Threshold conditions for integrated pest management models with pesticides that have residual effects

J. Math. Biol. (2013) 66:1–35DOI 10.1007/s00285-011-0501-x Mathematical Biology

Threshold conditions for integrated pest managementmodels with pesticides that have residual effects

Sanyi Tang · Juhua Liang · Yuanshun Tan ·Robert A. Cheke

Received: 15 December 2010 / Revised: 3 December 2011 / Published online: 29 December 2011© Springer-Verlag 2011

Abstract Impulsive differential equations (hybrid dynamical systems) can providea natural description of pulse-like actions such as when a pesticide kills a pest instantly.However, pesticides may have long-term residual effects, with some remaining activeagainst pests for several weeks, months or years. Therefore, a more realistic methodfor modelling chemical control in such cases is to use continuous or piecewise-con-tinuous periodic functions which affect growth rates. How to evaluate the effects ofthe duration of the pesticide residual effectiveness on successful pest control is key tothe implementation of integrated pest management (IPM) in practice. To address thesequestions in detail, we have modelled IPM including residual effects of pesticides interms of fixed pulse-type actions. The stability threshold conditions for pest eradica-tion are given. Moreover, effects of the killing efficiency rate and the decay rate ofthe pesticide on the pest and on its natural enemies, the duration of residual effective-ness, the number of pesticide applications and the number of natural enemy releaseson the threshold conditions are investigated with regard to the extent of depression

S. Tang is supported by the National Natural Science Foundation of China NSFC 10871122, 11171199and by the Fundamental Research Funds for the Central Universities GK201003001. R.A. Cheke isgrateful to the University of Greenwich for support under project E0171.

S. Tang (B) · J. LiangCollege of Mathematics and Information Science, Shaanxi Normal University,Xi’an 710062, People’s Republic of Chinae-mail: [email protected]; [email protected]

Y. TanDepartment of Mathematics and Physics, Chongqing Jiaotong University,Chongqing 400074, People’s Republic of China

R. A. ChekeNatural Resources Institute, University of Greenwich at Medway, Central Avenue,Chatham Maritime, Kent ME4 4TB, UK

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or resurgence resulting from pulses of pesticide applications and predator releases.Latin Hypercube Sampling/Partial Rank Correlation uncertainty and sensitivity anal-ysis techniques are employed to investigate the key control parameters which are mostsignificantly related to threshold values. The findings combined with Volterra’s princi-ple confirm that when the pesticide has a strong effect on the natural enemies, repeateduse of the same pesticide can result in target pest resurgence. The results also indicatethat there exists an optimal number of pesticide applications which can suppress thepest most effectively, and this may help in the design of an optimal control strategy.

Keywords Residual effects of pesticides · Pest control · IPM · Volterra’s principle ·Pest-natural enemy system

Mathematics Subject Classification (2000) 92D05 · 92D25 · 92D40

0 Introduction

In order to reduce harm caused by important pests of plants, animals and humans, theaim of integrated pest management (IPM) is to use a combination of low cost biolog-ical, cultural and chemical tactics that reduce pests to tolerable levels, with minimaleffects on the environment. IPM is a long-term control strategy (Tang and Cheke 2008;Van Lenteren 1995; van Lenteren 2000; Van Lenteren and Woets 1988).

Biological control, defined as the reduction of pest populations by natural ene-mies, is often a component of an IPM strategy (Parker 1971). Typically, it involvesan active human role such as increasing the number of natural enemies at criticaltimes, known as augmentation, usually through mass releases in a field or greenhouse(Neuenschwander and Herren 1988; Udayagiri et al. 2000).

However, repeated applications of pesticides may have unexpected consequencesbecause of Volterra’s principle. This was formulated when resolving anomalies infishery systems (Volterra 1926) and can be summarised by the statement that an inter-vention in a predator-prey system that removes predator and prey in proportion to theirpopulation sizes increases the prey population. Thus, when applied to pest systemswhen pesticides act not only on the pest species but also on their natural enemies, pestpopulations may be increased (often called resurgence) (Barclay 1982; Debach 1974;Roughgarden 1979; Ruberson et al. 1998). The classic example of this phenomenonconcerns the increase in cottony cushion scale insects Icerya purchasi, previouslyheld in check in citrus plantations in California by the introduction of the vedaliabeetle Rodolia cardinalis. After DDT was applied with the aim of further control, theresult was an increase in the scale insect population. More recent examples includethe rise of the Brown Planthopper Nilaparvata lugens as a rice pest in Asia (Heinrichsand Mochida 1985) and increases of Green Peach Aphid Myzus persicae followingsprays with pyrethrin against Colorado Beetle Leptinotarsa decemlineata leading toreductions in non-target organisms (Reed et al. 2001).

