PredictingtheSpreadof PlantDisease:Analysisof anInfinite-DimensionalLeslieMatrix Model for

Phytophthora infestans

JamesA. Powell�andIvanSlapnicar

�Departmentof MathematicsandStatistics

UtahStateUniversityLogan,Utah84322-3900USA

WopkevanderWerfCropandWeedEcologyGroup

WageningenUniversityBode98,Postbus4306700AK Wageningen

TheNetherlands

July19,2002

Abstract

A modelfor the sizeclassdistribution of plant diseaseon plant tissuesis developed,in-spiredby lateblight lesionsonpotatoandtomatocausedby Phytophthora infestans. In theab-senceof spatialdispersalthemodelbecomesaninfinite-dimensionalLesliematrix,andwhenspatialdispersalis consideredseveralelementsof theLesliematrix areconvolution operatorsaccountingfor thespreadof sporesandgenerationof new lesions.The maximumpredictedspeedat which lesionsspreadis calculatedby extendingthemethodof NeubertandCaswell[15], whichdeterminesmaximumpossiblefront speedfor propagationof invasionsof anage-structuredpopulation,to thecaseof infinite-dimensionalmatrices.Observedspeedsagreewithpredictedspeedsto within errorsresultingfrom convergenceto thestableagedistribution andto theasymptoticfront speed.

Keywords: Phytophthora, integrodifferenceequations,Lesliematrices,fronts, invasion,ratesofspread

AMS Subjects: 92D25,92D99

Running Head: Spreadof PlantDisease

Submittedto SIAM J. Applied Math

�Correspondingauthor, email: powell@math.usu.edu�onsabbaticalfrom Universityof Split, Croatia

1 Introduction

Plantdiseases,causedby fungi, bacteria,virusesandothermicroorganisms,area leadingcauseof

agriculturalcroploss. Oneof themostimportantplantdiseasesin theworld, in termsof damage

andcontrolcosts,is lateblight diseasein potatoesandtomatoes,causedby theoomycetePhytoph-

thora infestans(Hooker, 1981). Oomycetesarea distinct groupof plant pathogenswhich until

recentlywereregardedasfungi, but have now beenclassifiedasa distinct taxon,morerelatedto

algaethanto fungi. Epidemiologicallyhowever, with regardto thespreadof diseasein plantpop-

ulations,oomyceteshave muchin commonwith fungalpathogens.Their life cycle includesthe

samestepsof infectionof ahost,formationof biomass‘mycelium’ in thehost,spatialexpansionof

theaffectedarea‘lesion’ in thehost,andformationanddispersalof dispersalbodies‘spores’.For

the purposeof this paper, we will thereforespeakaboutfungi whenwe discussepidemiological

processesthat arerelevant to both fungi andoomycetes.The oomycetePhytophthora infestans

is taken as an exampleorganismbecauseof its practical importanceand becauseits life cycle

attributesarewell studied.

Thehostplantrangeof P. infestanscoversat least90 plantspecies,mostof themmembersof

theplant family Solanaceae[5]. A greatdealof researchis dedicatedtowardsbreedingresistant

potatoandtomatovarietiesandto developingnew fungicides. Today, crop lossesdueto potato

lateblight have beenestimatedat 10 to 15 percentof theglobalannualproduction[1]. Theeco-

nomicvalueof theselossesplusthecostof cropprotectionamountto 3 billion USdollarannually

[4]. Currently, the control of potatolate blight dependson the frequentuseof fungicides. De-

spitethis chemicalinput, lateblight epidemicsareincreasinglymoredifficult to control. A better

understandingof the epidemiologyof potatolate blight is neededto develop new, effective and

environmentallyfriendly controlstrategies.

Reproductive strategiesof fungi,includingthe taxonomicallydistinctbut ecologicallysimilar

oomycetes,arevariedin the extreme. A commonthemefor foliar plant pathogenicfungi is the

productionof airbornesporesfrom sporulatingbodies. Sporesare released,spreadwith wind

and/orrain andafter landingon (nearby)plantsurfacesthey potentiallycausenew lesions.Once

a lesionis initiated, thepathogenicfunguscolonisesthesurroundingplant tissueby sendingout

hyphaeandextractingnutrientsfrom this tissue.The lesiongrows at a relatively constantradial

rate.Severaldaysafteraregionof tissuehasbeencolonisedby hyphae,sporulatingbodiesdevelop

2

from thelocalmyceliumandsporescanbeproducedandreleasedfor sometime. After thisperiod,

thelocal myceliumdiesandsporulationstops.In themeantime thecolonisedarea,andtherefore

thelesion,hasexpanded.

For P. infestansthis generalpatternof latency, infectiousnessandsenescenceresultsin the

very typicalcircularlesionswith aninfectiveannulussomedistancebehindthe(invisible) leading

edgeof the lesionanddeadtissuesomeradialdistancebehindthat. Onemaythink of lesionsas

the basicinfection unit of P. infestans(ZadoksandSchein[27]). Releaseandspreadof spores

from anannularsporulatingregion insideeachlesion,followedby infection,is thebasicmodeof

propagationof P. infestansthroughacrop.Growthof P. infestansis stronglyinfluencedby thedaily

cycle of temperature,relative humidity of theair andleaf wetness.Similarly, dispersalof spores

and initiation of new lesionsexperiencesstrongdaily forcing from periodicity in temperature,

relativehumidityof theair, leafwetnessanddaily winds.

Below we will proposea discrete-time,continuous-spacemodelfor thedensityof lesionsin a

crop. In a spatiallyinvariantsetting,neglectingboundariesenforcedby finite leaf size,themodel

for the densityof lesionsbecomesan infinite-dimensionalLeslie matrix, andwhendispersalis

includedmany of thenonzero-entriesin thematrixbecomespatialconvolutionoperators.Analysis

of theLesliematrix indicatesthatall eigenvaluesareboundedby a largesteigenvalue,whosesize

canbe computedanalytically. The existenceof this largesteigenvalueallows us to computean

asymptoticboundon the rateof invasionof new lesionsinto uninfectedcrops. Theseresultsare

testedfor factoriallycrossedparametervariationsandtwo dispersalkernels;resultsarecompared

with known convergenceerrorsdueto the power methodapproximationandaccelerationof the

front.

2 Modeling the Population Dynamics of Fungal Invaders

2.1 Age Structure of Lesions

An individual lesionon a leaf grows at a measurableandwell-definedradialgrowth rate, ��� , per

day, andafteracertainlatency period( ��� fivedaysfor P. infestans) theinvadedareaof theleaf

sporulatesfor a certainnumberof days( � �� 1 day). Theprogressof an individual lateblight

lesionon a singleleaf is depictedin Figure1. During the infectiousperiodsporesarereleased

at a given rate, ��� , per areaper day, andthesesporesdisperse.Somefraction of sporeswhich

3

Spo

rula

ting

Are

aof l

esio

n(5

day

s be

hind

lead

ing

edge

)

4 mm/day

Lead

ing

edge

of

lesi

on (

invi

sibl

e)

Initi

al p

oint

of

infe

ctio

n

behi

nd s

poru

latin

gN

ecro

sing

tiss

ue

area

Edg

e of

pot

ato

leaf

Figure1: Diagramof progressof asinglelesionthroughapotatoleaf. Theactualfurthestlocationof hyphaein the lesion,denotedasthe dashedcircle, is invisible. Theedgeof the visible lesionis thesporulatingarea,indicatedabove betweenheavy solid circles,which emergesfrom theleafsurfacefive daysafter infectionby hyphaeandproducessporulatingbodies.In the radial regionbehindthe sporulatingareathe lesionhasusedup all leaf resources,leaving a visible necroticlesion.A typicalmaximuldaily growth ratefor a lateblight lesionis 4 millimetersperday.

