Global stability and optimisation of a general impulsive biological control model

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biological control model

Ludovic Mailleret, Frédéric Grognard

To cite this version:

Ludovic Mailleret, Frédéric Grognard. Global stability and optimisation of a general impulsive

bio-logical control model. [Research Report] RR-6637, INRIA. 2008, pp.28. �inria-00320578�

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Thème BIO

Global stability and optimisation of a general

impulsive biological control model

Ludovic Mailleret — Frédéric Grognard

N° 6637

Ludovi Mailleret

†

, Frédéri Grognard

‡

ThèmeBIOSystèmesbiologiques ProjetComore

Rapportdere her he n°6637Septembre200825pages

Abstra t: An impulsivemodelofaugmentativebiologi al ontrol onsisting ofageneral ontinuouspredator-preymodelin ordinarydierentialequationsaugmentedbyadis rete partdes ribingperiodi introdu tionsofpredatorsis onsidered. Itisshownthatthereexists aninvariantperiodi solutionthat orrespondstopreyeradi ationanda onditionensuring itsglobalasymptoti stabilityisgiven. Anoptimisationproblemrelatedtothepreemptive useof augmentativebiologi al ontrol is then onsidered. Itis assumed that thepertime unit budgetofbiologi al ontrol (i.e. the numberof predatorstobereleased)isxed and thebest deployment ofthis budgetis soughtafter in termsof releasefrequen y. The ost fun tiontobeminimisedisthetimetakentoredu eanunforeseenprey(pest)invasionunder someharmlesslevel. Theanalysis showsthat theoptimisationproblemadmitsa ountable innitenumberof solutions. An argumentation onsideringthe requiredrobustnessofthe optimisationresultisthen ondu tedanditisshownthat thebest deploymentistouseas frequent(andthusassmall)as possiblepredatorintrodu tions.

Key-words: Predator-preydynami s,impulsivemodel,globalstability,optimisation.

∗

This work was supported by the COLOR a tion LutIns& o funded by INRIA Sophia Antipolis MéditerranéeandbytheProje tIA2LfundedbytheSPEDepartmentofINRA.

†

INRA,UR880,F-06903,SophiaAntipolis,Fran e(ludovi .mailleretsophi a.i nra. fr) .

‡

Résumé : Un modèleimpulsionnel de luttebiologique est onsidéré; il est omposé d'un modèleproie-prédateurgénéralenéquationsdiérentiellesordinaires omplétéparunepartie dis rètedé rivantl'introdu tion périodique deprédateurs. Ilest démontréqu'ilexiste une solutionpériodiqueinvariantequi orrespondàl'éradi ationdesproiesetune onditionqui permet d'assurersastabilitéglobaleasymptotiqueestdonnée. Un problèmed'optimisation lié à l'utilisation préventivede la lutte biologique est alors onsidéré. Il est supposé que le budget par unité de temps de la luttebiologique ( 'est-à-direle nombre de prédateurs lâ hés)estxeetlameilleurerépartitionde ebudgetestre her héeentermesdefréquen e d'introdu tion. Lafon tionde oûtàminimiserestletempsprispourréduireuneinvasion imprévue de ravageurs (les proies) jusqu'à un niveau inoensif. L'analysemontre que le problèmed'optimisation onsidéréadmetunnombreinnidesolutions. Lapriseen ompte delarobustessené essairedelasolutionoptimaleindiquequelameilleurestratégie onsiste àutiliserdeslâ hersdeprédateurs aussifréquents(etdon aussifaibles)quepossible. Mots- lés: Dynamiqueproie-prédateur,modèleimpulsionnel,stabilitéglobale,optimisation

1 Introdu tion

Thebiologi al ontrol method intends toredu e harmful organisms populationsand relies ontheutilisationof theirnaturalenemies,thebiologi al ontrol agents. It anbeused to ontrol e.g. ve torsin ve tor-borne diseaseslikemalaria orinvasivespe ies(maythey be animals or plants), but it is mostly dedi ated to the ontrol of inse t pests, espe ially at theagri ultural roppingsystems ale[25℄. Threemain biologi al ontrolstrategies anbe identied:

ˆ onservation biologi al ontrol: bio ontrol agents are already present in the system andarefavouredthroughvariousmeans(habitatmanagement...).

ˆ lassi albiologi al ontrol:asmallpopulationofbio ontrolagentsisintrodu edon ein thesystemandalongtermequilibriumbetweenthepestsandthemselvesistargetted.

ˆ augmentativebiologi al ontrol: bio ontrol agentsarerepeatedlyintrodu edin order toeradi atethepestpopulation.

In this ontribution wefo us onthis last method, whi h is fairly well modelled by an impulsivesystemofordinarydierentialequations(a ontinuoussystemae tedbysudden state modi ations at some dis rete time moments). The intrinsi dynami s of the two intera tingpopulations(pests andbio ontrolagents) aredes ribedbythe ontinuous part of the systemand the repeated (periodi ) bio ontrol agents introdu tions by the dis rete one. Early examples of the use of impulsive models in biology are [7, 12℄ in hemostat modelling. Sin ethen,theyhavere eived onsiderableattentionintheareaofepidemiology [4, 13, 28℄, an er treatment [6, 5, 17℄, populations dynami s [20, 32℄ or integrated pest management[19,27,33,37℄,to itesome. Thoughtherearenumerousstudiesonbio-inspired impulsivemodels,toourknowledgeonlyafewoftheseis on ernedwithoptimisation: see [3, 9,24, 31, 34℄. Here wefo us both on aglobal onvergen e result for a rather general predator-preymodel with periodi impulsive predator introdu tions and on a problem of optimisation onhow to best deploy these introdu tions in the ontext of rop prote tion throughbiologi al ontrol. Ourresultsfollowandgeneraliseearlieronesthatwereobtained inamorerestri ted ontext[23℄.

Thispaperisorganisedasfollows. First,theimpulsivepredator-preymodelof augmen-tativebiologi al ontrol thatwill bepayedattentiontothroughout thepaperispresented. Thehypothesesassumedonthebiologi alpart( ontinuoussystem) arequalitativeso that they anen ompassalargenumberof lassi albiologi al/e ologi alfun tions. Theexisten e of aninvariant periodi prey-eradi ationsolution isproved and asu ient ondition that ensures itsglobal asymptoti stability is given. Then, we fo us on an optimisation prob-lem in the perspe tive of preemptive/preventativeuse of augmentative biologi al ontrol whi h parti ularlysuits theprote tionofhigh valuedagri ultural rops likee.g. or hards, vegetables,mushroomsorornamentals[2,11,14,15,16,29℄. Itisassumedthataxed bud-get ensuringpest eradi ation isallowed andthat onlyits deployment(a tradeo between frequent-smallorrare-largereleases)maybemodied. The ostfun tion to be minimised

is the time taken to redu e an unforeseen prey(pest) invasion under someharmless level and theinput isthe impulsivereleasefrequen y. It isrst shown that thelo al optimisa-tion problem has a ountable innite numberof solutionsand, in a se ond step using an argumentation based onthe needed robustness of ourresult, it is proved that the lo ally optimumsolutionisto hooseasfrequent(andthusassmall)aspossiblereleases. Itisalso shown that this strategy givesa sub-optimal solution of theglobal optimisation problem. Moreover,thelargerthefrequen y, thelowerthe ostfun tion,so that evenifone annot a hieveaverylargefrequen y,itisalwaysinterestingto hoosethelargestpossibleone. A MonteCarlolikenumeri alsimulationillustratesouranalyti aloptimisationresultandthe arti leis on ludedbysomere ommendationstobiologi al ontrolpra titionersthatfollow fromouranalysis.

