The Effects of Partial Crop Harvest on Biological Pest Control

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Control

Sapna Nundloll, Ludovic Mailleret, Frédéric Grognard

To cite this version:

Sapna Nundloll, Ludovic Mailleret, Frédéric Grognard. The Effects of Partial Crop Harvest on

Bio-logical Pest Control. [Research Report] RR-6284, INRIA. 2007, pp.27. �inria-00171093v2�

inria-00171093, version 2 - 12 Sep 2007

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Thème BIO

The Effects of Partial Crop Harvest on Biological

Pest Control

Sapna Nundloll — Ludovic Mailleret — Frédéric Grognard

N° 6284

Sapna Nundloll

∗

,Ludovi Mailleret

†

, Frédéri Grognard

∗

ThèmeBIOSystèmesbiologiques ProjetsComore

Rapportdere her he n°6284Septembre200727pages

Abstra t: Inthispaper,theee ts ofperiodi partial harvestingofa ontinuouslygrown rop on augmentative biologi al ontrol are analyzed. Partial harvesting an remove a proportionofbothpestsandbiologi al ontrolagents,soitsinuen eonthe ontrole ien y annotbeapriorinegle ted. Animpulsivemodel onsistingofageneralpredator-preymodel in ode, augmentedbyadis rete omponent to depi treleasesof biologi al ontrol agents and theperiodi partial harvesting is used. Theperiods aretaken asinteger multiples of ea h other. A stability ondition forpest eradi ationisexpressed astheminimal valueof thebudgetperunit timetospendonpredators. We onsiderthepartialharvestingperiod tobexedbyboththeplant'sphysiologyandmarket for esso that theonlymanipulated variable is the releaseperiod. It is shown that varying the releaseperiod with respe t to theharvestperiodinuen estheminimalbudgetvaluewhentheformeris arriedoutmore oftenthanthelatterandhasnoee twhenreleasestakepla easoftenasorlessfrequently thanthepartial harvests.

Key-words: Predator-preydynami s; partial harvest; inundativebiologi al ontrol; im-pulsive ontrol;stability;minimal budget

∗

INRIA-COMORE,2004routedesLu ioles,BP93,F-06902SophiaAntipolisCedex,Fran e.

†

Résumé : Ce do ument montre les eets de la ré olte partielle périodique sur la lutte biologique inondative dans le as de la prote tion d'une plantation à roissan e ontinue. Commelaré oltepartielleestsus eptibledepréleverunepartiedesravageursainsiquedes auxiliairesdeluttebiologique,onnepeutignorersoninuen esursone a ité.

Le modèle étudié onsiste en un modèle proie-prédateur en EDO lassique dé rivant l'intera tionbiologiqueentrelesdeuxpopulations,auquels'ajouteunepartiedis rèterepr±entant les phénomènes, intrinsquement dis rets, liés à la ré olte et à la lutte biologique. Les paramètresde ré oltepartielle sontsupposés xésparlaphysiologiede laplante ainsique lademandeé onomique; lapériodede lâ herdesauxiliaires et lebudgetinvesti sontdon lesseulsparamêtresmodiablesdusystème. L'exigen edestabilitédel'étatdusystèmeoù lesravageurssontabsentsnous donne unbudgetminimal d'auxiliairesàutiliser parunité detempsqui peutêtrefon tiondelapériodedeslâ hers. Nousdémontronsquelapériode deslâ hersinuesurlebudgetminimal quandellealieu plussouventquelaré olte,mais n'aau uneetquandellealieumoinsfréquemment.

Mots- lés : Dynamiques proie-prédateur; ré olte partielle; lutte biologique inondative; ommandeimpulsive;stabilité;budgetminimal

1 Introdu tion

Biologi al ontrolistheredu tionofpestpopulationstoharmlesslevelsthroughtherelease oftheirnaturalenemies. Thelatter anin ludebothparasiti andpredatoryspe ies,whi h are deployed at sele ted lo ationsthroughout the rop and, whereverpossible, to spe i parts ofindividual plantswhere thepest is likelyto atta k. Su essful ontrol proje tsin the eld haveinvolved the use of only one predatoryspe ies su h asin [3, 9℄, as well as more omplexbiodiverses hemessu hasthosesuggestedby[14,8,2,17℄andthereferen es therein. The target pest spe ies and the setting, i.e. where the rop is grown, usually determines the type of ontrol required, namely whether pest eradi ation is ne essary or not. Foranexhaustivelistofdenitions andappli ations,wereferthereaderto[5,16℄.

Inthis report, we onsider theprote tionof ontinuously grown ropswhi h havezero toleran etopestinvasions. Therearetwoaspe tsinthistypeof ulture.

ˆ Firstly, inundative ontrol whi h is aprophyla ti method of pest ontrol yields the most satisfatory results when implemented (see [7, 20, 6℄ for theoreti al/simulatory studies and [4, 8, 9, 1℄ for real life experiments). A al ulated numberof predators are repeatedly inje ted into the e osystem, independently of the dete tion of pest inse tsinthegreenhouse. Su hpopulationsarenotallowedtothriveand onsistonly of individuals whose main sour e of subsisten e is thepest inse t, in the absen e of whi h,they(the predatorinse ts) rapidlydieout. Thefrequen yof thereleasesand thenumberofpredatorsinje tedea htimeensuresthataminimal'sentry'population ispresenttoredu e thedamage ausedbythepestsontheiratta k.

ˆ Se ondly, overtheir growingperiod, these rops are partially harvested onaregular basis. Sin e itis known that harvestsare likelyto inuen e, even ounterintuitively, predator-preydynami s[18,15℄,ithastobetakenintoa ountintheformulationof theproblem.

We onsiderthesimplestditrophi asewherebyonepredatoryspe iesisusedto eradi- ateapestpopulation. Ourmodel onsistsofODEsaugmentedbyadis rete omponentto in orporate theee t ofpartial harvestand releases thatby theirverynature aredis rete phenomena. Thisisa lassi alformulationthatisusedwidelyintheliteraturewhere impul-sivedynami sarestudied. Examplesare[12, 15℄in the ontextofagri ulturale osystems, [19℄inepidemiology,[10℄inpulsed hemotherapyto itesome. Fewpapersintheliterature onimpulsive rop prote tionhoweverseemto fo us onstability ofthepest-free state: yet thisisofpra ti alimportan eespe iallyforhighvalued rop ultures.

