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Ecological Modelling 222 (2011) 2084–2092

Contents lists available at ScienceDirect

Ecological Modelling

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

mechanistic model to predict transgenic seed contamination in bee-pollinatedrops validated in an apple orchard

ebecca C. Tysona,∗, J. Ben Wilsona, W. David Laneb

Barber School of Arts and Sciences, University of British Columbia Okanagan, 3333 University Way, Kelowna, BC V1V 1V7, CanadaPacific Agri-Food Research Centre, 4200 Hwy 97, Summerland, BC V0H 1Z0, Canada

r t i c l e i n f o

rticle history:eceived 13 May 2010eceived in revised form 10 March 2011ccepted 27 March 2011vailable online 27 April 2011

iffusion modelsolation distanceollen dispersalransgenic croputcrossingathematical model

a b s t r a c t

The adventitious presence of transgene containing seed in conventional crops is an issue of considerableinterest; a model to predict levels will aid regulators and help to address concerns of farmers and con-sumers. While outcrossing levels have been described in crops such as rape that are wind-pollinated,or both wind- and insect-pollinated, much less is known about pollen dispersal in exclusively insect-pollinated crops. In this paper, we develop a mechanistic model for pollen movement that is based on adescription of bee movement through space. Our model is a system of diffusion-based partial differentialequations that we use to predict percent transgenic seed at arbitrary distances from a transgenic sourceunder different planting scenarios. We present a two-pronged study, in which the mathematical mod-elling work is informed by experimental work. The latter was carried out in an apple orchard with a rowof 200 transgenic source trees carrying the GUS marker gene. Fruit from neighbouring conventional treeswas gathered at distances ranging from 3.5 m to 183 m, and the seeds were extracted and germinated.Percent transgenic seed at each location was determined by testing the seedlings for the presence of theGUS marker gene. We use the experimental data to validate and parameterize the model, and then runmodel simulations to determine expected percent transgenic seed in various linear landscapes. We find

that the percent transgenic seed in neighbouring conventional trees and orchards is a function of the sizeof each orchard block and the distance between them. The model explicitly shows the effect of overlap-ping transgenic and nontransgenic pollen distributions in setting seed distributions, and also shows thevalue of buffer rows in reducing outcrossing levels in neighbouring crops. The model parameters can beadjusted to suit particular crops and locations, and may be useful for determining plausible distributions

ngs n

on transgenic seed planti

. Introduction

Land planted with genetically modified crops has increasedapidly in recent years (Council for Agricultural Science andechnology, 2007). Simultaneously, widespread concern about thenintended spread of transgenes remains a public policy issueindering further development of transgenic crops (Rieger et al.,002; Chandler and Dunwell, 2008; Morris et al., 1994). Cur-ent regulations in Europe require that any crops with more than.9% presence of transgenes be labeled transgenic (Council forgricultural Science and Technology, 2007; European Parliament,003), and thresholds as low as 0.1% have been suggested (Huygen

t al., 2003). It is important then, that methods are available to pre-ict the expected level of outcrossing in conventional crops grown

n proximity to transgenic ones.

∗ Corresponding author. Tel.: +1 250 807 8766; fax: +1 250 807 8004.E-mail address: rebecca.tyson@ubc.ca (R.C. Tyson).URL: http://people.ok.ubc.ca/rtyson (R.C. Tyson).

304-3800/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2011.03.039

eeded to allow for an adventitious presence of, for example, 0.9%.© 2011 Elsevier B.V. All rights reserved.

Plant pollen is dispersed either by wind or insects, usuallybees. For plants using both dispersal methods, the relative impor-tance of each is variable and not well-described (Hüsken andDietz-Pfeilstetter, 2007), though pollinator-mediated pollination isthought to be the dominant mechanism and effective over longerdistances, at least for oilseed rape (Hüsken and Dietz-Pfeilstetter,2007). Most of the existing research on pollen gene flow fromtransgenic crops has been carried out with oilseed rape, and withexclusively wind-pollinated plants such as corn (Beckie et al., 2006).Empirical work has shown that most pollen falls within a distance ofless than 20 m from the parent plant (Hüsken and Dietz-Pfeilstetter,2007; Reboud, 2003) (though see (Messeguer et al., 2006)).

There is evidence that pollen spread is typically leptokurtic(Morris, 1993), but this may be an artefact of the observationmethods used (Osborne et al., 1999). Indeed, Rieger (Rieger et al.,2002) did not observe a leptokurtic decline, or even an exponen-

tial decline, though their results are controversial since they didnot check for adventitious presence of transgene in the plantedseed. The amount of outcrossing between transgenic and conven-tional crops depends on the distance between the transgenic and

R.C. Tyson et al. / Ecological Modelling 222 (2011) 2084–2092 2085

Fig. 1. Diagram of the study site showing the location of transgenic trees, conventional trees, and transects along which apple seeds were gathered. The zoom panel in thelower right shows R, the half-canopy width of the row of transgenic trees. The study site consisted of 4 blocks planted with apple trees, shown as rectangles in the diagramand labeled as blocks B1–B4. The transgenic trees occurred as a single row of 200 trees in the longest orchard block. In addition to the transgenic trees, which were of typegala, Orchard block B1 consisted of a single cultivar (Cripp’s Pink, also called Pink Lady), while the other blocks were composed of a large number (≈50) of mixed cultivars.T B4. Thl ered a

cbPfoa

t(crpttr

he transgenic trees and cross-compatible with all of the other trees in blocks B1–abeled A–D and I–IV, were placed across the orchard blocks, and apples were gath

onventional crop blocks, the nature of any vegetation in the gapetween them, and the size of the area planted (Hüsken and Dietz-feilstetter, 2007; Messeguer et al., 2006; Reboud, 2003). Otheractors such as flowering time and weather conditions also affectutcrossing level; we focus here on the primary factors of distancend flower presence.

