f geographic distribution and relative abundance of the

FORUM

Geographic Distribution and Relative Abundance of the Invasive Glassy-Winged Sharpshooter: Effects of Temperature and Egg Parasitoids

ANDREW PAUL GUTIERREZ,1,2 LUIGI PONTI,1,3 MARK HODDLE,4 RODRIGO P.P. ALMEIDA,5

AND NICOLA A. IRVIN4

Environ. Entomol. 40(4): 755Ð769 (2011); DOI: 10.1603/EN10174

ABSTRACT The capacity to predict the geographic distribution and relative abundance of invasivespecies is pivotal to developing policy for eradication or control and management. Commonly usedmethods fall under the ambit of ecological niche models (ENMs). These methods were reviewed andshortcomings identiÞed. Weather-driven physiologically based demographic models (PBDMs) areproposed that resolve many of the deÞciencies of ENMs. The PBDM approach is used to analyze theinvasiveness of the polyphagous glassy-winged sharpshooter (Homalodisca vitripennis [Germar]), apest native to the southeastern United States and northeastern Mexico that extended its range intoCalifornia in 1989. Glassy-winged sharpshooter vectors the pathogenic bacterium, Xylella fastidiosa(Wells) that causes PierceÕs disease in grape and scorch-like diseases in other plants. PBDMs forglassy-winged sharpshooter and its egg parasitoids (Gonatocerus ashmeadi Girault and G. triguttatusGirault) were developed and linked to a PBDM for grape published by Wermelinger et al. (1991).Daily weather data from 108 locations across California for the period 1995Ð2006 were used to drivethe PBDM system, and GRASS GIS was used to map the simulation results. The geographic distributionof glassy-winged sharpshooter, as observed, is predicted to be largely restricted to the warm areas ofsouthern California, with the action of the two egg parasitoids reducing its abundance �90%. Theaverage indispensable mortality contributed by G. triguttatus is �1%. A temperature-dependentdevelopmental rate model for X. fastidiosa was developed that suggests its geographic range is alsolimited to the warm inland areas of southern California. Biological control of glassy-winged sharp-shooter further decreases the pathogenÕs relative range. Climate warming scenarios of �2�C and �3�Csuggest that the distribution and severity of glassy-winged sharpshooter andX. fastidiosawill increasein the agriculturally rich central valley of California. The utility of holistic analyses for formulatingcontrol policy and tactics for invasive species is discussed.

KEY WORDS grape, glassy-winged sharpshooter, Homalodisca vitripennis, invasive species, GIS

Invasive species cause in excess of $140 billion in lossesannually in the United States (Pimentel et al. 2000).Experience suggests these problems are often handledin a crisis mode by state and federal agencies that lackthe capacity to predict the potential geographic rangeand relative abundance of invasive species under cur-rent and climate change scenarios. This capacity ispivotal to developing sound policy for their eradica-tion or control and management. Predicting the geo-graphic distribution and dynamics of invasive species

in time and space has been a difÞcult recurring prob-lem (Mills and Getz 1996), and the complexity mayincrease with global climate change.

It is well known that the local phenology, dynamics,and abundance of ectothermic species are largely de-termined by weather and interacting species, withclimate (i.e., long run weather) and interacting spe-cies determining the native geographic range (Andre-wartha and Birch 1954, Walther 2002, Wellington et al.1999). Examples of the effects of weather (principallytemperature) on invasive species and their biologicalcontrol include cottony cushion scale (Icerya purchasiMaskell) (Quezada and DeBach 1973), olive scale(Parlatoria oleae (Colvee)) (Huffaker and Kennett1966, Rochat and Gutierrez 2001), klamath weed (Hy-pericum perforatum L) (Huffaker and Kennett 1959),and red scale (Aonidiella aurantii (Maskell))(DeBach and Sundby 1963).

Among the methods used to predict the geographicdistribution of species are ecological niche models(ENMs). ENMs may be physiological index models(Fitzpatrick and Nix 1968, Gutierrez et al. 1974,

1 Center for the Analysis of Sustainable Agricultural Systems (CA-SAS Global), 37 Arlington Ave., Kensington, CA, USA 94707.

2 Corresponding author: Division of Ecosystem Science, College ofNatural Resources, University of California, 329 Mulford Hall, Berke-ley, CA 94720-3114 (e-mail: [email protected]).

3 Laboratorio Gestione Sostenibile degli Agro-Ecosistemi(UTAGRI-ECO), Agenzia nazionale per le nuove tecnologie,lÕenergia e lo sviluppo economico sostenibile (ENEA), CentroRicerche Casaccia, Via Anguillarese 301, 00123 Roma, Italy.

4 Department of Entomology, University of California, Riverside,CA 92521.

5 USA Department of Environmental Science Policy and Manage-ment, University of California, Berkeley, CA 94720-3114.

0046-225X/11/0755Ð0769$04.00/0 � 2011 Entomological Society of America

Sutherst et al. 1991), statistical models (e.g., general-ized linear models, generalized additive models, gen-eralized boosted models, genetic algorithm for rule-set prediction, principal components analysis (seeEstrada-Pena 2008, Mitikka et al. 2008), or modelsbased on artiÞcial intelligence concepts (Phillips et al.2006). These approaches attempt to characterize theecological niche of a species based on weather andother factors in the speciesÕ native range, and then touse the ENM to estimate the potential invasiveness ofthe species in new areas. DeÞciencies of ENMs in-clude difÞculty incorporating trophic interactions(Davis et al. 1998), aggregate weather data may beused that miss important short-term weather effects,the assumed native range may be in error, and differ-ent ENM approaches may give different results.

Many of these deÞciencies may be resolved usingphysiologically based demographic models (PBDMs)that may include the bottom-up effects of plantgrowth and development, and the top-down action ofnatural enemies (e.g., Gutierrez et al. 2005, 2007, 2009,2010, Ponti et al. 2009). Instead of beginning with theassumed native range of an invasive species, PBDMscapture the weather-driven biology and populationdynamics of the invasive species and of its interactingspecies. Observed daily weather or climate scenariosare used to drive the model dynamics of the species ofinterest across time and geographic space.

In this paper, we use the invasive glassy-wingedsharpshooter, Homalodisca vitripennis (Germar) inCalifornia to explore the utility of the PBDM ap-proach. We Þrst review glassy-winged sharpshooter inCalifornia.