2 Model Des ription and Analysis

2.1 A general impulsive biologi al ontrol model

Inthispaper,weassumethatthetri-trophi system( rop-pests-bio ontrol agents) may be well represented by its bi-trophi simpli ation representing the prey (pest) predator (bio ontrol agent) intera tions. The underlying assumption is that the prey population remainsatmoderatelevelssothatthe ropisnotlimitingandthusdonotneedtobetaken intoa ountinthemodel. We onsiderthesimple aseinwhi hnointerspe i intera tions withinthepredatorpopulationae tsthepredatorpreyintera tion,thoughsomeintheprey populationmayo ur(seeRemark1below). A ordingto lassi alpredator-preymodelling, wegetthefollowingtwo-dimensionalmodel:



˙x = f (x) − g(x)y

˙y = h(x)y − my

(1)

x

denotingthepreysand

y

thepredators.

f (x)

denotesthepreys'growthvelo ity,

g(x), h(x)

and

m

denotes thepredators' fun tional response,numeri al response and mortalityrate, respe tively. Sin ebiologi alpro essesarealwaysdi ulttomodel,weonlyassumegeneral qualitativehypothesesonthefun tions

f (.), g(.)

and

h(.)

:

Hypotheses 1 (H1): Let

f (.), g(.)

and

h(.)

be lo ally Lips hitz fun tions on

R

+

su h that: (i)

f (0) = 0

(ii)

g(0) = 0, g

′

_{(0) > 0}

and

∀x > 0, g(x) > 0

(iii) the fun tion

f (x)

g(x)

isupper boundedfor allpositive

x

(iv)

h(0) = 0

and

∀x > 0, h(x) > 0

Remark 1: Alargepartofthe predator-preyfun tions en ountered in theliteraturet these hypotheses: (H1-i) only indi ates that no spontaneousgeneration of pests (prey) is possible;

f (.)

mightbe onstant,linear,logisti ormodelanAlleeee tet ...(H1-ii)means thatthereisnoprey onsumptionasthepreyleveliszeroandthatifsomepreyarepresent, thepredatorareabletondand onsumethem. Moreover,

g(x)

issupposedtobein reasing at the origin. As a onsequen e

g(x)

may bea Holling I orII fun tion.

g(x)

might even benon-monotoni (in aseof e.g. preygroup defen e abilities)like in Holling IV, but in this asefrom (H1-iii)

f (x)

must be of type IVtooorbe ome negativefor high

x

values (likeforlogisti growth). Noti ehoweverthatHollingIIIlikefun tionalresponse(i.e. those withnullderivativeat

x = 0

)fall beyondthes opeofthis study. Nohypothesesaremade onthenumeri alresponse

h(x)

ex ept(H1-iv) whi h isquitenatural. Indeed,asitwill be learerinthefollowing,ourargumentationismainlybasedonthefa tthat

g(x)

ispositive forpositive

x

.

Wenowmodeltheaugmentativebiologi al ontrolpro edureasthe

T

-periodi releases of biologi al ontrol agents (with

T > 0

). Releases are, by their very nature, dis rete phenomena: everytime

nT

(

n ∈ N

) somepredators are instantly added into thesystem. Letussupposethat thereis axedrateofpredators

µ

(i.e. numberofpredatorsperunit time)tobereleasedinthesystem. Noti ethatsu haquantity

µ

isadire t measureofthe ostsofthebiologi al ontrolmethodoversometimeperiod. Hen e inthefollowingitwill bereferredto aswellasthereleaserateorthebiologi al ontrolbudget.

This results in the following: at ea h time period

T

,

µT

predators are added to the predatorpopulation

y

,yieldingthedis retepro ess:

∀n ∈ N, y(nT

+

_{) = y(nT ) + µT}

(2)

Combining (1) and (2) yields the following impulsive biologi al ontrol model, similar to,but moregeneralthantheonespresentede.g. in [18,19℄:







˙x = f (x) − g(x)y

˙y = h(x)y − my

∀n ∈ N, y(nT

+

_{) = y(nT ) + µT}

(3)

Remark 2: It is quiteeasy to he kfrom (H1) that theproposed model is, asrequired forbiologi almodelsingeneral,anon-negativesystem(seee.g. [22℄),i.e. thenon-negative orthantof thestate spa e is positivelyinvariantby system (3) . Then everynon-negative initial onditions

x

0

and

y

0

at initial time

t

0

(that only make sense with respe t to the onsideredproblem)produ ethroughmodel (3)non-negativetraje tories

x(t)

and

y(t)

for allpositivetime.

2.2 Global onvergen e

Inthisse tionweshowthatprovidedthepredatorreleaserate

µ

isgreaterthansomevalue, augmentativebiologi al ontrolis ableto driveanypest populationto zero. This resultis summarisedinto thefollowingTheorem.

Theorem1 UnderHypotheses1,model (3)possessesapestfree"

T

-periodi solution:

(x

p

(t), y

p

(t)) =

0,

µT e

−m(t

mod T )

1 − e

−mT

!

(4)

whi h islo allyasymptoti allystable ifandonlyif:

µ >

mf

′

₍₀₎

g

′

₍₀₎

(5)

andgloballyasymptoti allystable if:

µ > S

, sup

x≥0

mf (x)

g(x)

(6)

Remark 3: Notethatfrom(H1-iii)

S

asdened in(6)doesexist. Proof: Werstfo usonthepestfreeset:

{(x, y) ∈ R

2

+

, x = 0}

,whi his learlyinvariant bysystem(3)through(H1-i)and(H1-ii). Withinthisset,a ordingto(H1-iv), system(3) be omes:







˙x = 0

˙y = −my

∀n ∈ N, y(nT

+

_{) = y(nT ) + µT}

(7)

thatyields:

y((n+1)T

+

_{) = y(nT}

+

_)e

−mT

_+µT

. Itis learthatthesequen e

(y(nT

+

₎₎

n∈N

has asingleandgloballystableequilibrium

y

⋆

_{= µT /(1 − e}

−mT

₎

. Then,system(7)traje tories globally onvergestowardsthe

T

-periodi solution:

∀t ∈ (nT, (n + 1)T ], (x(t), y(t)) =



0,

µT e

−m(t−nT )

1 − e

−mT



whi h is indeed the sameas(4), sothat the existen eof the

T

-periodi pest free solution forsystem(3)hasbeenproved.

