Submitted 18 November 2016Accepted 23 March 2017Published 26 April 2017

Corresponding authorWayne E Thogmartinwthogmartinusgsgov

Academic editorAnn Hedrick

Additional Information andDeclarations can be found onpage 14

DOI 107717peerj3221

Copyright2017 Thogmartin et al

Distributed underCreative Commons CC-BY 40

OPEN ACCESS

Density estimates of monarch butterfliesoverwintering in central MexicoWayne E Thogmartin1 Jay E Diffendorfer2 Laura Loacutepez-Hoffman3Karen Oberhauser4 John Pleasants5 Brice X Semmens6Darius Semmens2 Orley R Taylor7 and Ruscena Wiederholt8

1Upper Midwest Environmental Sciences Center US Geological Survey La Crosse WI USA2Geosciences and Environmental Change Science Center US Geological Survey Lakewood CO USA3 School of Natural Resources and the Environment and Udall Center for Studies in Public PolicyUniversity of Arizona Tucson AZ USA

4Department of Fisheries Wildlife and Conservation Biology University of Minnesota St Paul MN USA5Department of Ecology Evolution and Organismal Biology Iowa State University Ames IA USA6 Scripps Institution of Oceanography University of California San Diego La Jolla CA USA7Department of Ecology and Evolutionary Biology University of Kansas Lawrence KS USA8 Everglades Foundation Palmetto Bay FL USA

ABSTRACTGiven the rapid population decline and recent petition for listing of the monarchbutterfly (Danaus plexippus L) under the Endangered Species Act an accurate estimateof the Eastern migratory population size is needed Because of difficulty in countingindividual monarchs the number of hectares occupied by monarchs in the overwin-tering area is commonly used as a proxy for population size which is then multipliedby the density of individuals per hectare to estimate population size There is howeverconsiderable variation in published estimates of overwintering density ranging from69ndash609 million haminus1 We develop a probability distribution for overwinter densityof monarch butterflies from six published density estimates The mean density amongthe mixture of the six published estimates was sim279 million butterflies haminus1 (95CI [24ndash807] million haminus1) the mixture distribution is approximately log-normaland as such is better represented by the median (211 million butterflies haminus1) Basedupon assumptions regarding the number of milkweed needed to support monarchsthe amount of milkweed (Asclepias spp) lost (086 billion stems) in the northern USplus the amount of milkweed remaining (134 billion stems) we estimate gt18 billionstems is needed to return monarchs to an average population size of 6 ha Considerableuncertainty exists in this required amount of milkweed because of the considerableuncertainty occurring in overwinter density estimates Nevertheless the estimate is onthe same order as other published estimates The studies included in our synthesisdiffer substantially by year location method and measures of precision A betterunderstanding of the factors influencing overwintering density across space and timewould be valuable for increasing the precision of conservation recommendations

Subjects Conservation Biology Ecology Entomology Mathematical BiologyKeywords Mixture distribution Monarch butterfly Uncertainty modeling Danaus plexxipusDensity estimation

How to cite this article Thogmartin et al (2017) Density estimates of monarch butterflies overwintering in central Mexico PeerJ5e3221 DOI 107717peerj3221

INTRODUCTIONlsquolsquoI can see no other escape from this dilemma (lest our true aim be lost forever) thanthat some of us should venture to embark on a synthesis of facts and theories albeit withsecond hand and incomplete knowledge of some of themmdashand at the risk of makingfools of ourselvesrsquorsquo (Schroumldinger 1944 1)

Monarch butterflies overwintering in the high-elevation Oyamel fir (Abies religiosa)forests of centralMexico form spectacular aggregations thought to number in themillions ofindividuals per hectare (Urquhart amp Urquhart 1976 Brower 1977) The cool temperaturesof these high-elevation sites allow monarchs to slow their metabolism conserving lipidreserves for the approximately 5-month wintering period Clustering in densely packedcolonies on the lower branches of Oyamel fir trees also minimizes mortality during coldand rainy winter nights (Anderson amp Brower 1996 Brower et al 2009 Williams amp Brower2015) and increases humidity thus reducing evaporation and desiccation as the dry seasonadvances (Brower et al 2008) In early spring migration of monarchs over much of easternNorth America resumes from this location a multi-generational migratory phenomenonseen in few other insects

Since winter 1994ndash1995 World Wildlife Fund-Mexico (WWF) in collaboration withthe Mexican Secretariat of Environment and Natural Resources (Secretariacutea de MedioAmbiente y Recursos Naturales SEMARNAT) the National Commission for ProtectedAreas (Comisioacuten Nacional de Aacutereas Naturales Protegidas CONANP) and the MonarchButterfly Biosphere Reserve (MBBR) have monitored the overwintering populationThe winter monitoring consists of estimating the area over which these densely packedcolonies occur (Calvert amp Brower 1986 Garcia-Serrano Reye amp Alvarez 2004 Slaybacket al 2007 Vidal Loacutepez-Garciacutea amp Rendoacuten-Salinas 2014 Vidal amp Rendoacuten-Salinas 2014Rendoacuten-Salinas amp Tavera-Alonso 2014) Occupied trees are mapped in each colony andthe perimeter of the colony is measured The enclosed area is then calculated in hectaresoccupied and used as an index of population size