4

Parameter Description Nominal Value (Units)��� SporulationIntensity ����� (Spores/meter� /day)��� Latency Period 5 (days)� � InfectiousPeriod 1 (day)� infect Probabilityof infectionperlandedspore ��� � �� intcpt Probabilityof interceptionperdispersedspore ��� ������ Radialgrowth rateof lesions ��� ��� � ! (meter/day)�#"$� LeafAreaIndex 5 (meter� crop/meter� soil)% Meandispersaldistancefrom parentlesion 1 (meter)

Variable Description (Units)&Ageof Lesion (days)' Dayof Simulation(independentvariable) (days)(*)+ Densityof Lesionsof age

&onday ' (number/meter� )" + Areaof a lesionof age

&days (meter� )��" + Newly grown areafor a&-day-oldlesion (meter� )

Table1: Parametersandvariablesof thePhytophthora infestansinvasionmodel.Nominalvaluesaregleanedfrom [6] aswell asestimatesprovidedby field researchers[23], usingtheruleof thumbthateachparentlesionproducesabouttendaughterlesionsin idealcircumstances.

settlefrom theair areinterceptedby leaves(with probability � intcpt), andof theseinterceptedspores

a fraction, � infect, successfullygerminatesandinfectsthe plant (provided it doesnot land on area

alreadyoccupiedby a lesion).Theparametersof themodelandnominalvaluesarelistedin Table

1.

Whena lesionis&

daysold, theareathat it addsis thedifferencebetweentheareait is, " + ,- & ����./� andtheareait will becomeon thenext day, " +10 � ,- &32 ��./�4���5� . Thus,��" +10 � 6" +10 ��7 " + , ��� �98 - &:2 �;. � 7 & �=< -?> &@2 �;. , ��� �BA >5, & ��� �DCConsequently, whenalesionis six or moredaysold, theareawhichis sporulatingis theareawhich

wasaddedto thelesion ��� daysago. Since(E)+ is thedensityof lesionsof age

&dayson day ' ,

thenumberof sporesproducedby theselesionsis

SporesProduced ( )+ �F���G�H��" + � IKJ A ( )+ �F���G� >5, ��� � - & 7 ���L.NM - &PO ��L. CThis is an idealizationbasedon the assumptionthat leavesof the plant aremuchlarger thanthe

lesions;stabilityof ourresultsto relaxationof thisassumptionwill beinvestigatedin latersections.

5

Assumingthat all dispersalhappenslocally, the numberof sporesarriving is the numberof

sporesproducedandamodelfor reproductionof lesionscanbewritten( ) 0 �� � intcpt �Q� infect �R� unocc- ( )� M ( )� M�S�S�ST.P� - SporesArriving .( ) 0 �� ( )�

...( ) 0 �+ ( )+ ���

...

Thecombinationof probabilitiesin thefirst line is theprobabilityof thecompositeeventthat(first)

a sporelandson a leaf andis not subsequentlyknockedoff ( � intcpt), that(second)thesporeis able

to germinateandpenetratetheouterskin of the leaf ( � intcpt), andthat (third) thesporehaslanded

on leaf areanot currentlyoccupiedby a lesion( � unocc). Probabilisticparametersaresetusingthe

‘rule of thumb’ that1 parentlesionproducesa net10 daughterlesionsin theNetherlandsin ideal

circumstances[23]. Theprobabilityof a sporelandingon unoccupiedleaf areacanbecalculated

from theratioof thetotal leafareaandthetotalareaoccupiedby lesions,� unocc- ( )� M ( )� MDS�S�SU.#WVGX5Y[Z �#"\� 7^][_+1` � (E)+ " +�#"$� Ma�;bcdVeX5YRZf� 7 >5, ���K��#"$� _g+1` � ( )+ & � M4�;b C (1)

Thenumberof sporesproducedthepreviousdayis givenby

SporesProduced d�#�G� _g+1` I�J 0 � ( )+ ��" + � I�Jh ikj lAreaof InfectiousLesions

A ���G� >5, ��� � _g+1` IKJ 0 � ( )+ - & 7 ���L. C2.2 The Effects of Dispersal

To investigatethe spreadof lesionsthrougha crop onemustincludedispersaleffectswhich de-

scribehow sporesproducedin onelocationarrive at a differentlocation. Dispersalcanoccurby

wind, by raindrops‘splattering’ [17], or evenballistically by pressurizedexpulsionfrom sporan-

gia,andmodelscanrangefrom relatively simpleprobabilisticdescriptionsto solutionof turbulent

diffusion equationsin andabove the crop [9]. We will adoptherethe descriptive, probabilistic

approachandintroducea dispersalkernel, m -on . , which is theprobabilityof a sporeproducedatn p� beingdispersedto the location n . To determinethe densityof spores,� -qn . , arriving at a

6

location n , givena spatialdistributionof sporeproduction,� -qn . , oneevaluatestheintegral� -on .��r _� _ m -on 7ts .=� - s .vu s defwmyxP� COnemaythink of thisassummingtheprobabilitiesthatsporesproducedat location s , thenumber

of which is givenprobabilisticallyas � - s .vu s , will dispersethedistance-on 7zs . to thelocation n .Mathematicallywewrite thisastheconvolution, �tWm�x{� .

To includedispersalin theage-structuredmodelweneedto interpret( )+ asthespatialdensity

of lesionswhichare&

daysold onday ' andupdatethe‘SporesArriving’ to includetheeffectsof

dispersalfrom all spatiallocations.Thisgives

SporesArriving ����|� _g+1` IKJ 0 � - m�x ( )+ ./" + � I�J A ���|� >5, ��� � _g+1` I�J 0 � - & 7 ���L.=mpx ( )+ CWriting }~ ) - ( )� M ( )� M�S�S�S ( )+ M�S�S�ST.k� thespatio-temporaldynamicsaregovernedby a nonlinear

Lesliematrixwith dispersaloperations:}~ ) 0 � B �e� K x }~ )�� M (2)

where� is theinfinite dimensionalmatrix

B ��������������� � � � � �F� unocc

> ��� unocc � �F� unocc S�S�S - & 7 ����./�F� unocc S�S�S� � � � � � � � � � S�S�S� � � � � � � � � � S�S�S� � � � � � � � � � S�S�S� � � � � � � � � � S�S�S� � � � � � � � � � S�S�S...