Wenow on entrateonthestabilityof thepest freesolution(4)forsystem(3),that is tosaywedonotrestri tourselvestothepestfreeset. Werst hangesystem(3)variables to onsiderthedeviationsfromthepestfreesolutionthat aredenoted

(˜

x, ˜

y)

sothat:

(˜

x(t), ˜

y(t)) = (x(t), y(t)) − (x

p

(t), y

p

(t))

thatyields:



_{˙˜x = f(˜x) − g(˜x)(˜y + y}

p

(t))

˙˜y = h(˜x)(y

p

(t) + ˜

y) − m˜

y

(8)

Werstinvestigatelo alasymptoti stability(LAS)oftheperiodi solution

(x

p

(t), y

p

(t))

. Assuming

(˜

x, ˜

y)

aresmall,wegetfrom (8)atrstorderin

x

˜

and

y

˜

:



_{˙˜x = (f}

′

_{(0) − g}

′

_(0)y

p

(t))˜

x

˙˜y = h

′

_(0)y

p

(t)˜

x − m˜

y

whi hisalinearsystem(in

x

˜

and

˜

y

)with

T

-periodi oe ients. TheLASof

(x

p

(t), y

p

(t))

forthe original system(3) is equivalent to theLAS of

(0, 0)

forsystem (9). The lassi al approa h (Floquet Theory) is to ompute the dis rete dynami al system that maps the variables at some time to the variables one period of the periodi oe ients later. The originisthenasymptoti allystableifandonlyiftheobtaineddis retedynami alsystemis asymptoti allystable(seee.g. [1℄forthetheoryand [19℄foranappli ation omparableto ours). Here,weget:



˜

x

˜

y



(n + 1)T

+

=

e

R

(n+1)T

nT

f

′

(0)−g

′

(0)y

p

(τ )dτ

₀

†

e

−mT

!

_

˜

x

˜

y



nT

+

thematrixbeinglowertriangular,the

†

termdoesnotinuen ethesystem'sstabilitysothere is no need to ompute it. Clearly

e

−mT

belongs to

(0, 1)

, and

e

R

0

T

(f

′

_(0)−g

′

_(0)y

p

(τ ))dτ

ispositive. Then,

(0, 0)

isLASfor(9)(i.e.

(x

p

, y

p

)

is LASfor(3))i thislatter oe ientislowerthanonei.e. :

Z

T

0

(f

′

_{(0) − g}

′

_(0)y

p

(τ ))dτ < 0

⇔

f

′

(0)T <

Z

T

0

g

′

₍₀₎

µT e

−mτ

1 − e

−mT

dτ

⇔

µ >

mf

′

₍₀₎

g

′

₍₀₎

(10)

sothat theLASof

(x

p

, y

p

)

partof theTheoremisshown.

Wenowfo usontheglobalasymptoti stability(GAS)ofthepest freesolution. From nowon,weassumethatsystem(3)isinitiatedat

(x

0

, y

0

)

attime

t

0

≥ 0

, i.e. system(8)is initiatedat

( ˜

x

0

, ˜

y

0

) = (x

0

, y

0

− y

p

(t

0

))

attime

t

0

≥ 0

.

Letus onsiderthefollowingfun tion:

G(˜

x) = m

Z

x

˜

κ

1

g(s)

ds

for some

κ

> 0

. Sin e from (H1-ii)

g(x)

is positive for positive

x

,

G(.)

is an in reasing fun tion from

x = 0

˜

, where it goes to

−∞

sin e

g(.)

is lo ally Lips hitz on

R

+

. In the followingweinvestigate

G(˜

x(t))

behaviourastime

t

goestoinnity. Wehave:

G(˜

x(t)) − G(˜

x

0

) =

m

Z

x(t)

˜

˜

x

0

1

g(s)

ds

=

m

Z

t

t

0

˙˜x(τ)

g(˜

x(τ ))

dτ

=

Z

t

t

0

 mf (˜

x(τ ))

g(˜

x(τ ))

− m(˜

y(τ ) + y

p

(τ ))



dτ

(11)

Consideringsystem(8)togetherwith(H1-iii)and Remark2,weget:

sothat,throughstandardarguments:

˜

y(t) ≥ min(0, ˜

y

0

)e

−m(t−t

0

)

(12) Using(11)and thedenition of

S

in(6),wethenhaveforall

t ≥ t

0

:

G(˜

x(t)) − G(˜

x

0

) ≤

Z

t

t

0



S − my

p

(τ ) − m min(0, ˜

y

0

)e

−m(τ −t

0

)



dτ

=

Z

(⌊

t0

_T

⌋

+1

)

T

t

0

(S − my

p

(τ )) dτ

+

 t

T



−

 t

0

T



− 1

 Z

T

0

(S − my

p

(τ )) dτ

+

Z

t

⌊

t

T

⌋

T

(S − my

p

(τ )) dτ + min(0, ˜

y

0

)



e

−m(t−t

0

)

_{− 1}



(13)

sin e

y

p

(t)

is

T

-periodi . Itis learthat therst, third andfourth termof therighthand sideof (13)areupperbounded. Now,supposethat(6)holds,theninasimilarwaythanin (10):

Z

T

0

(S − my

p

(τ )) dτ < 0

sothat,sin e



t

T



goestoinnityas

t

does,therighthandsideof (13)goesto

−∞

. Then, ondition(6) implies:

lim

t→+∞

G(˜

x(t)) = −∞ ⇔

t→+∞

lim

x(t) = 0

˜

One annowprovethat

˜

y

onvergestozeroaswell. Indeed,from (12):

˜

y

0

≤ 0 ⇒

y(t) ≥ ˜

˜

y

0

e

−m(t−t

0

)

sothat,either

y(t)

˜

onvergestozeroininnitetimefrombelow,oritrea hesthepositively invariantregionwhere

y ≥ 0

˜

innitetime. Uptoinitialtime

t

0

translation,wethenhaveto onsiderthepositive

y

˜

0

only. Sin e

˜

x

onvergestozeroand

h(0) = 0

, itis learthat there existsatime

t

f

su hthat:

∀t > t

f

, h(˜

x(t)) ≤

m

2

⇒ ∀t > t

f

, ˙˜

y ≤ h(˜

x)y

p

(t) −

m

2

y

˜

Sin e

h(˜

x)y

p

(t)

goestozeroas

t

goestoinnity,sodoes

y

˜

. Wehaveshownthatif ondition (6)holds,

(0, 0)

isgloballyattra tivefor(8) ,i.e.

(x

p

(t), y

p

(t))

isgloballyattra tivefor(3). Itistheneasily he kedthroughtheHospitalrule,(H1-i)and(H1-ii)that:

lim

x→0

+

f (x)

g(x)

=

f

′

₍₀₎

g

′

₍₀₎

Then, ondition(6)impliesthatthene essaryandsu ient ondition(5)forLASof

(x

p

, y

p

)

holdstrue,whi h on ludes theproof.



Theorem1ensuresthatwhenthe ondition(6)isveried,theextin tionoftheprey(pest) populationisGASonthepositiveorthantofthestatespa e. Ifthelo al ondition(5)were notsatised,thepestfreesolutionwouldthennomorebestable. Usingbifur ationtheory for impulsivesystems proposed by [17℄, onethen should be able to perform abifur ation analysis, similar to what is done e.g. in [19, 26℄ to show the ri h dynami s that system (3)would produ e. However,asstatedin theintrodu tion,withrespe ttoourappli ation we on entratehereonthe pest eradi ationproblemonly. Inthe asewhere ondition(5) holdstrue,but ondition(6)doesnot,thepest freesolutionisstilllo allystablebut,sin e ourglobal stability ondition(6) is onlysu ient, we annotrule outthe possibility that it is alsoglobally stable. Su h areleaserate hasthe advantageof beingsmaller thanthe onerequiredfor GAS sothat itwould be heaperfor thegrowerto hooseaoneassu h: it allows good ontrol of limited pests invasions. The pri e to pay is howeverthe risk of onsiderable ropdamagesifalargepestinvasion o urs.