Inourwork,weattemptto giveane onomi dimensionto thesolutionofourproblem by dening the releases in terms of the number of predators to invest in over a budget period. Using Floquet Theory as presented by [19℄, we are able to express the stability onditionastheminimalnumberofpredatorsperbudgetperiodrequiredtodrivethepests to zeroat a givenreleasefrequen y. [13℄ showedhowthis numbervaried withthe release period hosen. The worst ase s enarioof pest atta k o urring at an intermediate stage betweentwopredator releaseswas onsidered and the optimalreleasepoli y whi hwould

guaranteethemoste ientprote tionagainstsurgesinthepestpopulationwas al ulated. Inparti ular,itisshownthatthehigherthefrequen iesofpredatorrelease,thesmallerthe time intervaloverwhi h thepest populationwasabove athreshold ommonly referredto astheE onomi InjuryLevel[21℄-andhen ethelowerthedamagein urredbythe rops. Inlinewiththeworkof[13℄,weinvestigatehowthefrequen yofreleasesistobevaried with respe t to the (xed) harvesting frequen y to minimise the minimal budget value. We onsider the harvest period as a referen e sin e it is set by market onstraints. The ee t of partial harvesting is similar to that of pesti ide usage proposed by [12℄ in their IntegratedPest Management(IPM)strategy. Ourmodel departsfrom thelatter'sin three ways. Firstly, boththe predatorand pest populations are subje ted to partial harvesting when this o urs. Se ondly, hypotheses made on the fun tions governing the population hangesare weakand anen ompass mostofthedensity-dependentfun tions proposed in the literature. Finally, oneperiod is taken as the integermultiple of other. This feature is keyin solvingforthe stability onditionto obtainthe minimal budgetvalue. The ase wherethefrequen iesarenotthesameisin luded.

Itisshownthatforagivenharvestperiod,whenreleasestakepla elessoftenorasoften as harvests, the minimal budget is at a al ulated value whi h is independent of release period. However,when releases takepla e moreoften than harvests, theminimal budget required always ex eeds this value. This result runs ounter with that obtained by [13℄: mergingthetwoseemstoindi atethattheharvestfrequen yisathresholdthatshouldnot beex eededwhenreleasingpredatorsfore ientbiologi al ontrol.

In the rst se tion of this arti le, the system model is presented. The mathemati al analysis of the system's stability and the formulation of the stability ondition in terms of the minimal budget are presented in the next se tion. A brief interpretation of the mathemati alresultsfollows. Finally,we on ludewithadis ussionontheirimpli ations.

2 Model des ription

Themodelwepresent onsistsofa ontinuousparttodepi tthepredator-preyintera tion. We onsider the aseat theonsetof pest invasionwhere the rop -thepest food supply -isinabundan e. Be auseofthis, atthisstage,itissu ienttomodel onlythepest

x

and predator

y

spe ies.























˙x

=

f

(x) − g(x)y

˙y

=

h(x)y − dy

x(nT

_h

+

)

=

(1 − αx)x(nTh)

∀n ∈ N

y(nT

_h

+

)

=

(1 − αy)y(nTh) + δ (nTh

mod

Tr) µTr

∀n ∈ N

y(mT

+

r

) =

(1 − δ (mTr

mod

Th) αy)y(mTr) + µTr

∀m ∈ N

(1)

The rst twoequations govern the intrinsi predator-prey intera tion o urring in the system. The three ones depi t the impulsive phenomena that we onsider with harvest takingpla eat

nTh

andreleasesat

mTr

.

Inthe ontinuouspart,thefun tionsdis ussedarenotspe iedsotheyarerepresentative ofasmanysystemsas possible. Onlythefollowinghypothesesare made.

Hypothesis 1 Let

f

(x)

,

g(x)

and

h(x)

belo ally Lips hitz ontinuousin

R

+

su h that ˆ

f

(0) = 0

ˆ

g(0) = 0

,

g

′

_{(0) > 0}

and

g(x) > 0 ∀x > 0

ˆ

h(0) = 0

and

h(x) > 0 ∀x > 0

ˆ

f (x)

g(x)

and

g(x)

x

areupper boundedfor

x

≥ 0

f

(x)

isthegrowthvelo ityorfeedinginputofthepests. Itrepresentsthegrowthfun tion ofthepestspe iesandinourmodel,italsoen ompassesanynon-predatorylossesofthepest population(e.g. logisti growth). We assumethat the predator population is neverlarge enoughforintra-predatorintera tiontotakepla esothefun tionalandnumeri alresponses anbeexpressedsolelyintermsofthepreynumbers,i.e. as

g(x)

and

h(x)

respe tively.

Weassumethatpestgrowthrate,thefun tionalandnumeri alresponsesareallnilwhen thee osystemispest-free.

Thefun tional responseisin reasingfor smallpest populationlevels. Wealso onsider that, in the presen e of pests, predation always takespla e with a negativeimpa t on

x

(

g(x) > 0

)and apositiveimpa ton

y

(

h(x) > 0

). Note that onditions anbeindu edas mu h by thepredator inse t foragingabilities per se asthey anbefa ilitated by pla ing the predator inse ts at known lo ations on the plant where the pests are most likely to atta k. In lassi al density dependent models,

g(x)

is bounded orlinear, so that

g(x)

x

is

always bounded. The boundedness of

f (x)

g(x)

means that there is no value of

x

where the pest growth

f

(x)

overwhelmingly dominates the predation

g(x)

, whi h would render the biologi al ontrol impossible.

Partial rop harvestsand predatorreleases o urrespe tivelyevery

Th

and

Tr

.

αx

and

αy

represent the respe tive proportions of the prey and predator populations ae ted at ea hharvest. Theseparametersareallowedbedierentsin einreality,itisverylikelythat ea hspe iestendstoo upydierentpartsoftheplant. Wealsoassumethattheinse tsare uniformly distributed throughout our plantation so that the ee t of partial harvesting is dire tly orrelatedwiththenumberofplantsharvested. Weassumelinearmaturationofthe ropsotheproportionof ropsharvestedea htimeandhen einse tsremovedis onsidered asxed. The

δ

-fun tionisdened thus toidentify instantsof simultaneouspartial harvest andpredatorrelease.

δ(θ) =



1

if

θ

= 0

0

otherwise

(2) Finally, we presume that we have axed budget of predatorsover a designated time period that is distributed evenlyamong thereleases that are arried out.

µ

refersto the totalnumberofpredatorspur hasedpertimeunit. Expressing

Tr

inthesameunits asthe budgetperiodgivesthe ontrol

µTr

asthenumberofpredatorsreleasedevery

Tr

.

3 Mathemati al analysis

In our analysis, we restri t ourselvesto the ase where either one of the periods (release orpartial harvests) is theintegermultiple ofthe other. Note howeverthat the model (1) formalismismoregeneral. Westudythesystemintheabsen eofpests,i.e. when

x

= 0

. In additionofbeinginvariant,itisthetargetstateofoursystem. Thestabilityofthesystem aroundthat state istherefore of interest. Our analysis takespla e separatelyfor the ase whenreleasesaremorefrequentthanharvests,andwhentheyarelessfrequent.