This monitoring of the Eastern population of monarch butterflies (Danaus plexippus) inNorth America suggests large declines in the wintering population size over the last decadeand a half (Semmens et al 2016) The largest population size recorded since monitoringbegan in the early 1990s was 1819 ha in winter 1996ndash1997 Since this peak in abundancemonitoring suggests that the population has declined by over 90 (Brower et al 2012Vidal amp Rendoacuten-Salinas 2014 Rendoacuten-Salinas amp Tavera-Alonso 2014) to a record low of067 ha in winter 2013ndash2014 (Rendoacuten-Salinas amp Tavera-Alonso 2014) These declines inabundance are believed to be due in large part to declines in habitat availability in thebreeding range of the north-central United States principally through loss of commonmilkweed (Asclepias syriaca) in agricultural crops (Pleasants amp Oberhauser 2013 Pleasants2015 Pleasants 2017) as well as forest degradation in the Mexican overwintering habitat(Brower et al 2016)

In 2014 due to concerns over these overwintering population declines the US Fishand Wildlife Service was petitioned to list monarchs as a threatened species under the

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 218

Endangered Species Act (Center for Biological Diversity et al 2014 docket number FWS-R3-ES-2014-0056) The agency subsequently initiated a status review to determine whetherlisting for the entire species was warranted The White House announced a strategic goalof increasing the eastern population of the monarch butterfly to 225 million butterflies by2020 (Pollinator Health Task Force 2015) This 225 million butterfly goal was motivatedin part by the premise that 225 million butterflies equated to 6 ha of habitat occupiedby monarch butterflies in overwintering sites (Pollinator Health Task Force 2015) or 375million butterflies haminus1 The magnitude of this target for the Eastern migratory populationof monarch butterflies has important implications for the estimated level of restorationeffort (ie increasing milkweeds) needed to sustain the population in eastern NorthAmerica Therefore accurate determination of the overwintering population size not justthe area over which it occurs in winter is a critical step in determining the magnitude ofthe conservation challenge

To translate from colony extent to population numbers estimates of the area occupiedby overwintering aggregationsmust bemultiplied by an estimate of density (monarchsunitarea) Current understanding of the overwintering densities of monarch butterflies in theseaggregations comes from a handful of published sources principally Brower et al (1977)Brower et al (2004) and Calvert (2004) Brower et al (1977) and Tuskes amp Brower (1978)used density estimates from capture-mark-recapture for California overwintering coloniesfor a rough estimate of abundance inMexico Theymultiplied the density estimate for SantaCruz California which was 95000 butterflies haminus1 by 15 to account for the difference inarea covered by Mexican and Californian colonies and again by 10 to account for theirsuggestion that California colonies were 10 of the density of the Mexican colony Theirsuggestion of 1425 million monarch butterflies occupying 15 hectares in one location inMexico was deemed lsquolsquoa conservative estimatersquorsquo of 95 million monarch butterflies haminus1 Itwas not until nearly a quarter-century later that attempts at calculating density using on-sitemeasurements made in Mexico were published Calvert (2004) used two approaches withcapture-mark-recapture data to estimate population densities of 21 to 100millionmonarchshaminus1 with higher densities occurring later in the season when the colony had contractedAt a different colony Calvert (2004) measured monarch density on a sub-sample of treebranches and trunks and generated an estimate of 103 million monarchs haminus1 Brower etal (2004) took a different tack extrapolating the density of monarchs killed during a winterstorm (in January 2002) at two sites and obtained estimated densities of 53 and 73 millionmonarchs haminus1 for the two sites for a mean estimate of 65 million monarchs haminus1 Thismean estimate was subsequently revised down to 50 million monarchs haminus1 (Slayback etal 2007) to be more conservative (L Brower pers comm 2016) Obviously considerablevariation exists in the estimates of overwintering densities which has important policyramifications for their use by groups working together to chart a strategy for protectingmonarchs including the US Fish and Wildlife Service other partner agencies in the USMexico and Canada and non-governmental actors

Accurate estimation of the density of monarch butterfly populations overwinteringin Mexico provides critical information for determining the abundance of the Eastern

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 318

Table 1 Densities (in millions haminus1) of monarch butterflies overwintering in central Mexico bymethod and source with the estimated standard deviation

Method Publication Date of study Density SD

Petersen capture-mark-recapture Calvert (2004) Late-Dec 1985 69 12Jolly-Seber capture-mark-recapture Calvert (2004) Early-Jan 1986 338 13Storm mortality-based (Zapatero) Brower et al (2004) Mid-Jan 2002 184 201Storm mortality-based (Conejos) Brower et al (2004) Mid-Jan 2002 159 244Petersen capture-mark-recapture Calvert (2004) Late-Jan 1986 609 12Branch extrapolation Calvert (2004) ca Early-Feb 1979 103 21