It istobenotedthat thestability onditions(5)and (6)onthereleaserate

µ

areboth independent of therelease period

T

. Therefore, on ea releaserate

µ

fullling astability onditionis hosen, pest eradi ation ould bea hievedindependently of the hoi eof the releaseperiod

T

. Indeed, both infrequentand largereleases(

T

and

µT

large)orfrequent and small releases (

T

and

µT

small) would ultimately drive the pest population to zero. Thoughthereisnodieren einthelongterm,dierentvaluesofthereleaseperiod

T

might howeverresultindierentout omesforthetransientdynami softhesystem. Hen einthe following, wewill takeadvantageof theadjustableparameter

T

toaddressthe pra ti ally importantquestion: howto best" (inasense thatwill bedetailed inthesequel) deploya givenbiologi al ontrol budget

µ

toeradi ate pest outbreaks?

3 Optimal Choi e of the Release Period in the Preemp-tive Case

3.1 Statement of the optimisation problem

Throughoutthefollowing,weassumethatthe onditionforstabilityofthepestfreesolution holds true. We fo us on preemptive/preventativeuse of biologi al ontrol agentsreleases: wesupposethatbiologi al ontrolagentsarereleasedin anti ipationofpestsoutbreaks,so thatnaturalenemieswouldbeabletoghtthepestrightatthetimeoftheirinvasion. This departsfrommost lassi albiologi al ontrolpro edureswhi hareofamorefeedba k"type: bio ontrol agentsarereleasedon ethepests havebeendete tedonly. Su hapreventative approa hhashoweverbeenpayedattentiontosin eitappearsasa hievingmorea eptable pest ontrol, espe ially for high valued rops that are very sensitive to the slightest pest outbreak (see [11, 29, 10℄ for theoreti al/simulatorystudies and [8, 16, 15, 2, 14℄ for real

life experiments and a biologi al perspe tive). Hen e in the following, we suppose that preventativereleasesarebeingperformedandthatthesystemisintheinvariantset

{˜

y ≥ 0}

asapest population

x

0

invadesthe ropat someunforeseenmoment

t

0

.

Our goalis to providere ommendations onhowto deploy(i.e. howto hoose

T

)su h apreemptivebiologi al ontrolstrategyto minimise ropdamagesdue toapestoutbreak. Toevaluate ropdamagesindu edbythepests,weusethethe on eptofE onomi Injury Level"(EIL) thathasbeenintrodu edfromtheearlybasesoftheoreti albiologi al ontrol [30℄. EIL(denoted

x

¯

inthesequel)isdened asthelowest(positive)pestpopulationlevel thatwill ausee onomi lossesonthe rop".

x

¯

isthena onstantparameterofourproblem. For a given pest outbreak

x

0

> ¯

x

, we onsider that the damage ost

J

is an in reasing fun tionofthetimespentbythepestpopulationabovetheEIL

x

¯

,i.e. :

J =

Z

t

f

t

0

γ(τ )dτ

with

t

f

dened su hthat

x(t

f

) = ¯

x

and

γ(.) > 0

. Itis learthatminimising

J

isequivalent tominimising

Π = (t

f

− t

0

)

sothatwewillonly on entrateonthislatterinthefollowing. We are left with two more problems. The rst one is that, for given

x

0

and

T

, the damage ost

J

(or

Π

) strongly depends on the pest outbreakinstant

t

0

, i.e. for a given

T

, thesameinvading pest level

x

0

yields dierentdamage ostdepending onthe valueof

t

0

. Without loss of generality,from now onwewill onsider that

t

0

∈ [0, T )

. We seek to provide the safest biologi al ontrol pro edure so that we will on entrateon thedamage ostfortheworst

t

0

,ensuringanyother

t

0

would resultin smallerdamage osts. Hen e in thefollowing,weseekthe

T

thatminimisesthedamage ostforitsworst

t

0

i.e. weseekto minimise

max

t

0

∈[0,T )

Π

.

Theremainingproblemisthatouranalysis annotbea hievedforthegeneralmodel(3). It ishoweverpossibleto arryit outforalo al approximationofthe

˙x

partofthe model about

x = 0

as well asfor anupper-bound of

˙x

. Indeed, onsider thelo alapproximation of

˙x

near

(x

p

, y

p

)

. From(9),and onsideringthe hangeofvariable

z

1

=

m

g

′

(0)

ln

x

x

¯

(whi h isobviouslyin reasingin

x

),weget:

˙z

1

= S

l

− my

p

(t)

where

S

l

=

mf

′

₍₀₎

g

′

(0)

≤ S

.

Now we onsider the general

˙x

equation of model (3) under the hypothesis that the systemisin thepositivelyinvariantset

{˜

y ≥ 0}

. Wegetforall

t ≥ t

0

:

˙x ≤ f (x) − g(x)y

p

(t)

Letusmakethe hangeofvariable

z

2

= m

R

x

¯

x

1

g(x)

dx

(whi hisin reasingin

x

),weget:

˙z

2

≤ S − my

p

(t)

Inthesequelwewillthenfo usonthesystem:

˙z = σ − my

p

(t)

(14)

with

z

standingfor

z

1

or

z

2

and

σ

for

S

l

ortheoriginal

S

. Throughequation (14) wewill thus study in one step the exa t lo al approximation of our system as well as an upper bound ofit,yieldingalo allyoptimalresultaswellasagloballysup-optimalone.

3.2 Preliminary results

Tostateourmainoptimisationresult,werstneedsomepreliminary omputations. Sin e weassumethat thestability onditionofthepestfreesolutionholdstrue,wehave:

Hypotheses 2 (H2): Thebiologi al ontrol budgetis hosen su hthat:

µ > σ

Werstshowthat providedthereleaseperiod

T

isnottoolarge,thepestpopulationis de reasingfor

t ≥ t

0

.

Proposition 1 Thereexists areleaseperiod

T

ˆ

su hthat if

T < ˆ

T

,thepest populationis de reasingfor

t ≥ t

0

.

Proof: Considerequation(14) . Provided

min

t

y

p

(t) >

σ

m

,

z

isde reasingfor

t ≥ t

0

;then sodoesthepest population

x

. Noti ethat:

min

t

y

p

(t) =

µT

e

mT

_{− 1}

isade reasingfun tionof

T

with:

lim

T →0

+

min

_t

y

p

(t) =

µ

m

>

σ

m

sin ethestability onditionholds.

ThenProposition 1holdstruewith

T

ˆ

solutionof:

µ ˆ

T

e

m ˆ

T

_{− 1}

=

σ

m

(15)



Proposition1haspra ti alinterestregardinggrowers on erns: indeeditisnot interest-ingtoinvestinpreventativereleasesofbiologi al ontrolagentsandallowaninvadingpest populationtoproliferateafterthemomentofinvasion. Itismu hmoredesirableto hoose apro edurethat willalwaysmakethepest populationde reasingaftertheinvasion. Thus inthesequelweassumethat:

Hypotheses 3 (H3): Thereleaseperiodis hosen su hthat

T ∈ (0, ˆ

T )

.