Weshowthatintheabsen eofpestsattheinitialtime,thepredatorpopulation onverges towardsapositiveperiodi solution. Wethendemonstratethat whenpreysarepresentat theinitialtime, onvergen eofthepredatorpopulationalsotakespla etothatsameperiodi solution,whilethepreysgoextin tprovidedsome onditionontheparametersisveried. 3.1 Pest-free stabilityanalysis

Releasesmorefrequentthan harvests Proposition 1 Let

Th

= kTr

where

k

∈ N

∗

and Hypotheses 1 be satised. Then, in the absen eof pests,model(1)possessesa globally stableperiodi solution

(xph

(t) , yph

(t)) =







0, y

∗

_e

−d(t

mod T

h

)

_{+ µTre}

−d(t

mod T

r

)

⌊

t

mod Th

_Tr

⌋−1

X

j=0

e

−jdT

r







(3) where

y

∗

=



1−e

−

_dTh

1−e

−

dTr



(1 − αy

) + αy

1 − (1 − αy)e

−dT

h

µTr

(4)

Proof: When

Th

= kTr

, in the absen e of pests and using Hypotheses 1, thesystem is simpliedto























˙x

=

0

˙y

=

−dy

x(mT

r

+

) =

(1 − δ (m

mod

k) αx)x(mTr)

y(mT

+

r

)

=

(1 − δ (m

mod

k) αy)y(mTr) + µTr

∀m ∈ N

(5)

The pest populationstays nil sin ein the absen e of pests, theirpopulation does not hangeeither. Thesolution

xph(t) = 0

Ontheotherhand,thepredatorpopulationwillvarya ordingtothenumberof preda-torsmanually inje tedinto the systemand, sin ethepopulationisnon-zero, a ordingto thepartialharvestee t. Theabsen eoftheirsour eoffoodwill auseanexponentialde ay ofthe population. We demonstratethatthese for es will provokethe predatorpopulation to rea h aperiodi pattern ofperiod equalto

Th

, whi h weshall referto asthe period of referen e. Theinstantfollowinga oin idingpartialharvestandreleaseistakenasthepoint ofreferen e.

ToproveProposition 1, werst show byindu tion that thepredator populationright afterarelease anbeexpressedintermsofthepointof referen easfollows

y(nTh

+ iT

+

r

)

= y(nT

h

+

)e

−idT

r

+ µTr

i−1

X

j=0

e

−jdT

r

(6) where

i

∈ [0, 1, . . . , (k − 1)]

Itis seenthat(6)isvalid for

i

= 0

sin eitisequalto

y(nT

_h

+

)

= y(nT

_h

+

)e

0

_{+ µTr}

−1

X

j=0

e

−jdT

r

_{= y(nT}

+

h

)

Nowsupposethat(6) holdsfor

i

= q

where

q

∈ [0, 1, . . . , k − 2]

,i.e.

y(nTh

+ qT

+

r

) =

y(nT

h

+

)e

−qdT

r

+ +µTr

q−1

X

j=0

e

−jdT

r

Wewill now show that (6) is valid for

i

= q + 1

. We al ulate

y(nTh

+ (q + 1)T

+

r

)

from

y(nTh

+ qT

+

r

)

using(5),thensubstituting(3.1)into(7)asfollows

y(nTh

+ (q + 1)T

+

r

)

= y(nTh

+ qT

r

+

)e

−dT

r

+ µTr

=



y(nT

_h

+

)e

−qdT

r

_{+ µTr}

q−1

X

j=0

e

−jdT

r





e

−dT

r

+ µTr

= y(nT

_h

+

)e

−(q+1)dT

r

_{+ µTr}

q

X

j=1

e

−jdT

r

_{+ µTr}

= y(nT

_h

+

)e

−(q+1)dT

r

_{+ µTr}

q

X

j=0

e

−jdT

r

(7)

sothat (6)holdstruefor

i

∈ [0, 1, . . . , k − 1]

.

Toevaluatetheevolutionof

y

a ordingtotheperiodofreferen e

Th

,weneedto al ulate thevalueof

y((n + 1)T

+

h

)

,whi h isequivalentto

y(nTh

+ kT

+

r

)

, in termsof

y(nT

+

h

)

. At

thispointhowever,partialharvestingtakespla ebeforepredatorrelease;sowerstexpress itintermsof

y(nTh

+ (k − 1)T

+

y

(n + 1)T

h

+

=

y

(nTh

+ (k − 1)T

+

r

) e

−dT

r

(1 − αy) + µTr

=



y(nT

_h

+

)e

−d(k−1)T

r

_{+ µTr}

k−2

X

j=0

e

−jdT

r





e

−dT

r

(1 − αy) + µTr

=

y(nT

_h

+

)e

−dT

h

_{+ µTr(1 − αy)}

k−1

X

j=1

e

−jdT

r

_{+ µTr}

=

y(nT

_h

+

)e

−dT

h

_{+ µTr}



(1 − αy

)

k−1

X

j=0

e

−jdT

r

_{+ αy}





Notethatthesummationterm analsobeevaluated sothesequen eisexpressibleas

y

(n + 1)T

_h

+

=

y(nT

_h

+

)e

−dT

h

_{+ µTr}



(1 − αy)

1 − e

−dT

h

1 − e

−dT

r

+ αy



(8) Inthislineardynami alsystem,the oe ientof

y(nT

+

h

)

,

e

−dT

h

islessthanonein magni-tude,sothesequen ewill onvergetoalimit,theequilibriumof(8). Thisequilibriumyields (4)andthe onvergen eof

y(t)

toaperiodi solution

yph(t)

basedon

y

∗

.

Nowthat wehave establishedthe existen eof the periodi solution

yph(t)

, we seek to formulateit. Wefo usonareferen eperiodover

nTh

< t

≤ (n + 1)Th

duringwhi h

yph(t)

is pie ewise ontinuous,withthe ontinuous omponentsseparatedbypredatorreleases. The ontinuousintervalsaredenedover

nTh+iTr

< t

≤ nTh+(i+1)Tr

where

i

∈ [0, 1, . . . , k−1]

. Foragivenvalueof

t

, thevalueof

i

iseasily identiedasbeing

i

= ⌊

t mod T

h

T

r

⌋

. Thevalue

of

y

ph(t)

isthenoftheform

yph(t) = yph(nTh

+ iT

r

+

)e

−d(t

mod T

r

)

and,from(6)with

y(nT

+

h

) = y

∗

,wehavethat

yph(nTh

+ iT

r

+

) = y

∗

e

−idT

r

+ µTr

i−1

X

j=0

e

−jdT

r

sothat

yph(t) =



y

∗

_e

−idT

r

_{+ µTr}

i−1

X

j=0

e

−jdT

r





e

−d(t

mod T

r

)

=

y

∗

_e

−d(t

mod T

h

)

_{+ µTr}

_e

−d(t

mod T

r

)

i−1

X

j=0

e

−jdT

r

=

y

∗

e

−d(t

mod T

h

)

_{+ µTre}

−d(t

mod T

r

)

⌊

t

mod Th

_Tr

⌋−1

X

j=0

Thisisofthesameformas proposedin (3),thereby ompleting ourproof.



Theformofthe

yph

fun tion isillustratedonFigure1.

0

T

r

2T

r

T

h

= 3T

r

4T

r

5T

r

2T

h

7T

r

8T

r

3T

h

y

p

h

(t

)

t

Figure1: Formoftheperiodi solution

y

ph(t)

inthe asewhere

k

= 3

. Releasesofpredators areapparentatevery

mTr

instant,whilethe umulativeee tofharvestandreleaseleads toanapparentsmallerreleaseatevery

nTh

instant. Betweenthoseinstants,thepopulation de aysexponentiallysin eithasnopreytofeedon.

Releasesless frequent than harvests

When harvesting is more frequent than the releaseof predators, we havea similar result abouttheexisten eofaperiodi solution.