migratory population Uncertainties in these density estimates may arise from manydifferent sources such as natural variability in the monarchrsquos response to environmentalconditions including their own population numbers and variability in environmentalstressors over time and space (Williams amp Brower 2016) Incomplete knowledge regardinga situation or variable often occurring as a result of measurement (or observation) errorunstated assumptions or extrapolations also contributes uncertainty in density estimatesMethods have been developed for estimating variables when faced with stochastic variationand incomplete knowledge (McLachlan amp Peel 2000 Zadeh 2002 Pearson 2011) Based onthe form of available information probability theory can be used to incorporate parameteruncertainty and variability into an expected value distribution (Shapiro 2009) The expectedvalue distribution describes the distribution around the expected value or the weightedaverage of all possible outcomes Here we used finite mixture distribution modeling toderive the expected distribution of monarch overwintering density from the estimates ofCalvert (2004) and Brower et al (2004) We then applied the median estimated abundancefrom that distribution to a corrected time series of overwinter abundance (Semmens et al2016) to understand the magnitude of population change since systematic monitoring forwintering monarchs began in the early 1990s We also propose an environmental correlateto changing within-season density Additionally we used our estimated density to estimatethe amount of milkweed needed to sustain the 6-ha goal population

METHODSWe calculated the fuzzy random variable for monarch density using two general stepsFirst for each of six available estimates of density we calculated the uncertainty aroundthe estimated central tendency (estimated mean or reported value) and then modeledthis using a lognormal distribution (except in one case where extreme values required anextreme value distribution) Patterns in species abundance are often lognormal (Sugihara1980 Limpert Stahel amp Abbt 2001) This resulted in five lognormal distributions and oneextreme value distribution each centered on the original point estimate of density (Table1) We then combined the six distributions to estimate a new combined distribution ofdensity incorporating putative levels of uncertainty in the underlying reported estimatesThe six available estimates of density come from two sources four estimates from Calvert(2004) and two from Brower et al (2004)

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 418

Calvert (2004) used the Petersen and Jolly-Seber capture-mark-recapture methodsfor calculating mean and 95 confidence interval estimates of overwinter density fromthe capture records of tagged butterflies in late December 1985 and mid-January 1986The reported confidence intervals were asymmetrical We approximated the populationstandard deviations on the natural log scale from these confidence intervals according toσ =

(log(mean)minuslog(lower 95 limit )

211 +log(upper 95 limit)minuslog(mean)

211

)2 where 211 is the t critical

value under the assumption of a small sample size Using the reported mean and estimatedstandard deviations we fit both gamma and lognormal distributions (Wilks 2006) thelognormal distribution fit the published confidence intervals best (matched most closely)and was used in the analysis

Calvert (2004) also estimated overwinter density from a sample of monarch butterfliescollected from branches and tree trunks in 1977 He used 12 branches of varying sizeto regress monarch abundance against branch size and then used data on tree structureto estimate the average size and number of branches for trees of different sizes He thenmeasured the number of monarchs on 17 different tree trunks He summed lsquolsquocrownmonarchsrsquorsquo (on branches) and lsquolsquotrunk monarchsrsquorsquo to obtain an estimate of monarchs pertree This branch-based estimate of density was reported without a measure of associatedvariance Therefore we treated Calvertrsquos branch-based estimate as a Fermi approximation(Machtans amp Thogmartin 2014) a rough estimate for a difficult-to-estimate quantity andinferred the variance to be a function of the number of parameters in the branch and trunkcalculations We assumed this estimate was correct within a factor of two this range givesthis estimate similar precision as the other January estimates of density Thus given thisassumption we calculated the upper limit in the crown and trunk estimates as 2

radicntimesmean

of the crown and trunk estimate respectively where n equaled the number of parametersused in the calculation of the estimate The lower limit was similarly calculated but withthe inverse of 2

radicn as the multiplier Based on our reading of Calvert (2004) we surmised

there were five parameters in the crown estimate (diameter at breast height crown massbranches per crown tree density monarch weight) and three in the trunk estimate (surfacearea of a column monarch sample density tree density) Once the lower and upper limitsin the estimates were established the standard deviation was estimated as above for Calvertcapture-mark-recapture data

Brower et al (2004) reported estimated densities from two colonies (Zapatero andConejos) in mid-January 2002 and thereafter assumed a midpoint as the nominal meandensity of colonies We were provided the original data used in extrapolating stormmortality observations to hectare-scale density estimates Details of data collection areprovided by Brower et al (2004) but briefly the data comprise of counts made of deadand moribund individuals observed in 29 02 m times 02 m plots in each of the two coloniesThe mortality data are highly skewed particularly those data from the Conejos colony(Appendix S1) for example gt1000 dead and moribund monarch butterflies were countedin a single 004 m2 plot For the Zapatero colony we fit a lognormal distribution to theobserved counts whereas for the highly skewed Conejos colony we fit a generalized extremevalue distribution The generalized extreme value distribution is characterized by meanE[X ] = ζ minusβ [1minus0(1minusκ)]κ and variance Var[X ] = β2(0[1minus2κ]minus02[1minusκ]

)κ2

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 518

(Wilks 2006 87) where κ is a shape parameter ζ is a location or shift parameter and βis a scale parameter Because these distributions provide the expected count for a 004 m2

area we then extrapolated this distribution to the hectare scale (by multiplying by 250000)to make them commensurate in scale with the other published estimates of density