......

......

......

......

......

�/������������� M (3)

K is thematrixcomposedof dispersalkernels,

� �������������� � � � � m -on .�m -on .�m -on . S�S�S�m -qn .�S�S�S� -qn . � � � � � � � � � S�S�S� � -qn . � � � � � � � � S�S�S� � � -qn . � � � � � � � S�S�S� � � � -on . � � � � � � S�S�S� � � � � -qn . � � � � � S�S�S...

......

......

......

......

......

� ������������ M (4)

and the operationof element-by-elementmultiplication (Hadamardproduct)is denotedby ‘ � ’,while theconvolution ‘ x ’ is takenelementby element.Thecompositeconstant,�� >;, � intcpt � infect �#����� � M (5)

7

is thenetnumberof new lesionsproducedin anunoccupiedenvironmentby an �� 2 � -dayold

lesion (the youngestlesionwhich is infectious). Nonlinearity is introducedinto the systemby� unocc, whichmustbecomputedonadaily basisfor eachlocationusingformula(1).

3 An Upper Bound for the Speed of Invasion

3.1 Review of the Minimum Wave Speed Calculation

Wesummarizehere(andadoptthenotationof) argumentspresentedby NeubertandCaswell[15]

for finite Lesliematriceswith dispersal,whicharein turnbasedonresultsof Weinberger[25, 26],

Kot et al. [12, 13] andNeubertet al. [16]. Estimatingthespeedof thewave of invasion,or front,

turnson analyzingthelinearizationof (2). For sufficiently small(*)+ (for example,in advanceof

themaininfestation),� unocc A � andthedynamicscanbewritten}~ ) 0 � A ��� K x }~ )5� M (6)

where� is thelinearizationof B, �� ����VJ unocc� � B CSufficiently far in advanceof thefront, thespatialshapeof solutionsmaybeapproximated}~ )���� ���¡  }¢ Mwhere }¢ is a vectordescribingtherelative abundancesin differentage-classesof lesions,eachof

whichdropsoff exponentiallyata rate, £ , in thedirection,n , in advanceof thefront. If a front has

formedandis travelingatadistance¤ periteration,then}~ ) 0 � -on .�¥}~ ) -on 7 ¤�. ��� �¡¦4���¡  }¢ Mandsubstitutinginto (6),� �¡¦4���¡  }¢ 8 A ��� K x � ���¡  � < }¢ � ���q B§A � M - £;.©¨ }¢ C (7)

Here ª - £5. is themoment-generatingmatrixcomputedby element-by-elementintegrationofª - £;.� r _� _ � �¡« � - s .vu s C (8)

8

To seewhy, consideroneof thenonzeroelementsof K in thefirst row:myx � ���¡  �r�_� _ � ���©¬T ��­«a® m - s .vu s � ���¡  re_� _ � �¡« m - s .vu s � ���q 5¯ - £5.°Mwhere ¯ - £;. is the(scalar)momentgeneratingfunctionfor thedispersalkernel m .

Cancellingcommonfactorsin (7) givesaneigenvalueproblem� �¡¦ }¢ §A � M - £;.©¨ }¢ def H - £;. }¢ C (9)

Suppose± -?² . has(countable)eigenvalues³ � - £5.°M°³ � - £;.NMDS�S�S , non-increasinglyorderedby magni-

tude.Theminimumwavespeedconjecture is thatthespeedof thewaveof invasionis smallerthan¤ ´ , where ¤ ´ Ve��µ¶¸· � ·@¹�º �£ ��µ - ³ � - £;.¸.¼»M (10)

where ½£ is themaximum£ for whichall elementsof ª - £5. aredefined.

Therearetwo perspectivesto takeon theapplicabilityandinfluenceof ¤ ´ , theminimumwave

speedperspective andthe dynamicperspective. From the minimum wave speedperspective, ¤ ´providesanoverestimateof all possiblespeedsfor frontsarisingfrom compactlysupportedinitial

conditions.Theargumentcanbesummarizedasfollows. Giventhat thenonlineargrowth rateis

non-negativebut alwaysboundedaboveby thelineargrowth rate,it is clearthatthenormof solu-

tionsto thenonlinearsystemis boundedby thenormof solutionsto thelinearsystem.Any finite,

compactlysupportedinitial conditioncanbeboundedabove by somespatialtranslateof � ���¡  }¢ ,

for all £ . Sincethe linearizeddynamicsmapsexponentialsolutionsto exponentialsolutions,the

nonlinearevolution from compactinitial conditionsis boundedabove by the linear evolution of

suitablytranslated� ���q  }¢ , independentof £ . Thesecanbe written (asabove) astranslatingso-

lutions with given speeds.The slowestof thesemust thereforeprovide an over-estimateof the

progressof nonlinearly-evolving fronts with compactinitial data. For finite matricesthis argu-

mentwasquiteelegantlystatedrecentlyby NeubertandCaswell[15]. In many, but not all, cases

it canalsobe shown that fronts accelerateto the minimum speed,in which caseit becomesthe

asymptoticspeedof fronts.

A related,dynamicperspective suggeststhat the ‘minimum’ speedshouldbe the asymptotic

front speed.This perspectiveharksbackto Kolmogorov et. al. [11], but wasstatedin thecontext

of dynamicsby DeeandLanger[3] andPowell et. al. [19, 18]. In a travelling frameof reference,

9

¾ n 7 ' ¤ , thesolutionto thelinearizedequationcanbewritten}~ ) 6¿GÀ ��� º � �­Á ) ¦/ H -qÃkÄ . ½}~ ¶ » M (11)

where¿GÀ���� denotestheinverseFouriertransform, ½}~ ¶ is theFouriertransformof theinitial data

andH is asin thediscussionabove, but evaluatedwith thesubstitution£�Å ÃkÄ . Asymptotically,

usingthepowermethod,theintegrandin (11)canbewritten� �­Á ) ¦/ H -¡Ã¼Ä . ½}~ ¶ dÆ � � �­Á ) ¦/ ³ ) � -qÃkÄ . ½ }� � -fÄ . 2 S�S�S­dÆ �ÈÇ Y�É § 'cÊ ��µ - ³ � -¡Ã¼Ä .¸. 7 à ¤ ÄÌË ¨ ½ }� � -?Ä . 2 S�S�SÍMwhere ³ � is thelargestmagnitudeeigenvalueand ½ }� � theassociatedeigenvector. Thus}~ ) A �>5, r _� _ � ÁÎÂaÏ Ç Y�É § 'cÊ ��µ - ³ � -¡Ã¼Ä .¸. 7 à ¤ ÄÌË ¨ ½ }� � -fÄ .Ðu Ä C (12)

The integral in (12) canbeevaluatedby methodof steepestdescentsto get a further asymptotic

approximation;thestationarypoint is givenby therootofuu Ä § ��µ - ³ � -qÃkÄ .=. 7 à ¤ Ä ¨ setd� C (13)

If Ä ´ is thestationarypoint solving(13),anassociatedspeedfor thetravelling frameof reference,¤ ´ , is chosensothatthewaveneithergrowsnorshrinksin this frameof reference,thatis