Proposition 2 Let

Tr

= kTh

where

k

∈ N

∗

and Hypotheses 1 be satised. Then, in the absen eof pests,model(1)possessesa globally stableperiodi solution

(xpr

(t) , ypr

(t)) =



0, y

∗

_e

−d(t

mod T

r

)

_{(1 − αy}

₎

⌊

t

mod Tr

Th

⌋



where

y

∗

=

µTr

1 − (1 − αy)

k

_e

−dT

r

(10)

Proof: When

Tr

= kTh

, in the absen e of pests and using Hypotheses 1, thesystem is simpliedto























˙x

= 0

˙y

= −dy

x(nT

+

h

)

= (1 − αx)x(nTh)

y(nT

_h

+

)

= (1 − αy)y(nTh) + µTrδ

(n

mod

k)

∀n ∈ N

(11)

Aspreviouslyexplained,

xpr(t)

issolvedfortriviallyasbeing

xpr(t) = 0

We provethat the predator population again rea hes a periodi solution. This time, however,theperiodofreferen eis

Tr

. Thepointofreferen eistheinstantaftera oin iding harvest and release. We show by indu tion that the population after a harvest an be expressedas

y(mTr

+ iT

_h

+

) = y(mT

+

r

)e

−idT

h

(1 − αy

)

i

(12) where

i

∈ [0, 1, . . . , (k − 1)]

.

Itis seenthat(12)isvalidfor

i

= 0

sin eitresumesto

y(mT

+

r

) =

y(mT

r

+

)e

0

(1 − αy)

0

Suppose(12)holdsfor

i

= q

where

q

∈ [0, 1, . . . k − 2]

,i.e.

y(mTr

+ qT

_h

+

) =

y(mT

+

r

)e

−qdT

h

(1 − αy

)

q

(13) Wewillnowshowthat(12)isvalidfor

i

= q + 1

. We al ulatethevalueof

y

when

i

= q + 1

in terms of

y(nTh

+ qT

+

r

)

, knowingfrom

˙y = −dy

in (11)it will bean exponential de ay withtheadded omponentfortheharvest. Wethensubstitute(13)inand obtain

y(mTr

+ (q + 1)T

_h

+

) =

y(mTr

+ qT

_h

+

)e

−dT

h

_{(1 − αy)}

=

y(mT

+

r

)e

−qdT

h

(1 − αy)

q

e

−dT

h

(1 − αy

)

=

y(nT

_h

+

)e

−d(q+1)T

h

_{(1 − αy)}

q+1

(14)

Thisis learlythesameformgivenfrom theexpressionin (12),therebyvalidating it.

y

∗

isgivenasthexedpointofthesequen erepresentingpost-releaseinstants. Therefore, using(12)for

i

= k

andmodel(11),wenext al ulate

y((m + 1)T

+

r

)

as

y((m + 1)T

+

r

) = y(mTr

+ kT

h

+

)

= y(mT

+

r

)e

−kdT

h

(1 − αy)

k

+ µTr

= y(mT

+

r

)e

−dT

r

(1 − αy)

k

+ µTr

(15)

Inthis lineardynami al system,the oe ient of

y(mT

+

r

)

,

e

−dT

r

_{(1 − αy)}

k

islessthan oneinmagnitude,whi h onrmstheexisten eofthexedpoint

y

∗

towhi h thesequen e onverges. This equilibrium yields (10)and the onvergen e of

y(t)

to aperiodi solution

ypr(t)

.

Nowthat we haveestablishedthe existen e of the periodi solution

ypr(t)

, we seek to formulateit. Wefo usonareferen eperiodover

mTr

< t

≤ (m+1)Tr

duringwhi h

ypr(t)

is pie ewise ontinuous,withthe ontinuous omponentsseparatedbyharvests. Theintervals of ontinuityspan

mTr

+ iTh

< t

≤ mTr

+ (i + 1)Th

where

i

∈ [0, 1, . . . , k − 1]

. Foragiven valueof

t

,the valueof

i

iseasily identiedasbeing

i

= ⌊

t mod T

r

T

h

⌋

. Thevalueof

ypr(t)

is thenoftheform

ypr(t) = ypr(mTr

+ iT

_h

+

)e

−d(t

mod T

h

)

and,from(12)with

y(mT

+

r

) = y

∗

,wehavethat

ypr(mTr

+ iT

_h

+

) = y

∗

_e

−idT

h

_{(1 − αy)}

i

sothat

ypr(t)

=

y

∗

e

−idT

h

_{(1 − αy)}

i

e

−d(t

mod T

h

)

=

y

∗

e

−d(t

mod T

r

)

_{(1 − αy)}

⌊

t

mod Tr

Th

⌋

whi h isexa tlytheexpressiongivenin(9)and ompletestheproof.



Theformofthe

ypr

fun tion isillustratedonFigure2.

3.2 Global stability analysis

Sin e we will study the onvergen e of the solutions to

(0, yp(t))

(where the

p

subs ript stands aswell for

ph

or

pr

), it will be onvenientto des ribe the systemin terms of the deviation oordinateswithrespe ttothereferen eperiodi solution:

˜

x(t) =

x(t) − xp(t)

˜

y(t) =

y(t) − yp(t)

Thisyields

˙˜x = f(x) − g(x)y

=

f

(˜

x) − g(˜

x)(˜

y

+ yp(t))

(16) and

˙˜y = h(x)y − dy − h(xp)yp

+ dyp

0

T

h

2T

h

T

r

= 3T

h

4T

h

5T

h

2T

r

7T

h

8T

h

3T

r

y

p

r

(t

)

t

Figure 2: Form of the periodi solution

ypr(t)

in the ase where

k

= 3

. Harvests are apparentatevery

nTh

instant,whilethereleaseofpredatorsdominatestheharvestatevery

mTr

instant. Betweenthose instants, thepopulationde ays exponentiallysin e it hasno preytofeedon.

The impulsive ee ts on

x

˜

are obviously un hanged ompared to those on

x

. On the otherhand,thereleaseee ts on

y

disappearin

y

˜

; indeed,wehave

˜

y(mT

r

+

) = y(mT

r

+

) − yp(mT

r

+

) = y(mTr) + µTr

− (yp(mTr) + µTr) = ˜

y(mTr)

Theharvestingimpulsesarepreservedintheexpressionof

y

˜

˜

y(nT

h

+

) = y(nT

h

+

) − yp(nT

h

+

) = (1 − αy)y(nTh) − (1 − αy)yp(nTh) = (1 − αy)˜

y(nTh)

Inthesequel,wewill performaglobalandalo alstabilityanalysis. Forthelatter,we willneed the omputation ofthe linearapproximationofthedeviation systemaroundthe periodi solution

(0, yp(t))

:



_{˙˜x = (f}

′

_{(0) − g}

′

_(0)yp(t))˜

_x

Releasesmorefrequentthan harvests

We will rst prove our result in the ase where releases take pla e more often than har-vests. We obtain twodierent onstraints for the Lo al Asymptoti Stability (LAS) and Global Asymptoti Stability (GAS) of the periodi solution in system (1). The latter is obviously strongerthan theformer, but is su ientin the ase where pests outbreaksdo notimmediatelytakelargeproportions.