In practice, farmers and regulators need to know isolation dis-ances resulting in predictable levels of transgene containing seedRong et al., 2010). Research with oilseed rape indicates that someharacteristics of pollen dispersal are predictable. Since oilseedape pollen is dispersed by both wind and a host of natural insect

ollinators, it is difficult to separate the effects of different pollina-ion agents, and limits the extent to which the results can be appliedo other types of crops, particularly bee-pollinated ones. The oilseedape research however, suggests that pollen from other plants will

e areas between the orchard blocks were empty of floral resource. Eight transects,t 3.5 m (12 ft) intervals along these transects.

be dispersed according to patterns that can be quantified. A mecha-nistic model for pollen dispersal based on these observed patternsand that can be used to predict outcrossing levels would be veryuseful to growers and regulators (Rong et al., 2010).

In this paper, we examine the bee-mediated spread of trans-genes in a crop that is exclusively pollinated by bees; wind isnot a pollination agent. We develop a mechanistic model for bee-mediated pollen movement, and so our approach is general andcan be used to predict bee-mediated pollen movement in a varietyof crops. Most outcrossing models to date are empirical and focuschiefly on wind-mediated pollen dispersal (Beckie and Hall, 2008),

and so the framework we present here represents a significant stepforward. As a practical application, we place our work in the con-text of outcrossing in apple. Only a few reports have described theextent of outcrossing or spread of pollen by wild or honey bees

2 Modelling 222 (2011) 2084–2092

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Table 1Total percent transgenic seed found in transects A–D. Data shown is combined datafrom two years of observations (2002 and 2003). Percent transgenic seed is deter-mined from the total number of transgenic seeds collected at distance x on transectsA–D divided by the total number of seeds collected at distance x on transects A–D,yielding one value for each distance. The field study was designed with imperialmeasurements (ft) and so we report the distances in both the original imperial andequivalent metric units.

Transects A–D

Distance from transgenicsource (m (ft))

Number of seeds insample

Percent transgenicseed

3.5 (12) 4065 20.067.5 (24) 3431 12.4811 (36) 3473 6.86

086 R.C. Tyson et al. / Ecological

n fruit orchards (Reim et al., 2006; Soejima, 2007; Ish-Am andisikowitch, 1998; Kron et al., 2001). The data to date are fairlyoarse, and sample sizes small, limiting the development of accu-ate predictive models. This paper describes a two-pronged study,here we use a database totalling about 38,000 seeds to informmodel that predicts outcrossing levels of bee-pollinated crops

n general, and is validated on apple in particular. We show howhe model can be used to predict outcrossing rates at discrete posi-ions or within recipient orchards neighbouring a transgenic pollenource.

. The empirical study

.1. Method

The development of transgenic seed resulting from dispersalf apple pollen by honey bees was measured in an isolated applerchard in eastern Washington State in an area suitable for com-ercial apple production. The nearest alternative apple pollen

ource was an orchard 5 km away from the study site, and the inter-ening landscape consisted largely of fields used for annual crops oray. The fruit-bearing trees were present in four rectangular blocks,ith the longest one containing a row of 200 transgenic trees 115 m

ong. The physical layout of the fruit-bearing trees (conventionalnd transgenic) in the orchard is illustrated in Fig. 1.

The orchard consisted of a mixture of many apple cultivarslanted in parallel rows in four distinct blocks. The trees presentedfairly continuous canopy of blossoms although trees or groups of

rees were missing in some rows. The area between blocks was bareround and did not support bee foraging. Honeybees (Apis mellif-ra) were rented and placed in the bare ground between blocks inhe first year of sampling; in the second year no honeybees weremported and so pollination was carried out by local domestic oreral bees. In both years the number of bees present was sufficiento result in a full commercial crop, and the conditions were suit-ble for producing a normal commercial crop throughout the area.loom time is through the last half of April with temperatures from

ows of 0–5 and highs of 18–25 ◦C. Rain is uncommon in this area.The presence of the transgenic GUS marker gene in the donor

rees (the row of transgenic trees) was confirmed by a third partyefore the trees were put in the field. In addition, germinated seedrom the donor plants, tested for the GUS gene in the year of thetudy, confirmed the presence of the gene in the expected 1:1atio plus a small additional frequency explained by a low level ofelf pollination. Transgene presence was confirmed using the GUSresence test described below (Section 2.1.1). The transgenic treesere of type Gala and were cross-compatible with the conventional

rees. These trees were the only transgenic cultivar in the area thatould produce transgenic pollen. The receiving cultivar in Block Aas Cripp’s Pink (also called Pink Lady). The cultivars in the other

hree blocks were a diverse mixture of about 50 major and minorommercial fruiting cultivars. All were Malus domestica or wereompatible and could cross-pollinate with it. Similarly, the conven-ional trees of various cultivars were cross-compatible with eachther. Apple does not develop seed when self-pollinated, though aow level of self-pollination appears to occur as the blossoms agend the self-incompatibility mechanisms degrade. The transgenicrees had a single copy of a marker gene (GUS), and so only half theollen grains from the transgenic trees contained the chromosomeith the marker gene.