Real § ��µ - ³ � -¡Ã¼Ä ´ .¸. 7 à ¤ ´ Ä ´ ¨Ñw� C (14)

Working throughthe algebra,onefinds that thesolutionsto (13, 14) correspondexactly to (10),

usingthesubstitutionÃkÄ ´{Å £�´ .As pointedout by DeeandLangerandlaterby Powell et al., theseequationshave a dynamic

interpretation.Thequantitybeingmaximizedin (13) is theexponentialgrowth rateof aparticular

Fourier modein a frameof referencetravelling with speed¤ . Thus,the stationarypoint, Ä ´ , is

thatmodewhich hasmaximalgrowth rate;themethodof steepestdescentsbecomesa statement

that the asymptoticfront solution is that solution which grows from the most unstableFourier

modein theensembledescribingtheinitial data.Thechoiceof ¤ ´ via (14) is thenjust diagnosing

the speedattachedto the mostunstablemodeusinga stationaryphaseargument. The dynamic

viewpoint suggeststhat the minimal wave speed,¤ ´ , shouldnot only be an upperbound,but

alsothe asymptoticspeedobvserved,sinceit is connectedto the growth andpropagationof the

10

mostunstablewave componentof thesolution. Moreover, anasymptoticform for thesolutionis

predicted,}~ ) - ¾ . ��� ÁÎÂ°Ò¼Ï Æ � -fÄ ´4.Ó > ' , Ç Y�É § ��µ - ³ � -¡Ã¼Ä ´ .¸. 7 à ¤ ´ Ä ´ ¨FÔÕ ³vÖ Ö� -¡ÃkÄ ´°.³ � -¡Ã¼Ä ´ . 7Ø× ³�Ö � -qÃkÄ ´°.³ � -qÃkÄ ´ .�Ù �/ÚÛ �#ÜÝ ½ }� � -fÄ ´ . 2�Þ C Þ C (15)

Incorporatingthe factorofÓ ' from thedenominatorof (15) into theexponentindicatesthatob-

servedfrontsshouldconvergefrom below to theasymptoticspeed,¤�´ , as¤ observed W¤ ´ × � 7 ��µ - ' .> ' Ä ´ Ù M (16)

a resultwhichwewill useto analyzeourobservationsbelow.

3.2 Determination of Maximum Eigenvalue

Calculatonof ¤ ´ is on firm groundwhenthematricesinvolvedarefinite. For thefungalproblem,

however, thematricesconcernedareinfinite dimensionalandcalculationof themaximumeigen-

valueof H - £;. is notstraightforward.Recallingthatwehavetaken ¯ - £;. to bethe(scalar)moment

generatingfunctionfor thedispersalkernel m -qn . , andtaking ���ßwà to bedefinite,wecanwrite

H - £5.#�������������� � � � � � ¯ - £;. > � ¯ - £;. � � ¯ - £;.�S�S�S - & 7 àá.=� ¯ - £;. S�S�S� � � � � � � � � � S�S�S� � � � � � � � � � S�S�S� � � � � � � � � � S�S�S� � � � � � � � � � S�S�S� � � � � � � � � � S�S�S...

......

......

......

......

......

� ������������ M (17)

where� is definedby (5).

In orderto analyzethespectrumof H - £;. , we considerthelinearoperatorH â�ã � Å ã � defined

by -qn � M n � M n ! M�S�S�ST.#Å ×�ä _g `vå -?Ä 7 à�. n  M n � M n � M n ! MDS�S�S Ù M ä defw� ¯ - £;.NM (18)

whereãf� is theBanachspaceof all realsequencesx def -qn � M n � M n ! M�S�S�ST. suchthat ]çæ n  æ­è6é . The

matrix H - £;. is therepresentationof H in thestandardbasise  def � ÁΠ, where� ÁΠis theKronecker

symbol.Ourchoiceof thespaceãf� is natural,sincethetotalnumberof lesionsandsporesis always

finite. Thedomainof H is

DomH çê x ëQã � â3ììììì _g `vå -fÄ 7 àá. n  ììììì èíéïî M11

which alsoreflectsthe previousnaturalassumption,sincethe summationin the definition in the

domainis proportionalto thetotalnumberof sporesproduced(large,but finite). TheoperatorH is

unbounded,invertible,andits inverseis theleft-shift operatorH ��� definedby-on � M n � M n ! MBS�S�SU.�Å -qn � M n ! M n�ð M�S�S�ST. CClearly, H ��� is bounded,soH is aclosedoperator, andwecanuseclassicalanalysisof closed

operators(see[2, ñ 2.6and ñ 2.7]). For each³ , theoperatorH ò def H 7 ³�� is definedby-qn � M n � M n ! M{S�S�SU.#Å ×v7 ³ n � 2 ä _g `vå -?Ä 7 à�. n  M n ��7 ³ n � M n �7 ³ n ! M n !7 ³ n�ð MBS�S�S Ù C (19)

Thepointspectrumof H is theset � %�- H . of all points ³ for whichH ò hasnoinverse.Eachelement

of � %�- H . is theeigenvalueof H. For eacheigenvalue ³ , eachx ë DomH suchthatH ò x ï� , is

thecorrespondingeigenvector. Thus,equatingtheright handsideof (19) to zerogivesn  w³ n  0 � M Ä ï��M > M � MDS�S�S C (20)

For thefirst component,by using(20)andinduction,wehave7 ³ n � 2 ä _g `vå -?Ä 7 à�. n �³ ÂÍ��� d� C (21)

From(20) it followsthatany non-trivial solutionof H ò x w� mustsatisfy n �ôód� , so(21) implies

7 ³ 2 ä _g `vå -fÄ 7 àá. �³ ÂÍ��� 7 ³ 2 ä �³ ! _g ` � ij  0 � d� CSincewearelookingfor thelargesteigenvalue,weconfineourselvesto thecaseæ ³ æ O � . By using

differentiationof geometricseries,wehave7 ³ 2 ä �³ ! �- ³ 7 �;. � w� CTherefore,theeigenvaluesof H arethezerosof thepolynomial³ å 7 > ³�õ 2 ³ ð 7zä w�ÈM (22)

whichalsosatisfy æ ³ æ O � . From(20)weseethatthecorrespondingeigenvectorsare

x ÷ö­�áM �³ M �³ � M �³ ! MDS�S�Sùø C12

It is obvious that x ë�ãf� , but since æ ³ æ O � also implies ] -fÄ 7 àá.=ú æ ³� æôè�é , we also have

x ë DomH. Interestingly, therootsof thepolynomial(22) canbecomputedexactly in termsof

radicals(e.g.by Mathematica), andthey all lie outsidetheunit circle. Inspectingall six roots,we

seethatthelargestmagnituderootof thepolynomial(22) is

³ � - £5.� �� 2 > �?û¼!� � > 2 >áü Ó ä 26ý ����þ Ó ä 2 üá>Kÿ ä � �?û¼! 2 � > 2 >áü Ó ä 2Wý ���áþ Ó ä 2 ü�>�ÿ ä � �?û¼!� S > �?û¼! C (23)