Recently, many models concerning IPM, the optimal timing of pesticide applica-tions and the timing of natural enemy releases have been proposed (Tang et al. 2008;Tang and Cheke 2008; Tang et al. 2009, 2010). For example, Tang et al. (2010) devel-

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Integrated pest management models with residual effects 3

oped a hybrid impulsive pest-natural enemy model in which pulsing actions such asspraying pesticides and releasing natural enemies were considered. Three cases wereinvestigated: (a) spraying pesticides more frequently than releasing natural enemies;(b) spraying pesticides less frequently than releasing natural enemies; and (c) releas-ing natural enemies frequently and spraying pesticides only when pest densities reachtheir economic threshold (ET). For the first two cases, the stability threshold con-ditions for a pest eradication periodic solution were provided. Moreover, the effectsof times of spraying pesticides (or releasing natural enemies) and control tactics onthe threshold conditions were investigated with regard to the extent of depression orresurgence resulting from pulses of pesticide applications.

If chemical pesticides have a short residual effect on their target, repeated applica-tion is often required to suppress a pest, which can cause undesirable changes suchas pesticide resistance. Meanwhile, biological pesticides are generally more envi-ronmentally friendly, but often lack residual effects and can be strongly influencedby environmental factors. A longer, effective residual control can reduce pesticideapplication frequencies if pesticides with long-lasting residual effects such as orga-nochlorine compounds are used (Residual effects of pesticides 2010). For example,some insecticides used against the bed bug Cimex lectularius can have residual effects1 week to 4 months after application, micro-encapsulated formulations of the pyre-throid lambda-cyhalothrin can be effective against vectors of malaria Anopheles spp.9 months after indoor sprays on walls and some compounds are active against termitesfor years (Raupp et al. 2001; Schmutterer 1988; Zacharda and Hluchy 1991), but long-residual pesticides may also reduce the effectiveness of natural enemies (Zacharda andHluchy 1991).

Therefore, it is important to know how such antagonistic pesticides affect the out-comes of pest control measures, and how the short-term or long-term residual effectsof pesticides on both pests and natural enemies affect the success or failure of pestcontrol. One approach to understanding the range of possible ecological interactionsamong pests, natural enemies and pesticides is to develop population models (Barclay1982; Barlow et al. 1996; Tang and Cheke 2005; Tang et al. 2008; Tang and Cheke2008; Tang et al. 2010). To address these questions, we propose models dealing indetail with the killing efficiency rate and decay rate of pesticides, i.e. the residualeffects of pesticides on pests or natural enemies or both. A piecewise or step functionwas employed to describe the residual effects of pesticides (i.e. pulsing actions), andthe threshold conditions for the stability of pest-free periodic solutions were obtained.The relations between the threshold values and the number (timing) of pesticide appli-cations or the number (timing) of releases of natural enemies were addressed in partic-ular. Our results indicate that there exists an optimal number of pesticide applicationswhich maximizes the threshold values, and consequently is more effective at control-ling the pest. The residual effects of pesticides (i.e. killing efficiency rate and decayrate) on the threshold conditions were also addressed. We explored the parameterspace by performing an uncertainty analysis and sensitivity analysis using the Latinhypercube sampling (LHS) method, and evaluating partial rank correlation coeffi-cients (PRCCs) (Blower and Dowlatabadi 1994; Marino et al. 2008) for various inputparameters against threshold conditions, and then the most significant parameters weredetermined.

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The modelling methods and results provide an optimal control rule which can beused to determine (a) the optimal spray dosages of pesticides; (b) optimal spray times;(c) the residual effects of pesticides on the success of pest control; (d) the best timingand effectiveness of pesticide applications and (e) a good residual pest control spraystrategy.

1 Pest growth model with pulses of chemical control

The logistic growth model has been widely used in many fields as a basic model ofpest growth. Chemical control is one of the components of an IPM strategy, and it isvery useful because it can kill a significant proportion of a pest population quicklyand there is a variety of ways to model the effects of chemical control with the logisticmodel. One of the simplest ways is to assume that the pesticide kills the pest instantly,thus giving a pulsing type action, and recently this type of model has been investigatedintensively (Liu and Chen 2004a,b; Tang and Chen 2003; Tang and Cheke 2005; Tanget al. 2008; Tang and Cheke 2008; Tang et al. 2010).

It is interesting to note that the residual effects of pesticides play key roles insuccessful pest control. Therefore, a realistic and appropriate method for studyingchemical control is to assume that the pesticide effects are modelled by continuous orpiecewise-continuous periodic functions which affect the growth rate in the logisticmodel (Panetta 1995). Such periodic functions make the growth rate fluctuate suchthat it decreases significantly, or even takes on negative values, when the pesticideis present, but increases again during the recovery stage. Thus it follows from themodelling ideas proposed by Bor Jeffrey (1995) and Panetta (1995) that we have thefollowing model

d P(t)

dt= r P(t)[1 − b(t) − ηP(t)], (1)

where the killing efficiency rate b(t) is a periodic function with period T, r representsthe intrinsic growth rate and η is the carrying capacity parameter.

Denote 〈b(t)〉 = 1T

∫ T0 b(t) dt . It follows from the results proposed by

Bor Jeffrey (1995) and Panetta (1995) that the zero solution of model (1) is glob-ally stable if 〈b(t)〉 ≥ 1, and there exists a globally stable T -periodic solution if0 ≤ 〈b(t)〉 < 1. These results indicate that the pest dies out if the mean killing effi-ciency rate 〈b(t)〉 ≥ 1, and it oscillates periodically if the mean killing efficiency rate0 ≤ 〈b(t)〉 < 1. For further discussion of the threshold condition see Panetta (1995).