Glassy-Winged Sharpshooter Invasion of California

Glassy-winged sharpshooter is a polyphagous sub-tropical-tropical species native to the southeasternUnited States and northeastern Mexico (Adlerz 1980,Triapitsyn and Phillips 2000, Turner and Pollard 1959)that extended its range into California in 1989 (So-rensen and Gill 1996). Glassy-winged sharpshooterand native leafhoppers vector the pathogenic bacte-rium, Xylella fastidiosa (Wells et al. 1987), that causesPierceÕs disease (PD) in grape (Vitis vinifera L.) andscorch-like diseases in other plants (e.g., oleandersand almonds) (Hopkins and Purcell 2002, Purcell1997). Different grape varieties are cultivated in dif-ferent ecological zones of California. Before the in-vasion of glassy-winged sharpshooter, outbreaks of PDoccurred in coastal and central California vineyards(Redak et al. 2004). Increases of PD in southern Cal-ifornia have been attributed to glassy-winged sharp-shooter (Almeida and Purcell 2003, Purcell and Saun-ders 1999).

Glassy-winged sharpshooter develops on numeroushost plants (Lauziere and Setamou 2009) where itfeeds on nutrient-poor xylem (Mizell et al. 2008). Twoor more generations per year occur in southern Cal-ifornia (Blua et al. 2001, Turner and Pollard 1959).Glassy-winged sharpshooter lacks a diapause stagewith development continuing throughout the year as

weather and host plant conditions allow. Glassy-winged sharpshooter normally overwinter as repro-ductively inactive adults (Hummel et al. 2006, Turnerand Pollard 1959) with adults able to ßy at ambient airtemperatures �11�C and feed at temperatures�13.3�C (see Johnson et al. 2006). Citrus is a majoroverwintering host in California (Perring et al. 2001).

To limit the spread of glassy-winged sharpshooter,an area-wide control program using insecticides andquarantine was implemented (California Departmentof Food and Agriculture [CDFA] 2005), with a suc-cessful biological control program using egg parasi-toids (Gonatocerus ashmeadi Girault (GA) and G.triguttatus Girault (GT)) (both Hymenoptera: My-maridae) initiated in 2000 (Pilkington et al. 2005). GAwas likely self-introduced (Vickerman et al. 2004), butwas also mass reared and redistributed by the CDFA.Sporadic establishment of GT in southern Californiaoccurred because of the CDFAs classical biologicalcontrol program (Pilkington and Hoddle 2007a,2007b). Despite these control efforts, glassy-wingedsharpshooter established in southern California andthe southern Central Valley (Blua et al. 2001), and hasbeen detected in north central California (CDFA2003).

The goals of our study were to assess the potentialdistribution and relative abundance of glassy-wingedsharpshooter, its parasitoids and PierceÕs disease inCalifornia, and to examine the biological control ofglassy-winged sharpshooter by GA and GT. We alsoexamine the hypothetical effects of a warmer climateon glassy-winged sharpshooter distribution and abun-dance. To do this, we developed a holistic PBDMsystem of the interactions of grapevine, glassy-wingedsharpshooter, and its egg parasitoids.

Method

The System Model for Grape, Glassy-WingedSharpshooter, and Two-Egg Parasitoids. The systemmodel is modular consisting of sixteen (n � 1, 16)linked functional age-structured population dynamicsmodels that may be in units of numbers, mass or both.Each sub-model is based on the distributed matura-tion-time dynamics model proposed by Vansickle(1977, see appendix). Wermelinger et al. (1991) de-veloped the grapevine model we used. It is a canopymodel consisting of subunit models for the mass ofleaves (n� 1), stem (2), shoots (3), and root (4), andthe mass and number of fruit clusters (5, 6). Themodels for glassy-winged sharpshooter consist of age-mass structured models for unparasitized eggs (7, 8),nymphs (9, 10), and adults (11, 12). Each parasitoidmodel consists of submodels for immature stages inparasitized eggs and for free-living adults [GA (13,14); GT (15, 16)]. The underpinning modeling con-cepts are found in Gutierrez and Baumgartner (1984)and Gutierrez (1992), and the mathematics of thetime-invariant and time-varying distributed matura-tion-time dynamics models used here are found inVansickle (1977), Gutierrez (1996), and DiCola et al.(1999). A brief review of the dynamics model is pre-

756 ENVIRONMENTAL ENTOMOLOGY Vol. 40, no. 4

sented in the appendix. The model for the pathogenX.fastidiosa is a temperature-dependent developmental(growth) rate model. The time step in all of the dy-namics models is a day of variable length in physio-logical time units as appropriate for each species,stage, or both.

Data from the literature used to formulate and pa-rameterize the models were of varying degrees of com-pleteness and quality, and reinterpretation of the dataand judgment in their use were required. The biologysummarized below was incorporated in the systemmodel (see Appendix).Grape. The model for grapevine (variety cheninblanc) (Wermelinger et al. 1991) was used withoutmodiÞcation to capture the bottom-up plant effects onglassy-winged sharpshooter, the egg parasitoids, andtheir interactions (Fig. 1a). The effects of PD on grapewere not included in the system model (see below).The model captures the phenology of winter dor-mancy, bud break, veraison and harvest, and dry mat-ter growth and development of vegetative and fruitsub units. The ßow of energy (dry matter) in thesystem is illustrated in Fig. 1b. Water and nutrients areassumed nonlimiting, and hence only daily maximumand minimum temperatures and solar radiation (calcm�2 d�1)are required for theplant.Full details of thegrape model are found in Wermelinger et al. (1991, seealso Gutierrez et al. 1985), and hence the model is notreviewed further.Glassy-WingedSharpshooter.The biology of glassy-

winged sharpshooter and its two-egg parasitoids isillustrated in Figs. 1b and c. Among the factors esti-

mated for the insect species were temperature thresh-olds, nonlinear temperature-dependent developmen-tal rate functions, duration of life stages, maximum percapita age-speciÞc fecundity, temperature-dependentscalars for reproduction and longevity, temperature-dependent mortality rates, and sex ratios (Table 1).Rate of development of glassy-winged sharpshooter.

The effects of temperature (T) on the rate of devel-opment of all of the species in the system, includingthe pathogen, were modeled using equation 1 (Briereet al. 1999). This model has a minimum number ofparameters: constants a and b, a lower threshold (�L),and an upper threshold (�U) where the function be-gins to decline to zero.

The rate of development of the glassy-wingedsharpshooter egg stage (RGW,egg�T� � 1/days) wasestimated in the range 16.7Ð35�C (Al-Wahaibi andMorse 2003) (Fig. 2a) yielding thresholds �L� 11.9�Cand �U � 35.75�C.