Apples were gathered at stations 9 m apart along the 8 tran-

ects illustrated in Fig. 1, in two consecutive years. The numberf apples obtained per station varied with lower numbers of fruitor none at all) being collected if fruit was not available close tohe designated station location. In Block A there was consistency

15 (48) 3060 4.218 (60) 2829 1.67

with about 60 fruit or 13 kg of apples collected per station. In theremaining blocks, fruit gathered per station was highly variable.Gene transfer is via seed, not fruit, as fruit is derived from mater-nal tissue. To assess gene flow then, apple seeds were extractedfrom the fruit. The number of seeds per apple varies with a num-ber of factors with the most important being cultivar, conditions offlower development, and any flower damage that occurred (from,for example, frost). Seed number varied from 0 to about 20 per fruit.Golden Delicious apples, for example, average around 10–12 seedsper fruit, while Red Delicious apples average at closer to 5 seedsper fruit.

The apple seeds were germinated in the laboratory using stan-dard methods used by apple breeders. The seeds were first stratifiedin damp sand at 4 ◦C for 8–10 weeks in the dark. The flats were thenmoved to the greenhouse and held at 20 ◦C to promote germina-tion. When the seedlings were at the cotyledon stage (4–6 cm high),the seedlings from each sampling station were pooled, a 0.5 cmsegment of each hypocotyl was harvested, then assayed for thepresence or absence of the marker gene as described below (Sec-tion 2.1.1). The percent transgenic seed at each station was thenrecorded. Percent germination was 95–100% for all samples.

2.1.1. GUS presence testThe presence of the GUS marker gene was assessed using a

standard method (Jefferson et al., 1987; Sambrook and Russell,2001) that is very widely used and highly regarded for its consis-tency and accuracy in tissue such as that used in the experimentsdescribed herein. The tissue sample from each plant was incubatedin buffer, and then a colourless substrate was added. The reactionwas allowed to develop over 24 h with incubation at 22 ◦C. The pres-ence of the gene was indicated by a dark blue colour throughoutthe plant tissue compared to a pale yellow/white/gray colour in thecontrols and negative samples.

2.2. Empirical results

The experimental results are presented in Tables 1 and 2 andin Fig. 2. For each seed assayed, x was the straight-line distancealong the transect between the sampling station and the nearestpoint on the row of transgenic trees. For transects A–D this dis-tance is measured along the current transect. For transects I–IV, xis the distance between the sampling station and the “Referencepoint” shown in Fig. 1. Note that in this case the distance x is mea-sured along a straight line that is different from the transect linesI–IV drawn in Fig. 1. We expect that the proportion of seed from

conventional trees with the marker gene describes how bees act asfunctional pollenizers, resulting in gene flow from the transgenicto the conventional trees.

R.C. Tyson et al. / Ecological Model

Table 2Total percent transgenic seed found in transects I–IV. Data shown is combined datafrom two years of observations (2002 and 2003). Percent transgenic seed is deter-mined from the total number of transgenic seeds collected at distance x on transectsI–IV divided by the total number of seeds collected at distance x on transects I–IV,yielding one value for each distance. The field study was designed with imperialmeasurements (ft) and so we report the distances in both the original imperial andequivalent metric units.

Transects I–IV

Distance from transgenicsource (m (ft))

Number of seeds insample

Percent transgenicseed

18 (30) 623 0.1618 (60) 383 0.0027 (90) 641 0.00

37 (120) 809 0.0046 (150) 1606 0.2555 (180) 2064 0.0564 (210) 2959 0.1773 (240) 1940 0.0582 (270) 2068 0.48

101 (330) 2394 0.08110 (360) 1602 0.06119 (390) 1015 0.10128 (420) 769 0.00137 (450) 1243 0.24146 (480) 30 0.00155 (510) 30 0.00165 (540) 666 0.00

santtisi

FtbcfopItc

174 (570) 125 0.00183 (600) 115 0.00

In Fig. 2, note the rapid decline in the occurrence of transgeniceed over the first 30 m from the transgenic source, followed bynoisy tail with very low frequency of transgenic seed. A simpleonlinear regression fit of the exponential model y(x) = a e−bx + c tohe data gives the curve shown in Fig. 2. Data was pooled across

he two years of the study since the occurrence of transgenic seedn the far-distance transects (transects I–IV) was a rare event, ando pooling was necessary in order to obtain sufficient experimentalnformation about the tail of the distribution.

0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

30

35

40

45

Per

cent

Tra

nsge

nic

Seed

Distance (metres)

ig. 2. Plot showing the percent GUS presence found in seed as a function of dis-ance (asterisks and circles) and the best-fitting exponential model as determinedy nonlinear regression (solid curve) (the exponential model and data are highlyorrelated with R2 = 0.72, p < 0.01). The percent GUS presence at each distance wasound by taking the total number of transgenic seeds divided by the total numberf seeds gathered at that distance. The data from all transects and both years wasooled to obtain the data shown. Circles represent data gathered along transects

–IV, and asterisks represent data gathered along transects A–D. The data was fito the curve y = a e−bx + c where the best-fitting values were a = 41.0, b = − 0.169 and= 0.133.

ling 222 (2011) 2084–2092 2087

While the exponential curve gives a reasonable fit to the data(exponential model and data are highly correlated with R2 = 0.72,p < 0.0001), the difficulty with this fit is that it provides no mech-anistic explanation for the shape of the outcrossing curve. Wetherefore cannot extrapolate the exponential model to other sit-uations, such as different planting scenarios or distances beyondthe 200 m limit of the study. In order to resolve this issue, we turnto studies of bee behaviour and a mechanistic modelling approachbased on understanding how bees move pollen through space.