The killing efficiency rate b(t) can be defined as an exponential distribution func-tion or exponential function of pesticide dosage because high pest mortality requiresa large dosage of pesticide (Bor Jeffrey 1995), i.e.

b(t) = 1 − exp(−k D) or b(t) = exp(k D),

where k is a positive constant and D is the dosage of the pesticide.As noted in the introduction, most pesticides have a residual effect, and this effect

decays with time (t) because of the degradation of the pesticide and natural clearance

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(Schaalje 1990). In general, the killing efficiency rate b(t) may be modified by incor-porating the decay effect of the pesticides as a negative exponential function suchas

b(t) = me−δt

where m = 1 − exp(−k D) (or exp(k D)), δ is the positive decay parameter (termedthe decay rate). Therefore, if we repeatedly apply pesticides at time point nT (where nis a positive integer and T is the period of pesticide application), then the killing effi-ciency rate b(t) with residual effects can be described by the following exponentiallydecaying piecewise periodic function

b(t) = me−δ(t−nT ), nT ≤ t < (n + 1)T, (2)

and then we have

〈b(t)〉 = 1

T

T∫

0

b(t) dt = m

δT[1 − e−δT ]. (3)

Remark 2.1 The killing efficiency rate b(t) can take on various periodic (period T )forms including the step function of the form

b(t) ={

m, nT ≤ t < T1 + nT,

0, T1 + nT ≤ t < (n + 1)T,(4)

where T1 is the duration of pesticide residues. The forms of the two different typesof killing efficiency rates are shown in Fig. 1. Note that taking the limit of the b(t)function defined by (2) as δ → 0 yields the step function (4), and 〈b(t)〉 = T1m

T . Notethat all results obtained in the following related to b(t) function defined by (4) can beobtained similarly for b(t) function defined by (2). For simplicity, we will focus onthe exponentially decaying piecewise periodic function b(t) defined by (2) in the restof this paper.

2 The IPM model with residual effect of pesticides

The comprehension of interactions amongst pests, natural enemies and pesticides canbe enhanced with mathematical modelling (Barclay 1982; Barlow et al. 1996). Mod-elling advances regarding IPM strategies include analyses of continuous and discretepredator-prey models (Barclay 1982; Tang and Chen 2002; Tang et al. 2005, 2008,2009). In addition, discrete host-parasitoid models have been used to examine fourcases involving the timing of pesticide applications when these also led to the deathof the parasitoids (Beddington et al. 1978; Waage 1982; Waage et al. 1985).

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0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time

Kill

ing

effic

ient

rat

e w

ith r

esid

ual e

ffect

s

Fig. 1 Plots of killing efficiency rate with residual effects for the step (dashed line) and piecewise functions(solid line)

In the present work, we first take the simplest case where in each impulsive period Tthere is a pesticide application, so the killing efficiency rate function can be formulatedby (2) or (4), and further in each impulsive period T there is an introduction constant Rfor the natural enemies which does not depend on the sizes of the populations. Theseassumptions result in the following pest-natural enemy model

⎧⎪⎪⎨

⎪⎪⎩

d P(t)dt = r P(t)[1 − b(t) − ηP(t)] − β P(t)N (t),

d N (t)dt = λβ P(t)N (t) − d N (t), t �= nT,

N (nT +) = N (nT ) + R, t = nT,

(5)

where N (t) is the population size of the natural enemy at time t and the functionb(t) is defined by (2) or (4), β denotes the attack rate of the predator, λ representsconversion efficiency and d is the predator mortality rate. The same model (5) butwithout any residual effects of the pesticides on the pest (i.e. only an instantaneouskilling efficiency was considered) has been extensively investigated (Tang and Chen2002, 2003; Tang et al. 2008, 2009, 2010). In the following section, we will focus onthe existence of the pest-free periodic solution and its stability threshold conditions.

2.1 Pest free periodic solution

One of the main purposes of investigating system (5) is to determine how to implementan IPM strategy such that the pest population goes extinct eventually. To do this, wefirst need to consider the following subsystem.

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⎧⎪⎨

⎪⎩

d N (t)dt = −d N (t), t �= nT,

N (t+) = N (t) + R, t = nT,

N (0+) = N (0).

(6)

Lemma 2.1 The sub-system (6) has a positive periodic solution N∗(t) and for everysolution N (t) of (6) we have |N (t) − N∗(t)| → 0 as t → ∞, where N∗(t) =R exp(−d(t−nT ))

1−exp(−dT ), t ∈ (nT, (n + 1)T ], n ∈ N , N∗(0+) = R

1−exp(−dT )and

N = {0, 1, 2, . . .}.Proof The solution of (6) at any impulsive interval (nT, (n + 1)T ] is given by

N (t) = N (nT +)e−d(t−nT ), nT < t ≤ (n + 1)T .

At time point (n + 1)T , a constant number R of natural enemies is released and

N ((n + 1)T +) = N (nT +)e−dT + R.