RGW,egg�T� �a � �T � �L�

1 � b�T��U�

�0.0085�T � 11.9�

1 � 2.4T�35.75 [1]

RGW,egg�T� was scaled for use for all other glassy-winged sharpshooter life stages because data for themwere unavailable.Glassy-winged sharpshooter longevity. Using data

from Setamou and Jones (2005) and the lower thresh-old �L (equation 1), we estimated the average devel-

Fig. 1. The biology of the grape/glassy-winged sharpshooter/pathogen/two parasitoid system: (a) the basic tri-trophicrelationships, (b) dry matter ßow in grape and to glassy-winged sharpshooter, and (c) the stage speciÞc relationships betweenglassy-winged sharpshooter and its two parasitoids (G. ashmeadi (GA) and G. triguttatus (GT)).

August 2011 GUTIERREZ ET AL.: ANALYSIS OF GLASSY-WINGED SHARPSHOOTER SYSTEM 757

opmental times in dd for nymphs (N� 792 dd�11.9�C)and adults (A � 1,876 dd �11.9�C) (Table 1). Son etal. (2009, equation 2 estimated the mean time in days(d) to 50% adult survivorship (i.e., mortality) at dif-ferent temperatures, with 81.75 d at 25.25�C (�Tmax)being the maximum. Normalizing the data, averageadult survivorship declines with age (Fig. 2b), and isconcave on temperature (0 � �GW�T� � 1) with themaximum at Tmax and zero at 0 and 40�C (Fig. 2c,equation 2i with parameters �1 � �2).

�GW�T� � exp1 � �Tmax � T�/� � �1� � e�Tmax�T�/��2��

� exp1 � �25.25 � T�/� � 7.05�

� e�25.25�T�/��7.05�� [2i]

Expected average nymph and adult longevity (sub-scripts N and A) change daily with temperature andcan be expressed as

N�T�t�� N � �GW�T�t�� [2ii]

and

A�T�t�� A � �GW�T�t�� [2iii]

Incorporating variable longevity in the model re-quires using the time-varying form of the distributedmaturation time model (Vansickle 1977).

The mortality rate (GW�T�) of adults d�1 at tem-perature T was also estimated from Son et al. (2009)(equation 3).

Table 1. Parameters for the glassy-winged sharpshooter/parasitoid/disease system

Function Parameter GWSS G. ashmeadi G. triguttatus X. fastidiosa

Rate of development at temp TR�T� � a�T � �L�/�1 � b�T��U��

a 0.0085a 0.005b 0.005c 0.00865e

�L 11.9 9.25 10.5 12.5b 2.4 4.5 4 2.75�U 35.75 33.85 34.1 32.25

Duration of life stages (dd) Egg (107.5)f Egg-pupae (198)b,c Egg-pupae (204)d

Nymphs (792�N) Adult � (241�GA) Adult � (222�GT)Preovip (216) (see text for �GA) (see text for �GT)Repro-adults (1660�P)See text for �N, �P

g

Temp scalars Tmin 15.86h 12.5b,c 14.5d

T � 1 � �T� Tmin � Tmid

Tmid� Tmax 32.5 34.5 34

Max age (x � dd) speciÞc per capitafecundity/d at 27�C

0.167x

1,0029xf,h

T,GA � 110e�0.0161xb T,GT � 110e�0.0273xd

Preoviposition period 216 ddf �1 d �1 dSex ratio (females /males) 1:1f 2:1b 2:1d

Egestion ratei � 0.83 Ñ ÑDemand rate (mg d�1)j Nymphs �0.0003e0.006x� � dd Ñ Ñ

Adults �0.0003e0.006�x�792�� ddRespiration rate (mg mg�1d�1)i Nymphs-adults 0.1 � 20.1�T Ñ Ñ

Delay parameter ki 25 25 25

a Al-Wahaibi and Morse 2003, b Pilkington and Hoddle 2006a, cChen et al. 2006a, d Pilkington and Hoddle 2007a, e Feil and Purcell 2001,f Setamou and Jones 2005, g Son et al. 2009, h Johnson et al. 2006, and Leopold et al. 2004, i A. P. Gutierrez (assumption), j Brodbeck et al. 2004.

Fig. 2. The thermal biology of glassy-winged sharpshooter: (a) the developmental rate of eggs on temperature (Al-Wahaibi and Morse 2003), (b) adult survivorship on age in dd �11.9�C at 25�C, (c) the effects of temperature on normalizedadult longevity, (d) the effects of temperature on adult mortality d�1 (data for b, c, and d are from Son et al. 2009), (e) theage speciÞc oviposition proÞle for glassy-winged sharpshooter on age in dd �11.9�C at 25�C (Setamou and Jones 2005), and(f) the normalized effect of temperature on gross fecundity (Johnson et al. 2006).


0 � GW�T� �1

81.75�GW�T�� 1 [3]

The mortality rate data are indicated by symbol (●),while the dashed line is equation 3 (Fig. 2d). Adults donot feed in the range 0�C� T � 11.9�C and the mor-tality rate is �0.1 d�1. At T� 0�C and T� 40�C, adultssurvive less than the sampling interval of a day, andhence it is not surprisingly the data for T � 0�C areunderestimated by the Þtted function. A better Þt forGW�T� at low temperatures (the solid line, Fig. 2d)was obtained by adjusting the Þtting parameters inequation 2i (i.e., �1 � 4.49 and �2 � 6.049).Reproduction in glassy-winged sharpshooter. Seta-

mou and Jones (2005) reported a sex ratio of 1:1 andan average fecundity of 194 eggs/� at 27�C whenreared on cowpea. However, Leopold et al. (2004)reported a range of 500-1100 eggs/�, and hence weadopted a more realistic mean value of 800. To capturethis biology, we used the age-speciÞc (x) ovipositionproÞle from Setamou and Jones (2005) but scaled itfour-fold, and Þt the function proposed by Bieri et al.(1983) (equation 4i). The preoviposition period is 216dd�11.9�C, after which fGW(x) increases linearly to 22.7eggs d�1 at age x� 374.2 dd�11.9�C, and then declinesslowly to zero (Fig. 2e).

fGW� x� �0.167x

1.0029x[4i]

We could add stochasticity to the model to accom-modate the observed range of fecundities, but littleadditional clarity would accrue. Reproduction occursin the range 13.9Ð32.5�C (Johnson et al. 2006), and anormalized symmetrical scalar function (0 �GW,ovip(T)� 1, equation 4ii) with a maximum at23.2�C was assumed (Fig. 2f, see Gutierrez 1992, 1996).