The six density methods differed substantially in their reportedmeans Our estimates arebased on measurements from three years (1979 1985ndash1986 2002) These estimates (asidefrom one) are presumptively drawn from lognormal distributions Elementary probabilitytheory cannot describe the distribution of the sumof lognormals (Dufresne 2004) Thus werelied on mixture distribution modeling Absent any data on the precision of the differentmethods we developed a mixed probability density function g as an equal-weighted sumof k component densities g (χ |xmicroσ )= π1f (χ |micro1σ1)+middotmiddotmiddotπk f (χ |microkσk) where k isequal to the six distributions described above We created this distribution by drawing106 samples randomly from each distribution and then combining the drawn samplesMeasures of central tendency and 25 and 975 quantile estimates were derived fromthe resulting mixture distribution

Temperature and humidity play an important role in monarch overwintering behaviorand roost suitability (Anderson amp Brower 1996 Brower et al 2008 Brower et al 2009Brower et al 2011) For instance freezing dew on the surface of exposed butterflies canlower their supercooling resistance ostensibly because ice crystals invade the spiracles ofthe butterflies providing nucleation centers in the supercooled body fluids which thenmay freeze (Anderson amp Brower 1996) With the published density estimate ascribed tothe reported day of year (either the midpoint or endpoint) we quantified the observationthat monarch butterflies pack more tightly with decreasing winter temperature (Brower etal 2011 Vidal amp Rendoacuten-Salinas 2014) by regressing the observed pattern of density todaily mean temperature and daily mean dew point (for the closest location for which thesedata were available Toluca Mexico for the period 1977ndash2014 US National Climatic DataCenter (NCDC) Global Summary of the Day httpswww7ncdcnoaagovCDOcdodownloaded Tue Dec 22 094255 EST 2015) Dew point is the temperature (varyingaccording to atmospheric pressure and humidity) below which water condenses dew pointis often correlated with minimum temperature especially when humidity is high

Once we derived the relevant density estimate we calculated total abundance ofmonarchs and the associated milkweed (Asclepias spp) required to sustain them NailStenoien amp Oberhauser (2015) used Monarch Larva Monitoring Program egg density andsurvival data to estimate the number of milkweed plants needed to produce an adultmonarch migrating to Mexico Their calculations (see their equation 2) resulted in anestimate of 285 milkweed stems per monarch necessary to produce one adult for the fallmigration With this estimate from Nail Stenoien amp Oberhauser (2015) we translated the6 ha overwintering goal for monarchs into numbers of milkweed stems under averageclimatic conditions (Pleasants 2017)

We also applied the derived density estimate to the time series of overwinteringabundances to estimate potential change in monarch population size through timeSemmens et al (2016) used a state-space formula for estimating a corrected time seriesof the areal estimate of the overwinter population size This state-space formula enabled

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 618

Figure 1 Presumptive distributions for estimates of overwinter monarch butterfly density in centralMexico (A) Presumptive distribution for structurally defined density used by Calvert (2004) The verticalgray line is the published estimate (B) Three capture-mark-recapture methods for estimating density asreported by Calvert (2004) pertinent to the mid-January to early February sampling period Vertical graylines are the published estimate (C) Presumptive distributions for storm-mortality method of density es-timation used by Brower et al (2004) from two colonies Zapatero (solid line) and Conejos (dashed line)median densities are depicted in gray

estimation of the underlying true state of the population corrected for observation noiseWe multiplied the corrected estimate of areal overwinter population size by the median ofthe mixture distribution to calculate the annual monarch butterfly population abundance(given constant annual density)

RESULTSReconstructed distributions matched the means reported by Calvert (2004) the Januarycapture-mark-recapture distributions showed high levels of uncertainty and overlappedeach other but were significantly larger than the January branch-basedmethod (Fig 1) TheJanuary branch-based method roughly coincided with the mean of the December capture-mark-recapture distribution The Brower et al (2004) storm-mortality approach and theCalvert (2004) January capture-mark-recapture methods reported results two to four timeshigher than the Calvert (2004) branch and December Petersen capture-mark-recapturemethods (Fig 1)

The mixture distribution was roughly lognormal in shape (Appendix S2) with apronounced spike due largely to the branch-based and December Petersen capture-mark-recapture distributions (Fig 2) The mean and median of this distribution were 278and 211 million butterflies haminus1 respectively (25 quantile= 24 975 quantile= 807million butterflies haminus1)

Density regressed against temperature dew point and day of year most stronglysupported a negative relationship between density and dew point (Table 2 and Fig 3A)

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 718

Figure 2 Amixture distribution equally combining the individual distributions from the Jolly-SeberDecember and January Petersen Branch and Brower stormmortality methods (means of the underly-ing distributions are denoted by the blue lines)

Table 2 Density of monarch butterflies overwintering in central Mexico regressed against daily meantemperature daily mean dew point and day of year where 12 Dec= 1 20 Dec= 8 1 Jan= 21 and1 Feb= 52 K is the number of parameters AICc is the small-sample Akaikersquos Information Criterion1AICc is the difference between the best model and the focal model (AICciminusminAICc) AICc ω is themodel weight or conditional probability of the model relative to the other models in the model set and LLis the log-likelihood

Variables K AICc 1AICc AICc ω LL

Dew point 3 6390 0 087 minus2295Day of year 3 6886 496 007 minus2543Temperature 3 6940 550 006 minus2570Temperature+ Dew point 4 9239 2849 0 minus2220Dew point+ Day of year 4 9344 2954 0 minus2272Day of year+ Temperature 4 9857 3467 0 minus2528