We now seekthe momentof invasion

t

0

that maximises thedamage osts

J

(i.e. that maximisesthedamagetime

Π

)foragiveninvadingpestpopulation

x

0

> ¯

x

( orresponding to

z

0

> 0

)andagivenreleaseperiod

T ∈ (0, ˆ

T )

. Noti ethatwiththe

z

formalism,wehave

Π = t

f

− t

0

with

z(t

0

) = z

0

and

z(t

f

) = 0

. Wehave:

Lemma1 [[23℄℄Suppose

z

0

> 0

and

T ∈ (0, ˆ

T )

arexed,thenoneofthefollowingholds: (i)

∃k ∈ N, (µ − σ)kT = z

0

then

Π = kT

(ii)

∃k ∈ N, max

t

0

Π(t

0

) = Π(t

∗

0

) = (k + 1)T − t

∗

0

(i.e.

z((k + 1)T ) = 0

)

We will rst show ase (i) and prove that, otherwise, either ase (ii) holds or

Π

is maximumat

t

∗

0

= 0

, thislatter asebeingimpossible. Proof: Suppose that

Π(t

0

)

is maximum for

t

∗

0

∈ (0, T )

and su h that

(t

∗

0

+ Π(t

∗

0

)) ∈

(kT, (k + 1)T )

forsomeinteger

k

. That isto saythat wesupposethat

Π(t

0

)

ismaximum forboth

t

0

and

t

f

= (Π(t

0

) + t

0

)

beingstri tlywithintwomomentsthataremultiplesof

T

.

Nowpi k

t

m

0

< t

∗

0

< t

M

0

in

(0, T )

su h that

(t

0

+ Π(t

0

)) ∈ [kT, (k + 1)T )

for all

t

0

∈

[t

m

0

, t

M

0

]

. Withinthisset,dene

θ(t

0

)

as:

Π(t

0

) = kT − t

0

+ θ(t

0

)

(16) Integrating(14)between

t

0

and

t

0

+ Π(t

0

)

,weget:

z(t

0

+ Π(t

0

))

= z

0

+ σΠ(t

0

) − m

Z

t

0

+Π(t

0

)

t

0

y

p

(τ )dτ

= z

0

+ σΠ(t

0

)

−

mµT

1 − e

−mT

Z

T

t

0

e

−mτ

_{dτ + (k − 1)}

Z

T

0

e

−mτ

_{dτ +}

Z

θ(t

0

)

0

e

−mτ

_dτ

!

= z

0

+ σΠ(t

0

) −

µT

1 − e

−mT



e

−mt

0

_{− e}

−mθ(t

0

)

_{+ k(1 − e}

−mT

₎



(17)

whi h is,fromthedenitionof

Π

,equalto

0

. From(16),wehave:

dΠ

dt

0

(t

0

) =

dθ

dt

0

(t

0

) − 1

(18)

Usingthisanddierentiating(17)withrespe tto

t

0

yields:

dθ

dt

0

(t

0

) =



mµT

1−e

−mT



e

−mt

0

_{− σ}



mµT

1−e

−mT



e

−mθ(t

0

)

− σ

(19)

Tohaveamaximumof

Π

at

t

∗

0

,itisrequiredthat:

dΠ

dt

0

(t

∗

0

) = 0 ⇔

dθ

dt

0

(t

∗

0

) = 1

sothat from(19), wemust have

θ(t

∗

0

) = t

∗

0

. Then, from(17)and (16) ,theinteger

k

must besu hthat:

(µ − σ)kT = z

0

and

Π = kT

doesnotdependon

t

0

,whi hshow ase(i). Otherwise

Π(t

0

)

hasnoextremum at

t

∗

0

verifying

t

∗

0

∈ (0, T )

and

(t

∗

0

+ Π(t

∗

0

)) ∈ (kT, (k + 1)T )

.

Two ases remain to be studied: either

Π

is maximum for

t

0

= 0

, or

t

∗

0

is su h that

(t

∗

0

+ Π(t

∗

0

)) = (k + 1)T

. Theremaining ases

t

∗

0

= T

and

(t

∗

0

+ Π(t

∗

0

)) = kT

mightbestudied by

k

reparametrisation.

We on entraterstonthe ase

t

∗

0

= 0

. Thenfrom(18)and(19)therightderivativeof

Π

at

t

0

= 0

is:

dΠ

dt

0

(t

0

= 0

+

) =



mµT

1−e

−mT



− σ



mµT

1−e

−mT



e

−mθ(t

0

)

− σ

− 1

(20)

FromProposition 1,wehave

∀T ∈ (0, ˆ

T )

:

σ <

mµT

(1 − e

−mT

₎

e

−mT

Sin e by denition

θ(t

0

) ≤ T

boththe numerator and denominator of the fra tion in the righthandsideof (20)arepositive,theformerbeinglargerthanthelatter.

dΠ

dt

0

(t

0

= 0

+

₎

is thenpositivesothat

Π

hasa(lo al)minimumat

t

0

= 0

,rulingoutthe ase

t

∗

0

= 0

. Wenowfo usonthe asewhere

t

∗

0

issu hthat

t

∗

0

+ Π(t

∗

0

) = (k + 1)T

. Thenfrom(16),

θ(t

∗

0

) = T

andtheleftderivativeof

Π

at

t

∗

0

is:

dΠ

dt

0

(t

∗−

0

) =



mµT

1−e

−mT



e

−mt

∗

0

− σ



mµT

1−e

−mT



e

−mT

_{− σ}

− 1

In averysame way asin the previous ase, sin e

T < ˆ

T

weshowthat this derivative is positive. Similarlyone anshowthattherightderivativeof

Π

at

t

∗

0

isnegative. This anbe performedthroughthereparametrisationof

k

as

(k + 1)

whilenoti ingthatthis orresponds to

θ(t

∗

0

) = 0

.

Thenforthe onsidered

z

0

and

T

,

Π

ismaximumfor

t

∗

0

su hthat

(t

∗

0

+ Π(t

∗

0

)) = (k + 1)T

, whi h ompletestheproofof ase(ii).



Lemma1isquitenatural. Indeed,fromthelinearityandperiodi ityof (14) ,thede rease thattakespla ebetweentimes

T

and

kT

isindependentof

t

0

. Theworst aseshouldthen

ontain the end of the rst time intervalwhere the predatorsare s ar e, rather than the beginningofthelastintervalwhere theyareabundant.

Inthefollowingwelook forthereleaseperiod

T ∈ (0, ˆ

T )

that minimisestheworst ase damagetime:

max

t

0

Π

.

3.3 Optimal Choi e of the Release Period

Wenowinvestigatewhi h

T ∈ (0, ˆ

T )

minimises the worst asedamage time

max

t

0

Π

(i.e. thatminimisestheworst asedamages

max

t

0

J

). Wegetthefollowingresult: Theorem2 Suppose

z

0

> 0

isxed. Let:

T

1

=

z

0

µ − σ

(21)

Thenthere exists

n

0

∈ N

∗

su hthat

min

T ∈(0, ˆ

T)

max

t

0

Π = T

1

isrea hed at

T = T

n

=

T

1

n

forall integer

n > n

0

.

