e c o l o g i c a l m o d e l l i n g 2 0 9 ( 2 0 0 7 ) 323–332
avai lab le at www.sc iencedi rec t .com
journa l homepage: www.e lsev ier .com/ locate /eco lmodel
dynamical model of honeydew droplet production byooty-beech scale insects (Ultracoelostoma spp.) in Newealand Nothofagus forest
lex Jamesa, Roger Dunganb, Michael Planka,∗, Ryoko Itoa
Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New ZealandSchool of Biological Sciences, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
r t i c l e i n f o
rticle history:
eceived 19 June 2006
eceived in revised form
0 June 2007
ccepted 29 June 2007
ublished on line 20 August 2007
eywords:
eech forest ecology
oneydew production
hloem feeding
a b s t r a c t
Sugar-rich honeydew production by phloem-feeding sooty beech scale insects (Ultra-
coelostoma spp.) is a keystone ecological process in New Zealand beech (Nothofagus) forest. A
dynamical model to describe the formation of individual honeydew droplets is derived using
simple fluid dynamics. The model assumes that the insects play little part in regulating
flow, and that flow is driven by phloem hydrostatic pressure and regulated by environmen-
tal conditions affecting water evaporation from droplets. The model predicts different types
of behaviour depending on local environmental conditions. In dry conditions, evaporation
rates are high and honeydew droplets can become so concentrated that further excretion
is prevented and droplet formation ceases. In humid conditions, the high concentrations
required to prevent droplet formation do not occur and droplets continue to grow indefi-
nitely. The model is fitted to new data and the data are used to estimate values for the rate
oiseuille flow
oft-scale insects
of flow of phloem sap through the insect and for the concentration required to prevent fur-
ther excretion. Model predictions provide a close qualitative match to field data. However,
rates of fluid flow from the trees through the insects would need to be much greater than
the model assumes to produce values for droplet production that match field data. This
suggests that the insects are playing an active role in regulating the flow rate.
. Background
notable feature of the beech (Nothofagus spp.) forests ofhe northern South Island of New Zealand is the inter-ction between beech trees and sap-sucking sooty beechcale insects in the genus Ultracoelostoma (Margaridoididae).
econd- and third-instar females and second-instar males ofltracoelostoma construct and inhabit tests on the trunks andranches of beech trees (Morales et al., 1988). They insert theirtylets into phloem cells and feed on the phloem sap, which isich in photosynthetically derived carbohydrates but contains
only low concentrations of proteins (Grant and Beggs, 1989).The insects excrete excess carbohydrate in solution througha waxy tube connected to their anus. This carbohydrate solu-tion, called honeydew, forms droplets at the ends of the tubes.
Honeydew excretion (“production”) by scale insects isa keystone ecological process in 15% (1 million ha) of NewZealand’s remaining indigenous forests (Beggs and Wardle,2006) and an important seasonal food source for native birds
and insects (Murphy and Kelly, 2001; Barlow et al., 2002).Droplets of honeydew that fall or wash from trees supportdiverse communities of sooty mould (up to seven distinctspecies (Hughes, 1976)) that provide feeding and breeding
Insects of honeydew-producing age inhabit tests, nest-like
324 e c o l o g i c a l m o d e l l
sites for fungus-feeding arthropods (Didham, 1993). Honey-dew also has important conservation implications for NewZealand beech forest ecosystems, as the abundant carbohy-drate resource supports enormous populations of aggressivelyinvasive social wasps (Vespula germanica and Vespula vulgaris).During summer, the wasps out-compete native animals foraccess to the honeydew resource, which can dramaticallyreduce the breeding success of an endangered forest parrot(the kaka Nestor meridionalis; (Beggs, 2001; Beggs and Wardle,2006)).
Despite the demonstrated importance of honeydew inbeech forest ecosystems, little is known about what controlsthe availability of honeydew at ecosystem scales, or whatregulates rates of honeydew produced by individual insects.Although recent work has demonstrated that honeydew pro-duction varies widely between seasons (Beggs et al., 2005),and is strongly regulated by day-to-day changes in weatherconditions (Dungan and Kelly, 2003), mechanisms regulatingproduction by individual insects are not clearly understood.
The insects face some interesting limitations to feedingand excretion that are amenable to analysis using tech-niques of fluid dynamics. Engineering approaches have beenapplied to the phloem and nectar feeding of a diverse rangeof insects, including butterflies (Kingsolver and Daniel, 1993),aphids (Mittler, 1967; Loudon and McCulloch, 1999), andxylem-feeding cicadas (Novotny and Wilson, 1997). Beechscale insects have important differences to previously mod-elled species that may control their feeding behaviour and, byextension, the broader dynamics of beech forest ecosystemproductivity. For example, aphids are relatively short lived,and a significant limitation to their feeding behaviour is theselection of suitable feeding sites and the mechanical probingof host plant tissue (Tjallingii, 1995). Similarly, bark beetlesexpend significant amounts of time searching for a suscep-tible host tree (this has been modelled by Byers, 1996). Incontrast, scale insects are relatively long lived (up to two yearsWardhaugh et al., 2006) and, unlike most commonly studiedphloem feeding insects that feed on leaves and non-woodyplant parts, the insects are sessile and feed on woody tis-sues separated from soft tissues by up to 20 mm. Once settled,scale insects do not move, and appear to feed constantly fromthe same phloem cells, thus providing a unique modellingproblem.