Inordertostatethefollowingtheorem,werstneedtodenethefun tion

µ

h

(S, r) = d



S

+

ln (1 − αx)

rTh



1

1 −



α

y

(

1−e

−

dTh

)

1−(1−α

y

)e

−

dTh







e

−

dTh

_k

k



1−e

−

dTh

_k







Thisfun tionisin reasingin

S

and

r

be ausethesignofthepartialderivativesisdetermined bythesignofthelastfa tor,whi h anbeshowntobepositive. Indeed,thisfa torispositive when

αy

1 − e

−dT

h

1 − (1 − αy) e

−dT

h

!





e

−

dTh

k

k



1 − e

−

dTh

_k







<

1

andwehave

αye

−

dTh

_k

_{≤ αy}

and

1 − (1 − αy) e

−dT

h

_{> αy}

,so that

αy

e

−

dTh

_k

1 − (1 − αy) e

−dT

h

<

1

Also,

k



1 − e

−

dTh

_k



_{≥ 1 − e}

−dT

h

sin ebothsidesoftheinequalityhavethesamevaluein

Th

= 0

,and

d

dTh



k



1 − e

−

dTh

k



= de

−

dTh

k

≥ de

−dT

h

₌

d

dTh

1 − e

−dT

h

whi h showsthat

µ

h

(S, r)

isin reasingin

S

and

r

. Theorem1 When

Th

= kTr

with

k

∈ N

∗

, the solution

(x(t), y(t)) = (0, yph(t))

of (1) is LASi

µ > µ

h

 f

′

₍₀₎

g

′

₍₀₎

, g

′

₍₀₎



(19) andisGASif

µ > µ

h



sup

x≥0

f

(x)

g(x)

,

sup

x≥0

g(x)

x



(20)

Proof: Westartwiththeproofofglobal onvergen eunder ondition(20). Inthisproof, we will rstshow that

x

˜

goes to zero,from whi h wewill derivethat

y

˜

goesto

0

also (so that

y(t)

onvergesto

yph(t)

).

Let theinitial ondition forsystem (16)-(17) be

(˜

x0,

y0)

˜

at time

t0

= 0

+

, that is after theharvestandthepredatorreleasethattakepla eattheinitial time. Analyzing(17)and notingthat

yph(t) + ˜

y

= y(t) ≥ 0

,wehave

˙˜y ≥ −d˜y

sothat

y(t) ≥ min(0, ˜

˜

y0)e

−dt

.

Inordertoanalyzethe

˙˜x

equation,wedenethefun tion

G(˜

x) =

Z

˜

x

x

0

1

g(s)

ds

(21)

whi h aneasilybeseentobeanin reasingfun tionof

x

˜

sin e

g(s) > 0

. Sin ewealsohave that

g(s) <



sup

x≥0

g(x)

x



s

,itisstraightforwardthat

lim

˜

x

→0

>

G(˜

x) = −∞

. Inordertoshowthe ex-tin tionofthepestswewillthenprovethat

G(˜

x)

goesto

−∞

as

t

goestoinnity. Therefore, wewritethe

G

dynami s:

dG(˜

x)

dt

=

1

g(˜

x)

˙˜x

=

f (˜

_g(˜

x)

_x)

− ˜

y

− yph(t)

≤

f (˜

_g(˜

x)

_x)

− min(0, ˜

y0)e

−dt

_{− yph(t)}

Wewillnow onsidertheevolutionof

G

betweentwosu essiveharvests,thatistheevolution of

G

betweenthetimes

nT

+

h

and

(n + 1)Th

foragiven

n

:

G(˜

x((n + 1)Th)) ≤ G(˜

x(nT

_h

+

)) +

Z

(n+1)T

h

nT

+

h

 f (˜

x(s))

g(˜

x(s))

− min(0, ˜

y0)e

−ds

_{− yph(s)}



ds

Sin e noimpulse is presentinside theintegral,we an dropthe

+

supers ript in its lower extremity.

Wewillnowanalyzehowtheharvestthattakespla eattime

(n + 1)Th

impa ts

G

. We have

G(˜

x((n + 1)T

_h

+

))

=

R

˜

x((n+1)T

_h

+

)

x

0

1

g(s)

ds

=

R

˜

x((n+1)T

h

)

x

0

1

g(s)

ds

+

R

˜

x((n+1)T

h

+

)

˜

x((n+1)T

h

)

1

g(s)

ds

≤

G(˜

x(nT

_h

+

)) +

R

(n+1)T

h

nT

h

h

_{f (˜}

_x(s))

g(˜

x(s))

− min(0, ˜

y0)e

−ds

− yph(s)

i

ds

+

R

˜

x((n+1)T

h

+

)

˜

x((n+1)T

h

)

1

g(s)

ds

(22)

Thelasttermrepresentstheinuen eofharveston

G

and aneasilybeapproximated be- ause

˜

x((n + 1)Th) > ˜

x((n + 1)T

_h

+

) = (1 − αx)˜

x((n + 1)Th)

. Denoting

Sg

= sup

x≥0

f (x)

g(x)

and

rg

= supx≥0

g(x)

_x

,wehave

Z

(1−α

x

)˜

x((n+1)T

h

)

˜

x((n+1)T

h

)

1

g(s)

ds

≤

Z

(1−α

x

)˜

x((n+1)T

h

)

˜

x((n+1)T

h

)

1

rgs

ds

=

ln(1 − αx)

rg

(23)

Introdu ing (23) into (22) then yields a bound on the appli ation between su essive momentsafter harvest.

G(˜

x((n+1)T

_h

+

)) ≤ G(˜

x(nT

_h

+

))+

Z

(n+1)T

h

nT

h

 f (˜

x(s))

g(˜

x(s))

− min(0, ˜

y0)e

−ds

_{− yph(s)}



_ds+

ln(1 − αx)

rg

(24) We annowevaluateanupper-boundfor

G

atanytime

t

≥ 0

. Dening

l

astheinteger partof

t

T

h

,wehave:

G(˜

x(t)) − G(x0) ≤

R

t

0

h

_{f (˜}

_x(s))

g(˜

x(s))

− min(0, ˜

y0)e

−ds

_{− yph(s)}

i

_ds

_{+ l}

ln(1−α

x

)

r

g

≤

R

t

0

Sg

− min(0, ˜

y0)e

−ds

_{− yph(s) ds + l}

ln(1−α

x

)

r

g

=

−

R

t

0

min(0, ˜

y0)e

−ds

_ds

₊

R

t

lT

h

[Sg

− yph(s)] ds + l

R

T

h

0

[Sg

− yph(s)] ds

+l

ln(1−α

x

)

r

g

=

min(0,˜

y

0

)

d

(e

−dt

− 1) +

R

t

lT

h

[Sg

− yph(s)] ds + l

R

T

h

0

[Sg

− yph(s)] ds

+l

ln(1−α

x

)

r

g

Thersttwotermsarebounded(therstoneisobviousandthese ondoneisupper-bounded by

SgTh

). We then have to analyze the third one, whi h hasbeen obtained throughthe periodi ityof

yph(t)

andthefourthin ordertoknowif

G(˜

x(t))

goesto

−∞

when

t

goesto innity. Infa t,itsu estohave

Z

T

h

0

[Sg

− yph(s)] ds +

ln(1 − αx)

rg

<

0

toa hievethis. Itismore leanlyrewrittenintheform

Z

T

h

0

yph(t)dt > SgTh

+

ln(1 − αx)

rg

(25)

In order to obtain (20), we are now left with the omputation of

R

T

h

0

yph(t)dt

, whi h is detailedin Proposition3oftheAppendix:

Z

T

h

0

yph(t)dt =

µTh

d



1 −

αy

1 − e

−dT

h

1 − (1 − αy) e

−dT

h

!





e

−

dTh

k

k



1 − e

−

dTh

_k











(26)

Introdu ing(26)into(25)thenyields(20),whi h showsthat thislast onditionis su- ientforhaving

x

˜

goingto

0

as

t

goesto

∞

.