Sincein deriving this expressionwe have assumedthat æ ³ æ O � , we concludethat (23) givesthe

largesteigenvalueof the operatorH - £5. from (17). Since ä p� ¯ - £5. , expression(23), together

with (10), allows for predictionof ratesof invasionas a function of parametersdescribingthe

fecundity, dispersal,andinfectiousnessof P. infestans. In addition,sincethemaximumeigenvalue³ � - £;. behaveslike � - ä �?û å . from (23), we alsoconcludethat the predictedupperbound ¤ ´ from

(10) is stablein thesensethatsmallchangesof theparametersfrom Table1 or entriesin thematrix

causeonly smallchangesin ¤�´ .3.3 Finite Dimensional Case

In generalonemay thereforeexpect speedsof the nonlinearinvasion,governedby the infinite

system(2), to approachspeedspredictedfor the linear system,(6), using the minimum-speed

methodologyfrom the previoussections.An additional,new, wrinkle occursbecauseof the age

structure:invasionsareinitialized with lesionsof age1, andthe actualdynamicprogressof the

diseaseis modelledby applicationof finite operators,whosenumberof entriesgrows by onefor

eachdayfollowing theinceptionof theinvasion.Consequently, whenconsideringtheobservability

of thepredictedwavespeedtherearetwo convergenceissuesto consider. Thefirst is thetraditional

issueconcerningtherateat which nonlinearfrontsof fixeddimensionalityapproachtheminimal

wavespeed.Thesecond,novel issueconcernstherateat which thefinite dimensionaleigenvalue,

presumablycontrollingthespeedof propagationin theage-structuredpopulation,approachesthe

largesteigenvaluein theinfinite system.

Let H � - £;. betheleading�y��� submatrixof H - £;. from (17),whereä is definedby (18). Let³ ¬ � ®� - £;. denotethe largestpositive eigenvalueof H � - £;. . To estimateeffect of reductionto finite

dimensionfor thelinearizedcase,weneedtocompute³ ¬ � ®� - £;. , andcompareit to ³ � - £;. . First,since

H � - £5. is non-negative andirreducible,by the Perron-FrobeniusTheorem[14, Theorem9.2.1] it

13

follows that the absolutelylargesteigenvalueof H � - £;. is real andpositive. Thus ³ ¬ � ®� - £;. exists

for every � and it is equalto the spectralradiusof H � - £;. . For example,Figure2 shows the

eigenvaluesof H � ¶k¶ - £;. for ä ��� . In Figure2 we seesix distinct eigenvalues(therearefive

distincteigenvaluesfor odd � ), andtherestof theeigenvaluesarecloseto theouterborderof the

unit circle.

−1.5 −1 −0.5 0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5

Figure2: Eigenvaluesof H � ¶k¶ - £;. for ä ¥��� . Note the six eigenvaluesoutsidethe unit circle,converging to thesix rootsof thepolynomial ³ å 7 > ³ õ 2 ³ ð 7zä d� in theinfinite case.

Let usprove thatthesequenceof largesteigenvalues,Ê ³ ¬ � ®� - £;. Ë , is convergentfor ä fixed.We

do thisby proving thatthesequenceis boundedandincreasing.Let

� � ��v� X�� × �áMD�áM��áM���M��áM �ä M �> ä M �� ä M�S�S�SNM �- ' 7 àá. ä Ù Mandset

H � - £;.� � ���� H � � � CThefirst row of H � - £5. is - �ÈMa�ÈM4�ÈM4��M4�ÈM��áMD�áM��áM�S�S�S¸M��;.NMthefirst sub-diagonalis ö­�áM���M��áM��áM ä M > M � > M � � M à� M�S�S�SÍM � 7 à� 7 � ø|Mandtheremainingelementsof H � - £5. arezero. By applyingGersgorin’s Theorem[14, Theorem

7.2.1] columnwise,it follows that all eigenvaluesare includedin the union of discswhich are

centeredatzeroandhave radii� � d� � d� ! 6� ð �áM � õ ä M �  ÄÄ 7 � 2 �áM Ä > M � M�S�S�SNM� 7 à C14

Therefore, æ ³ Á - H � - £;.¸. æ � VGX5Y Ê ä M � Ë M à ��M > M�S�S�S�� CSincethematricesH � - £;. andH � - £5. haveidenticaleigenvaluesthesequenceÊ ³ ¬ � ®� - £;. Ë isbounded.

Further, let � � - ³:M4£5. bethecharacteristicpolynomialof H � - £;. . SinceH � - £5. hastheform of the

companionmatrix, it is easyto seethat� � - ³:M4£5.#d³ � 7zä ³ � � å 7 > ä ³ � ��� 7 � ä ³ � � � 7 S�S�S 7 - � 7 � .�³ 7 - � 7 à�. CBy inductionwe have ��� 0 � - ³@M°£;.#�³$� � - ³@M°£;. 7 - � 7 �­. CSince��� - ³ ¬ � ®� - £;.NM°£;.#w� , wehave��� 0 � - ³ ¬ � ®� - £;.NM°£;.#�³ ¬ � ®� - £;. S;� 7 - � 7 �­. è � CTherefore,� � 0 � - ³:M4£5. hasa realzerowhich is greaterthan ³ ¬ � ®� - £5. . It follows that thesequenceÊ ³ ¬ � ®� - £;. Ë is increasingand,sinceit is alsobounded,convergent.By comparingtheseresultswith

thoseof Section3.2, it is obviousthat ³ ¬ � ®� - £;.\Å ³ � - £;. . This convergenceis very fast,asshown

in Figure3 for ä ï�áM����ÈM > � .

0 20 40 60 80 1001.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

m

λ 1(m) (s

)

ρ=1

ρ=10

ρ=20

Figure3: Convergenceof ³ ¬ � ®� - £;. (denotedby � ) to ³ � - £;. (solid) for ä =1,10and20. Here �is both the numberof days(generations)sincesimulationinceptionandthe orderof the matrix.Convergenceis rapidin all cases,sothatby thetwentiethgenerationof aninfestationfor practicalpurposesthefinite andinfinite valuesarethesame.

Two questionsremainto beanswered:

15

� How muchis theinvasionspeed,¤ non-linear, obtainedfrom therealisticmodel(2) overestimated

by theinvasionspeed,¤ linearized, obtainedfrom thelinearizedmodel(6)?

� How well doesthe theoreticalboundfor invasionspeed¤ ´ approximatethe speedof the

linearizedmodel ¤ linearized, giventhattheinfinite matrix is anapproximationto theiterationof

operatorsof finite, but growing, dimension?

Thesequestionsareaddressednumericallybelow.