It is thought that scale insects insert their mouthparts intoindividual phloem vessel elements (this feeding mode hasbeen demonstrated for aphids Tjallingii, 1995). It is presumedthat the force for the phloem sap to flow from the trees tothe insects is provided by positive hydrostatic pressure withinindividual phloem sieve elements, which is actively main-tained as part of the trees’ phloem transport pathways (vanBel, 1995). It is possible that the insects play little role in thisprocess, as feeding may not require any effort to actively suckfluid from phloem cells. However it is not clear how or if theinsects are actively regulating this flow, and how this regula-tion is affected by the environmental factors that are known todetermine rates of honeydew production (Dungan and Kelly,
2003).
Because of the demonstrated importance of honeydewin New Zealand’s beech forest ecosystems, it is importantthat factors that may limit honeydew production are under-
2 0 9 ( 2 0 0 7 ) 323–332
stood, or how changes in environmental conditions (such aspredicted increases in summer drought from anthropogenicclimate change in areas where honeydew is abundant Kennyet al., 2002) may alter the amount of honeydew present inforests. Forest-scale honeydew production is the product ofdroplet production by individual insects, so an understandingof the processes regulating honeydew droplet formation willstrengthen models of honeydew availability.
A dynamical model of honeydew droplet produc-tion that includes important rate-limiting steps in thetree–insect–environment continuum is presented. It is ini-tially assumed that the insect plays no active role in the flowmechanism, and that fluid flow is determined by the pressuregradient from the phloem to the atmosphere, modified bythe morphological and environmental characteristics of thetree–insect–atmosphere continuum. By comparing valuesfor honeydew production predicted by the model to valuesobtained in the field, this assumption may be explicitlytested, and the extent to which insects are actively control-ling production rates assessed. The aim is not to developa fully resolved predictive model of honeydew production,but to elucidate the role of the insect within the system, inparticular whether or not the insect is actively involved inregulating the flow rate. This will lead to a better appreciationof the important factors affecting honeydew production atthe level of an individual insect and of the wider ecosystem.
Loudon and McCulloch (1999) applied the Hagen–Poiseuilleequation to calculate flow rates in the feeding tubes of fluid-feeding insects. The theoretical flow rates were compared toexperimental data, and it was found that the theoretical val-ues were very accurate for the vast majority of insects. Thisis the method used to calculate flow rates in the presentmodel. Since Loudon and McCulloch (1999), there have beenrelatively few examples of models of fluid feeding in theliterature. Sperry et al. (2005) used the Hagen–Poiseuille equa-tion to model flow rates through xylem conduits in plants;Siekmann et al. (2001) carried out a statistical analysis offactors affecting the feeding behaviour and survival of par-asitic wasps. The approach taken in this paper differs fromSiekmann et al. (2001) in that it uses a mechanistic rather thana statistical description. In other words, the model is derivedfrom fundamental equations governing the relevant trans-port processes (fluid flow and water evaporation). Kingsolverand Daniel (1993), Loudon and McCulloch (1999) and Sperryet al. (2005) have offered mechanistic models of phloem andxylem transport. However, ours is the first model that cou-ples the feeding process to the excretion process, by explicitlyconsidering flow rates in the anal tube, and the effects onthese of droplet formation and evaporation at the end ofthe tube.