So by induction we get

N (t) = N (0)e−dt + R(e−d(t−nT ) − e−dt )

1 − e−dT, nT < t ≤ (n + 1)T . (7)

Denote Xn = N (nT +), then we get the following difference equation of impulsivepoint series

Xn+1 = Xn exp(−dT ) + R,

which has a unique positive fixed point X∗ = R1−exp(−dT )

. Thus, if we let N∗(0+) =X∗ = R

1−exp(−dT ), then there is a periodic solution of (6), denoted by N∗(t), and

N∗(t) = R exp(−d(t − nT ))

1 − exp(−dT ), nT < t ≤ (n + 1)T . (8)

For the stability of N∗(t), it follows from (7) and the formula of N∗(t) that

|N (t) − N∗(t)| =∣∣∣∣N (0)e−dt − Re−dt

1 − e−dT

∣∣∣∣ → 0 as t → ∞,

and then the results of Lemma 2.1 follow. This completes the proof of Lemma 2.1.

Based on the results of Lemma 2.1, there exists a ’pest-free’ periodic solution ofsystem (5) over the n-th time interval t0 = nT < t ≤ (n + 1)T,

(0, N∗(t)) =(

0,R exp(−d(t − nT ))

1 − exp(−dT )

)

, (9)

and we have the following threshold dynamics.

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Theorem 2.1 Let

R0 = 1

T

T∫

0

[

b(s) + β

rN∗(s)

]

ds, (10)

then (i) the pest-free periodic solution (9) of system (5) is globally asymptoticallystable if R0 > 1; (ii) the system (5) is permanent if R0 < 1.

The proof of Theorem 2.1 is given in Appendix A. Therefore, for the piecewiseperiodic function b(t) given by (2), we have

〈b(t)〉 = 1

T

(n+1)T∫

nT

[

b(s) + β

rN∗(s)

]

ds

= 1

T

(n+1)T∫

nT

[

me−δ(s−nT ) + β

r

R exp(−d(s − nT ))

1 − exp(−dT )

]

ds

= m

δT

[1 − e−δT

]+ β R

rdT.

If follows from the Theorem 2.1 that the threshold condition (denoted by R PW0 ) which

guarantees the globally asymptotically stability of pest-free periodic solution (9) ofsystem (5) is as follows

R PW0

.= m

δT[1 − e−δT ] + β R

rdT> 1. (11)

In particular, let R = 0, then the pest-free periodic solution (0, 0) is globally stableif m

δT [1− e−δT ] > 1, which means that if the mean killing efficiency rate with a decayrate due to a pesticide application over period T is larger than unity, then the pestspecies eventually goes to extinction.

Similarly, if the decay rate δ tends to infinity (biological control only), then thecondition which guarantees the global stability of the pest eradication periodic solu-tion becomes r <

β RdT , which means that if the intrinsic growth rate of the pest is less

than the mean predation rate over period T, then the pest species will become extincteventually. However, for an IPM strategy (here both chemical and biological controltogether) inequality (11) indicates that an integrated control strategy is more effectivethan any single control strategy.

Furthermore, bifurcation analyses shown in Fig. 2 indicate that the model (5) withstep or piecewise b(t) function may exhibit very complex dynamical behaviour suchas period doubling bifurcations and multiple attractors co-existing for a wide rangeof parameters, e.g. the two attractors with quite different pest amplitudes that cancoexist for two different killing efficiency rate functions (Fig. 2c, d). These results

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0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

δ

P(t

)

(b)

14 15 16 17 180

2

4

6

8

T(T1=T/2)

P(t

)(a)

0 2 4 60

1

2

3

4

5

P(t)

N(t

)

(c)

0 1 2 3 4 50

2

4

6

8(d)

P(t)

N(t

)T=15 δ=0.2

Fig. 2 Complex dynamic behaviour of the model (5) for step and piecewise b(t) functions. a Bifurcationdiagram for step function with bifurcation parameter T . The other parameters are as follows: m = 1.2, T =3, r = 1, η = 0.02; b bifurcation diagram for piecewise function with bifurcation parameter δ. The otherparameters are as follows: m = 1.2, T = 3, r = 0.5, η = 0.02. c Two coexisting attractors at T = 15.d Two coexisting attractors at δ = 0.2

show that the initial densities of pest and natural enemy populations can affect the out-come of classical biological control (Jones et al. 1999) and, also, that small randomperturbations due to environmental noise, such as pesticide applied at different dos-ages and the numbers of natural enemies released, will result in switch-like transi-tions between the two attractors [not shown here, for more details see Tang et al.(2010)].

A typical feature of bifurcation diagrams is the occurrence of sudden changesin the types of the attractors, and these sudden changes are usually related to peri-odic windows in the middle of the chaotic range of attractors (see Fig. 2a, b). Froman ecological point of view, we see from Fig. 2a, b that the period of pesticideapplications and residual effects of pesticides may destabilize stable dynamics intomore complex dynamics with increasing pesticide application period T or increasingdecay rate δ.