3. The model

Bee behaviour is complex and involves direct communica-tion between conspecifics, return trips to the hive, and complexresponses to the environment (Burns and Thomson, 2005; Osborneet al., 2001; Morris et al., 1994). Recently, Tyson et al. (2011) devel-oped a diffusion model for bee movement which we use as a basisfor our model of pollen movement. We note here that the modeldoes not include any effects of temperature, rainfall, or directionalbias in bee behaviour, and also assumes complete flowering syn-chrony between all of the cultivars in the orchard. While thesefactors do have an effect on pollen dispersal and thus outcrossinglevels, they chiefly act to modulate the effects of distance and com-peting pollen. A model that considers all of the myriad details thataffect outcrossing would be so complex, that it would lose someof its usefulness as an educational model that can explain generalpatterns in experimental results (Okubo and Levin, 2001; Conway,1977). In the interest of developing a useful yet mathematicallytractable model, we assume constant weather and maximum syn-chrony conditions, and focus on the effect of distance and amountof competing pollen. Following the format presented in Tyson et al.(2011), we are led to consider two subpopulations of pollen: motilepollen and stationary pollen.

Motile pollen is all pollen still in parent flowers or attached todispersing bees. Dispersing bees are divided into harvesters andscouts (Tyson et al., 2011), where the harvesters are engaged insmall-scale diffusive movement from flower to flower, while thescouts are engaged in large-scale advective movement from oneflower patch to another. We assume that motile pollen is attachedto both harvesters and scouts, but that pollen can only be picked upand deposited by harvesters. Each bee picks up pollen when in har-vesting mode, and then that pollen travels through space as the beerelocates through a sequence of harvester and scout movements asdescribed by Tyson et al. (2011). Eventually, the pollen is depositedon another flower during an episode of harvester movement bythe bee. In this way, motile pollen becomes stationary pollen atconstant rate ˇ.

Stationary pollen is pollen that has been transported from itsparent flower and deposited onto another flower. We assume thatthis pollen is no longer available to the motile pool. Only stationarypollen can lead to seed formation.

Production of motile pollen depends on flower presence.Accordingly, we define a function F(�x) which is equal to 1 wher-ever flowers are present, and zero everywhere else. We assume thatnew flowers are uniformly distributed, and that the pollen densityper flower is constant. New motile pollen thus enters the pool ofdiffusing pollen at constant rate ˛F(�x).

With these assumptions, we arrive at the mathematical equa-tions for pollen movement:

∂Pd(�x, t)∂t

=

diffusion︷︸︸︷D∇2Pd +

production︷︸︸︷˛F(�x) −

deposition︷︸︸︷ˇPd −

movement mode switching︷︸︸︷�(Pd − Pa) ,

(1a)

2 Model

weaatdPet(fptd

ptsvpgvttaaeFdsoao

˚

wnoP˚

dttpo

088 R.C. Tyson et al. / Ecological

∂Pa�(�x, �, t)∂t

= −

advection in direction �︷︸︸︷v(cos(�), sin(�)) ·

(∂Pa�

∂x,

∂Pa�

∂y

)

+

movement mode switching︷︸︸︷�

(Pd

2�− Pa�

)(1b)

∂Pa(�x, t)∂t

=

advection in all directions �︷︸︸︷∫ 2�

0

∂Pa�(�x, �, t)∂t

d� , (1c)

∂Ps(�x, t)∂t

=

deposition︷︸︸︷ˇPs, (1d)

here Pd(�x, t) is the density of motile pollen still in parent flow-rs or attached to harvesters, Pa(�x, t) is the density of motile pollenttached to scouts, and Ps(�x, t) is the density of stationary pollen,t location �x and time t. The parameters D and v are, respectively,he isotropic diffusion rate and the advection speed. The advectionirection is given by �. The resulting stationary pollen distributions(�x, t) determines the distribution of pollen from a source of flow-rs arranged as described by F(�x). We use the term “pollen shadow”o describe this distribution. Equation (1) is the same as equation6) in Tyson et al. (2011), except for the addition of the equationor stationary pollen (1d), and the terms describing deposition androduction of motile pollen. The detailed development of the modelerms related to bee movement (rather than pollen production andeposition) is explained in Tyson et al. (2011).

It is chiefly the distribution of transgenic seed, rather thanollen, that interests us. We must therefore determine how theransgenic pollen shadow relates to the distribution of transgeniceed. This relationship will depend on the arrangement of con-entional trees and the amount of competing conventional pollenresent. Since apple trees cannot self-pollinate, any pair of pollenrains are competitive only if they are from different apple culti-ars. At a given flower then, we define all “viable stationary pollen”o be all pollen that has been deposited on the flower’s stamen andhat can potentially fertilize the eggs. Which pollen grains actuallyre successful in reaching the eggs depends on a number of factors,nd the pollen grains that arrive first often do not have the high-st success rate (Yoder et al., 2009; Visser and Verhaegh, 1980).or simplicity, we assume that the amount of transgenic seed pro-uced at location �x is proportional to the amount of total viabletationary pollen deposited at �x that is transgenic. Since only halff the pollen from the transgenic source carries the transgene, themount of transgenic pollen deposited at �x is half the total amountf pollen from the transgenic source. Mathematically, we have

T (�x) = (1/2)PTs(�x)PTs(�x) + PNs(�x)

(2)

here PTs and PNs are, respectively, the density of transgenic andontransgenic viable stationary pollen at �x, and ˚T (�x) is the amountf transgenic seed produced at �x. The spatial distributions PTs(�x) andNs(�x) are obtained by solving equations (1). We call the distributionT (�x) the transgenic “seed shadow”.