2. The model
2.1. Description of the insects
structures that eventually cover most of their body, and whichthey construct from wax filaments extruded from abdominalpores. Mature insects remain within these structures for most
e c o l o g i c a l m o d e l l i n g 2 0 9 ( 2 0 0 7 ) 323–332 325
Fig. 1 – (A) Anal tubes of Ultracoelostoma scale insects protuding from sooty mould-covered bark of a Nothofagus tree trunk.Individual insects are obscured within their wax tests, but droplets of honeydew can be clearly seen. These anal tubes arec.10 mm long. (B) Scanning electron microscope image of a mature female Ultracoelostoma scale insect. The anal tubeconnects to the heavily sclerotised anal ring at point A. The insect’s body (B) is largely featureless, although vestigial legscan be seen above the point where the stylets (S) emerge from the body. (C) Scanning electron microscope images of crosssections of typical Ultracoelostoma anal tube (upper panel) and mouthparts (lower panel). The four interlocking styles of them maxs mag
ofutmapaeh1go(uws3ps
2
Pir
outhparts can be seen. The feeding tube is formed by twotylets, the upper of which was damaged in preparing this i
f their lives. Thus protected from predatory animals, androm desiccation, the insects insert their mouthparts into thenderlying trunk or branch tissue of the host tree, and pene-rate through several layers of bark cells to the phloem. The
outh parts comprise four interlocking stylets (two maxillarynd two mandibular) which form a tube through which thehloem sap flows to the insect. The mouth parts are very thinnd long relative to the insect’s body size. Analysis of scanninglectron microscope (SEM) images shows that the mouthpartsave an internal diameter of 2–5 �m and can be more than0 mm long. In comparison, mature female scale insects areenerally less than 5 mm in diameter. The internal morphol-gy of soft scale insects is described in more detail by Foldi1997). Honeydew is excreted through a wax anal tube, that isp to 10 times body length in extreme cases, and considerablyider than the tube formed by the stylets. SEM images have
hown these tubes have internal diameters of approximately0 �m and are 10–25 mm long (see Fig. 1). For modelling pur-oses the insect can be considered to comprise three distinctections; the mouth parts, a body cavity and an anal tube.
.2. Modelling flow rate
resuming that both the mouth and anal tubes are cylindrical
n shape, flow along them will be purely laminar and the flowate can be described as
��Pa4
8�L, (1)
illary stylets, which are enclosed by two mandibulare.
where L and a are the length and internal radius of the tube,�P is the drop in pressure along the tube, and � is the fluid’sdynamic viscosity. Consider two such tubes (e.g. the mouthtube and anal tube), with lengths and radii L1, L2 and a1, a2
respectively, connected in series with a total pressure drop �P.If there are no significant boundary effects at the intersectionpoint, the flow rate is
Q0 = ��Pa41a4
2
8�(a42L1 + a4
1L2). (2)
The flow rate predicted by Eq. (2) is most strongly limited bythe dimensions of smallest tube in the series, the mouth-parts, as these have an internal diameter approximately 10times narrower than the anal tube. Using the parameter valuesgiven in Table 1, Eq. (2) predicts a flow rate of approximatelyQ ≈ 7.5 × 10−4 �l/s. This gives a production rate of 47 �l perinsect per 24 h. Assuming the phloem sap has a gravimetricconcentration of 16% sucrose (Pate et al., 1998) this equatesto 10 mg of sucrose per insect per day. Treating the insect’sbody as a third tube has a negligible additional effect on flow,due to the relatively wide body diameter in comparison to thenarrow mouth and anal tubes.
2.3. Flow limitations
Anecdotal and quantitative field observations provide someguidance as to the attributes of the system that represent crit-ical limits to honeydew flow and droplet production. Dungan
326 e c o l o g i c a l m o d e l l i n g 2 0 9 ( 2 0 0 7 ) 323–332
Table 1 – Parameters used in the model with values, units and references (where appropriate)
Description Value Units Reference
VariablesS sucrose content of the droplet – kg –V volume of a drop – m3 –
r radius of the droplet(
3V
4�
)1/3
m –
Parametersa1 radius of anal tube 15 × 10−6 m Dataa2 radius of mouth tube 2 × 10−6 m DataCin volumetric concentration of sucrose in
the inlet flow167 kg m−3 Pate
C� volumetric concentration of sucrose inthe droplet (at �% gravimetricconcentration)
– kg m−3 Moller
C100 density of sucrose 1518 kg m−3 MollerC� gravimetric stopping concentration – kg m−3 –D diffusion coefficient of water and air 2.42 × 10−5 m2 s−1 BirdL1 length of anal tube 0.02 m DataL2 length of mouth tube 0.005 m Datams molecular weight of sucrose 324 g mol−1 –mw molecular weight of water 18 g mol−1 –RH relative humidity – % DataT temperature 15–20 ◦C Datax0 saturated vapour pressure of water (mole
fraction)0.00603 × 107.5T/(237.3+T) – Monteith
x∞ vapour pressure of environment (molefraction)
RH x0 – –
�P pressure gradient 1 × 106 Pa Gould� dynamic viscosity of the inlet flow 1.66 × 10−3 Pa s Bubnik�w density of water 1000 kg m−3 –
Groupings
˛0 evaporation rate (uncorrected) D�x0 − x∞1 − x∞
m3 s−1 –
˛ evaporation rate ˇV1/3
(C100V − S
C100V − ıS− RH
)m3 s−1 –
ˇ evaporation rate constantD
1/x0 − RH
(3�2
4
)1/3
m2 s−1 –
ı mole fraction conversion 1 − C100mw
�wms– –
k concentration correction factorC�
C� − Cin– –
Q0 flow rate (uncorrected)��P a4
1a42
8� (a42L1 + a4
1L2)m3 s−1 –
Q flow rate(
1 − S
C� V
)kQ0 m3 s−1 –
. (200
The references are Bird (1960); Bubnik and Kadlec (1995); Gould et al(1998). For data references see Section 3
et al. (unpublished data) have shown that honeydew produc-tion, in terms of the dry mass of honeydew carbohydrates,increases with warmer air temperature, but decreases withincreasing air saturation deficit. Similarly, it is commonlynoted that the sugar concentration of individual dropletsincreases the longer droplets are left on the end of the tubes,and that droplets can become so concentrated and viscousthat flow rates are inhibited. Gaze and Clout (1983) suggestthat, at concentrations greater than 70%, the droplets become
tacky and may inhibit further flow, but flow rapidly resumesonce these concentrated droplets are removed. Clearly, evap-oration of water from standing droplets is a significant limitto droplet production.