When dew point is nearest 0 C monarch density is predicted to be greatest Temperatureand dew point in central Mexico are both lowest in mid-January (Fig 3B)

Using the mixture distribution from the full set of density-estimation methods andassuming constant annual density during winter monitoring the time series of overwinterpopulation size suggested monarch butterflies may have numbered 310 million individualsin winter 1996ndash1997 and dropped to as low as 37 million in winter 2013ndash2014 nearly anorder of magnitude difference (Table 3 and Fig 4) The mean annual abundance over this20-year period was 119 million butterflies (95 CI [69ndash212] million)

A 6-ha population goal using the median of the six-estimate mixture distributionequated to a mean of 127 million monarchs (6 ha times 211 million haminus1) Assuming 285milkweed stems are needed to produce a single monarch (Nail Stenoien amp Oberhauser2015) 127 million monarchs equaled sim362 billion stems (127 million monarchs times 285milkweedmonarch)

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 818

Figure 3 (A) Overwinter density of monarch butterflies as a function of mean daily dew point (C) (B)Boxplots (median and 1st and 3rd quartiles with 95 confidence interval whiskers) of observed dailydew points for each winter month over the period 1977ndash2015

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 918

Table 3 Observed and fitted area of monarch butterflies overwintering in central Mexico with associ-ated predicted population size (in millions of individuals) and 95 credible interval

Start year Observed Ha Fitted Haa 50 250 9750

1993 623 679 1554 725 34261994 781 824 1863 954 37511995 1261 1200 2671 1330 52981996 1819 1380 3103 1524 60611997 577 677 1558 818 32081998 556 602 1366 713 26741999 897 695 1512 1027 22712000 383 515 1104 656 15992001 936 700 1516 1036 23592002 754 511 1088 755 19742003 1112 532 1128 780 22942004 219 291 653 451 9832005 591 415 903 626 14822006 687 464 1004 693 16692007 461 418 909 629 14012008 506 337 724 503 12212009 192 252 564 378 8282010 402 372 825 554 11952011 289 318 716 463 10352012 119 204 466 251 6872013 067 159 371 169 5422014 113 217 509 228 757

NotesaSee Semmens et al (2016) for details and credible intervals

DISCUSSIONBased upon reasonable assumptions regarding distributional form and characteristics ofavailable published data we suggest the preponderance of evidence supports a medianoverwintering density of monarchs of 211 million butterflies haminus1 (25 quantile =24 million butterflies haminus1 975 quantile = 807 million butterflies haminus1) The fewobservations we have suggest monarch density changes with dew point likely increasing indensity from late-December to a peak inmid-January as the temperature cools This changeover time in density coincides with the observations of Vidal amp Rendoacuten-Salinas (2014)Vidal amp Rendoacuten-Salinas (2014) suggested clusters of overwintering monarchs disaggregatewhen temperature increases which is coherent with the temperature-density relation wereport

Our analyses of the Brower et al (2004) samples for Conejos and Zapatero coloniesled to considerably different conclusions than had we used the estimate they reportedIf we had used their 50 million haminus1 estimate as one of five (not six) samples in ouranalyses we would have estimated a mixture distribution with a mean and median of 457million butterflies haminus1 respectively (code for analysis available in Code SupplementalInformation 1) This estimate is twice our estimatedmedian (211million haminus1) The reason

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 1018

1Similar to a return to the 1998ndash1999overwinter population which was 602ha and 137 million monarchs [71ndash267million] (Table 2)

Figure 4 Annual population size (with 95 CI) by year winter starts for monarch butterflies over-wintering inMexico The black line and associated blue confidence band depict patterns in annual abun-dance according to the full mixture distribution (ie mean density of 207 million haminus1) The dashed grayline is an upper-end quasi-extinction risk threshold (025 ha) described by Semmens et al (2016)

Brower et al (2004) concluded a density estimate of 50 million haminus1 rather than the 159and 183 million haminus1 we report is because they used the mean to characterize the skeweddistributions of their samples a mean skewed to higher values by a few outlying estimatesSamples from both the Conejos and Zapatero colonies are small (n= 29) and skewed bya few very large counts (Appendix S1) For sufficiently large samples the distribution ofthe sample mean is approximately normal according to the Central Limit Theorem At thesample sizes reported by Brower et al (2004) we cannot be assured that the Central LimitTheorem holds The preponderance of the observed data and the mass of the resultingdistributions fitted to those data are considerably smaller than the mean (eg median forConejos colony was 86 butterflies per sample versus a mean of 290 butterflies per sample)As a result the values we drew from the fitted distributions for Conejos and Zapaterocolonies led to a lower expected density than if we had used the higher published estimate

The North American Climate Clean Energy and Environment Partnership Action Plan(Trudeau Obama amp Nieto 2016) identified the restoration target of a 6-ha overwinterabundance of monarch butterflies as a goal to be achieved by Canada the US and Mexicoby 2020 This 6-ha target is equivalent to approximately 127 million monarchs1 (witha putative 95 confidence interval of 14ndash484 million) according to estimated medianoverwinter density One of the central outstanding questions for conserving monarchs ishow much milkweed must be planted to create sufficient habitat to support this targetpopulation size Pleasants (2017) suggested the loss ofmilkweed in theNorth Central regionof the US between 1999 and 2014 amounted to 862 million stems with an additional 134billion stems remaining Pleasants (2017) argued that milkweed in corn and soybean fieldsproduced 39 times more monarch eggs than milkweeds in non-agricultural habitat (see