Sin e

x

˜

goes to zero, there exists a nite time

tf

after whi h

h(˜

x) ≤

d

2

for all times. Therefore,afterthistime,wehave

˙˜y = h(˜x)(yph(t) + ˜

y) − d˜

y

≤ h(˜

x)yph(t) −

d

2

y

˜

Wehaveseenthat

h(˜

x)yph(t)

goestozeroas

t

goestoinnity;sodoesalso

y

˜

.

Inordertohavetheglobalasymptoti stability,weareonlyleftwiththelo alasymptoti stability to prove. Inorder to do that, we onlyhave to onsider the dis retesystem that maps the state at time

nT

+

h

onto the state at time

(n + 1)T

+

h

with respe t to the linear equation(18)andthedis retepart. Aftersome omputations,weobtain:



˜

x

˜

y



(n + 1)T

_h

+

= B



˜

x

˜

y



nT

_h

+

(27) where

B =





(1 − αx)e

R

nTh

(n+1)Th

f

′

_(0)−g

′

_(0)y

ph

dτ

₀

‡

(1 − αy

)e

−d

R

nTh

(n+1)Th

dτ





Note that

‡

is a term that we do not use in our analysis, therefore is not expressed. Indeed,sin ethematrixistriangular,itisstable if

|B11| < 1

,i.e.

Z

(n+1)T

h

nT

h

yphdτ >

f

′

_(0)Th

_{+ ln(1 − αx)}

g

′

₍₀₎

(28)

Similarly to what was doneearlier, it anbeshownthat (28) is equivalent to (19), so thatthene essaryandsu ient onditionforlo al stabilityisproven.

It is dire tlyseen that (28) issatisedwhen (25)isbe ause

µ

h

(S, r)

is in reasingin

S

and

r

andwehave

f

′

(0)

g

′

₍₀₎

= lim

x

→0

≥

f

(x)

g(x)

≤ sup

x≥0

f

(x)

g(x)

and

g

′

_{(0) = lim}

x

→0

≥

g(x)

x

≤ sup

x≥0

g(x)

x

(29)

This ompletestheproofofglobalstability,sin ewehaveshownglobal onvergen eand lo alstabilitywhen(20)issatised.



Releasesless frequent than harvests

Ifwenow onsiderthe asewherepredatorsreleasestakepla elessoftenthanharvests,we alsoobtainglobalandlo al stabilityresultsbasedonthefollowingfun tion

µ

r

(S, r) = d



S

+

ln(1 − αx)

rTh

 1 − (1 − αy)e

−dT

h

1 − e

−dT

h

whi h isin reasingin

S

and

r

sin ethelast fra tionispositiveand

αx

≤ 1

. Theorem2 When

Tr

= kTh

with

k

∈ N

∗

, the solution

(x(t), y(t)) = (0, ypr(t))

of (1) is LASi

µ > µ

_r

 f

′

₍₀₎

g

′

₍₀₎

, g

′

₍₀₎



(30) andisGASif

µ > µ

_r



sup

x≥0

f

(x)

g(x)

,

sup

x≥0

g(x)

x



(31)

Proof: This proof does not depart very mu h from the one of Theorem 1. The only dieren eisthatthereferen eperiod isnow

Tr

. Weusethesamefun tion

G(˜

x)

asin(21) andananalysisidenti altotheoneoftheprevioustheoremleadsto

G(˜

x(mTr

+ (l + 1)T

_h

+

)) ≤

G(˜

x(mTr

+ lT

_h

+

))

+

R

mT

r

+(l+1)T

h

mT

r

+lT

h

h

_{f (˜}

_x(s))

g(˜

x(s))

− min(0, ˜

y0)e

−ds

_{− ypr(s)}

i

_ds

₊

ln(1−α

x

)

r

g

whi hisexa tly(24)sin eitisdepi tingthebehaviourofthemodelbetweentwoharvesting instants.

Extendingthisto thewhole

Tr

interval,weobtain

G(˜

x((m + 1)T

+

r

))

≤ G(˜

x(mT

r

+

)) +

R

(m+1)T

r

mT

r

h

_{f (˜}

_x(s))

g(˜

x(s))

− min(0, ˜

y0)e

−ds

− ypr(s)

i

ds

+k

ln(1−α

x

)

r

g

Wenowseethat thisexpressionisidenti al to(24)withtheex eptionof thepresen eof a

k

fa torandtheexpressionof

ypr(t)

,whi h omesfrom(9)insteadof(3). Condition(25)thenbe omes

Z

T

r

0

ypr(t)dt > SgTr

+ k

ln(1 − αx)

rg

(32)

andthe omputationof

R

T

r

0

ypr(t)dt

with

ypr(t)

asin(9)yields:

Z

T

r

0

ypr(t)dt =

µTr

d

1 − e

−dT

h

1 − (1 − αy

)e

−dT

h

(33) Thisleadsto onditon(31)(the omputationof(33)isdetailedinProposition4in the ap-pendix). Global onvergen eof

(˜

x,

y)

˜

to

(0, 0)

isthen on ludedbyusingthesameargument asin theproofofTheorem 1toshowthe onvergen eof

y

˜

to

0

.

The lo al stability ondition(30) then dire tly arisesfrom the analysis ofthe stability ofthedis retelinearized systemthat maps

y(mT

˜

+

r

)

onto

y((m + 1)T

˜

+

r

)

.



˜

x

˜

y



(m + 1)T

r

+

= B



˜

x

˜

y



mT

_r

+

(34)

where

B =

(1 − αx)

k

_e

R

(m+1)Tr

mTr

f

′

_(0)−g

′

_(0)y

pr

dτ

₀

e

R

mTr

(m+1)Tr

h

′

_(0)y

pr

dτ

_{(1 − αy)}

k

_e

−d

R

_mTr

(m+1)Tr

dτ

!

(35)

Again, sin ethesystemmatrix islowertriangular, forstability wesimply requirethat

|B11| < 1

and

|B22| < 1

. Thelatteryieldsatrivial ondition,sowe al ulate theformer:

Z

(m+1)T

r

mT

r

yprdτ >

f

′

_(0)Tr

_{+ k ln(1 − αx)}

g

′

₍₀₎

(36)

whi h dire tlyleadsto ondition(30)forlo alstability andtheproof ofglobal stabilityis also ompletebe ause

µ

r

isin reasingin

S

and

r

and(29)isstillsatised.