In order to match our theoretical seed shadow with the dataescribed in Section 2.2, we need to know the pollen shadow from

ransgenic trees planted in a line of finite length. We consider thiso be approximated by an infinitely long line of transgenic treeslanted in a homogeneous domain of floral resource (that is, a rowf transgenic apple trees planted in an orchard of conventional

ling 222 (2011) 2084–2092

apple trees). In this case, the problem reduces to a one-dimensionalone. Consequently, pollen can only advect in two directions, � = 0and � = �, and so advecting pollen is reduced to two populations:Pr (right-moving) and Pl (left-moving). The pollen dispersal model(1) in this case becomes

∂Pd

∂t= D

∂2PTd

∂x2+ ˛F(x) − ˇPd − �(Pd − Pr − Pl), (3a)

∂Pr

∂t= −v

∂Pr

∂x+ �

(Pd

2− �Pr

)(3b)

∂Pl

∂t= v

∂Pl

∂x+ �

(Pd

2− �Pl

)(3c)

∂Ps

∂t= ˇPd, (3d)

with transgenic floral resource distributed according to

F(x) ={

1, |x| ≤ R,

0, |x| > R.(4)

The length 2R is the width of the transgenic tree canopy centredat x = 0. Our model (3) is the same one given in equation (12) inTyson et al. (2011), with the addition of a fourth equation for thestationary pollen, and the arbitrary width R of the transgenic sourceF(x).

Bees disperse quickly through an orchard, so we assume thatthe pollen distributions rapidly equilibrate to steady state func-tions denoted P∗

d(x), P∗

r (x) and P∗l(x). We assume further that it is the

steady state solutions that are relevant for determining the percenttransgenic seed at any location. Following the method described inTyson et al. (2011), we obtain the general solution for P∗

das

P∗d(x) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

c1e�1x + c2e−�1x + c3e�3x + c4e−�3x, if x < −R

c5e�1x + c6e−�1x + c7e�3x + c8e−�3x + ˛

ˇ, if |x| ≤ R,

c9e�1x + c10e−�1x + c11e�3x + c12e−�3x, if x > R,

(5)

The coefficients ci are found by applying the boundary conditions atx = ± ∞ and the matching conditions at x = ± R. Notice that P∗

d(0) →

˛/ˇ as R increases, and so as the width of the flowering regionincreases, the density of motile pollen inside that region maximizesat ˛/ˇ. This can be verified mathematically by solving the steady-state system with R → ∞.

We now turn to the solution of Eq. (3d). Since we assume thatpollen deposited onto a non-parent flower cannot be transferredfrom that flower to another one, Ps is only created and not removed.Thus, Ps must necessarily increase with time, and there is no steady-state solution for Ps. Furthermore, the initial conditions for bothmotile and stationary pollen affect the solution for Ps(x, t). In orderto develop some sense of how Ps(x, t) behaves, we assume that themotile pollen is initially distributed according to the steady-statesolution. In particular, we have Pd(x, 0) = P∗

d(x). Then the distri-

bution of pollen is a function of x only, and we can immediatelysolve Eq. (3d). We obtain Ps(x, t) = ˇP∗

d(x)t + c(x). Since Ps(x, 0) = 0

(initially, all pollen is motile), we must have c(x) = 0, and so

Ps(x, t) = P∗d(x)t. (6)

Note that only harvester bees can deposit motile pollen ontoanother flower, so it is only diffusing pollen, P∗

d(x) that contributes

to stationary pollen, Ps(x, t).The same calculations apply to the pollen shadow from a con-

ventional source. In the presence of multiple pollen types, wedenote the transgenic diffusing and stationary pollen shadowsby P∗

Td(x) and PTs(x, t) respectively, and the diffusing and station-

ary conventional pollen shadows by P∗Nd

(x − xN) and PNs,i(x − xN, t),

R.C. Tyson et al. / Ecological Modelling 222 (2011) 2084–2092 2089

Fig. 3. Plot showing the percent transgenic seed (circles) as a function of distance,and the curve of best fit (solid curve) obtained using Eq. (8). The best fitting solu-tion curve (R2 = 0.98) was determined by minimising the least squares differencebetween the model and data for a large range of parameter values. For the near-distance measurements (0–10 m) we used the data from transects A–D, and forthe long-distance measurements (¿10 m) we used the data from transects I–IV (ourrationale is presented in Section 2.2). The model and data are highly correlated:R2 = 0.988, p < 0.01. The levels of conventional (dash-dotted curves) and transgenic(dotted curve) pollen predicted by the model are also shown, as well as the totalamount of pollen (solid curve). The y-axis represents “pollen level” for the PT , PN

and PT + PN curves, and percent transgenic pollen for the 100PT/(PT + PN) curve. Themodel results were obtained for a 3.5 m wide transgenic source from x = 0 m to 3.5 m.There were two patches of conventional trees in the model domain (as per the dataaax

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m

˚

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4

ttevmawfi

Table 3Parameter values for the model (3) obtained by fitting the steady-state equations tothe field data. The time unit is unknown because the field data was all obtained atsteady state.

Parameter name Description (units) Fitted value

R Radius of tree canopy (m) 1.75D Pollen diffusion rate (m/time2) 8.5˛ Rate of pollen production (1/time) 1ˇ Rate of pollen deposition (1/time) 0.1� Rate of pollen transfer between advecting 0.01

long minimum-distance lines from sampling stations to the transgenic source), andjacent block 18 m wide from x = 3.5 m to 22.5 m and the second wider block from= 40 m to 200 m. The rest of the domain is empty of resource.

here xN is the location of the centre of the conventional orchard.he widths RT and RN of the transgenic and conventional orchardsre not necessarily the same. With this notation, the percent trans-enic seed that we would expect at any position x is given by (2)hich with (6) becomes

Tseed(x) = (1/2)PTs(x)tPTs(x)t + PNs(x − xN)t

= (1/2)P∗Td

(x)

P∗Td

(x) + P∗Nd

(x − xN). (7)

he time dependence in the stationary pollen distributions can-els out of each term and a steady-state solution is obtained. Thushe percent transgenic seed at any location is equal to the percentransgenic diffusing pollen at that location.