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3 Different frequencies of spraying insecticide and releasing natural enemy

An understanding of the residual effects of pesticides on natural enemies is crucial forIPM, given that pesticides tend to be harmful to most natural enemies (Ruberson et al.1998). Pesticides may often be more toxic to natural enemies than they are to theirtarget since predators and parasitoids must search for their prey, spending time andenergy in travelling over the surface of plants, thereby increasing the probability thatthey will contact the insecticide. Such behaviour may contribute to pest resurgencesof the sort mentioned in the Introduction (Debach 1974).

In order to avoid the effects of pesticides on natural enemies, chemical controland biological control can be applied at different times. However, all models [suchas model (5)] developed before assumed that all control tactics are applied simulta-neously (Tang et al. 2010). In the present work, we assume that at each impulsive pointτn there is a pesticide application that not only kills the pests, but also can kill naturalenemies. In addition, at each impulsive point λm a constant number R of the natu-ral enemies is introduced, with R being independent of the sizes of the populations.It is thus possible to rank the different patterns of insecticide applications in termsof their dynamic effects in relation to the timing of natural enemy releases. Thesemodifications result in the following model:

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

d P(t)dt = r P(t)[(1 − b1(t)) − ηP(t)] − β P(t)N (t),

d N (t)dt = λβ P(t)N (t) − [d + b2(t)]N (t), t �= λm,

N (λ+m) = N (λm) + R, t = λm,

(12)

where bi (i = 1, 2) represent the residual effects of pesticides on the pest and naturalenemies, respectively, which will be defined in more detail below.

From a practical point of view and following the ideas proposed by Tang et al.(2010), we consider two different cases in terms of the timing of IPM applications, asfollows.

Case 1 Pesticide applications more frequent than releases of natural enemies.Assume λm+1 − λm ≡ TN for all m(m ∈ Z), where Z = {1, 2, . . .} and TN is the

period of releasing natural enemies. For this case the model (12) is said to be a TN

periodic system if there exists a positive integer kp such that

τn+kp = τn + TN

holds true for all n ∈ Z . This implies that in each period TN , kp pesticide applicationsare made.

Based on the discussions in the previous sections, we now know that the thresholdcondition for the step killing efficiency rate function is a special case of the piecewisekilling efficiency rate function. Therefore, from now on we will only consider thepiecewise function and provide the threshold condition for the stability of the pest-free periodic solution. Meanwhile, due to the periodicity of the killing efficiency ratefunction with period TN , we need to consider the killing efficiency rate in any giveninterval [hTN , (h + 1)TN ), which can be described as follows

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b j (t) =

⎧⎪⎨

⎪⎩

m j e−δ j (t−(h−1)TN −τk p )

, t ∈ [hTN , hTN + τ1),

m j e−δ j (t−hTN −τi ), t ∈ [hTN + τi , hTN + τi+1),

m j e−δ j (t−hTN −τk p )

, t ∈ [hTN + τkp , (h + 1)TN )

(13)

for j = 1, 2 and i = 1, 2, . . . , kp − 1.Case 2 Natural enemy releases more frequent than pesticide applications.

Assume τn+1 − τn ≡ TP for all n(n ∈ Z), where TP is the period of pesticideapplications. For this case the model (12) is said to be a TP periodic system if thereexists a positive integer kn such that

λm+kn = λm + TP

holds true for all m ∈ Z . This implies that in each period TP , kn natural enemy releasesare applied. Thus, the killing efficiency rate for this special case can be modified asfollows

b j (t) = m j e−δ j (t−hTP ), hTP ≤ t < (h + 1)TP , j = 1, 2. (14)

The main purposes of the following are to focus on Cases 1 and 2, and investigatethe effects of the timing of the application of IPM tactics and the duration of theresidual effects of pesticides on pest control.

3.1 Detailed dynamical analysis of Case 1

Note that if the conditions provided in Case 1 hold true, then system (12) is a TN -period model. In this subsection we will focus on Case 1 and provide a thresholdcondition which guarantees the stability of the pest-free periodic solution, and carryout a sensitivity analysis of varying parameter values on the threshold condition byemploying PRCCs (Blower and Dowlatabadi 1994; Marino et al. 2008) and LatinHypercube Sampling (LHS) methods (Mckay et al. 1979). Furthermore, the relatedbiological implications are addressed accordingly.

3.1.1 Threshold condition for the stability of the pest-free periodic solution

It follows from Appendix B that for Case 1 the following subsystem

{d N (t)

dt = −[d + b2(t)]N (t), t �= hTN ,

N (hT +N ) = N (hTN ) + R, t = hTN

(15)

has a globally stable periodic solution which is given by

N TN (t) = Y ∗ exp

⎡

⎢⎣

t∫

hTN

−(d + b2(s)) ds

⎤

⎥⎦ , t ∈ (hTN , (h + 1)TN ], (16)

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where b2(t) is given by (2) and Y ∗ is given by (B.2).Appendix B also shows that there is a pest-free periodic solution, (0, N TN (t)), of

system (12) over the h-th time interval hTN < t ≤ (h + 1)TN , which is also globallyattractive if the threshold value RTN

0 > 1, where

RTN0 = m1

δ1TN

kp+1∑

i=2

(1 − e−δ1(τi −τi−1)) + βY ∗

rTN

TN∫

0

B(s) ds

.= Term1 + Term2. (17)

The detailed calculation of the second integration of the above threshold value isprovided in Appendix B.