5); Moller et al. (1996); Monteith and Unsworth (1990) and Pate et al.
To incorporate this effect it is assumed that the flow of solu-tion into the droplet decreases linearly as the concentrationof the droplet rises. Using data from standard chemical tables(Lide, 2003), gravimetric concentrations (�%, grams of sucroseper 100 g solution) are converted to volumetric concentrations(C�, kg sucrose per m3solution) with an empirical correctionfactor as
C� = 9.57� + 0.056�2.
It is assumed that phloem sap has fixed sucrose concentrationCin and initially flows unimpeded into the droplet at rate Q0.However, as the concentration of the droplet increases aboveCin, the flow rate Q decreases linearly from Q0 until, at concen-
g 2 0
tr
Q
Tiebcad
2
Odvtwod
˛
wramd“o
pU
x
Acrcpf
(
w
ı
auc
˛
e c o l o g i c a l m o d e l l i n
ration C� , the flow stops entirely. This gives a corrected flowate of
= H
(1 − S
C� V
)1 − S/(C� V)1 − (Cin/C� )
Q0
= H
(1 − S
C� V
)(1 − S
C� V
)kQ0. (3)
he volume of the droplet, in m3, and the sucrose content,n kg, are V and S, respectively. The Heaviside function, H(x),nsures the flow rate is zero if S/V ≥ C� thus disallowing flowack up the tube at concentrations greater than the stoppingoncentration. The value of C� , the concentration of honeydewt which flow stops, will be determined from experimentalata in Section 3.
.4. Evaporation rate
nce the solution has left the tube and started to form aroplet on the end, its small size and large surface area toolume ratio mean it is subject to a relatively large evapora-ion rate. By considering Fick’s Law of diffusion on a sphericalater droplet in a quiescent gas, and following the methodol-gy of Bird (1960), the rate of evaporation ˛0 of water from theroplet can be estimated as
0 = D�rx0 − x∞1 − x∞
,
here D is the diffusion coefficient of water and air, r is theadius of the drop, x0 is the saturated vapour pressure of airnd x∞ is the vapour pressure of the surrounding air, botheasured as mole fractions. Note that the evaporation rate is
ependent on the droplet size and this is a formulation of theD-squared law” of evaporation where the diameter squaredf the droplet varies linearly with time.
Values for x0 depend strongly on the ambient air tem-erature, T, and can be estimated following Monteith andnsworth (1990) as
0 = 0.00603 × 107.5T/(237.3+T).
s water evaporates from the droplet it will become moreoncentrated. This rise in concentration will reduce the evapo-ation rate. Raoult’s law states that the vapour pressure of eachomponent in an ideal solution is dependent on the vapourressure of the individual component and the mole fraction,
, of the component present in the solution.
x0)soln = fx0 = V − S/C100
V + S/�w(mw/ms − �w/C100)x0 = C100V − S
C100V − ıSx0,
here
= 1 − C100mw
�wms≈ 0.916
nd �w is the density of water and mw and ms are the molec-
lar mass of water and sucrose, respectively. This gives theorrected evaporation rate
= H
(C100V − S
C100V − ıS− RH
)ˇV1/3
(C100V − S
C100V − ıS− RH
),
9 ( 2 0 0 7 ) 323–332 327
where RH = x∞/x0 is the relative humidity and the evaporationparameter, ˇ, is
ˇ = D
1/x0 − RH
(3�2
4
)1/3
.
The Heaviside function disallows the possibility of condensa-tion of water from a humid atmosphere onto the droplet.
2.5. A dynamical systems based model
It is a simple step to bring the flow rate and the evaporationrate together to form a dynamical model for the volume of adroplet, V, and the sucrose content, S
dV
dt= Q − ˛,
dS
dt= QCin,
with initial conditions V(0) = S(0) = 0 where S(t)/V(t) → Cin ast → 0.