0 � GW,ovip�T� � 1 � �T � 13.9 � 23.2

23.2 � 2

� 1

[4ii]

The combined effects of age (fGW(x)) and temper-ature (GW,ovip(T)) on per capita reproduction arecaptured by equation 4iii.

EGW� x,T� � GW,ovip�T� � fGW� x� [4iii]

The Egg Parasitoids. Several solitary mymarid en-doparasitoids attack glassy-winged sharpshooter eggsin the SE United States and NE Mexico (Triapitsynand Phillips 2000). One of these, GA, was accidentallyintroduced to California (Sorensen and Gill 1996,Vickerman et al., 2004), while GT was actively intro-duced in 2000 (Pilkington et al. 2005). Data on bothparasitoids (Pilkington and Hoddle 2006a, 2007a, Irvinand Hoddle 2007), and additional data for GA (Chenet al. 2006a, 2006b) were used to parameterize theparasitoid models.Developmental rates. Data on the rate of develop-

ment of the egg-pupal period on temperature for GAand GT are illustrated in Figs. 3a and b. Using the formof equation 1, we estimate the lower and upper ther-

mal thresholds for GA (9.25�C and 33.85�C), and GT(10.5�C and 34.1�C) (equation 5i, 5ii).

RGA�T� �0.005�T � 9.25�

1 � 4.5T�33.85 [5i]

RGT�T� �0.005�T � 10.5�

1 � 4T�34.1 [5ii]

The lower and upper thresholds for glassy-wingedsharpshooter are �2.65 and 1.9�C higher than for GA,and 1.4 and 1.65�C higher than for GT (Fig. 2a vs. Figs.3a and b). The mean egg-pupal (subscript I) and adult(A) developmental times for the two parasitoids indegree-days above their respective threshold areI,GA � 198dd�9.25�C, A,GA � 241dd�9.25�C, andI,GT � 204dd�10.5�C and A,GT � 222dd�10.5�C.Reproduction. Estimates of net parasitoid repro-

ductive rates vary among authors, but the underlyingnormalized patterns are similar (Figs. 3c and d). Thepreoviposition period of both parasitoids is less than aday, and the per capita age-speciÞc fecundity proÞles(fp(x)) for each parasitoid (subscript p � GA, GT)declines on age (x) at rates rGA� 0.0161, rGT� 0.0273(equation 6i).

fp� x,T � 27�C� � 110e�rpx [6i]

As for glassy-winged sharpshooter, the effects oftemperature on the fecundity for each parasitoid(Figs. 3c and d) were captured by scalar functions(0 � ovip,p�T� � 1, equation 6ii, 6iii).

ovip,GA�T� � 1 � �T � 12.5 � 23.5

23.5 � 2

[6ii]

ovip,GT�T� � 1 � �T � 14.5 � 24.25

24.25 � 2

[6iii]

Fig. 3. The thermal biology of two-egg parasitoids ofglassy-winged sharpshooter: GA (a, c, e) and GT (b, d, f): (a,b) rates of development, (c, d) normalized eggs/female, and(e, f) normalized mean adult female longevity (data fromPilkington and Hoddle (2006a, 2007a; symbol ● ) and Chenet al. (2006b; symbol �)).


Ignoring species subscripts, the combined effects offp(x) and ovip,p�T� on age-speciÞc per capita fecun-dity (F(x,T)) are captured by equation 6iv.

Fp� x,T� � ovip,p�T� � 110e�rpx [6iv]

Realized fecundity also varies because of the effectsof temperature on adult longevity (Fig. 3e and f)(Pilkington and Hoddle 2006a, 2007a), and competi-tion among females and feeding (Irvin and Hoddle2007, 2009; Irvin et al. 2007). Only the effects of tem-perature and competition are included in the model.Adult longevity. The patterns of normalized average

adult longevity on temperature (�GA and �GT, Figs. 3eand f) are similar to those for the developmental rates(equation 1, and hence the same function was used).

�GA�T� �0.0317 � �T � 2�

1 � 3.5T�34 [7i]

�GT�T� �0.0518 � �T � 7�

1 � 2T�31.5 [7ii]

The data extrapolated to zero, suggest that GA ismore cold and heat tolerant than GT, and Þtted equa-tions 7i, ii are used to scale the maximum average adultlongevity (A,GA, A,GT) for the effects of daily tem-perature (equation 8i, ii).

A,GA�T�t�� A,GA�GA�T�t�� [8i]

A,GT�T�t�� A,GT�GT�T�t�� [8ii]

As with glassy-winged sharpshooter, time varyingchanges in parasitoid longevity have demographic con-sequences that require the time-varying form of thedistributed maturation-time model (Vansickle 1977).Inter-specific competition. Irvin and Hoddle (2005)

and Irvin et al. (2006) found that GA parasitized 45%more eggs of age 1Ð6 d than did GT, and in compe-tition studies, overall parasitism by GA was up to 76.0%higher. Exposing both parasitoids simultaneously tohigh egg densities for 24 h resulted in 23% higherparasitism by GA, while parasitism rates were the samewhen low egg densities were exposed for 1 h. Neitherparasitoid distinguishes between parasitized and un-parasitized eggs.

A type III functional response model (Gutierrezand Baumgartner 1984, Gutierrez 1996), that incor-porates competition in the exponent, was used to es-timate the total number of host eggs attacked (Na)because of the simultaneous action of both parasitoids(equation 9).

NaGA,GT � �DGAPGA � DGTPGT�

�1 � exp� � � GA � GT�H

DGAPGA � DGTPGT�� [9]

Ignoring age structure for clarity of presentation,PGA and PGT are female parasitoid densities, DGA andDGT are the per capita demands for hosts (equation6iv),H is the density of unparasitized plus parasitizedeggs, and GA and GT are search functions (e.g., 0� � so�1 � e�sP� � so with so � 0.6, and pa-

rameters sGA � 0.025 and sGT � 0.0125 that reßectdifferences in search efÞciency ( GA � GT).Na is Þrst corrected for the proportion of previously

attacked eggs, and the new attacks are distributed toeach parasitoid species in proportion to their demandcorrected for their search efÞciency. In cases of superand multiple-parasitism, the oldest supernumerary isassumed to survive.Weather Data.We assume nonlimiting water (irri-

gation) and nutrients for grape, and hence only dailymaximum and minimum temperatures (�C), and solarradiation (cal cm�2 d�1) were used as inputs for theplant model. Daily rainfall (mm), daily runs of wind(kmd-1)andrelativehumiditywerealsoavailable, andcould be used to run a water balance model (Ritchie1972) for site-speciÞc applications. However, soilmoisture holding characteristics were not available atthe scale of our study. Maximum and minimum tem-peratures were the inputs for the insect models. Theterm weather will refer only to these variables.Weather data from 108 locations in California for theperiod 1995Ð2006 (http://www.ipm. ucdavis.edu/)were used to run the system model. The few missingweather records at each location were estimated bylinear interpolation within the data or by substitutingvalues from a near by location.