Comment

Aswehaveseen,when the ondition(20)or(31)issatised,theextin tionof thepestsis GAS.Whenthelo al ondition(19)or(30)isnotveried,theextin tionofthepestsisnot stableandabifur ationanalysissimilartowhatisdonein[10,12℄wouldshowthepresen e ofalimit y lewhen

µ

is losetothelimit. When

µ

satises ondition(19)or(30)only,the pests extin tionislo allystableandwe annotruleoutthatitisgloballystable(sin e our global onditionisonlysu ient). Su habudgethastheadvantageofbeingsmallerthan theonethatguaranteesglobalstability. Itallowsforgood ontroloflimitedpestinvasions; howeverthe ultureisatriskofbeingdestroyedbyalargepest outbreak.

Sin e, in both ases, the onditions for lo al and global stability are identi al up to twodierent parameters, any analysis of the onsequen es of oneof those onditions will immediatelytranslatetotheother. Theinterpretationof onditions(19)-(20)and(30)-(31) willbegivenin thenextse tion.

4 Interpretation of results Itiseasytoseethat

µ

r

isindependentof

Tr

. Theinuen eof

Tr

on

µ

h

istri kiertoidentify sowe shall analyse it mathemati ally rst. We then present graphi ally the variation of both

µ

r

and

µ

h

withrespe tto

Tr

foratypi alsetofparametervalues,andattenpttogive apra ti alinterpretationoftheseresults.

4.1 Mathemati al analysis Werstneedtonotethatwhen

S

+

ln(1−α

x

)

rT

h

<

0

,foranyofthelo alorglobal ondition,the onditionistriviallyveried. Indeed,itimpliessimplythat nobiologi al ontrolisneeded forexterminatingthepests;infa t,thepartialharvestingisee tiveenoughforthispurpose

(as

αx

islargeenough). Wenowevaluatehowthereleasefrequen yinuen estheminimal budgetwhenthis onditionisnottrivial.

We have already seen that

µ

r

is independent of

Tr

. We will now study the latter's inuen eon

µ

h

.

Theorem3 Let

T

h

= kTr

where

k

∈ N

∗

.

The minimalbudget ismonotoni allyde reasingwith respe ttothe releaseperiod

T

r

for non-negativevalues of

µh

,i.e.

∂µ

h

∂Tr

<

0

(37)

.

Proof: Knowing that

Tr

is equalto

T

h

k

, it is possibleto identify the sign of

∂µ

r

∂T

r

noting that

∂µ

h

∂Tr

=

∂µ

h

∂k

∂k

∂Tr

=

∂µ

h

∂k

 −k

2

Th



So sgn



_∂µ

h

∂Tr



= −

sgn



_∂µ

h

∂k



(38)

µ

_h

isexpressedastheprodu toftwodistin tiveparts,oneofwhi hisindependentof

k

and whi h, forthenon-trivialstability ondition,ispositive,

S

+

ln(1 − αx)d

rTh

>

0

where

S

and

r

are theparameters requiredfor thelo al and global onditions,as dened previously.

These ondpartisviewedasa ompositefun tion of

k

sothat(38) anbeevaluatedas

sgn



_∂µ

h

∂Tr



= −

sgn





∂

∂k





1

1 −



α

y

(1−e

−

dTh

)

1−(1−α

y

)e

−

dTh



σ(k)









(39)

where

σ(k) =



e

−

_dTh/k

k

(

1−e

−

_dTh/k

₎



. Then, weget sgn



_∂µ

h

∂Tr



=

−

sgn







α

y

(1−e

−

dTh

)

1−(1−α

y

)e

−

dTh



1 −



α

y

(1−e

−

dTh

)

1−(1−α

y

)e

−

dTh



σ(k)

2

∂σ

∂k







=

−

sgn

 ∂σ

∂k



=

−

sgn



e

−dT

h

/k

k

2

_{(1 − e}

−dT

h

/k

)

2

 dT

h

k

− 1 + e

−dT

h

/k



=

−

sgn

ke

−dT

h

/k

_{+ dTh}

_{− k}

(40) Sin e

∂

∂k

(ke

−dT

h

/k

_{+ dTh}

_{− k) =}



1 +

dTh

k



e

−dT

h

/k

_{− 1}

≤ 0

andusingl'Hospital'sRule

lim

k→∞



ke

−dT

h

/k

_{+ dTh}

_{− k}



= dTh

+ lim

k→∞

 e

−dT

h

/k

_{− 1}

1

k



= dTh

+ lim

k→∞

dT

h

k

2

e

−dT

h

/k

−

_k

1

2

!

= 0

wededu ethatsgn

ke

−dT

h

/k

_{+ dTh}

− k
> 0

Therefore, sgn



_∂µ

h

∂Tr



<

0

(41)



We an dedu e that we hit the smallest minimal value for the budget for the largest possible

Tr

in this ase that orresponds to when

k

= 1

. This happens when the release frequen yequalsthepartialharvestfrequen y.

4.2 Dis ussion

Figure3representstheanalyti alresultsobtainedin thepreviousse tionsfora hosenset of parameters. The plot in ludes the twostudied ases: either oneof thepartial harvest andthereleaseperiodisanintegermultipleoftheother.

0.35

0.4

0.45

0.5

µ

T

h

2T

h

3T

h

4T

h

5T

h

T

h

T

r

2

T

_h

4

T

h

8

Figure 3: Variation of the minimal numberof predatorsrequired perbudget year

µ

as a fun tion of releaseto harvest period ratio. Parameters are giventhe values (in arbitrary units):

αx

= αy

= 0.5

,

d

= 1

, andtherate of growth

f

′

₍₀₎

, fun tionalresponse

g

′

₍₀₎

and numeri alresponse

h

′

₍₀₎

withrespe ttotimewhenthee osystemispest-free,i.e.

xp(t) = 0

, areallequalto1.

Under this setof possible s enarios,in reasingthefrequen yof releasebeyond the fre-quen yof harvestrequires that thetotal numberof predatorsto invest in be higherthan that when releases take pla e less or as often as partial harvests. In the latter ase, to ensurepesteradi ation,thetotalbudgetofpredatorsto investin isxed, independentlyof thereleaseperiod.

Theseresultsimplythatitis learlyless ostlytoprote tagreenhouse ultureforlower frequen iesofrelease. Ofadditionale onomi interest,inthis ase,thebiologi altreatment is always ombined with partial harvesting, so that there is little or no extra ost linked to the presen e of workers on-site. However, we re all that [13℄ previously demonstrated that thehigher thereleasefrequen y, thesmaller theworst- ase damages. Combiningthe resultsfrombothstudiesseemstoindi atethat themostprotablereleasestrategyamong thepossibilitiesthat havebeen onsidered istheonewhere releasesare syn hronizedwith thepartialharvests.

5 Con lusion

Theresultsobtainedinthispaperforthestabilityofthesystemareyetanother onrmation that inundative ontrol an be an ee tive means of suppressing low pest invasions in a greenhouse. Thisrequiresthatasu ientnumberofpredatorsareintrodu edinthesystem asin,forinstan e,[7, 9℄.