Introducing the following dimensionless variables (indi-cated by hats),
V =(
kQ0
ˇ
)3
V, S = C100
(kQ0
ˇ
)3
S, t = (kQ0)2
ˇ3t,
C� = C100C� , Cin = C100Cin, (4)
the model in its fullest form is
dV
dt= H
(1 − S
C� V
)(1 − S
C� V
)− H
(V − S
V − ıS− RH
)V1/3
×(
V − S
V − ıS− RH
),
dS
dt= H
(1 − S
C� V
)(1 − S
C� V
)Cin. (5)
It is assumed that Cin < C� < 1, i.e. that the concentrationof the fluid in the anal tube is lower than the concentrationat which inflow is blocked; and this in turn is lower than theconcentration of a droplet of pure sucrose. Henceforward thehats are omitted.
The quantitative behaviour of the system (i.e. how quicklythe drop grows and the size and sugar content it eventuallyreaches) depends on the flow rate Q0, and evaporation param-eter ˇ. Although ˇ can be determined from empirical data,values for Q0 are derived by assuming that the insect is a pas-sive conduit through which the sap flows in accordance withphysical laws. In Section 3, the experimental data will be usedto estimate the “best-fit” value of Q0.
In contrast, the qualitative behaviour is independent of ˇ
and Q0, as they do not appear in the nondimensionalisedmodel (5). Rather, the important factors are the environmen-tal conditions (i.e. relative humidity), the physical properties ofthe sucrose solution and the stopping concentration C� . Thesevalues can be found in the literature (see Table 1), apart from
C� , which is estimated from the data in Section 3. The differ-ent classes of qualitative behaviour that can occur will nowbe identified for a range of values of this parameter and of therelative humidity RH.
i n g
A
328 e c o l o g i c a l m o d e l l
For the system to reach a steady state, the following twoconditions must be satisfied
1 − S
C� V≤ 0 and
V − S
V − ıS≤ RH.
The first of these is satisfied when the concentration of thedroplet has increased to C� , and so the tube is blocked andflow stops. The second condition is satisfied when the vapourpressure at the surface of the droplet is equal to the vapourpressure of the ambient air, resulting in zero evaporation. Bothconditions occur when the concentration S/V exceeds somethreshold value. The zero flow condition occurs if S/V ≥ C� ,and the zero evaporation condition occurs if
S
V≥ C∗ = 1 − RH
1 − ıRH.
Clearly this threshold concentration for zero evaporationvaries with relative humidity (from C∗ = 1 for RH = 0 to C∗ = 0for RH = 1). Thus the behaviour of the system falls into twodistinct types, depending on which of the two threshold con-centrations is the lower. If C� < C∗, i.e.
RH < RH∗ = 1 − C�
1 − ıC�, (6)
then the solution will reach a steady state, otherwise thedroplet will continue to grow indefinitely.
Three different types of behaviour may be identified. Theseare not dependent on the flow or evaporation rates, but arewholly a function of the relative humidity and the stoppingconcentration C� . Fig. 2 shows examples of these three typesof behaviour:
Low Humidity. When the ambient relative humidity is low,i.e. C∗ > C� , the droplet concentration will rise as waterevaporates, until it reaches C . At this point the sucrose
�
level has reached a steady state and the volume has reachedits maximum value. The volume will then decrease, and theconcentration increases until S/V = C∗. Then the droplet isat a steady state with a high concentration and the tube
Fig. 2 – Schematic time series illustrating the different solution rsucrose content. The bottom row shows the droplet concentratiosteady state. In cases B and C the concentration reaches a steady
2 0 9 ( 2 0 0 7 ) 323–332
is blocked until the droplet is removed by either wind orpredators.
B Medium Humidity. For higher humidity values, i.e. C∗ <
C� , the evaporation rate is slower. In these conditions thedroplet and its concentration grow but will never reach thehigh concentrations needed to block the anal tube. The hon-eydew solution will continue to flow into the droplet until itgrows to such a size that it can no longer be supported by theanal tube and falls off. If the zero evaporation concentrationC∗ is greater than the input concentration Cin, then again C∗
will be the limiting concentration for the droplet: C → C∗ ast → ∞. However, even though the concentration, S/V, mayreach a steady state, the droplet volume and sucrose con-tent will not. These will continue to grow as time tends toinfinity.
C High Humidity. If the relative humidity is so high that thezero evaporation concentration C∗ is less than the inputconcentration Cin, then there will never be any evapo-ration, and the volume and sucrose content will simplygrow linearly with time at constant concentration Cin. Thethreshold humidity separating this from case B is
RH = 1 − Cin
1 − ıCin.
For the standard parameter values given, this case willoccur if the relative humidity is above 98.9%. Thus caseC, where the concentration of the droplet never increasesabove Cin, will only occur in extremely humid conditions.