Global climate models predict California will be-come warmer (1.8Ð3�C or more), but not drier, asprecipitation as rain rather than snowfall will occurat higher elevations (http://meteora.ucsd.edu/cap/pdfÞles/CA climate Scenarios.pdf). A full study ofclimate change effects using climate model data is be-yond the scopeof this study, andhenceweexamineonlythe hypothetical effects of 2� and 3�C increases in dailyaverage temperature on potential glassy-winged sharp-shooter distribution and relative abundance.Simulation and GIS. The system model is modular,

and Boolean variables in a setup Þle were used todetermine the combinations of species in the differentruns. Batch Þles were used to run the model acrosslocations for the period of available weather data. Thesame initial population densities and other conditionswere assumed at all locations (Table 1). Many of theinitial Þeld studies on glassy-winged sharpshooter oc-curred at Temecula, CA, and hence detailed simula-tion results for this location are presented to illustratesystem dynamics. The system model was coded inBorland Delphi Pascal.

The geo-referenced simulation data were written toÞles by year, and the data were mapped for elevationsbelow 750 m using the open source Geographic Re-sources Analysis Support System (GRASS) GIS soft-ware (GRASS Development Team, 2010). Inverse dis-tance weighting interpolation of the data using thev.surf.idw GRASS module was used in mapping. Thepatterns in the maps reßect not only the site-speciÞceffects of weather on the dynamics, but also the lo-cation and distances between weather stations. Thesystem was assumed equilibrating to local conditionsduring 1995, and hence the simulation data for thisyear were not included in the summary analyses. We


caution that the predicted densities are indices ofrelative favorability.

Multiple regression across all years and locationswas used to summarize the simulated effects on log10

total new glassy-winged sharpshooter adults y�1 of thetwo parasitoids, and season long cumulativedda�11.9�C

and ddb�11.9�C. The parasitoids were included in theregression model as presence-absence dummy vari-ables (0, 1). All four combinations of parasitoids in-cluding the null set were examined. Only independentvariables and interactions with slopes signiÞcantly dif-ferent from zero (P � 0.05) were retained in themodel. Marginal analysis of the regression model (�y/�xi) was used to estimate the average magnitudeand direction of factors affecting glassy-winged sharp-shooter abundance.

Results

System dynamics at Temecula, CA. Glassy-wingedsharpshooter densities and dynamics vary greatlyacross years and locations in California. We selectedTemecula in southern California, where heavy infes-tations of glassy-winged sharpshooter occurred, to il-lustrate the richness of the system dynamics (Fig. 4,see Fig. 5c). The tick-marks on the x-axis indicating 1January are reference points to help the reader visu-alize seasonality. Maximum and minimum tempera-tures for the 10-year period are shown in Fig. 4a.

In the absence of the parasitoids, simulated glassy-winged sharpshooter populations cycle and reach lev-els of 1,200Ð1,600 nymphs per vine during summer.Populations decline with cooling fall-winter temper-atures that affect reproduction and survival (Fig. 4b).

The peak densities are similar to those recorded in theÞeld before biological control (M. Hoddle, unpub-lished data).

With the addition of the parasitoids, glassy-wingedsharpshooter levels decline 20Ð30 fold (Fig. 4c) withthe contribution of GA being roughly 10-fold greaterthan that of GT (Fig. 4d). The combined percentparasitism of eggs reaches 80Ð90% during midsummerand fall, and accords well with Þeld data showing thatthe majority of egg masses are attacked (Krugner et al.

Fig. 4. The simulated dynamics of glassy-winged sharpshooter and two-egg parasitoids at Temecula, CA, during 1995through 2005: (a) daily maximum and minimum temperatures (�C), (b) glassy-winged sharpshooter egg and nymph densitiesbefore biological control, (c) unparasitized eggs, and nymphs and adults densities after the introduction of the two-eggparasitoids, and (d) glassy-winged sharpshooter eggs parasitized by G. ashmeadi (____) and by G. triguttatus (- - -) with theshaded area being the percentage of eggs parasitized by both parasitoids.

Fig. 5. Distribution of wine grape and glassy-wingedsharpshooter in California: (a) the major wine grape areas,(b) the observed distribution of glassy-winged sharpshooter(Pilkington and Hoddle 2007a, 2007b), and (c) the predicteddistribution of mean glassy-winged sharpshooter new adultsplant�1 y�1 in the absence of biological control. Temecula,CA (● ) is indicated in Fig. 5c.


2009). The high initial Þeld densities of glassy-wingedsharpshooter required 3 yr to suppress to levels nowobserved in the Þeld. The delay in control was likelybecause of the outward dispersal of adult parasitoids,and the immigration of glassy-winged sharpshooteradults from surrounding high-density populations. Tomodel these dynamics would require a meta-popula-tionmodel (seeGutierrezet al. 1999)of the spatial andtemporal dispersal dynamics of all of the species (Parket al. 2006). The data for this are not available.Regional Analysis. The regional analysis has two

parts: (1) mapping the summary variable for glassy-winged sharpshooter density before and after biolog-ical control by egg parasitoids across California, and(2) a multiple regression or marginal analysis of thesimulation data.Glassy-winged sharpshooter before biological control.

The major wine grape regions in California are shownin Fig. 5a, while the recorded distribution of glassy-winged sharpshooter is illustrated in Fig. 5b (seePilkington and Hoddle 2007a, b). In the absence ofbiological control, the predicted distribution of glassy-winged sharpshooter is similar to its recorded distri-bution in California (Fig. 5b vs. 5c), with the majorwine producing areas of northern and central Cali-fornia predicted to be largely unsuitable (Fig. 5a vs.5c). The highest mean densities of glassy-wingedsharpshooter adults are predicted in desert areas andthe mild coastal region of southern California, withlower densities predicted in the southern and north-ern reaches of the Great Central Valley (Fig. 5c). Thecoastal and near coastal regions of northern California

are predicted to have relatively low glassy-wingedsharpshooter populations.Glassy-Winged Sharpshooter after biological control.