4 Numerical Tests

In numericalsimulationsweconsideredtwo typesof dispersalkernelsin (4): theGaussiankernel,m -on .� �% Ó >;, � ���ÝÝ�� Ý M (24)

andtheLaplacekernel m -qn .� �>�% � � � � �� M (25)

where % is themeandistancetraveledby sporesin meters(nominallysetto 1 meter).Thesetwo

kernelsareamongthe mostcommonlyusedfor dispersalstudies.The Guassianform describes

a processof randomdispersionin the horizontaldirectionassporesfall from a given height to

theground;theLaplacekerneldescribesthenet resultsof a randomhorizontaldiffusion in time

coupledwith a constantrateof precipitationof sporesto the ground. Convolution anddispersal

wereimplementedusingFastFourierTransformsandthepropertythat the transformof thecon-

volution is theproductof the transforms.In all simulations4096grid pointswereused;thesize

of thespatialdomainwas ��þá� � ¤ ´ meters,where ¤�´ is themaximumpredictedvelocity. Given

initial conditionsstartingin thecenterof thedomain,this gave enoughspacesothat in 60 ‘days’

of simulationa developingfront had1.5timesasmuchroomto propagateasthemaximumspeed

linearprediction.Boundaryconditionsweretakento beperiodic.

Eachsimulationwasperformedin thenon-linearcaseusing(2), where � unocc from (3) wasre-

computedin eachstep(eachday) using formula (1), and in the linearizedcaseusing (6). The

behavior of � unocc andshapeof typical fronts for parametersasin Table1 andboth Laplaceand

Gaussiankernelsis depictedin Figure4, for bothlinearandnonlineargrowth rates.In bothcases,� unocc wasusedto diagnoseanddepictthe locationof the front; that is, to determinethe location

16

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (meters)

P(o

ccup

ied)

Laplace Kernel − Front Propagation

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (meters)

P(o

ccup

ied)

Gaussian Kernel − Front Propagation

Figure4: Evolution of the front from initial conditions( ¶� ��� ð M æ n æ�è � M ( ¶´ � otherwise,

in the caseof the Laplacekernel (top) andGausskernel (bottom),with nominalparametersasgiven in Table1. The fraction of resourceoccupied, � 7 � unocc, is plottedhere. Time slicesareten daysapart,with the evolution of the nonlinearfront givenby solid lines andthe linear frontgivenby dottedline. Notice that during the last time slice small round-off errorshave grown inadvanceof the front; theseeventuallygrow anddominatethe solution. The nonlinearsolutionlooksmuchsmootherat thispointbecausecalculationof � unocc involvessummingoverageclasses,whichsmoothstheinstability.

17

of thewaveof invasioneach‘day’ wewould calculate� unocc (evenif it wasnotusedin thedynam-

ics, asin the linearsimulations)anddeterminethecurrentextentof the invasionby determining

which grid cell containedthatpoint where � unocc �� . Fromtheobtainedresultswe thendeduced

the speedsof invasion( ¤ non-linear and ¤ linearized, respectively) in both non-linearandlinearsettingsby

calculatingthedistancespropagatedover10daysat theendof thesimulation.

For eachsimulationwe alsocomputedtheupperboundof the invasionspeed¤ ´ from (10) as

follows: we multiplied the compositeconstant� from (5) andthe momentgeneratingfunction¯ - £5. from (8) to obtain ä from (18). This ä wastheninsertedinto (23) to obtain ³ � - £;. . Finally,³ � - £;. was insertedinto (10), and the minimum over £ wascomputed,giving ¤ ´ . The speedof

invasion ¤ ´ shouldmatchthespeedobtainedby thesimulationin the linearizedcase.According

to (8), themomentgeneratingfunctionis givenby¯ - £;. � � Ý��oÝÝfor theGaussiankernel(24),andby ¯ - £5. �� 7 % � £ � Mfor theLaplacekernel(25).

Figure5 shows an exampleof simulationwith nominalvaluesof parametersfrom Table1.

For thesevalues,the compositeconstant� from (5) and (17) is equalto � ��� C � à � � . The

simulationwas run with GaussianandLaplacekernel, respectively, with % � in both cases.

Solid curvesshow the progressof infection in the non-linearcases,anddashedcurvesshow the

progressin the linearizedcaseswith � unocc � in (3). For this examplethetheoreticalspeedsare¤ ´ �� C �v�;à meters/dayfor theGaussiankerneland ¤ ´ �� C ÿ � � meters/dayfor theLaplacekernel.

Weseethat,for bothkernels,thespeedsobtainedby linearization,¤ linearized, overestimatethespeeds

of thenonlinearmodel, ¤ non-linear, andthetheoreticalspeeds¤ ´ slightly overestimate¤ linearized. This is

quite interesting,asthe dynamicperspective on front propagationwould suggestthat ¤ ´ should

betheasymptoticspeedfor bothlinearandnonlinearfronts,andsimulationresultwith fixed-size

Lesliematricesindicaterapidconvergenceto thepredictedminimalwavespeed(see,for example,

NeubertandCaswell[15]).

To morecompletelyinvestigatethecomparativebehavior of ¤ non-linearversus¤ ´ and ¤ linearizedversus¤ ´ , we performeda seriesof simulationswith differentvaluesof parametersfrom Table1 in a

18

0 10 20 30 40 50 600

5

10

15

20

25

n (days)

For

war

d P

rogr

ess

(met

ers)

vnon−linear

=0.405 m/day, vlinearized

=0.408 m/day

0 10 20 30 40 50 600

10

20

30

40

50

n (days)

For

war

d P

rogr

ess

(met

ers)

vnon−linear

=0.902 m/day, vlinearized

=0.919 m/day

Gaussiankernel Laplacekernel

Figure5: Progressof an invasionwith meansporedispersaldistancesof 1 meterusingGaussian(left) and Laplace(right) kernels. Parametersare set to nominal valuesdescribedin Table 1.Invasionswereallowedto progresslinearly (unoccupiedresourcefraction, � unocc, setalwaysto 1,dashedlines)andnonlinearly(solid lines).Thepredictedspeedsare ¤ ´Bd� C �v�;à meters/dayfor theGaussiankerneland ¤ ´ ß� C ÿ � � meters/dayfor theLaplacekernel,which is a smalloverestimateof thelinearpropagationspeedsanda largeroverestimateof thenonlinearspeeds.

randomizedfactorialdesign.Thefirst threeparameters( ��� , �� and� � ) werekeptattheirnominal

values,while theremainingfiveparameterswerechosenasfollows:� infect ë Ê � C �á� ü à�M4� C �È�áM4� C �È� > à Ë M� intcpt ë Ê � C � ü à M4� C �áM4� C � > à Ë M��� ë Ê ���t��� � ! M > �t��� � ! M � �t��� � ! Ma���t��� � ! Ë M��"\� ë Ê � M4à�M ü Ë M% ë Ê � C à�M���M�� C à�M > Ë CThisgivesthetotalof 432simulationsfor eachkernel.In thesesimulations,thecompositeconstant� attainedvaluesin the interval � ë § � C � à � �vM���à C ü �áþá�K¨ , and the theoreticalboundfor invasion

speed¤ ´ attainedvalues¤ ´�ë § � C �á� ÿ þÈM4� C þ ü à�à5¨ for theGaussiankerneland ¤ ´�ë § � C > � ü � M > C �È��� � ¨for theLaplacekernel.Theresultsof simulationsaresummarizedin Figure6.