Because of uncertainty in the values of certain modelparameters, it is difficult to generate quantitative predictionsfor S and V. However, it is possible to predict the qualitativebehaviour of the system. Fig. 3 shows, for a range of values ofrelative humidity and the saturation constant C� , whether the
droplet: (A) blocks the anal tube and dries up; (B) continuesto grow, with a concentration that increases over time, untilit falls off the end of tube; (C) continues to grow, at constantconcentration, until it falls off the end of the tube.
egimes. Top row shows the volume of the droplet and then S/V. In case A the volume and sucrose content reach astate but the volume and sucrose levels continue to grow.
e c o l o g i c a l m o d e l l i n g 2 0
Fig. 3 – Behaviour of the system for different values of thesaturation constant C� and the relative humidity. Case A islow humidity: the droplet dries up a blocks the anal tube.Case B is medium humidity: the droplet continues to growuntil it is too large to be supported by the insect; theconcentration increases to a steady state over time. Case Cits
3
Miwsdhttwimw
FTb
s high humidity: the droplet continues to grow until it isoo large to be supported by the insect; the concentrationtays constant over time.
. Comparison to field data
easurements of the honeydew produced by individualnsects were made at a field site approximately 45 km north-est of Christchurch, New Zealand. (172◦15′E, 43◦12′S). The
ite was 410 m above sea level, and forest at the site wasominated by black beech (Nothofagus solandri). Samples ofoneydew produced by individual insects were collected onwo days, 31 January and 6 February 2006. On each day, threerees with heavy scale insect infestations (> 1000 insects m−2)
ere selected at random from a population of trees grow-
ng close to a walking track, average diameter (±1S.D.) at 1.4was 0.24 ± 0.06 m. On each tree, a band of bark 20 cm tall
as delimited approximately 1 m from the base of each tree.
ig. 4 – Collected data for each sample day. The top line is day 1emperature and humidity throughout the sampling period. (b) Vest fit. (c) Sucrose content with a line of best fit. (d) Concentratio
9 ( 2 0 0 7 ) 323–332 329
Within each band, a sample of 20 individual insects was cho-sen from those actively producing honeydew. Each insect wasassigned a unique identifier, and its location recorded. Overthe time when samples were taken, air temperature and rela-tive humidity were measured with a humidity probe (Humitter50Y, Vaisala, Woburn, MA) connected to a datalogger (CR10x,Campbell Scientific, Logan, UT). Air temperature and humid-ity were measured every 60 s, and averaged every 15 min forthe duration of measurements. Over the sampling period, val-ues for relative humidity and temperature ranged from 20–70%and 14–26 ◦C, respectively.
Samples of honeydew were collected, and the dry mass ofeach droplet determined using a variation of the gravimet-ric method described by Dungan et al. (2004). At the start ofeach measurement period droplets were cleared from eachmarked tube. At pre-determined times (420–6000 s on day 1,30–3540 s on day 2) after the initial clearance, the rate ofhoneydew production was determined by periodically remov-ing the honeydew droplets that had been produced sincethe droplets were first removed. Each droplet was collectedwith a separate glass microcapillary tube (1 �l capacity; Micro-caps, Drummond Scientific, Broomhall PA), the mass of whichhad been determined prior to sampling. The volume of eachdroplet was determined by measuring the length of the tubeoccupied. Tubes were immediately placed in pre-weighed alu-minium foil envelopes. The envelopes were transported backto the laboratory, where honeydew contained in the tubewas evacuated onto the opened foil envelope. Each sample(foil+tube+honeydew) was dried to constant mass in a forced-draught oven, for 72 h at 70◦C, and the dry mass of honeydewcalculated as the change in mass of the tube and envelopebefore and after sampling. The concentration of each dropletwas calculated from its volume and dry mass. Fig. 4 showstime series for the temperature and humidity, and plots ofdroplet volume, sucrose content and concentration against
age.
Two important model parameters, the stopping concentra-tion C� , and the sap flow rate Q0, are not currently known.The collected data may be used to estimate best-fit values for
(31st Jan 2006); the bottom line is day 2 (6th Feb 2006). (a)olume of the collected droplets against age, with a line ofn.
i n g
330 e c o l o g i c a l m o d e l l
these parameters. Note that the threshold value of the rela-tive humidity RH∗, given by Eq. (6), separating case A (dropletsdrying out) from case B (droplets continuing to grow) dependson C� and hence is not known a priori. To attempt to determineparameter values based quantitatively on the measured datawould be potentially inaccurate due to experimental error.Nevertheless, the qualitative behaviour of the data may be usedto obtain robust parameter values by gaining an estimate ofthe threshold relative humidity RH∗. It is argued that the datapresented in Fig. 4 behaves according to case B (i.e. the mediumhumidity regime) of the model, in which droplet volume andsugar content increase with droplet age, and concentrationreaches a plateau level (see Fig. 2(B)). This is seen by fitting lin-ear regression models to the volume and sugar content databy least squares: the lines of best-fit are shown in Fig. 4. Forthe volume data, the least squares fit has a correlation coeffi-cient of r2 = 0.707 for day 1 and r2 = 0.345 for day 2. Althoughthe data for day 2 are inconclusive, the data for day 1 stronglysuggest that volume is increasing with droplet age. For thesucrose data, r2 = 0.537 for day 1 and r2 = 0.542 for day 2, sug-gesting that sucrose content is also increasing with dropletage. A linear fit to the concentration data gives r2 = 0.15 forday 1 and r2 = 0.07 for day 2, indicating that there is no strongrelationship between concentration and droplet age. This isalso consistent with case B behaviour, in which concentration,after an initial increase (which may take place on a time scaleundetected by the data collection), reaches a steady state.