3.1.2 The effect of key parameters on the threshold value

What we consider in the following is how the residual effects of pesticides on the pestsand natural enemies, release constant R, timing of pesticide application τi (or kP ) andtiming of release period TN affect the success of pest control, i.e. the threshold valueRTN

0 . To address this question, firstly we fixed the release period TN , release constantR, the parameters r, β, λ, d, and varied the key parameters which describe the resid-ual effects of pesticides on the pest and natural enemies, i.e. mi and δi (i = 1, 2), andchose different numbers of pesticide applications over the period TN .

In Fig. 3, we see that the threshold value RTN0 is not monotonic with respect to the

number of pesticide applications kp, and the effects of all four parameters (mi and

δi , i = 1, 2) on the threshold value RTN0 are complex. Figure 3a shows the effects of

the killing efficiency rate with respect to the pest (i.e. m1) and the number of pesticideapplications (kp) on the RTN

0 , and the results indicate that the smaller the killing effi-ciency rate m1, the smaller the threshold value which follows, which will result in amore severe pest outbreak. The maximal threshold value RTN

0 depends on the killingefficiency rate and the number of pesticide applications. So in this case the optimalcontrol strategy is to apply pesticides two or three times over the period TN . In contrast,we found an opposite impact of the killing efficiency rate with respect to the naturalenemies (i.e. m2) and the number of pesticide applications (kp) on the RTN

0 as shownin Fig. 3b, i.e. the larger the killing efficiency rate m2, the smaller the threshold value.This shows that if the pesticide has a strong effect on the natural enemies, repeateduse of the same pesticide can result in target pest resurgence (Tang et al. 2010).

Similar effects of the decay rate of the pesticides with respect to the pest and naturalenemy species and the number of pesticide applications on the threshold values RTN

0are shown in Fig. 3c, d. All these simulations show that for a given releasing period,the number of applications of pesticides within this period and the killing efficiencyrate on both species and the decay rates are all crucial to pest control, and clarifythat (a) longer residual effectiveness on the insect pests and shorter adverse effects onthe natural enemies are beneficial for successful pest control (here the smaller δ1 andlarger δ2); (b) when the pesticide has a strong effect on the natural enemies we must

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1 2 3 4 5 6 7 8 9 100.85

0.9

0.95

1

1.05

1.1

kp

R0T

N

(a)

m1=0.64

m1=0.74

m1=0.84

1 2 3 4 5 6 7 8 9 10

0.94

0.96

0.98

1

1.02

1.04

1.06

kp

R0T

N

(b)

m2=0.4

m2=0.5

m2=0.6

1 2 3 4 5 6 7 8 9 100.9

0.95

1

1.05

1.1

1.15

1.2

1.25

kp

R0T

N

(c)δ

1=0.1

δ1=0.2

δ1=0.3

1 2 3 4 5 6 7 8 9 100.92

0.94

0.96

0.98

1

1.02

1.04

kp

R0T

N

(d)

δ2=0.55

δ2=0.65

δ2=0.75

Fig. 3 The effects of the number of pesticide applications, killing efficiency rates (m1, m2) and decay

rates (δ1, δ2) of pesticides on the threshold level RTN0 . Here the impulsive points τ1, τ2, . . . , τk p divide the

period TN into k p+1 equal small intervals and λi+1 −λi = TN for all i ∈ Z . The baseline parameter valuesare as follows: m1 = 0.74, m2 = 0.5, δ1 = 0.2, δ2 = 0.65, r = 0.5, β = 0.44, λ = 1, d = 0.15, R =3, TN = 20. a The effect of the killing efficiency rate m1 on R

TN0 ; b the effect of the killing efficiency rate

m2 on RTN0 ; c the effect of the decay rate δ1 on R

TN0 ; d the effect of the decay rate δ2 on R

TN0

carefully select the number of pesticide applications (two or three events in this case).The effects of all other parameters such as releasing period TN will be addressed later.

3.1.3 Sensitivity analysis of varying parameter values on the threshold condition

Sensitivity analysis was performed by evaluating the PRCCs (Blower and Dowlatabadi1994; Marino et al. 2008) for various input parameters against the threshold conditionRTN

0 , and then the most significant parameters (such as killing efficiency rates, decayrates, natural enemy releasing rate and its frequency) were determined. To do this, weemployed the LHS method, a type of stratified Monte Carlo sampling first proposedby Mckay et al. (1979) and later applied to deterministic mathematical models, inparticular by Blower et al. (1994). PRCC measures the influence of uncertainty inestimating the values of the input parameter on the imprecision in predicting the valueof the output variable (Blower and Dowlatabadi 1994; Marino et al. 2008; Mckay

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14 S. Tang et al.