The sizeof the error betweenobservation andpredictiondepictedin Figure6 reflectswhat

appearsto bea consistentoverpredictionof observedlinearandnonlinearspeeds,with thedegree

of overpredictionbeingapproximatelydoublefor nonlinearspeedsascomparedto linearspeeds.

19

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

v* (Predicted)

v Obs

erve

d

Gauss Kernel − Speed Comparison

Linear SpeedsNonlinear Speeds1:1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

v* (Predicted)

v Obs

erve

d

Laplace Kernel − Speed Comparison

Linear SpeedsNonlinear Speeds1:1

Figure6: Comparisonof predictedandobservedspeedsfor wavesof invasionwith andwithoutdensitydependentgrowth restrictionsfor bothLaplaceandGaussiandispersalkernels.Parametersarechosenin arandomfactorialdesigndescribedin thetext, with variationcenteredonthenominalvaluesdescribedin Table1. Observed linear andnonlinearspeedsaremarked with ‘ � ’ and ‘*’respectively. The solid line is the line ¤ observed ¤ predicted, indicatingperfectagreement.Resultsindicateaconsistentoverpredictionof observationby prediction,with agreaterdegreeof errorfornonlinearascomparedto linearpropagation.

20

Fromtheminimal speedperspective this is not soawful; afterall, ¤ ´ is anupperbound,but only

in relatively rarecaseshasit beenprovento betheasymptoticspeedof fronts.On theotherhand,

theagreementbetweenpredictionandobservationis generallysuperb(see,for example,Neubert

et al. [15, 16])– why shouldit belesssoin this case?And whatof thedynamicargument,which

suggeststhatfrontsshouldaccelerateto ¤ ´ ?Theexplanationfor thedegreeof observationliesin threeinterrelatedeffectsin oursimulation.

In thefirst place,thenetdailyper-capitagrowthratefor thenumberof fungallesionswasneverless

than1.5 in our simulations,andwasoftenaslargeas10, reflectingtheextremelyinvasive nature

of this pathogen.As a consequence,simulationsweredifficult to run for long periodsof time; at

somepoint smallround-off errorsin theneighborhoodof zerowould startto grow geometrically.

So, in practicewe wereunableto maintainsimulationsmuchbeyond 50 iterations,andrunning

longersimulationsto allow for greaterconvergencewasimpossiblebothbecauseof the extreme

instability of thezeropopulationstateaswell asthesizeof the transitionmatrices(which areas

largeasthenumberof days)ateachspatiallocation.

Confoundedwith this effect aretwo convergenceeffects,eachcontributing to theoverpredic-

tion. In the first placethereis the convergenceto the stabletravelling populationdistribution,

which is describedby thefirst neglectedtermsin thepower method.Thus,whenconsideringthe

evolutionof a front from compactinitial data,theasymptoticproblemshouldread� ) � Ò ¦ Ò }¢ H) - £ ´ . }¢ Æ � - £ ´ .=³ ) � }� � - £ ´ . 2 Æ � - £ ´ .¸³ ) � }� � - £ ´ . 2 S�S�S ³ ) � ZqÆ � - £ ´ . }� � - £ ´ . 2 Æ � - £ ´ . ³ ) �³ ) � }� � - £ ´ . 2 S�S�S b C

Here }� � - £;´a. canbeinterpretedastheasymptoticstablepopulationdistributionselectedby thewave

of invasion,while }� � - £ ´ . is theagestructureof the ‘ringing’ which occursaspopulationdistribu-

tionsconvergeto thestabledistributionalongafront,andtheratioof thelargestandsecond-largest

magnitudeeigenvaluesis therateof convergence.This is asymptoticallynegligible, but for finite

durationsimulations(like thosewe areforcedto run by theextremeinstability of thesystem)we

mayexpectanerrorin estimatingfront speedsproportionalto

�Power� æ }� � S }� � æ æ ³ � æ )æ ³ � æ ) C (26)

Thesecondconvergenceeffect is thenaturalaccelerationof the front to the asymptoticfront

speeddescribedby (16). Predictedby the steepestdescentmethodology, this canbe viewed as

21

the convergenceof the spatialshapeof the front to the asymptoticexponentialshapewhich is a

translateof ��� �¼Òf  . The speedconvergenceerror predictedby the steepestdescentsapproachis

(from 16)�

Speed� ��µ - ' .> ' Ä ´ C (27)

While�

Speed tendsto zeroas ' tendsto infinity, theconvergenceis slow, andagaintheconstraint

thatwe wereunableto simulatefor morethanseveral tensof iterationsmeansthat this errorcan

notbeneglected.

To investigatehow theseerrorsrelateto observederrorsin our simulationwe raneachof the

factoriallycrossedparameterstudiesfor aslongaspossible,diagnosingtheonsetof overwhelming

instability by the inevitablesuddenjump in the rateof progressof the front. In eachsimulation

the day at which the simulation‘broke’ wasdiagnosedby andrecordedas ' u­Æ s . During each

simulationthe forward progress( n�� � ) of the front wasdiagnosedasdescribedabove. Observed

speedswerethendiagnosedby¤ observed n��!� - ' u­Æ sc7 àá. 7 n��!� - ' u­Æ sL7 �;àá.��� CSimultaneously, the largesttwo eigenvaluesof the finite transitionmatrix, ³ � � - £�´4. and ³ � � - £;´a. ,werecalculated,with � evaluatedat thecenterof thespeedcalculationinterval, �� ' u Æ sc7 ��� .With this informationwe coulddeterminethesizeof the two errorcomponents,

�Power and

�Speed.

Theseerrorsarecomparedto theobservedrelativespeederror,¤ ´ 7 ¤ observed¤ ´ Min Figures7 and8.

5 Conclusion

We have shown in this paperhow to rationallyextendthe methodologyof NeubertandCaswell

[15], incorporatingagestructureanddispersalinto an integrodifferencepopulationmodel,to the

infinite dimensionalcaseof cropdiseaselesionspropagatingthroughacrop.Evenin theextremely

unstablecaseof latepotatoblight theagreementbetweenanalyticresultsandsimulationsarewell

within expectederrortolerances.Theminimumwavespeed,givenby theinfinite dimensionalver-

sionof 10, is anupperboundfor ratesof invasionprogationandseemsto betheasymptoticspeed

22

0 50 100 150 200 250 300 350 400 4500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Rel

ativ

e E

rror

Speed Errors − Gauss Kernel

Espeed

+ Epower

Epower

Espeed

Linear ObsNonlinear Obs

Figure7: Raw comparisonof observedrelative speederrorsin thenonlinear(‘*’) andlinear(‘ " ’)casesfor theGaussiankernel,with randomizedchoicesof parametricdata,centeredonthenominalvaluesin Table1. Theasymptoticerrorsizedueto agestructureconvergence,

�Power, is plottedas

‘ # ’, while asymptoticerror sizedueto speedconvergence,�

Speed, appearsas‘ $ ’. The total errorsize,

�Speed % �

Power is the solid line; resultsweresortedin termsof increasingtotal error. Thehorizontalaxis is the identifying index of the simulationandhasno units. Given that the actualerror relatesby anorderonefactorto theerrorsizesdepictedhere,both the linearandnonlinearfront speedsarewell within acceptableerrorbounds.