The effects of the two-egg parasitoids on the simulateddistribution and abundance of glassy-winged sharp-shooter are illustrated in Fig. 6. Note that the columnsin the Þgure indicate species and the rows speciescombinations. The addition of GA to the model doesnot change the distribution of glassy-winged sharp-shooter, but does cause a 90% or more decline in newadult glassy-winged sharpshooter densities every-where including the hot desert valleys of southernCalifornia (Fig. 5c vs. 6a). GA populations (e.g., cu-mulative parasitized eggs) are predicted highest insouthern California with moderate levels in the south-ern and northern Central Valley (Fig. 6b).

Adding GT (Fig. 6e) to the interaction of glassy-winged sharpshooter and GA appears to add little tothe control of glassy-winged sharpshooter (Fig. 6a vs.6c), but it does suggest competition between the twoparasitoids in most regions that causes a decline of�15% in GA densities across the state (Fig. 6d vs. 6b).GT is predicted rare in the hot desert regions of south-ern California, and is approximately 1⁄3 the density ofGA elsewhere (Fig. 6d vs. 6e). These differences arebecause of the fact that GA has a higher search rate(equation 5), is more cold and heat tolerant (Fig. 3evs. 3f), and is less aggressive with conspeciÞc femalesthan GT (Irvin and Hoddle 2005, Irvin et al. 2006, N.Irvin, unpublished). In the laboratory, some of thesedifferences resulted in GA producing up to 64% moreprogeny than GT (Irvin and Hoddle 2005).

Fig. 6. GIS maps of simulated cumulative glassy-winged sharpshooter new adult densities and cumulative eggs parasitizedby the two-egg parasitiods: (a) glassy-winged sharpshooter regulated by (b) GA, and (c) glassy-winged sharpshooterregulated by both (d) GA and (e) GT. Densities of the three species are average season long totals for years 1996Ð2005.Temecula, CA (● ) is indicated in Fig. 6a.


Multiple regression and marginal analysis. The sim-ulation data for log10 new glassy-winged sharpshooteradults (log10 Anew) on presence-absence dummy vari-ables (0, 1) for the parasitoids (GA� and GT�), andseason long cumulative dda�11.9�C and ddb�11.9�C

across all years and locations are summarized by equa-tion 10 and in Table 2. The goal of the analysis isheuristic, to assess the relative impact of the parasi-toids, and not prediction as measured by R2.

log10Anew � 0.324 � 0.0028dda�11.9�C

� 0.00073ddb�11.9�C � 3.787GA� � 2.430GT�

� 2.463GA� � GT�

R2 � 0.532, df � 3802, F � 866.57 [10]

Given the average effects of the other variables andusing mean values for the independent variables, theeffects of dda�11.9�C on log10Anew is large and positive(e.g., 0.0028 1696 � 4.75), while ddb�11.9�C has asmaller decreasing effect (�0.000727 486.6 ��0.35). The effects of GA� (�3.79 by 0.54 � �2.04)andGT� (�2.43 0.45 � �1.10) are negative, but theinteraction GA� GT� is positive (2.463 0.27 �0.665) suggesting competition.

Taking the antilog of equation 10 converts the com-ponents to yearly population growth and survivorshiprates. Average Anew survivorship because of GA�

alone is 10�2.04 � 0.0091, 10�1.1 � 0.0794 for GT�

alone, and 10�2.04�1.1�0.665 � 0.0034 with both para-sitoids. GT� increases average Anew mortality onlyslightly over GA� alone (e.g., (1Ð0.0034) - (1Ð0.0091) � 0.0058). The value 0.0058 is a measure ofGT� average indispensable mortality, with still smallervalues expected in hotter areas of the state. For ex-ample, the predicted statewide average density of GAis three-fold that of GT, but GA densities are eight-foldgreater in the hot desert region (Fig. 6d vs. 6e).The Distribution of the Pathogen X. fastidiosa. To

include the full dynamics of PierceÕs disease in themodel would require the development of a meta-population model composed of populations of plantswith the movement of vectors and disease betweenthem (e.g., Gutierrez et al. 1999). This is clearly be-yond the scope of this study. Important insights, how-ever, can be gained from a simpler analysis. For ex-ample, plotting the in vitro growth rate (RXf) for X.fastidiosaon temperature yields a familiar relationship(equation 11 with lower (12.50�C) and upper(32.25�C) thresholds (see Fig. 7a).

RXf �0.00865 � �T � 12.50�

1 � 2.75T�32.25 [11]

The range of favorable temperatures for the patho-gen is narrower (solid line, Fig. 7a) than that of glassy-winged sharpshooter which is 11.90�C to 35.75�C (seeFig. 2a).

A more meaningful metric for disease favorability isthe concave normalized growth index (PDI, see equa-tion 4.ii) that captures the thermal limits and theoptimum at the midpoint of the favorable range(dashed line, Fig. 7a). Mapping the 10-yr average PDI(Fig. 7b) shows that the area most favorable for thepathogen is similar to that for glassy-winged sharp-shooter in southern California, but it is dissimilar innorthern area where cold limits the pathogen (Feiland Purcell 2001; Purcell 1980, 1997) more than it doesglassy-winged sharpshooter (Fig. 5c vs. 7b).

Daugherty et al. (2009) found that inoculation ef-Þciency of glassy-winged sharpshooter is positivelyrelated to temperature and the number of infectiousvectors. We can capture this notion as the product ofPDI (Fig. 7b) and an index of normalized total newadults given the action of natural enemies (GWI, datafrom Fig. 6c). The interaction PDI GWI predicts afurther restriction of the geographic range for diseasefavorability in southern California because of the ac-tion of the egg parasitoids (Fig. 7b vs. 7c).

Table 2. Multiple regression analysis of the glassy-winged sharpshooter simulation data across locations and years

Variable MeanStandarddeviation

Correlationx vs y

RegressioncoefÞcient

Std. error ofreg. coef.

t-value

dda 1696 562.1 0.532 0.0028 0.00008 33.27ddb 486.6 360.2 �0.385 �0.00073 0.00013 �5.59GA� 0.539 0.499 �0.424 �3.787 0.097 �38.92GT� 0.454 0.498 �0.210 �2.430 0.108 �22.49GA� GT� 0.270 0.444 �0.235 2.463 0.146 16.89Dependent variablelog10 Anew 2.192 3.241

Fig. 7. The pathogen X. fastidiosa: (a) in vitro growthrate d�1 (RXf,

___) and normalized growth rate index (PDI,Ñ-) on temperature (Feil and Purcell 2001), and the distri-bution of (b) mean PDI, and (c) mean PDI GWI acrossCalifornia (see text for deÞnitions of PDI and GWI).