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 1118

Pleasants amp Oberhauser 2013) and that therefore the loss of Midwestern agricultural fieldswas especially hard hitting to monarchs he suggested that the loss of 850 million stems incorn and soy fields amounted to the equivalent of 331 billion non-agricultural stems ofmilkweed Similarly we estimate a 127 million monarch population would require 362billion stems of milkweed with 134 billion stems remaining in the landscape the milkweeddeficit could be as high as 228 billion stems orsim700 million more stems than was neededaccording to Pleasants (2017) to return the population to 6 ha of occupied overwinteringhabitat Alternatively if we subtract the estimated amount of milkweed needed for a 126million monarch population (ie 6 ha) from the equivalent needed for what Pleasants(2017) estimated remains (315 ha or sim70 million monarchs) we obtain 18 billion stems(95 CI [07ndash470] billion stems) or 200 million stems more than estimated by Pleasants(2017) These calculations assume a linear relationship between monarch and milkweedabundance but Pleasants (2017) demonstrated it was linear on the log-scale (ie Numberof milkweed stems = e012timesOW[ha]+724 where OW[ha] is number of hectares of butterfliesoverwintering in Mexico) indicating that our estimates of the milkweed deficit provideliberal upper bounds on what may be required

Clearly despite our best efforts at synthesizing the available information pertainingto overwinter density there remains considerable uncertainty in the estimated densitieswhich in turn influences uncertainty in subsequent calculations of population size andassociated levels of milkweed needed to sustain the species Mixture distributions areoften developed when data are believed to arise from more than one generation processor physical mechanism Densities of overwintering monarchs reported by Calvert (2004)and Brower et al (2004) may have differed for climatic seasonal behavioral populationsize or habitat-related reasons Our climate regression suggested approximately half of thevariation in density was attributable to variation in temperature-related climate variationRemaining variation is likely to be explained by other factors For instance Calvertrsquos studieswere conducted in 1979 and 1985ndash1986 whereas Brower et alrsquos data were collected in 2002Similarly Calvert collected data from the El Picacho and Sierra Chincua colonies whereasBrower et alrsquos findings came from the Zapatero (also known as Sierra Chincua) and LosConejos (also known as El Rosario) colonies these colonies occur across the CorredorChincua-Campanario-Chivati-Huacal (Garcia-Serrano Reye amp Alvarez 2004 Fig 2 inSlayback et al 2007) possibly contributing to environmental variation in density Furtherthe methods used to generate these density estimates differed substantially ranging fromcapture-mark-recapture methods to extrapolations based upon mortality estimates andstructural characteristics of Oyamel fir trees

At this time we have too little information to posit an advantage of one densityestimation method over another but it is likely that a combination nay a mixtureof reasons contributed to differences in estimates As a result the best we can do isacknowledge the extent of uncertainty in our estimates The mixture distribution wederived provides a reasonable articulation of the uncertainty associated with overwinterdensity estimates The magnitude of uncertainty in the estimated density suggests the meandensity is known within no better than a range of 13 to 3 times the expected value Ourestimates of uncertainty however may change with changes in assumptions If we had

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 1218

assumed the branch-based estimate of Calvert (2004) was correct within a factor of 3 4 ormore for instance the uncertainty in this estimate would have also contributed additionaluncertainty in the final estimates We assumed density might change both within andamong years stochastic variation around an unvarying mean density would not alter ourconclusions however Systematic change in mean density though especially as populationsize declines could have serious consequences on our inferences Longtime observers of theoverwintering colony (K Oberhauser pers obs 2016 and L Brower pers comm 2016)have suggested in recent years that monarchs are less densely packed on trees at the edgeof the colony compared to trees in the center Smaller colonies with a higher ratio of edgeto inner trees could lead to uniformly less dense colonies which in turn would result ina systematic decline in density with population size and thus invalidate our application ofdensity estimates to the current population as a result the observed decline in abundancewould be even steeper than we report These various assumptions point to the need forincreased understanding of factors contributing to variation in overwinter density

Precision of overwinter density estimates can be improved by measuring naturalvariability in species response to environmental conditions over time and space focusingon robust parameter measurement and estimation error and examining assumptions inmodels or extrapolations of these models Capture-mark-recapture methods such as thoseemployed by Calvert (2004) but replicated over years and locations seem to offer the mostpromising means of accomplishing this goal of improved overwinter density estimationCapture-mark-recapture is the most common method for estimating population sizein butterflies (eg Gall 1984 Bergman 2001 Baguette amp Schtickzelle 2003 Haddad et al2008) and its systematic use in the high-elevationOyamel fir forests of centralMexicowouldenable robust estimates of daily and total overwintering population sizes as well as survivaland emigration probability (Williams Nichols amp Conroy 2002) However capture-mark-recapture methods come at the considerable cost of disturbing overwintering individualsa practice that is currently disallowed and arguably not prudent given the small populationsize and the negative impacts of disturbance Increased disturbance of overwinteringindividuals quickens fat depletion disrupts the thermal advantages of communal roostingand exposes the butterflies to predation and colder temperatures if they are unable tofly back into trees The structural extrapolations employed by Calvert (2004) also resultin considerable disturbance Thus while estimating density during the late-Decemberreporting period for World Wildlife Fund-Mexico may be essential for understanding howmany monarchs to attribute to the area over which they occur accurate measurementsusing traditional approaches may come at considerable risk to the butterflies Less invasivepossibilities for estimating density such as colorimetric analysis of the intensity and areaof occupied trees (Williams amp Brower 2016) may prove to be a rewarding alternative