−1

−0.5

0

0.5

1

m1 m

2δ

1δ

2 r β d R T

N

PR

CC

s(a)

−500 0 500

−500

0

500

(b) [0.59245 , 0]

m1

R0T

N

−500 0 500

−500

0

500

(c) [−0.1193 , 0.00015593]

m2

R0T

N

−500 0 500

−500

0

500

(d) [−0.61232 , 0]

δ1

R0T

N

−500 0 500

−500

0

500

(e) [0.17969 , 1.0547e−008]

δ2

R0T

N

−500 0 500

−500

0

500

(f) [0.58844 , 0]

R

R0T

N

−500 0 500

−500

0

500

(g) [−0.67376 , 0]

TN

R0T

N

Fig. 4 PRCC results and PRCC scatter plots of the pest integrated control parameters including killingefficiency rate mi , decay rate δi (i = 1, 2), releasing constant R and period TN . The baseline parametervalues are as follows: m1 = 0.74, m2 = 0.5, δ1 = 0.2, δ2 = 0.65, r = 0.5, β = 0.44, d = 0.15, R =3, TN = 20, k p = 3. Here the impulsive points τ1, τ2, τ3 divide the period TN into 4 equal small intervalsand λi+1 − λi = TN for all i ∈ Z . a PRCC results with sample size 1,000, all parameters were variedsimultaneously. b–g PRCC scatter plots with sampling size 1,000 and significant p < 0.01. The title ofeach plot represents the PRCC value with the corresponding p-value

et al. 1979). We performed uncertainty and sensitivity analyses for all parametersin model (12) using LHS with 1,000 samples. A uniform distribution function wasused and tested for significant PRCCs for all parameters with wide ranges, such asm1 ∼ U (0.1, 2), δ1 ∼ U (0.1, 2), T ∼ U (5, 30), and the baseline values of all param-eters are given in the figure legend of Fig. 4. Unfortunately, we do not have realparameter values and we chose those parameter values just for illustrative purposes.

Figure 4a shows the PRCC results which illustrate the dependence of RTN0 on each

parameter, and PRCC scatter plots of residual effects of pesticides on both species(i.e. m1, m2, δ1, δ2) and the releasing constant R and period TN are given in Fig. 4b–g,respectively. We considered absolute values of PRCC > 0.4 as indicating an impor-

123


tant correlation between input parameters and output variables, values between 0.2and 0.4 as moderate correlations, and values between 0 and 0.2 as not significantlydifferent from zero. The positive sign of their PRCCs indicate that if the parametersare increased, the value of RTN

0 increases (and vice versa). The negative sign suggests

that, if increased, the value of RTN0 decreases (and vice versa). Therefore, the parame-

ters m1, δ2, β and R are responsible for decreasing the values of RTN0 , so increasing all

those parameters are beneficial for pest control. While the parameters m2, δ1, r, d andTN are responsible for lowering the values of RTN

0 , so increasing all those parameterswill result in a more severe pest outbreak.

The most significant control parameters are m1, δ1, R and TN (See Fig. 4b, d, fand g). It is noted that the PRCC values for m2 and δ2 are < 0.2, which indicates thatthe killing efficiency rate and the decay rate with respect to natural enemies are notsignificantly affected by the values of RTN

0 . Therefore, when pesticide applications aremore frequent than releases of natural enemies, the residual effects of pesticides onthe pests play a more important role than those for natural enemies.

3.2 Detailed dynamic analysis of Case 2

In this subsection we will focus on Case 2 and address similar questions to those raisedin the last subsection.

3.2.1 Threshold condition for the stability of the pest-free periodic solution

For Case 2, there are kn natural enemy releases during period TP , and the followingsubsystem

⎧⎨

⎩

d N (t)dt = −[d + b2(t)]N (t), t �= λm,

N (λ+m) = N (λm) + R, t = λm

(18)

has a globally stable periodic solution over the interval (hTP , (h + 1)TP ] (where h isa positive integer, see Appendix C), denoted by N TP (t), i.e.

N TP (t) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Y ∗C(t), t ∈ (hTP , hTP + λ1],Y ∗C(t) + R

∑ij=1 D j (t), t ∈ (hTP + λi , hTP + λi+1],

i = 1, 2, . . . , kn − 1,

Y ∗C(t) + R∑kn

j=1 D j (t), t ∈ (hTP + λkn , (h + 1)TP ],(19)

where Y ∗, C(t) and D j (t) are given in Appendix C.Furthermore, the pest-free periodic solution, (0, N TP (t)), of system (12) over the

h-th time interval hTP < t ≤ (h + 1)TP is globally attractive if the threshold value

123

16 S. Tang et al.

RTP0 > 1 with

RTP0 = m1

δ1TP(1 − e−δ1TP ) + β

rTP

TP∫

0

B(s) ds

.= Term1 + Term2, (20)

where

B(s) =

⎧⎪⎨

⎪⎩

E(s)e−bc Y ∗, s ∈ (0, λ1],E(s)[e−bc Y ∗ + RD(i)], s ∈ (λi , λi+1], i = 1, 2, . . . , kn − 1,

E(s)[e−bc Y ∗ + RD(kn)], s ∈ (λkn , TP ],(21)

and

E(s) = e[−ds+bce−δ2s )], D(i) =i∑

j=1

e[dλ j −bce−δ2λ j ].

The detailed calculation of the second integration of the above threshold value isprovided in Appendix C.