The relative humidity varied between 20% and 50% on day1 and between 50% and 58% on day 2. Thus for the environ-mental conditions to be in the medium humidity regime formost of the day, the threshold relative humidity RH∗ must beno higher than 40% (taking a conservative estimate). Eq. (6)
thus implies that
C� = 1 − RH∗
1 − ıRH∗ > 0.947,
Fig. 5 – The top line shows the model fitting for day 1, the bottomrange of gravimetric concentrations, C� , for each data set. For dayall other parameters as Table 1. (b) An example of the best-fit mosucrose content data.
2 0 9 ( 2 0 0 7 ) 323–332
which means that the gravimetric stopping concentrationmust be greater than 96%. Note that this lower bound, whichis a conservative estimate, is robust to experimental error inthe data values. It is based solely on the qualitative behaviourof the droplet data, which agrees most closely with case B ofthe model, and on the measurements of relative humidity.
To find a best-fit value for the flow rate, the sum of the leastsquares residuals for the volume and concentration weightedby the mean data values was used:
F = 1E(Vi)
∑d
(Vm(t) − Vd(t))2 + 1E(Si)
∑d
(Sm(t) − Sd(t))2,
where Vm(t) and Sm(t) are the model predictions at time t andVd(t) and Sd(t) are the data values gathered for droplets of aget. The weighted residual F was then minimised over Q0 fora selection of C� values. For any chosen value of C� , there isa best-fit value of Q0. There is little difference between thebest-fit lines obtained for the range of stopping concentrationsshown. Fig. 5 (a) shows these best-fit values as a multiple ofthe Poiseuille predicted flow rate, Eq. (2).
The best-fit Q0 values range from 10 to 60 times the valuepredicted by Poiseuille flow. Fig. 5(b) and (c) show the data withthe best-fit model solution. There is no visible difference forthe best-fit solutions at any of the C� values in the range.
4. Discussion
The model uses simple ideas from fluid dynamics to describeand predict the formation and behaviour of individual dropletsof honeydew produced by Ultracoelostoma scale insects thatare widespread in New Zealand’s Nothofagus beech forests.
By treating the insect as a series of connected pipes (mouth-parts, body and anal tube), and making the assumption thatthe insect exerts no control over the rate of flow, the model pro-duces qualitative descriptions of droplet behaviour that match
line day 2. (a) The best-fit flow rate for the appropriate1 RH = 40% and T = 23C; for day 2 RH = 55% and T = 16C;
del with the volume data. (c) The same model with the
g 2 0
fihgciwoms
ttwsvccFtdfath
osfldrtdeiuaio
sassssattidtd
tTlierh
e c o l o g i c a l m o d e l l i n
eld observations. Depending on conditions of atmosphericumidity, our model predicts three outcomes. Droplets eitherrow indefinitely at constant concentration when air is almostompletely saturated and evaporation is negligible; dropletsnitially grow but then shrink as water evaporates to the point
here they impede further inflow when air is particularly dry;r, when values for relative humidity are between approxi-ately 40% and 99%, droplets initially grow rapidly then more
lowly as water evaporates.Gaze and Clout (1983) noted that the honeydew concentra-
ions of droplets collected from individual insects on a singleree, i.e. subject to similar shade and humidity conditions,ere almost identical, whereas the concentrations seen over
everal months varied between 16% and 81%. This seasonalariation was attributed to changes in phloem sap sugar con-entration, whereas the model suggests that these variationsould easily be seen due to local environmental conditions.uture field work that aims to predict overall forest produc-ion should include this effect. It is commonly noted that onry days honeydew becomes so concentrated and viscous thaturther droplet formation is stopped. A combination of modelnd data analysis has shown that the concentration requiredo do this is at least 96% and that, for this to occur, the relativeumidity must be less than 40%.
Despite good qualitative agreement between model andbservation, comparison of model predictions with field datauggests that Poiseuille flow substantially underestimates theow rate. On two days in late summer with contrasting con-itions of air humidity and temperature, it was found thatates of fluid flow from the tree through the insect would needo be 10–60 times greater than predicted in order to produceroplets that behaved as observed. There are several possiblexplanations for this: (i) the model substantially overstates themportance of droplet evaporation; (ii) the model substantiallynderestimates the pressure gradient between the phloemnd the atmosphere; (iii) the insects are playing an active rolen regulating flow; or some combination of these factors. Eachf these possibilities is now discussed.