Proof: Consider

T

1

that is positive sin e

µ

is assumedto begreater than

σ

. It is then learthat thereexists aninteger

n

0

su hthat forallinteger

n > n

0

, T

n

< ˆ

T

. Now onsider a

T

n

with

n > n

0

;using(21),wehave:

(µ − σ)nT

n

= (µ − σ)T

1

= z

0

Su ha

T

n

orrespondsto ase(i)inLemma1,thuswehave

Π = T

1

whi hdoesnotdepend on

t

0

.

Nowweshowthat

Π = T

1

istheminimum(withrespe tto

T ∈ (0, ˆ

T )

)of

max

t

0

Π

. Pi k a

T 6= T

n

forall

n > n

0

. For su h a

T

, ase(i)ofLemma 1is ruledoutand a ordingto ase(ii), wehave:

∃k ∈ N, max

t

0

Π = Π(t

∗

0

) = (k + 1)T − t

∗

0

and

z((k + 1)T ) = 0

. Using(17)and

θ(t

∗

0

) = T

,wehave:

z

0

+ σ ((k + 1)T − t

∗

0

) − (k + 1)µT + µT

1 − e

−mt

∗

0

1 − e

−mT

= 0

(22) sothat:

(k + 1)T =

1

µ − σ



z

0

− σ t

∗

0

+ µT

1 − e

−mt

∗

0

1 − e

−mT



whi h yields:

max

t

0

Π = (k + 1)T − t

∗

0

= T

1

+

µ

µ − σ

 1 − e

−mt

∗

0

1 − e

−mT

T − t

∗

0



(23)

Noti ethat thefun tion



1−e

−mt

t



isde reasingof

t

,sothat,sin e

t

∗

0

< T

,wehave:

 1 − e

−mt

∗

0

1 − e

−mT

T − t

∗

0



> 0

Sin e from (H2),

µ > σ

, we haveshown that

max

t

0

Π

is greater than

T

1

for all period

T

su h that

∀n > n

0

, T 6= T

n

,whi h ompletestheproof.



Theorem 2 shows that for a given

z

0

we have a ountable innite number of release periods,the

T

n

,thatsolvesouroptimisationproblem.

However,wehavetwodi ultieswith Theorem2. On theonehand,throughequation (14) we studied both the lo al optimisation problem and the global suboptimal one. To solvethese two, weneed to hoose areleaseperiod

T = T

n

, whi h depends on

σ

. Inthe lo al optimisation ase,

σ

equals

S

l

=

mf

′

₍₀₎

g

′

₍₀₎

while in theglobal suboptimalone,

σ

equals

S = sup

_x≥0

mf

_g(x)

(x)

. Thereis noreasonwhy

S

S

l

would berational,so that Theorem2states that it is not possibleto hoose arelease period

T

that solves both the lo al and global optimisationproblem(what remainshowevertheobje tive).

On theotherhand, itis tobenoti ed that

T

1

, whi hdetermines theoptimalvaluesof thereleaseperiod,dependsonsomebiologi al"values,namelytheinvadingpestpopulation level

x

0

(i.e.

z

0

)andthe ombination ofparameters

σ

. This isquiteabigdrawba ksin e biologi alparametersareusuallybadly known andthe initialinvadingpest level

x

0

is not known at all. Clearlysu h model-basedoptimisation approa hesshould takeinto a ount theseun ertainties[21℄.

Hen e,inthefollowing,wewillshowhow

T

shouldbe hoseninordertoensurethatthe worst asedamage time

max

t

0

Π

remains lose to itsminimal value, independentlyof the valueof

x

0

and

σ

. Indoingso, wewill providean almost optimal hoi e of

T

that solves boththelo alandglobaloptimisationproblemandthatisrobusttothevalueof

x

0

andto parametersun ertainty.

3.4 Robustness Analysis

In the pre eding se tions, we wantedour hoi e of

T

to minimise the dieren e between the time that it takesto rea h

x

¯

from

x

0

for the worst possible hoi e of the initial time

t

0

. Inadditiontothat,wenowalso wantthis toberobustwithrespe t totheun ertainty ontheinitial ondition

x

0

andtothemodelparametersthatwewilldenote

p

inthesequel (

p

issimplythepair

(σ, m)

. Indeed,in thepoorlymeasured onditionsof rop ultureand with thea ompanyings ar ely knownmodels, it isfundamental fora ontrol method to be robust with respe t to these un ertainties. We then suppose that the true"

x

0

and

p

belongs tosome ompa tsets:

Hypotheses 4 (H4): The un ertainties onthe initial onditionandthe model parameters arebounded, with

z

0

∈ [z

0

, z

0

]

denoted

Z

and

p

thatbelongstoa ompa t set

P

.

We of ourse still assumethat theinvading populationis greater than theEIL

x

¯

(i.e.

z

0

> 0

) and that the stability ondition holds, i.e.

µ > max

p∈P

(σ)

. Our optimisation pro edurewillberobustprovidedour hoi eof

T

minimises

max

t

0

Π

fortheworst

z

0

and

p

hoi ewithin thesetsdenedin Hypotheses(H4).

Noti ethattheminimum(withrespe tto

T

)of

max

t

0

Π

isequalto

T

1

thatdoesdepend on both

z

0

and

p

. Thus in our robustness analysis it makes more sense to onsider the deviation of

max

t

0

Π

from

T

1

, ratherthan the absolute valueof

max

t

0

Π

. Hen e, werst fo usontheevolutionof:

max

(z

0

,p) ∈ Z×P



max

t

0

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



withrespe tto

T

. Wehave: Theorem3 Thefun tion:

T 7→

max

(z

0

,p) ∈ Z×P



max

t

0

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



(24)

isin reasingfor

T

smallerthansomepositive onstant

T

L

. Moreover:

lim

T →

0

+

_(z

max

0

,p) ∈ Z×P



max

t

0

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



= 0

(25)

Proof: Fromequation(23) ,wehave:

max

(z

0

,p) ∈ Z×P



max

t

0

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



=

max

(z

0

,p) ∈ Z×P



µ

µ − σ

 1 − e

−mt

∗

0

(T,z

0

,p)

1 − e

−mT

T − t

∗

0

(T, z

0

, p)



= max

p ∈ P



_µ

µ − σ

z

max

0

∈ Z

 1 − e

−mt

∗

0

(T,z

0

,p)

1 − e

−mT

T − t

∗

0

(T, z

0

, p)



Werst on entrateon:

max

z

0

∈ Z

 1 − e

−mt

∗

0

(T,z

0

,p)

1 − e

−mT

T − t

∗

0

(T, z

0

, p)



(26)

forxed

T

and

p

. Letusintrodu e:

T

l

(z

0

, z

0

, p) = min



ˆ

T (p),

z

0

− z

0

2(µ − σ)



with

T (p)

ˆ

denedinequation(15). Inthesequel,we onsiderthat

T

belongsto

0, T

l

(z

0

, z

0

, p)

. Ontheoneside,thisensuresthroughProposition1that,independentlyofthea tual param-eters,the pest population(i.e.

z

inour newvariables)alwaysde reases afterthe invasion time

t

0

.