If the level of competing pollen within the receiving orchard isaximal, then PNs(x, t) = (˛/ˇ)t and (7) simplifies to

Tseed(x) = (1/2)P∗Td

(x)

P∗Td

(x) + (˛/ˇ)(8)

hich is the minimum possible value. Note that we are only inter-sted in transgenic seed outside the domain of transgenic trees, ando we restrict our attention to | x |>RT.

. Predicting percent transgenic seed

For the field trials described in Section 2.1 we can assume thathe level of competing pollen was high. We thus fit our model (8) tohe data in Section 2.2 to determine the values of the model param-ters, with the restriction that RT = RN = 6. We found the parameteralues (D, � , v, ˇ and ˛) which gave the solution curve with the

inimum least squares difference between the theoretical solution

nd the data, and the largest proportion of solution points fallingithin the data ranges measured empirically. The data and best-tting model solution are shown in Fig. 3, and the parameter values

and diffusing pools (1/time)v Advection speed (m/time) 2.4

are given in Table 3. The parameters yielding the best-fitting modelsolution were determined through an exhaustive search of param-eter space. For each set of parameter values, the model solutionwas obtained numerically, and then the residual sum of squarescomputed as

SSerr =n∑

i=1

(yi − �Tseed(xi)) (9)

where the yi is the experimental observations at distances xi. Theparameter set that yielded the minimum residual sum of squareswas selected as the one defining the best-fitting model.

We note here that there is a discrepancy between in the near-distance data gathered in the A–D and I–IV transects. This suggeststhat the gap between orchard block B1 and the other orchard blocksaffected bee dispersal behaviour. The effect of gaps in floral resourcehas been investigated by other researchers (Morris et al., 1994;Bhattacharya, 2004; Reboud, 2003), and the results are inconclu-sive. Since we had high quality near-distance data from transectsA–D, we simply used that data for distances up to 30.5 m (100 ft),and then used the I–IV data for larger distances. This approachensures that we are using the maximum level of outcrossing con-tained in the data. We thus used Eq. (8) computed with the solutionsof (3) when the linear domain has a transgenic source at x = 0 m, flo-ral resource on 0 < x < 22.5 m and 40 < x < 200 m, and bare ground(no floral resource) elsewhere.

With the parameter values obtained from the data, we can use(7) to predict the percent transgenic seed in an arbitrarily wideconventional orchard at any distance L from a transgenic sourceof a given width. Since apples cannot cross-pollinate within a sin-gle cultivar, we must make some assumptions about the natureof the conventional and transgenic orchards. We first assumethat the transgenic orchard is of a different cultivar from allof trees in the conventional orchard, as this is the case thatcould lead to unintended cross-pollination of a transgenic blossomwith a conventional one. Second, we assume that the conven-tional orchard has sufficient pollenizing trees, or mixed cultivarsblooming simultaneously, so that all conventional pollen can beconsidered competing pollen for both the transgenic and conven-tional flowers.

In practice, the level of transgenic outcrossing in a conventionalorchard is determined by taking a random sample of harvestedapples from the orchard, and then testing for the presence of trans-genes. This is akin to measuring the average percent transgenic seedwithin the entire orchard. Mathematically we have

�Tseed(x) =∫ xN+RN

xN−RN

˚Tseed(x) dx, (10)

where the conventional orchard has width 2RN centred at xN. Thefunction ˚Tseed(x) is given by (7), where P∗

Tm and P∗Nm are given by

(5), with R replaced with RT or RN for the transgenic and conven-tional orchards respectively.

2090 R.C. Tyson et al. / Ecological Modelling 222 (2011) 2084–2092

0 20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80

90

100

N Non−Transgenic Trees (full orchard width)E Empty (no floral resource)

T Transgenic Trees (half−orchard width)Transgenic PollenNon−Transgenic PollenPercent Transgenic Seed

Distance (metres)

Pol

len

Lev

els

NE ET

Fig. 4. Pollen shadows for two orchards, one transgenic and the other conven-toc

cPatta

obrtsetlT

epla(

Fst(

Table 4Minimum isolation distance, as predicted by (7), between a transgenic orchard(200 ft wide) and a conventional orchard. The minimum distance depends on therelative size of the transgenic and conventional orchards. In the trap plant scenario,the conventional orchard is a single row of trees. The buffer is a zone on the edge ofthe conventional orchard that is nearest to the transgenic orchard. Fruit harvestedfrom trees within the buffer is not included in the conventional harvest, and so doesnot contribute to the outcrossing measurement.

Relative size ofconventionalorchard

Buffer width(m (ft))

Isolation distance required (m)

0.9% outcrossing 0.1% outcrossing

Trap N/A 558 1341

0.5 0 18 4603.5 (12) 15 4337.3 (24) 7.9 421

1.0 0 15 14103.5 (12) 2.4 4157.3 (24) 4.9 408

2.0 0 9.1 1320

ional, and the resulting predicted percent transgenic seed, in the case where halff the pollen from the transgenic orchard carries the transgene. The transgenic andonventional orchards are contiguous.

Sample simulation results are shown in Figs. 4 and 5. Theoncentrations of transgenic and conventional pollen, P∗

Tm = P∗Td

+∗Ta and P∗

Nm = P∗Nd

+ P∗Na respectively, drop quickly outside the

ppropriate source orchard. In the conventional orchard, percentransgenic pollen is highest near the edges of the orchard wherehe concentration of conventional pollen is low, and then drops tomuch lower value inside the conventional orchard.

If we set the tolerance level for percent transgenic seed at 0.9%r 0.1%, we can determine the minimum isolation distance requiredetween the transgenic source orchard and the conventionaleceiving orchard. The level of outcrossing will vary depending onhe width of the transgenic and conventional orchards. We con-ider the case of a large transgenic orchard paired with either anqually large conventional orchard, or a single conventional row ofrees. The former case is a continuous planting scenario, while theatter case is akin to the trap plant scenario used in field studies.he results are shown in Table 4.

