggeststhatthemodelwithbothcycliccomponentsisabetterdescriptionofthe
seasonaltemperaturechanges.Themagnitudeofthedifference,ΔAIC=50.20,suggeststhattheannualmodelhaslittlesupportrelativetothe
fullmodel.
APPENDIX E
MEAN ANNUAL TEMPERATURE DATA FOR ONTARIO AIRPORT
Weconductedaspectralanalysisofmeanannualtemperatures forOntarioAirport(ONT).Thedatavaluesaretheaveragesofthe12 mean
monthly temperaturesfor eachyear.The firstyearofcomplete data isfor1999,sothetimeseriesspans22yearsfrom1999through2021
(Figure A3, panel (a)).We used the same methodsasdescribedin section2.5forclimatedata: wedetrendedthedataby fit ting aquadratic
poly no mialusingl eas tsqu ar es,computedth es tan da rd iz ed re si duals ,andcomp ut edas mo ot hedno rm al iz ed sp ect rumfortheresidu alti me se -
ries.Becausethetimeseriesisrelativelyshort,weusedsmallerspansof3and3datapointsforthetwoiterationsofthesmoothingalgorithm.
Parameter Description
Estimate (±SE)
for full model
Estimate (±SE) for
annual model
a0centervalueofcycles(°C) 19. 5 0 ± 0 .12 19.4 9±0 .15
A1amplitudeofannualcycle(°C) 6.82± 0 .18 6.80± 0.21
ϕ1phaseofannualcycle(mo.) 1. 52±0.05 1.53±0.06
A2amplitudeofsemi-annualcycle(°C) 1.42± 0 .18 —
ϕ2phaseshiftofsemi-annualcycle(mo.) 0.48± 0 .12 —
TAB LE A 3 Parameterestimatesand
standarderrorsforfullandannualmodels
FIGURE A1 Monthlytimeseriesfornumbersofjuvenilesandsubadultsfor(a)theWGSand(b)theFSfromJanuar y2015throughMay
2021.Theshadedregionsrepresentspring(March–May)andfall(September–November)
JMMJ SN JMMJ SN JMMJ SNJMMJ SNJMMJ SNJMMJ SNJMM
0
2
4
6
8
Juveniles and subadults
Month
JMMJ SN JMMJ SN JMMJ SNJMMJ SNJMMJ SNJMMJ SNJMM
0
2
4
6
8
10
12
Juveniles and subadults
Month
2015 2016 2017 2018 2019 2020
2015 2016 2017 2018 2019 2020
(a)
(b)

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Wecomputed95%significancethresholdbygenerating2000randompermutationsoftheresiduals,obtainingasmoothednormalizedspec-
trumforeach,andusingthe95thpercentilesofthesesurrogatespectraforthe95%limits.FigureA3showstheONTtemperaturedataand
spectrum.Thefittedquadraticpolynomial(dashedlineinFigureA3,panel(a))isnearlylinearandsuggestsatrendofincreasingtemperatures.
Althoughthesignificancebandiswideduetotheshortlengthofthetimeseries,thespectrumfortheresidualsshowsasignificantpeakata
timescaleofabout7years(FigureA3,panel(b)).Thiscorrespondstothetimescalerangeofthelocalpeakinmeanwaveletpowerinthelower
rightcornerofFigure9.
FIGURE A2 (a)TimeseriesforobservedandfittedmeanmonthlytemperaturesfromOntarioAirport.(b)Plotsofthecomponentannual
andsemi-annualcyclesfromthefullmodel(equationA1).(c)Comparisonplotsofthefilteredtemperaturetimeseriesandthetheoretical
semi-annualcycle
SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDM
10
15
20
25
30
Monthly mean temperature (°C)
Observed
Fied
SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDM
10
15
20
25
30
Monthly mean temperature (°C)
Annual
Semiannual
SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDMJ SDM
15
20
25
Monthly mean temperature (°C)
Filtered
Semiannual
Month
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
(a)
(b)
(c)

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FIGURE A3 (a)Timeseriesforobser vedmeanannualtemperaturesfromOntarioAirport.Thedashedlineisthefittedquadratictrend
curvewhichisnearlylinear.(b)Smoothednormalizedspectrumforthestandardizedresidualsfromthetemperaturetimeseriesinpanel(a).
Dashedlineisthe95%significancethresholdforthenullhypothesisofnotimescaledependenceintheorderingoftheresiduals
2000 2005 2010 2015 2020
Year
17
18
19
20
21
Mean annual temperature (°C)
20 10 54
32
Timescale (years)
0.00.1 0.20.3 0.
40
.5
Frequency (cycles/year)
0.00
0.05
0.10
0.15
0.20
0.25
Normalized spectral power
(a) (b)