Ontheotherside,wehave:

z

0

− z

0

T (µ − σ)

> 2

Then,thereexists

n ∈ N

and

z

01

, z

02

∈ [z

0

, z

0

]

su h that:

z

01

T (µ − σ)

= n

and

z

02

T (µ − σ)

= n + 1

FromTheorem2,

T

isthus anoptimalperiod forboththeinitial onditions

z

01

and

z

02

. Werstshowby ontradi tionthat forall

z

0

∈ (z

01

, z

02

)

, ase(ii)ofLemma1holdsfor

k = n

. Indeedsuppose thatforagiven

z

0

∈ (z

01

, z

02

)

,wehave

k ≤ (n − 1)

,then:

max

t

0

Π = (k + 1)T − t

∗

0

≤

(n − 1 + 1)T − t

∗

0

≤

T

1

(z

01

, p) − t

∗

0

<

T

1

(z

0

, p) − t

∗

0

(27) sin efrom(21),

T

1

is anin reasingfun tion of

z

0

. (27)isnotpossiblesin efromTheorem 2:

max

t

0

Π ≥ T

1

(z

0

, p)

and

t

∗

0

≥ 0

. Therefore

k

annot be smalleror equal to

(n − 1)

for

z

0

∈ (z

01

, z

02

)

.

Nowsuppose that

k ≥ (n + 1)

forsome

z

0

∈ (z

01

, z

02

)

. From(22),wehave:

z

0

=

(µ − σ)kT + (µ − σ)T + σt

∗

0

− µT

1 − e

−mt

∗

0

1 − e

−mT

≥

(µ − σ)(n + 1)T − σ(T − t

∗

0

) + µT



1 −

1 − e

−mt

∗

0

1 − e

−mT



≥

z

02

+

 µ(T − t

∗

0

)

e

mT

_{− 1}



T

 e

m(T −t

∗

0

)

− 1

T − t

∗

0



−

σ

µ

e

mT

_{− 1}



(28)

Thebra kettedexpressionin (28)ispositivesin e:

e

m(T −t

∗

0

)

− 1

T − t

∗

0

≥ m

andfrom(15):

∀T ∈ (0, ˆ

T ), mT >

σ

µ

(e

mT

_{− 1)}

Hen e,(28) ontradi tsthehypothesis that

z

0

< z

02

so that

k

annotbegreater orequal to

(n + 1)

for

z

0

∈ (z

01

, z

02

)

. Thus, wehaveshown that forall

z

0

∈ (z

01

, z

02

)

, ase (ii)of Lemma1holdsfor

k = n

.

Sin e

T

isanoptimalperiodfortheinitial ondition

z

01

,weshould haveby ontinuity:

lim

z

0

→z

01

+

Π(t

∗

0

(T, z

0

, p)) = (n + 1)T − lim

z

0

→z

01

+

t

∗

0

(T, z

0

, p) = T

1

(z

01

, p)

Then:

lim

z

0

→z

+

01

t

∗

0

(T, z

0

, p) =

(n + 1)T − T

1

(z

01

, p)

=

(n + 1)T −

z

01

µ − σ

T = T

Asimilarargumentationon

z

02

yields:

lim

z

0

→z

02

−

t

∗

0

(T, z

0

, p) = 0

From(22),

t

∗

0

(T, z

0

, p)

isa ontinuousfun tionof

z

0

. Then,as

z

0

overs

[z

01

, z

02

]

,

t

∗

0

(T, z

0

, p)

rea hesitswholeintervalofdenition

[0, T ]

sothatwehave:

max

z

0

∈

[z

01

,z

02

]

 1 − e

−mt

∗

0

(T,z

0

,p)

1 − e

−mT

T − t

∗

0

(T, z

0

, p)



= max

t

∗

0

∈[0,T ]

 1 − e

−mt

∗

0

1 − e

−mT

T − t

∗

0



= max

z

0

∈ Z

 1 − e

−mt

∗

0

(T,z

0

,p)

1 − e

−mT

T − t

∗

0

(T, z

0

, p)



Dierentiating



1−e

−mt∗

0

1−e

−mT

T − t

∗

0



with respe tto

t

∗

0

, weshowthatit rea hesitsmaximum for:

ˆ

t

∗

0

=

1

m

ln



mT

1 − e

−mT



(29)

ˆ

t

∗

0

> 0

sin e

mT > 1 − e

−mT

. Inaddition,

t

ˆ

∗

0

< T

sin e

e

mT

_>

mT

1−e

−mT

. Then:

max

z

0

∈ Z

 1 − e

−mt

∗

0

(T,z

0

,p)

1 − e

−mT

T − t

∗

0

(T, z

0

, p)



=

1 − e

−m ˆ

t

∗

0

1 − e

−mT

T − ˆ

t

∗

0

=

1

m



mT

1 − e

−mT

− 1 − ln



mT

1 − e

−mT



, H(T, p)

H(T, p)

isanin reasingfun tionof

T

on

(0, T

l

(z

0

, z

0

, p))

sin e:

∂H(T, p)

∂T

=

e

−mT

_(e

mT

_{− 1 − mT )}

(1 − e

−mT

₎

2



1 −

1 − e

−mT

mT



> 0

Moreover,throughtheHospitalrule,wehaveforall

p ∈ P

:

lim

Letusintrodu e:

T

L

= min

p∈P

T

l

(z

0

, z

0

, p) > 0

Throughtheaboveanalysis,ithasbeenshownthat for

T ∈ (0, T

L

)

and

p ∈ P

,

H(T, p)

is anin reasingfun tion of

T

with

lim

T →0

+

H(T, p) = 0

.

Wenow omeba ktothefun tion denedin(24) . Wehave:

max

(z

0

,p) ∈ Z×P



max

t

0

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



= max

p∈P



µ

µ − σ

H(T, p)



Then,from

H(T, p)

properties,for

T

smallerthan

T

L

theworstdeviation of

max

t

0

Π

from

T

1

(a ordingto

(z

0

, p) ∈ Z × P

)isanin reasingfun tionof

T

. Moreover:

lim

T →

0

+

_(z

₀

_{,p) ∈ Z×P}

max



max

t

0

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



= 0

whi h on ludestheproof.



One andedu ethefollowingCorollaryfromTheorem3:

Corollary1 Let

T = min

p ∈ P

ˆ

T (p)

. Then:

inf

T ∈(0,T )

(z

0

,p) ∈ Z×P

max



max

t

0

∈

[0,T ]

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



= 0

(30) isa hievedfor

T = 0

+

.

Proof: FromTheorem 3:

inf

T ∈

(0,T

L

)

max

(z

0

,p) ∈ Z×P



max

t

0

∈

[0,T ]

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



= 0

(31)

andisa hievedfor

T = 0

+

sin etheargumentoftheinmumin (31)isin reasingof

T

and tendsto

0

as

T

does.