From the first two rows of Table 4 we see that outcrossing lev-ls are higher in the trap plant scenario than in the continuous

lanting scenario, with minimum isolation distances being much

arger in the former case. When outcrossing is being measured attrap plant, the level of competing pollen is low to non-existent

especially if the trap plant row contains only a single non-selfing

ig. 5. The predicted percent transgenic seed within the receiving orchard, for theame two orchards illustrated in Fig. 4. For this configuration, the average percentransgenic seed within the receiving orchard is less than 0.9%, but greater than 0.1%these thresholds are indicated on the figure as T1 and T2, respectively).

3.5 (12) 5.5 30.57.3 (24) 1.8 55.0

cultivar), and so the percent transgenic seed due to outcrossing ishigher than it would be for a large conventional orchard at the samelocation.

The last three rows of Table 4 demonstrate the effectivenessof buffer rows in reducing outcrossing levels. Buffer rows are treerows from which fruit are harvested separately, and so the out-crossing level in buffer row fruit is not averaged or mixed in withthe outcrossing level in the remaining orchard fruit. Buffer rowsin the continuous planting scenario reduce the minimum isolationdistance by a factor of 2–3 if the target outcrossing level is no morethan 0.9%. At the 0.1% outcrossing level, the same buffer rows aremuch less effective.

5. Discussion

Using a partial differential equation (PDE) framework, ourmodel predicts the distribution of pollen dispersed from a flow-ering crop, such as apple, that is pollinated exclusively by bees.Diffusion–advection equations have often been used to describedispersal of insects and other intelligent organisms (Okubo andLevin, 2001; Biesinger et al., 2000; Moorcroft and Lewis, 2006;Tyson et al., 2007) including bees (Morris, 1993), but only a fewconsider the advective and diffusing portions of the populationseparately (Skalski and Gilliam, 2003). Our results indicate thatthe harvester-scout model for bee dispersal (Tyson et al., 2011)can be successfully used to predict transgenic seed shadows forbee-pollinated crops.

While we present our model in the context of a transgenic out-crossing study carried out in an apple orchard, our model is basedon a mechanistic description of bee movement. While the bee dif-fusion and movement switching rates may vary from one crop toanother, or one location to another, the basic description of move-ment as a series of diffusive and advective motions will be validacross crops. The basic model then can be used for transgenic pollenmovement in any bee-pollinated crop, with movement parametersadjusted for each crop and location.

Our outcrossing results are consistent with earlier work on othertransgenic crops (Rieger et al., 2002; Reboud, 2003; Hüsken andDietz-Pfeilstetter, 2007), which show that the level of expected

transgenic outcrossing is highest at the near edge of the conven-tional crop, and decreases rapidly with distance from the transgenicsource. Note that we obtain this result without assuming any spe-cial behaviour by bees in the presence of a gap in resource. Field

Model

wflsctb

ptgorar0t4

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bofflactttpdamTbavotevtopr

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tptWw

R.C. Tyson et al. / Ecological

ork indicates that pollen flow across a gap is different from pollenow in the absence of a gap (Reboud, 2003; Morris et al., 1994) ando it is interesting to note that increased levels of transgenic seedan be expected at the near edge of the conventional crop even ifhe gap between the two orchards has no effect on bee foragingehaviour.

One strategy to reduce the unintended presence of transgenicollen in a conventional crop, is to separate the crop harvest intowo parts: harvest obtained from the crop rows nearest the trans-enic planting is separated from the rest. Consistent with workn oilseed rape, we find that the use of buffer rows considerablyeduces the isolation distance required, especially in the case of0.9% outcrossing tolerance level. A 3.5 m (12 ft) wide buffer row

educes the isolation distance by over 40% (see Table 4). At the.1% outcrossing tolerance level, the reduction in necessary isola-ion distance with a 3.5 m (12 ft) buffer row is much smaller, at only%.

These results relate to the case when the conventional andransgenic orchards are of comparable size. As the size of the con-entional orchard decreases, we find that the isolation distancencreases, due to the fact that the conventional orchard is pro-ucing less pollen to compete with the transgenic pollen. Whenhe conventional orchard is a single row wide, we are at the one-imensional analogue of the trap plant scenario. At this limit, wend the largest isolation distances.

In addition to relative size of the two orchards, and distanceetween them, there are other physical factors that can affectutcrossing levels. In particular, we discuss here the effects of dif-erences in flower density and flowering time. Differences in theower density of the two orchards behave in a similar way to rel-tive width of the two orchards (results not shown), as they bothorrespond to situations where one orchard produces less pollenhan the other one. Flower density per ground area is a function ofree age and cultivar characteristics such as biennial bearing, andree architecture, with tall spreading trees having more blossomer ground surface area than spindle or espaliered trees, simplyue to the size and shape of the tree. Differences in flowering timeslso occur, but the effects are more difficult to assess with ourodel, since our analysis is based on steady-state assumptions.

hat is, we assume that the bee and pollen distributions equili-rate quickly relative to the bloom time. This is a straightforwardssumption when flower density is constant over the entire inter-al. If the transgenic and conventional orchards have only partialverlap in their bloom time, then flower density changes from oneime period to another, and so equilibration must happen withinach distinct time period. As these are shorter than the time inter-al given by synchronous blooming, the ratio of equilibration timeo bloom time increases, and at some point we can no longer relyn our steady-state assumption. The aysnchrony level at which thisoint is reached depends on bee behaviour. This is a topic of futureesearch.