When collecting field data, it was assumed that the atmo-phere in the immediate vicinity of the droplets was the sames the bulk atmosphere in the forest understorey. It is pos-ible that the air immediately surrounding the droplets isignificantly cooler and more humid (and hence has loweraturation deficit) that the bulk atmosphere, due to heavyhade and efflux of water vapour from the tree stem (Orennd Pataki, 2001; Prinzing, 2003). However, model calcula-ions using bark–atmosphere humidity differences similar tohose reported by Prinzing (2003) suggest that underestimat-ng humidity still does not account for the large differences:ecreasing air saturation deficit in the model by 60% decreasedhe best-fit flow rate, but to not less that 7–8 times that pre-icted by Poiseuille flow.
It was assumed in the model that phloem pressure is inhe range 1–1.8 MPa, as measured by Gould et al. (2005).he internal pressure of individual phloem elements, particu-
arly in the woody stems of mature trees, has been measured
nfrequently, so may be higher than these values. How-ver, to produce rates of flow up to 60 times greater wouldequire that phloem pressure be 60 MPa, which is infeasiblyigh.
9 ( 2 0 0 7 ) 323–332 331
It is, therefore, proposed that the large differences betweenthe flow rates predicted by Poiseuille flow and those neces-sary to reproduce the observed dynamics of droplet formationshow that, rather than acting as a passive conduit, the scaleinsects are actively regulating their feeding behaviour andincreasing the apparent underlying flow rate. Two mecha-nisms may explain the higher than expected observations.First, it is possible that the insects are actively sucking thephloem sap from the trees, rather than using existing phloempressure to drive their feeding. Active sucking has beendemonstrated in other phloem-feeders (Mittler, 1967). How-ever Kingsolver and Daniel (1993) contend that, over extendedperiods of feeding by aphids, a balance between phloem intakeand honeydew production is obtained, suggesting that, overtime, honeydew production rates should match those pre-dicted by Poiseuille flow. Second, it is possible that insects arefeeding episodically, so rates of flow through the insects aregreater during feeding bouts. Such episodic feeding has beenshown in woolly aphids Eriosoma lanigerum feeding on apples(Sandanayaka and Hale, 2003). It is possible that the insects’excretion and feeding behaviour are uncoupled, and that theinsects accumulate honeydew for excretion during discreteevents, during which rates of flow through the anal tube aresignificantly elevated. This is supported by observations thatinsects that are definitely alive and active (as shown by thepresence of honeydew droplets) may not produce additionalhoneydew during an observation or sampling period (Kellyet al., 1992). By deliberately targeting insects that were pro-ducing honeydew, the field sampling may have only includedthose insects actively excreting, and producing droplets at arate greater than that predicted by Poiseuille flow. It wouldbe possible for the insects to accumulate honeydew and thenexcrete it at an elevated flow rate, without having to maintainan internal hydrostatic pressure as high or higher than that ofthe phloem, because the insect’s anal tube is much wider thatits mouthparts. Even a moderate hydrostatic pressure in theinsect’s body is still capable of producing a much higher flowrate through the relatively wide anal tube. Future modellingwork, in conjunction with data from experiments specificallydesigned to detect such behaviour, will address the possibilityof episodic feeding.
The model has been tested and calibrated by fitting toexperimental field data on droplet volume and sucrose con-tent. These data are sufficient to obtain a qualitative matchwith model predictions, and strongly suggest that the insectsare playing an active role in flow regulation. However, it is clearfrom Fig. 4 that the degree of experimental error in the data isquite large (in particular the fit of the model to the data fromday 2 is not as good as for day 1 due to the large amount ofscatter in the data set) and this provides a significant barrierto obtaining a model that is qualitatively correct. These prob-lems may be overcome in the future by collecting data undercontrolled laboratory conditions (although the improvementin experimental accuracy may be offset to some degree byremoving the object of interest from its natural environment).
To summarise, the model describes the qualitative
behaviour of honeydew droplets in a way that matches fielddata, and provides a mechanistic explanation for observations(e.g. Gaze and Clout, 1983) that tree-scale rates of honeydewproduction are regulated by environmental conditions that
i n g
r
A., Rickard, S., Atkinson, B., Fagan, L.L., Ewers, R.M., Didham,R.K., 2006. Vertical stratification in the spatial distribution
332 e c o l o g i c a l m o d e l l
strongly affect evaporation of water from droplets. The mostimportant conclusion of this work is that the insects are play-ing a more active role in the system than previously thought.Two possible mechanisms (active sucking and episodic feed-ing) by which the insects may actively regulate flow rates havebeen proposed, and these will be tested in future work usinga combination of modelling and empirical data.
Acknowledgements
Scanning electron microscopy was carried out by NeilAndrews, and microscope images were prepared by MattWalters.
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