Discussion

Calls for holistic analyses of agricultural and naturalsystems are often made (e.g., Sutherst and Bourne2009), but rarely does this occur. However, holisticapproaches are needed for predicting the geographicdistribution and relative abundance of invasive spe-cies and their control across ecological zones (e.g.,Mills and Getz 1996). The most common methodsused to assess such problems fall under the ambit ofENMs, but more holistic PBDMs, as used here, are alsobeginning to appear (see below). ENMs are com-monly formulated using weather and other abioticfactors from the native geographic range of the inva-sive species, and are then used to estimate the poten-tial distribution in extant and in new areas. ENMs maybe statistical (see Estrada-Pena 2008, Mitikka et al.2008)); applications based on artiÞcial intelligenceconcepts (Phillips et al. 2006); and physiological indexmodels (e.g., Fitzpatrick and Nix 1968, Gutierrez et al.1974, Sutherst and Maywald 1985). ENMs based onthese techniques assume the current species distribu-tion is the best indicator of its climatic requirements,the distribution is in equilibrium with current climate,and climate niche conservatism is maintained in bothspace and time (Beaumont et al. 2009). Other factorsmay adversely affect the predictions of ENMs: trophicinteractions, the use of average weather data that maymiss important and frequent short-term weatherevents, the assumed native range may be in errorleading to over or under estimates of the distribution(Davis et al. 1998), and different ENM approachesmay give different predictions. Roura-Pascual et al.(2009) compared Þve ENM methods to predict thepotential distribution of Argentine ants in the IberianPeninsula, and found differences in geographic pre-dictions and in the ability to identify areas of uncer-tainty regarding the antÕs invasive potential. Hickler etal. (2009) further cautioned that process-based rep-resentations of water balance in terrestrial ecosystemsshould be included in ENMs, and we might add, inPBDMs. A mechanistic soil water-balance model tocharacterize water effects on plant growth and fruit-ing proved critical in a PBDM analysis of the at-tempted biological control of the invasive annual yel-low starthistle (Centaurea solstitialis) by seed-headfeeding herbivores (Gutierrez et al. 2005). Despiteshortcomings, Thuiller et al. (2005) posits that ENMsmay provide powerful, unbiased Þrst-steps for screen-ing the potential range of invasive species in newareas.

ENMs and life-table statistics have been used toexamine the geographic distribution and invasivenessof the polyphagous glassy-winged sharpshooter, thedistribution of its two-egg parasitoids (GA and GT),and the potential distribution of the PD pathogen X.fastidiosa.Hoddle (2004) used the physiological-indexCLIMEX algorithm (Sutherst et al. 1991) to estimatethe geographic limits of glassy-winged sharpshooterand the pathogen. Unfortunately, the coarse grainpredictions of the CLIMEX model proved difÞcult to

compare with the Þne grain predictions of our PBDM(see below).

Pilkington and Hoddle (2006b, 2007b) used long-term average daily temperatures to estimate the num-ber of generations, and the net reproductive rates (Ro)of GA and GT in California. Using Ro as a metric, thepredicted area of highest favorability for both parasi-toids was the hot desert valleys of southern California,with the distribution of GA being much wider than forGT. The predictions for GA but not GT are similar tothose of our PBDM system (see below).A PBDM System for Grape/Glassy-Winged Sharp-shooter/Parasitoids.We propose that some of the de-Þciencies of ENM approaches may be overcome usingthe PBDM approach. Instead of starting from the as-sumed geographic distribution of a species, a PBDMsystem captures the weather-driven biology of theinteracting species, and uses it to map the dynamics(niche) of the species in time and space. A PBDMsystem can incorporate the bottom-up effects of plantgrowth and development, the top-down action of nat-ural enemies, and important physiology and behaviorof all of the species (e.g., Gutierrez et al. 2005, 2007,2009, 2010; Ponti et al. 2009).

Using daily maximum and minimum temperatureand solar radiation, the system model predicts that inthe absence of natural enemies, the most favorableareas for glassy-winged sharpshooter are coastal andnear coastal southern California, the southern desertarea, and to a lesser extent, portions of the lower andupper Central Valley (Fig. 5c). The major wine grow-ing regions of California (Fig. 5a) are predicted mar-ginal for glassy-winged sharpshooter (Fig. 5c). Insouthern California, the model predicts two to threeglassy-winged sharpshooter generations per season(Hummel et al. 2006) with peak glassy-winged sharp-shooter populations occurring in early to midsummer(Krugner et al. 2009). The geographic distribution ofglassy-winged sharpshooter in California is not af-fected by the two-egg parasitoids, but adult glassy-winged sharpshooter abundance is reduced �90%across the state (Fig. 5c vs. 6c). GA is the mosteffective parasitoid with a small average indispens-able mortality of �1% contributed by GT. The po-tential range of the pathogen X. fastidiosa isconsiderably less than that of glassy-winged sharp-shooter, but with the biological control of glassy-winged sharpshooter, the potential range of thepathogen is reduced still further to the desert re-gions of southern California (Fig. 7b and c).

The system model can be applied to other areasglobally where glassy-winged sharpshooter may in-vade. For example, glassy-winged sharpshooter in-vaded the tropical island of Moorea and neighboringislands of French Polynesia in 1999, where its numbersexploded in response to highly favorable weather, andaided by its toxicity to native generalist predators(Suttle and Hoddle 2006). However, the introductionof GA to Moorea in 2005 led to rapid control of glassy-winged sharpshooter (Grandgirard et al. 2009). Ourmodel, without modiÞcation, predicted these dynam-ics (not shown).


Other factors that may inßuence the dynamics ofglassy-winged sharpshooter include the level of aminoacids in host plants (Bi et al. 2005), diurnal and sea-sonal changes in nutrient-poor xylem (Mizell et al.2008), and plant water stress (Krugner et al. 2009). Wewere unable to include these factors in the model, butsuspect that their omission would not qualitativelyaffect our conclusions.Effects of warming temperatures. Temperatures are

expected to increase in California �1.8Ð3�C, but a fullstudy of the effects of climate warming on the grape/glassy-winged sharpshooter system is beyond thescope of this study (see Gutierrez et al. 2006). Here wepose a simpler hypothetical question: how wouldwarming temperatures in glassy-winged sharpshooterÕcurrent distribution affect its abundance?