CONCLUSIONSCombining the results of several studies conducted between 1979 to 2002 we concludean estimate of 211 million butterflies haminus1 is the most meaningful value when translatingarea occupied by overwintering monarchs into estimates of population size While this

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 1318

represents the best of our knowledge to date the number of studies estimating densities ofoverwintering monarchs is small and large discrepancies exist among various estimatesleading to considerable uncertainty A better understanding of the spatial and temporalfactors influencingmonarch densities in their overwintering colonies is needed to accuratelyunderstand monarch population size population viability and characteristics of theenvironment required for sustaining the species at desired levels of abundance Howeverwe acknowledge that this information may be difficult to attain and that continued carefulmonitoring of area occupied and non-intrusive estimation of relative density along withan understanding of the degree to which habitat restoration (and degradation Broweret al 2016) is occurring may provide our best understanding of the critical relationshipbetween milkweed availability and monarch numbers

ACKNOWLEDGEMENTSWe thank L Brower for data used in our analyses We thank A Agrawal L Brower AShapiro and an anonymous referee for comments greatly improving the quality of thiscontribution Any use of trade product or firm names are for descriptive purposes onlyand do not imply endorsement by the US Government The views expressed in this articleare the authorsrsquo own and do not necessarily represent the views of the US Fish and WildlifeService

ADDITIONAL INFORMATION AND DECLARATIONS

FundingThis work was supported by the John Wesley Powell Center for Analysis and Synthesisof the United States Geological Survey The funders had no role in study design datacollection and analysis decision to publish or preparation of the manuscript

Grant DisclosuresThe following grant information was disclosed by the authorsJohn Wesley Powell Center for Analysis and Synthesis of the United States GeologicalSurvey

Competing InterestsThe authors declare there are no competing interests

Author Contributionsbull Wayne E Thogmartin conceived and designed the experiments performed theexperiments analyzed the data contributed reagentsmaterialsanalysis tools wrotethe paper prepared figures andor tables reviewed drafts of the paperbull Jay E Diffendorfer Karen Oberhauser Darius Semmens and Ruscena Wiederholtconceived and designed the experiments wrote the paper reviewed drafts of the paperbull Laura Loacutepez-Hoffman and Orley R Taylor conceived and designed the experimentsreviewed drafts of the paper

Thogmartin et al (2017) PeerJ DOI 107717peerj3221 1418

bull John Pleasants conceived and designed the experimentsbull Brice X Semmens conceived and designed the experiments contributed reagentsmate-rialsanalysis tools

Data AvailabilityThe following information was supplied regarding data availability

The raw data has been supplied as elements within R statistical code which is includedas a Supplementary File

Supplemental InformationSupplemental information for this article can be found online at httpdxdoiorg107717peerj3221supplemental-information

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Brower LP Slayback DA Jaramillo-Loacutepez P Ramirez IM Oberhauser KSWilliamsEH Fink LS 2016 Illegal logging of 10 hectares of forest in the Sierra Chincuamonarch butterfly overwintering area in Mexico American Entomologist 6292ndash97DOI 101093aetmw040

Brower LP Taylor ORWilliams EH Slayback DA Zubieta RR Ramiacuterez MI 2012 De-cline of monarch butterflies overwintering in Mexico is the migratory phenomenonat risk Insect Conservation and Diversity 595ndash100DOI 101111j1752-4598201100142x

Brower LPWilliams EH Fink LS Slayback DA Ramirez MI Garcia MVL Zubieta RRWeiss SB Calvert WH ZuchowskiW 2011 Overwintering clusters of the monarch

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Brower LPWilliams EH Slayback DA Fink LS Ramirez MI Zubieta RR Limon-Garcia MI Gier P Lear JA Van Hook T 2009 Oyamel fir forest trunks providethermal advantage for overwintering monarch butterflies in Mexico Insect Conser-vation and Diversity 2163ndash175 DOI 101111j1752-4598200900052x

CalvertWH 2004 Two methods estimating overwintering monarch population sizein Mexico In Oberhauser KS Solensky MJ eds The monarch butterfly biology andconservation Ithaca Cornell University Press 121ndash127

CalvertWH Brower LP 1986 The location of monarch butterfly (Danaus plexippus L)overwintering colonies in Mexico in relation to topography and climate Journal ofthe Lepidopteristsrsquo Society 40164ndash187

Center for Biological Diversity the Center for Food Safety the Xerces Society forInvertebrate Conservation Brower L 2014 Petition to protect the monarchbutterfly (Danaus plexippus plexippus) under the Endangered Species Act Availableat httpwwwbiologicaldiversityorg species invertebratespdfsMonarch_ESA_Petitionpdf (accessed on 24 September 2015)