23

0 50 100 150 200 250 300 350 400 4500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Rel

ativ

e E

rror

Speed Errors − Laplace Kernel

Espeed

+ Epower

Epower

Espeed

Linear ObsNonlinear Obs

Figure8: Raw comparisonof observedrelative speederrorsin thenonlinear(‘*’) andlinear(‘ " ’)casesfor theLaplacekernel,with randomizedchoicesof parametricdata,centeredonthenominalvaluesin Table1. Theasymptoticerrorsizedueto agestructureconvergence,

�Power, is plottedas

‘ # ’, while asymptoticerror sizedueto speedconvergence,�

Speed, appearsas‘ $ ’. The total errorsize,

�Speed % �

Power is the solid line; resultsweresortedin termsof increasingtotal error. Thehorizontalaxis is the identifying index of the simulationandhasno units. Given that the actualerror relatesby anorderonefactorto theerrorsizesdepictedhere,both the linearandnonlinearfront speedsarewell within acceptableerrorbounds.

24

selectedfor wavesof invasion,assuggestedby thedynamicinterpretationsof DeeandLanger[3]

andPowell etal. [18, 19]. In fact,resultsin thispapersuggestthatthedynamicinterprationhasthe

additionalvirtueof accuratelydescribingtherateof convergenceto theasymptoticwavespeed.

We havealsodescribedamodellingapproachfor lesion-basedfoliar diseases,whichmayfind

potentialapplicationin any sort of invasionprocesswheregrowing patchesarethe basicunit of

infection. Examplesincludecheatgrassin theAmericanWest[22], thepathogenicfungi Botrytis

spp. which cause‘fire disease’in flower cropsandinfect field andgreenhousevegetables,small

fruits, ornamentalplants,flower bulbsandforesttreeseedlingsworld wide [10], andinsectssuch

astheSouthernPineBeetle(whichcreate‘spot’ infectionsin patchesof pineforest[20]) or gypsy

moth,which seemsto invadeby via spots[21]. By no meansis this thefirst attemptat modelling

spreadof lateblight andfungalpathogens(see,for examplevandenBoschet al. [24] andPielaat

andvandenBosch[17]), nor (moregenerally)agestructuredspreadin general(seeHengeveld[7]

andShigesadaandKawasaki[22] for reviews). But it is thefirst attemptthatwe know of to put

theconceptof anever-growing stagestructureon therelatively firm andsimplegroundof aLeslie

matrix formulation.

Thebiggestdrawbackin themodellingapproachusedhereis thedifficulty in accountingfor

two factors:finite leafsizeandcoalescenceof lesions.Thefirst factoris nottoodifficult to imagine

incorporating,thoughpossibletedious.At thecoarsestlevel whenlesionsgrow to theaveragesize

of a leaf they cangrow no more,which would amountto truncatingthe nonlinearLesliematrix

at anageclassof lesionscorrespondingto theareaof the largestleaves. At a slightly lesscoarse

scale,in plantswith a sizedistribution of leaves,onewould needto estimatetheprobabilityof a

lesionusingup all availableareaby theprobabilitythat it hadlandedon a leaf of its currentsize.

This would give a transitionprobabilityof smallerthanonefor lesionslarger thana certainsize,

which would dropto zerofor lesionsat the largestleaf size. Finally, whenlesionsgrow on finite

substratesthey will eventualyencounterboundaries,andwhile they maycontinueto grow thenew

growth areawill no longerbeannular. Consequentlysomeestimateof theprobabilityof observing

annulargrowth of area &('*) , which would thenalter the rateat which new lesionsareformed.

In all of thesecases,realistic incorporationof realistic leaf sizeswould ruin the specialmatrix

structurewhich allowed for analyticcalculationof the largestmagnitudeeigenvalue,thoughthe

theorypredictingtheexistenceof asingle,largesteigenvaluewould remainin place.

25

The secondfactor, coalescenceof lesionson a leaf, would be somewhat more difficult to

address.ShigesadaandKawasaki[22] outline a procedurefor approximatingthe rateat which

patchesof an invasive speciesrun into oneanother. The basicideais to modelthe processasa

summation,so thatwhentwo circularpatchesencounteroneanotherthey areapproximatedasa

new patchof sizeequalto thesumof thepreviouspatches.Knowing thedistributionof distances

at which patchesareestablishedadtheir radialgrowth rates,onecanpredictthemeantime until

patchesencounteroneanother. In our age-structureframework this would manifestasa new kind

of transitionprobability: amongall ageclasseswould bea classof transitionswhich would allow

a lesionof a givensizeto sumwith a lesionof any othersizeandcreatea new lesionin a size

classequalto thesumof the two. Theresultingtransitionmatrix would be lower triangular(ex-

ceptfor the top row, representingtheproductionof new lesions),andhave nonzeroentriesup to

thepoint wheremaximumdispersaldistanceandradialgrowth no longerallow for two lesionsto

coalesce(i.e. whenthe sizeof the lesionis greaterthanits capacityfor dispersal).At this stage

new theoreticaldifficulties areboundto be encountered;the infinite dimensionalversionof the

sizeclass/dispersalformalismwasrelatively easyto describein thecurrentcasedueto thesimple

form of theLesliematrix involved.

Both of thesefactors,however, clearly reducethe growth rateof the lesionpopulationand

thereforewouldslow downthewaveof invasion.Consequentlywemayexpectthat +-, ascalculated

above to remainanupperboundfor thespeedof invasions.It is thenparticularlyusefulsincethe

entirecalculationcanbe performedanalytically, given the simplealgebraicform of the largest

eigenvalue.This mayallow for relatively simpleevaluationof invasionthreatandcontrol for this

importantcropdisease.

26

Acknowledgements

Groundwork for this manuscriptwaslaid duringa sabbaticalvisit by Dr. Powell to Wageningen

Universityin theNetherlands,fundedin partby NSFgrantINT-9813421andby theWageningen

University ResearchSchool. Dr. van der Werf gratefully acknowledgesthe Dutch Technology

FoundationSTWfor providing visiting scientistgrantWBI.4958whichallowedhim to makecrit-

ical visits to Utah StateUniversity. Furthersupportwasprovided by Strategic ExpertiseDevel-

opmentProject620-33001-76of PlantResearchInternational,Wageningen.We all acknowledge

helpfulconversationswith Dr. GeertKessel,whoreadanearlierversionof thismanuscriptandpro-

videdmuchhelpfulbiologicalcommentary. Earlyideasonaprecursorof themodelpresentedhere

weredevelopedin lively discussionsamongWopke vander Werf, Walter Rossing,GeertKessel

andHansHeesterbeekin the framework of theaforementionedStrategic ExpertiseDevelopment

Activity.

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