The model predicts that increasing average dailytemperatures 2�C and 3�C would increase glassy-winged sharpshooter abundance throughout much ofCalifornia, except in the southern deserts where in-creased summer temperatures would approach its up-per thresholds for survival and reproduction (Fig. 8avs. Figures 8b and c). Glassy-winged sharpshooter andthe pathogen are predicted to increase in importantagricultural areas of California, especially in the Cen-tral Valley. Control of glassy-winged sharpshooter bythe egg parasitoids would not be affected.Comments on invasive species policy. Without ben-

eÞt of sound analysis, new invasive species tend to beapproached in a crisis mode that may result in largesums of money being spent on questionable controland eradication efforts (e.g., medßy, Ceratitis capitataWeidemann; Gutierrez and Ponti 2011). Glassy-winged sharpshooter is a case in point when it Þrstinvaded California. Fortunately, a successful biologi-cal control program using GA likely has helped solvethe problem. However, such good results may notoccur with other invasive species, and holistic ap-

proaches are needed to assess their invasive potentialand geographic range. Such analyses could serve as abasis for formulating sound ecological and economicpolicy for containment/eradication/management ef-forts, and for evaluating the efÞcacy of control pro-grams. This need was aptly illustrated by a prospectivePBDM analysis of the exotic light brown apple moth(Epiphyas postvittanaWalker) that was Þrst found incoastal north-central California in 2006. The PBDManalysis showed, as observed, that the geographicrange of the moth was largely near coastal, its popu-lations were low because of good natural control, thateradication was likely unachievable, and the proposedcontainment strategy was unrealistic (Gutierrez et al.2010). More recently, the European grape vine moth(Lobesia botrana (Den. and Schiff.)), a common pestof grape in the Mediterranean area, was discovered in2008 in Napa County in the heart of Northern Cali-forniaÕs wine country. It has since been found in eightadditional counties, some of which are in the CentralValley. A prospective PBDM analysis posits that Lobe-sia could spread statewide and beyond, eradication ofthe pest may not be possible, and pest managementprocedures may need to be developed and imple-mented (Gutierrez et al. submitted).

Thus, PBDMs can be used in a prospective mannerto gauge the potential geographic distribution of newpests in their native areas or in newly invaded ones,and with appropriate agronomic/ecological layers,PBDMs can be used to evaluate pest impact and man-agement options at the local and regional level. Un-fortunately, sound biological data to parameterize aPBDM for a new invasive species and the weather datato drive it may not be available at the beginning of theinvasion. However, given a modest scientiÞc infra-structure and budget, the necessary data can be gath-ered efÞciently using the model framework as a guide.In addition, climate model weather data are increas-ingly available, and can be used with the model toexplore the effects of climate change (Gutierrez et al.2006).

Acknowledgments

We thank the Editor E. Alan Cameron and two anonymousreviewers for their Augean Þfth task efforts in guiding thismanuscript to publication. We are grateful to M. Neteler(Fondazione Edmund Mach Ð Centro Ricerca e Innovazione,Trento, Italy; (http://gis.fem-environment.eu/) and an in-ternational network of co-developers for maintaining theGeographic Resources Analysis Support System (GRASS)software, and making it available to the scientiÞc community.

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Received 10 July 2010; accepted 2 June 2011.

Appendix: Brief Overview of the Distributed-Maturation-Time Dynamics Model

The physiologically based demographic model(PBDM) approach builds on the idea that all organ-isms are consumers and all have similar problems ofresource acquisition (inputs) and allocation (outputs)(Gutierrez 1992). This notion allows the use of thesame resource acquisition model and birth-deathdynamics models to describe all trophic levels includ-ing the economic one (Regev et al. 1998). For con-sumers such as grape and glassy-winged sharpshooter,the model for per capita resource acquisition (i.e., thesupply, S) is search driven by total consumer demands(D) in priority order for egestion, conversion costs,respiration (i.e., the Q10 rule in ectotherms), and re-production, growth, and reserves. The ratio 0�S/D�1is always less than unity because of imperfect con-

sumer search, and is used in the model to scale max-imal vital rates of species. Parasitoid seeks hosts (e.g.,eggs) as outlined in the text.

The Erlang distributed-maturation-time demo-graphic model is widely used to simulate the age struc-tured population dynamics of species. Details con-cerning the time invariant and time varying models arefound in Vansickle (1977) and DiCola et al. (1999, pp.523Ð524). The general form of the time invariantmodel for the ith age class of a population is as follows.

dNi

dt�kx

Ni�1�t� � Ni�t�� i�t�Ni�t�

[A1]

Ni is the density (mass or numbers) of the ith age class,dt is a change in time (e.g., a day),k is the number of age classes, is the expected mean developmental time,


x is the daily increment of physiological age com-puted using text equation 1, and

i�t� is the proportional age-speciÞc net-loss rateas modiÞed by temperature, age, net immigration,and mortality due natural enemies (i.e., the biologyoutlined in the text).

All life stages of insect species n can be included inone dynamics model or each life stage may be mod-eled separately. Reproduction by the adult ageclasses entering the Þrst age cohort of the egg stage(Ne,1(t)) as x0,n(t), with ßows between stages(ye,n(t), yl,n(t), yp,n(t)), and survivors exiting atmaximum adult age as yn(t) (Fig. A1a). (By analogy,

births in plants would be the production of newplant or subunits.) Ignoring stage subscripts, theßow rate (ri(t)) between age classes within a stage(Fig. A1b) depends on the number (mass) in theprevious age class, k, stage and x. The distribu-tion of stage maturation times is determined by thenumber of age classes k and the variance of matu-ration times (var) (e.g., k � 2/var) (Fig. A1c).The larger the value of k, the narrower is the Erlangdistribution of developmental times of cohort mem-bers. A value of k� 25 was selected that assumes anintermediate level of variability in developmentaltimes for all species and sub-stages.

Fig. A1. The distributed maturation time model: (a) ßow between life stages for species n, (b) ßow between age cohortswithin a life stage, and (c) distribution of developmental time given different numbers of age classes in a stage (Erlangparameter k) (see Fig. A1b).


f geographic distribution and relative abundance of the

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