Our studyaimed to providea ontrol strategyin the prote tionof ontinuously grown rops that are partially harvested on a regularbasis. Wedemonstrated that partial har-vestinghadanon-negligibleee tonbiologi al ontrolandneededtobetakenintoa ount whendevisinga ontrolstrategyin the aseofsu h rops.

We thus investigated the ombined ee ts of releasesand partial harvests in terms of the relative frequen iesof their implementation. We onsidered the asewhere these two eventso urredatperiodssu hthatonewastheintegermultipleoftheother,andwiththe twoevents oin idingover the longerperiod. In parti ular, we found when releases were asfrequentasorlessthanthepartial harvests,theminimal budgetdidnotdependonthe period of release but instead on the harvest parameters, the growth fun tion of the pest population,themortalityofthepredatorsandthefun tionalresponse.Whenreleaseswere morefrequentthanthepartial harvestshowever,theminimal budgetvaluein reasedwith thein reasingfrequen y ofthereleases,ex eedingthe onstantvalueobtainedfortheless frequent ase. Combined with the ndings of [13℄ whi h pointed out that higher release frequen iesled to theoptimal ontrol poli y, we on luded that forthe set of possibilities thatwasstudied,the urrentbeststrategyiswhenreleaseandharvestfrequen iesareequal. This approa hhas, however, its short omings. Sin ethe integermultiple fa toris key to al ulatingtheminimalbudgetwhi h wouldsatisfythestability onditions,itisnotyet generalisedtoothers enarioswhereneitherperiodistheintegermultipleoftheother. This wouldhappenforinstan eatotherrationalnon-integerratiosasirrationalones. Itishighly likelythat theseintermediateratios mightindu eother dynami sin the system. Whether theymightstabiliseit given evenlowerminimal budgetvaluesorfavour haosremains to beseen. Moreoverit would be interestingto extendtheresultsto the ase wherethe two ontrolsnever oin ideinspiteoffollowingaperiodi pattern. Thiswouldbeinthelineof thework,forinstan e, of[11℄, where pesti ide spraying-whi h isanalogous to harvests -andreleasesarenotsyn hronised.

Nevertheless, we onsider that our simpli ation already has its pra ti al e onomi al advantage. Indeed, oin idingperiods implylittleornoadditional ostsin urred interms of labour: the task of predator release an be assigned to workers in harge of partial harvesting. Field-testingisnowthenextsteprequiredto validatetheresultsofthispaper.

Appendix

Proposition 3 Let Hypothesis1hold, then

Z

T

h

0

yph(t)dt =

µTh

d



1 −

αy

1 − e

−dT

h

1 − (1 − αy) e

−dT

h

!





e

−

dTh

_k

k



1 − e

−

dTh

k











Proof: Inorder to omputetheintegral,wedes ribe

yph(t)

as

yph(iT

+

r

)e

−(t−iT

r

)

in ea h

timeinterval

[iTr,

(i + 1)Tr]

,with

yph(iT

+

r

)

givenby(6)when

y(nT

+

h

) = y

∗

. This yields:

R

T

h

0

yph(t)dt

=

k−1

X

i=0

yph(iT

+

r

)

Z

(i+1)T

r

iT

r

e

−d(t−iT

r

)

_dt

=

k−1

X

i=0



y

∗

e

−idT

r

_{+ µTr}

i−1

X

j=0

e

−jdT

r





Z

T

r

0

e

−dt

dt

=



y

∗ 1−e

_1−e

−

−dTr

kdTr

+ µTr

Pk−1

i=0

1−e

−

idTr

1−e

−dTr



1−e

−

dTr

d

=

y

_d

∗

(1 − e

−kdT

r

_{) +}

µT

r

d



k

−

1−e

−kdTr

1−e

−

dTr



=



1−e−dTh

1−e−dTr



(1−α

y

)+α

y

1−(1−α

y

)e

−

dTh

µT

r

d

(1 − e

−kdT

r

) +

µT

r

d



k

−

1−e

_1−e

−

−

kdTr

dTr



= µ

T

h

dk



(

1−e

−

_dTh

₎

_(1−α

y

)+α

y



1−e

−

dTh

_k

 

₍

_1−e

−

_dTh

₎



1−e

−

dTh

_k



₍

_1−(1−α

y

)e

−

dTh

)

+µ

T

h

dk



k



1−e

−

dTh

_k



₋

₍

_1−e

−

_dTh

₎



₍

_1−(1−α

y

)e

−

dTh

)



1−e

−

dTh

_k



₍

_1−(1−α

y

)e

−

dTh

)

= µ

T

h

dk



(

1−(1−α

y

)e

−

dTh

)

−α

y

e

−

dTh

k



(

1−e

−

dTh

₎



1−e

−

dTh

k



(

1−(1−α

y

)e

−

dTh

)

+µ

T

h

dk



k



1−e

−

dTh

k



−

₍

1−e

−

_dTh

₎



₍

_1−(1−α

y

)e

−

dTh

)



1−e

−

dTh

k



(

1−(1−α

y

)e

−

dTh

)

= µ

T

h

dk



−α

y

e

−

dTh

k



(

1−e

−

_dTh

₎

₊



_k



_1−e

−

dTh

_k

 

₍

_1−(1−α

y

)e

−

dTh

)



1−e

−

dTh

_k



₍

_1−(1−α

y

)e

−

dTh

)

=

µT

h

d



1 −



α

y

(

1−e

−

dTh

)

1−(1−α

y

)e

−

dTh







e

−

dTh

_k

k



1−e

−

dTh

k













Proposition 4 Let Hypothesis1hold, then

Z

T

r

0

ypr(t)dt =

µTr

d

1 − e

−dT

h

1 − (1 − αy

)e

−dT

h

Proof: Inorder to omputetheintegral,wedes ribe

ypr(t)

as

ypr(iT

+

h

)e

−(t−iT

h

)

in ea h

timeinterval

[iTh,

(i + 1)Th]

,with

ypr(iT

+

h

)

givenby(12)when

y(mT

+

r

) = y

∗

. Thisyields:

R

T

r

0

ypr(t)dt

=

k−1

X

i=0

ypr(iT

_h

+

)

Z

(i+1)T

h

iT

h

e

−d(t−iT

h

)

_dt

=

k−1

X

i=0

y

∗

e

idT

h

_{(1 − αy)}

i

Z

T

h

0

e

−dt

dt

=

y

∗ (1−e

−

_d

dTh

)

k−1

X

i=0

y

∗

e

idT

h

_{(1 − αy)}

i

=

µT

r

1−(1−α

y

)

k

e

−dTr

(1−e

−

dTh

₎

d

1−(1−α

y

)

k

e

−

kdTh

1−(1−α

y

)e

−dT h

=

µT

r

(1−e

−

dTh

)

d(1−(1−α

y

)e

−dT h

)



Contents

1 Introdu tion 3

2 Modeldes ription 4

3 Mathemati al analysis 6

3.1 Pest-freestabilityanalysis . . . 6 3.2 Globalstabilityanalysis . . . 11

4 Interpretationof results 18

4.1 Mathemati alanalysis . . . 18 4.2 Dis ussion . . . 20

5 Con lusion 22

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