Dufresne D 2004 The log-normal approximation in financial and other computationsAdvances in Applied Probability 36747ndash773 DOI 101017S0001867800013094

Gall LF 1984 The effects of capturing and marking on subsequent activity in Boloriaacrocnema (Lepidoptera Nymphalidae) with a comparison of different numericalmodels that estimate population size Biological Conservation 28139ndash154DOI 1010160006-3207(84)90032-6

Garcia-Serrano E Reye JL Alvarez BXM 2004 Locations and area occupied bymonarch butterflies overwintering in Mexico from 1993 to 2002 In Oberhauser KSSolensky MJ eds The monarch butterfly biology and conservation Ithaca CornellUniversity Press 129ndash133

Haddad NM Hudgens B Damiani C Gross K Kuefler D Pollock K 2008 Determiningoptimal population monitoring for rare butterflies Conservation Biology 22929ndash940DOI 101111j1523-1739200800932x

Limpert E Stahel WA Abbt M 2001 Log-normal distributions across the sciences keysand clues BioScience 51341ndash352DOI 1016410006-3568(2001)051[0341LNDATS]20CO2

Machtans CS ThogmartinWE 2014 Understanding the value of imperfect sciencefrom national estimates of bird mortality from window collisions Condor 1163ndash7DOI 101650CONDOR-13-1341

McLachlan GJ Peel D 2000 Finite mixture models InWiley series in probability andstatistics New York John Wiley and Sons

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Pearson RK 2011 Exploring data in engineering the sciences and medicine OxfordOxford University Press

Pleasants J 2015 Monarch butterflies and agriculture In Oberhauser KS Nail KRAltizer SM edsMonarchs in a changing world biology and conservation of an iconicbutterfly Ithaca Cornell University Press 169ndash178

Pleasants J 2017Milkweed restoration in the Midwest for monarch butterfly recoveryestimates of milkweeds lost milkweeds remaining and milkweeds that must be addedto increase the monarch population Insect Conservation and Diversity 1042ndash53DOI 101111icad12198

Pleasants JM Oberhauser KS 2013Milkweed loss in agricultural fields because ofherbicide use effect on the monarch butterfly population Insect Conservation andDiversity 6135ndash144 DOI 101111j1752-4598201200196x

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Rendoacuten-Salinas E Tavera-Alonso G 2014 Monitoeo de la superficie forestal ocupadapor las colonias de hibernacioacuten de la mariposa Monarca en diciembre de 2013World Wildlife Fund-Meacutexico Zitaacutecuaro Michoacaacuten Available at http assetspandaorgdownloadsmonitoreo_mariposa_monarca_en_mexico_2013_2014pdf (accessedon 07 August 2015)

Schroumldinger E 1944What is life The physical aspect of the living cell CambridgeCambridge University Press

Semmens BX Semmens DJ ThogmartinWEWiederholt R Loacutepez-Hoffman LDiffendorfer JE Pleasants J Oberhauser K Taylor O 2016 Quasi-extinction riskand population targets for the Eastern migratory population of monarch butterflies(Danaus plexippus) Scientific Reports 623265 DOI 101038srep23265

Shapiro AF 2009 Fuzzy random variables Insurance Mathematics and Economics44307ndash314 DOI 101016jinsmatheco200805008

Slayback DA Brower LP Ramirez MI Fink LS 2007 Establishing the presence andabsence of overwintering colonies of the monarch butterfly in Mexico by the use ofsmall aircraft American Entomologist 5328ndash39 DOI 101093ae53128

Sugihara G 1980Minimal community structure an explanation of species abundancepatterns American Naturalist 116770ndash786 DOI 101086283669

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Urquhart FA Urquhart NR 1976 The overwintering site of the eastern population ofthe monarch butterfly (Danaus p plexippus Danaidae) in southern Mexico Journal ofthe Lepidopteristsrsquo Society 30153ndash158

Vidal O Loacutepez-Garciacutea J Rendoacuten-Salinas E 2014 Trends in deforestation and forestdegradation after a decade of monitoring in the Monarch Butterfly BiosphereReserve in Mexico Conservation Biology 28177ndash186 DOI 101111cobi12138

Vidal O Rendoacuten-Salinas E 2014 Dynamics and trends of overwintering coloniesof the Monarch Butterfly in Mexico Biological Conservation 180165ndash175DOI 101016jbiocon201409041

Wilks DS 2006 Statistical methods in the atmospheric sciences 2nd edition BurlingtonAcademic Press

Williams EH Brower LP 2015 Microclimate protection of overwintering monarchsprovided by Mexicorsquos high-elevation Oyamel fir forests In Oberhauser KS Nail KRAltizer SM edsMonarchs in a changing world biology and conservation of an iconicbutterfly Ithaca Cornell University Press 109ndash116

Williams EH Brower LP 2016 A conservation concern how many monarchs are thereNews of the Lepidopteristsrsquo Society 5890ndash93

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Zadeh LA 2002 Toward a perception-based theory of probabilistic reasoning withimprecise probabilities Journal of Statistical Planning and Inference 105233ndash264DOI 101016S0378-3758(01)00212-9

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