To ompletetheproofofCorollary1wehavetostudy:

max

(z

0

,p) ∈ Z×P



max

t

0

∈

[0,T ]

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



for

T ∈

T

L

, T

. Noti ethatfrom

T

L

denition,wehave:

T

L

=

min

(z

0

,p)∈Z×P



min



ˆ

T (p),

z

0

− z

0

2(µ − σ)



= min



T ,

min

(z

0

,p)∈Z×P



_z

0

− z

0

2(µ − σ)



sothattheset

T

L

, T

isnon-emptyifand onlyif:

T > min

(z

0

,p)∈Z×P



_z

₀

_−z

₀

2(µ−σ)



. Suppose thatthisholdsandthat

T ∈

T

L

, T

,thenwehave:

max

z

0

∈Z

 1 − e

−mt

∗

0

(T,z

0

,p)

1 − e

−mT

− t

∗

0

(T, z

0

, p)



> 0

sin e otherwise it would be required that

∀z

0

∈ Z

there would exist

n ∈ N

su h that

T =

z

0

n(µ−σ)

whi his obviouslynotpossible. Then,

∀T ∈ [T

L

, T )

:

max

(z

0

,p) ∈ Z×P



max

t

0

∈

[0,T ]

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



=

max

p∈P



_µ

µ − σ

max

z

0

∈Z

 1 − e

−mt

∗

0

(T,z

0

,p)

1 − e

−mT

− t

∗

0

(T, z

0

, p)



thislatterbeingpositive,whi h on ludestheproof.



With the help of Theorem 3, we know that provided

T

is not too large, the smaller

T

is hosen, theless will be theworst ase (a ording to parametersand initial ondition un ertainty) deviationof

max

t

0

Π

from itsminimal value

T

1

. It hasmoreoverbeenshown thatas

T

tendsto

0

, theworst asedeviation of

max

t

0

Π

from itsminimalvaluegoesto

0

aswell.

We on ludefrom this analysis that therobust hoi e of

T

, asdened bythe riterion (30) onsists in hoosing

T = 0

. However,it isobviouslynotpossibleto a hievesu ha

T

in pra ti alappli ations: itwould onsistin ontinuouslyreleasingpredatorsinto the rop ulture. Asanalternative,andsin e theargumentofthe inmum in (30) isanin reasing fun tionof

T

for

T ∈ (0, T

L

)

,itisthenadvisedtotake

T

assmallaspossibletoa hievethe bestpossiblerobustness.

Alastremarkshouldbemadeabouttheneedforrobustness. Inthepre edinganalysis, wehave on entratedonasinglemodelintheform:

˙z = S − y

p

(t)

Weshould rememberthat this model overstwo ases: alo al linearisationof thesystem (whi h yields a ertain value of

S

) and a system that globally upper-bounds the a tual nonlinear system (whi h yields another value of

S

). At the very least, the hoi e of

T

that wemade should work well in both those ases. We an then on lude that, even in the unrealisti ase where the initial onditions and the parameters are verywell-known, one should make a hoi e of

T

that is su iently robust, so that both global and lo al approximationsare overed,i.e. oneshouldtake

T

assmallaspossible.

4 Numeri al Simulation

To illustrate the robustness resultstated in Theorem 3 we performed aMonte Carlo like numeri alsimulation:animpulsivepredator-preymodeloftheform(3)was onsideredwith

onstanta priori hosenparameters forthe ontinuous partof the model and

µ

verifying ondition(6). A triplet of theremaining parameterswasrandomly hosen: rst arelease period

T ∈ (0, T

L

)

,thepestinvasionmoment

t

0

∈ [0, T )

andthepestinvasionlevel

x

0

> ¯

x

. The value of

T

L

is easy to ompute from (15) sin e in this ase the parameter set

P

is redu edto asingletonandthatwehave hosenalarge

Z

interval. Thetimetorea h

x

¯

,

Π

, wasthen omputedaswellasitsdeviation from

T

1

. Thepro esshasbeenrepeateduntil a setof

2.10

5

dierentsimulationswasobtainedandtheresults,thedeviationof

Π

from

T

1

, wereplotted againstthetimeperiod

T

.

Theresultsofthisnumeri alworkispresentedonFigure1,ea hofthedot orresponding to one of the

2.10

5

simulations. What a tually orresponds to Theorem 3 is the upper envelopeof the data set only (sin e itis the maximum with respe t to parametersof the worst ase,a ordingto

t

0

,of

Π

).

Figure1: Graphi alillustrationofTheorem3. Ea hdot orrespondstothedeviation of

Π

fromtheminimalworst ase

T

1

foroneofthe

2.10

5

randomly hosentriplet

T ∈ (0, T

L

), t

0

∈

(0, T )

and

x

0

> ¯

x

.

Asit anbenoti edthesmallerthe hoi e of

T

,thesmallerthepositivedeviationof

Π

(thusofthe ropdamages)fromitsoptimalvalue

T

1

. Thepri etopayishoweverthatsu h a hoi eredu esalsothenegativedeviationof

Π

from

T

1

. Indeedananalysissimilartothe

oneperformedinSe tion3,butfo usingon:

min

(z

0

,p) ∈ Z×P



min

t

0

Π(t

0

, z

0

, p)



− T

1

(z

0

, p)



wouldshowthatthisisade reasingfun tionof

T

withlimit

0

as

T

tendsto

0

+

. Hen e,if onefeelslu ky,a hoi eof

T

smallmaynotbethebestonesin eone anobtain,fortunately, smaller

Π

(thussmaller ropdamages)ifthea tual

t

0

isnottheworstone.

5 Con lusion

Inthis ontributionwestudied arathergeneralpredator-preymodelwith periodi impul-sive additionsof predators(releases)that is relevantforbiologi al ontrol modelling. The impulsivereleasesofthebiologi al ontrolagentsweremodelledonthebasisofaxed bud-getto bespentpertime period (releaserate). Assu hit allowedto omparethedierent releasesperiod as the same overall amount of predators was used. In a rst step, it was shown that,providedthe releaserateof bio ontrol agents washigh enough,the biologi al ontrol programallows to eradi ate theprey(pest) population, whateverthevalue ofthe releaseperiod. Thenthee ien yofdierentreleaseperiods(foragivenreleaserate)were omparedontheassumptionthat thebiologi al ontrolprogramis ondu ted ona preven-tativebasis. It wasshownthat the time needed to eradi ate an unforeseen pest invasion o urring at aworst moment had a minimum that ould be a hieved through the hoi e ofa ountable innitenumberof releaseperiod. However,these releaseperiod valueswere stronglydependentonthemodelparametersandarobustnessanalysisofthisoptimisation problem showedthat the smallerthe releaseperiod, the lessthe timeneeded to eradi ate thepestinvasion,fortheworstinvasionmomentanda ountingforparameterun ertainty. Hen e,asanadvi etobiologi al ontrolpra titioners,ourstudyindi atesthatitislessrisky toreleasefrequentsmallamountsofbio ontrolagentsratherthanmassiveinfrequentones. This result (frequent releases of small doses yield better ontrol) is similar to the ones of [35,36,11℄thatwereallobtainedfromsimulatorymodels. Su habio ontrolstrategyisalso advo atedby[2℄ fromreallife experimentsandby[29℄onthebasisofasimulatorymodel. However,toourknowledge,this ontributionis therstto re ommendsu hastrategyon thebasisofamathemati alanalysis.

In a broader ontext, we believe that the te hniques used in this work may also be of interest for optimisation problems on impulsive models from other applied elds: for instan e, optimisationof va inationstrategies in epidemiology [28,4, 13℄ oroptimisation of hemotherapyin an ertreatments[17,6,5℄.

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