Because of the slow rate of decay in the long tails of the seedhadow distribution, achieving an outcrossing tolerance level of.1% requires isolation distances that are two orders of magnitude

arger than the distances required for the 0.9% outcrossing toler-nce level. The computed isolation distances (396–427 m) are wellithin a honeybee or large bumblebee’s potential range (Walther-ellwig and Frankl, 2000), and so it is certainly possible that appleollen could be carried between flowers several hundreds of metrespart.

Our results are obtained based on the simplifying assumptionhat there is either sufficient variety of apple types, or sufficient

resence of pollenizing trees (such as crabapple) in the conven-ional orchard so that there is plenty of competing pollen there.

e made this assumption for two reasons. First, our analyticalork showed that as orchard size increases, the maximum pollen

ling 222 (2011) 2084–2092 2091

density rapidly saturates at a constant level. Second, the field sitewas a specialized orchard with many different apple cultivars, andso, given the simplifications inherent in collapsing the system toa one-dimensional model, the assumption of saturating levels ofcompeting pollen made sense.

One factor which we did not address is the density of foragingbees. This factor could have a significant effect on predicted out-crossing levels, and should be investigated in a future model. It isknown that in a year when the weather is unseasonally cool dur-ing blossom time, the foraging bees are not as active and fruit setis decreased (personal communication with orchardists). This shouldnot affect outcrossing levels, unless it also affects the distances overwhich bees forage, or the amount of time they spend foraging eachday. A decrease in either of these parameters would reduce theamount of travel between neighbouring plantings, and thus alsodiminish the expected level of outcrossing.

We have completely ignored the effect of prevailing winds in ourmodel, chiefly because it is uncertain how bee foraging behaviour isaffected by wind. There is currently no conclusive work on the effectof moderate winds on the directions bees choose to forage in, or onthe distances over which foraging activity occurs. There is alwaysthe possibility that strong gusts could “blow a bee off course” andover a larger distance than one might otherwise expect, but theseare rare events and not well quantified.

6. Conclusions and future directions

The shape, and particularly the decay rate, of the pollen distribu-tion tails is a crucial part of our isolation distance computations, butthe tails are very difficult to measure empirically. Indeed, a naivefitting of our data to a standard exponential function yields tails ofconstant width, suggesting that the likelihood of outcrossing neverdecreases to zero even as the distance between flowers increasesto infinity. An alternative approach of fitting the data to an arrayof diffusion-advection models using AICc goodness of fit criteria,always selected the pure diffusion model as the best, even thoughthat model misses the data in the tails altogether. Both extremesare unsatisfactory, and so we arrived at our predicted decay rate byfitting the field data to a decaying function that was derived froma mechanistic model of bee behaviour (Tyson et al., 2011). Morework is needed in characterizing the tails of pollen shadows frominsect-pollinated crops, though there are significant challenges interms of the sample sizes required to detect outcrossing levels aslow as 0.9%, let alone 0.1%.

Our most immediate goal is to study the full two-dimensionalmodel, so that we can test the effects of geometry. Work on oilseed rape (Messeguer et al., 2006) has shown that geometry playsan important role, and so the level of outcrossing should dependon the relative lengths of the nearest edges, and the orientationof each orchard. We could also use the two-dimensional modelto study the effect of planting type on outcrossing level. That is,we could explicitly include pollenizer trees, both in the transgenicand conventional orchards, and observe their effect on the out-crossing levels. We are currently in the process of studying thetwo-dimensional model.

This paper represents the first serious effort at modelling insect-mediated pollen dispersal. All transgenic crops approved to date arepollenized by both wind-mediated and insect-mediated pollina-tion, or are self-fertile and thus self-pollinating. There are however,many economically important horticultural crops for which insect-pollination is much more important than wind-pollination. A

detailed understanding of insect-mediated pollen dispersal andpotential levels of outcrossing is an important step in the devel-opment of regulations and protocols surrounding transgenic crops.Our approach can be used to give a first-order estimate of expected

2 Model

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vp(cvtgtb

nwtifm(d

A

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R

B

B

B

B

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092 R.C. Tyson et al. / Ecological

utcrossing levels between neighbouring plantings. This work ishus likely to contribute mostly to providing insight to regulatorsssessing the environmental risk of outcrossing and secondly toonditions on where the material might be grown given certainsolation distances and buffer-type barriers.

An important contribution from the model is that it can pro-ide an estimate for outcrossing levels based on bee movementarameters (diffusion (D), advection (v) and switching coefficients�)) rather than extensive (and expensive) measurements of out-rossing distributions in the field. The tails of any distribution areery difficult to measure empirically, and yet it is these very tailshat are so important in the development of policy for new trans-enic plants. Therefore, a model that can predict the shape of theseails based on more easily measured parameters (in this case, beeehaviour parameters) is very useful.

Our model also is very helpful in providing mechanistic expla-ations for field measurements of outcrossing. In particular, theork presented in Table 4 shows with mathematical rigour that

rap plant scenarios overestimate required isolation distances. Themportance of the rare bee that travels far enough to carry pollenrom a source to a distant trap plant is difficult to asses experi-

entally. Mathematically however, we can translate bee behaviourand pollen dispersal) into expected percent transgenic seed at anyistance, given the amount of competing pollen available.

cknowledgements

The authors would like to thank Neal Carter, president, Okana-an Specialty Fruits for many helpful discussions and for hisnancial support of the experimental work. Many thanks alsoo Shaun Strohm and Andria Dawson for their help with thetatistical tests of goodness of fit. Two anonymous reviewers pro-ided detailed and helpful comments which have improved theanuscript. This work was supported by grants from NSERC (RT),ITACS (RT), UBC Okanagan (RT & JBW) and Agriculture and Agri-

ood Canada (WDL).

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