3.2.2 Comparing the effects of key parameters on RTN0 with RTP

0

Here we compare the RTN0 with RTP

0 , and investigate how different control tacticsimplemented in Case 1 and Case 2 affect the two threshold conditions. To do this,we noted that both RTN

0 and RTP0 are separated into two terms: the first terms in both

threshold conditions show the contributions of chemical control and residual effectsof pesticides on the pest over the period TN or TP ; the second terms in both formu-lae describe the contributions of chemical and biological control, residual effects ofpesticides on the natural enemies and releasing effort over the period TN or TP .

Similarly, we fixed all other parameters and let kp(kn) vary, for different period

TN = TP , as shown in Fig. 5, we see that the first term in the RTP0 is a constant as

kn increases, while the first term in RTN0 is an increasing function as kp increases.

However, due to the residual effects of pesticides on the natural enemies in the sec-ond term, RTN

0 is a decreasing function when the number of pesticide applications

increases, which results in a non-monotonicity of RTN0 (as shown in Fig. 3). But as

the number of natural enemy releasing events increases, the second term in RTP0 is

a monotonic function as kn increases, which results in a strictly increasing RTP0 (see

Fig. 5). In summary, the longer the period TN (or TP ), the smaller the threshold values,and consequently a more severe pest outbreak will occur. All these results also indicatethat when the residual effects of pesticides on the natural enemies are strong, the useof biological control tactics is optimal. If it is necessary to use pesticides to control the

123


1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

TN

=T

P=

20

1 2 3 4 5 6 7 8 9 100

1

2

3

1 2 3 4 5 6 7 8 9 100

1

2

3

1 2 3 4 5 6 7 8 9 100.2

0.3

0.4

0.5

0.6

TN

=T

P=

10

Term1

1 2 3 4 5 6 7 8 9 100

1

2

3

4

Term2

1 2 3 4 5 6 7 8 9 100

1

2

3

4

R0T

N(R0T

P)

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

kp(k

n)

TN

=T

P=

30

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

kp(k

n)

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

kp(k

n)

Case 1Case 2

Fig. 5 Comparing the effects of parameter space on the threshold values for Case 1 (RTN0 ) with Case 2

(RTP0 ). Here the impulsive points τ1, τ2, . . . , τk p divide the period TN into k p+1 equal small intervals and

λi+1 − λi = TN for all i ∈ Z in Case 1, and the impulsive points λ1, λ2, . . . , λkn divide the period TPinto kn+1 equal small intervals and τi+1 − τi = TP for all i ∈ Z in Case 2. The baseline parameter valuesare as follows: m1 = 0.74, m2 = 0.5, δ1 = 0.2, δ2 = 0.65, r = 0.5, β = 0.44, d = 0.15, and we chooseR = 3 for Case 1 and R = 1 for Case 2

pest, our results obtained here or Volterra’s principle indicate that the number of pes-ticide applications need to be carefully selected (see Fig. 3). Using the same methodwe can address different effects of all the other parameters on RTN

0 and RTP0 .

When natural enemy releases are more frequent than pesticide applications (i.e.Case 2), the most significant control parameters which affect the threshold value RTP

0are the releasing constant R and the releasing period TP (as shown in Fig. 6), becausethe biological control dominates the threshold condition, seen in more detail in Fig. 6and Fig. 3.

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18 S. Tang et al.

−1

−0.5

0

0.5

1

m1 m

2δ

1δ

2 r β d R T

P

PR

CC

s(a)

−400 −200 0 200 400

−500

0

500

(b) [0.19846 , 2.4335e−010]

m1

R0T

P

−400 −200 0 200 400

−500

0

500

(c) [−0.094495 , 0.0027792]

m2

R0T

P

−400 −200 0 200 400

−500

0

500

(d) [−0.16475 , 1.6127e−007]

δ1

R0T

P

−400 −200 0 200 400

−500

0

500

(e) [0.16751 , 9.9269e−008]

δ2

R0T

P

−400 −200 0 200 400

−500

0

500

(f) [0.74159 , 3.22e−175]

R

R0T

P

−400 −200 0 200 400

−500

0

500

(g) [−0.68876 , 0]

TP

R0T

P

Fig. 6 PRCC results and PRCC scatter plots of the pest integrated control parameters including the killingefficiency rate mi , decay rate δi (i = 1, 2), releasing constant R and period TP for Case 2. The baselineparameter values are as follows: m1 = 1.2, m2 = 0.5, δ1 = 0.3, δ2 = 0.2, r = 1.5, β = 0.6, d =0.15, R = 1, TP = 10, kn = 3. Here the impulsive points λ1, λ2, λ3 divide the period TP into 4 equalsmall intervals and τi+1 − τi = TP for all i ∈ Z . a PRCC results with sample size 1,000, all parameterswere varied simultaneously. b–g PRCC scatter plots with sampling size 1,000 and significant p < 0.01.The title of each plot represents the PRCC value with the corresponding p-value

4 Discussion

According to Volterra’s principle (Volterra 1926), if pesticides usually act not only onthe pest species but also, with even stronger impact, on their natural enemies, then,although the pesticide does indeed combat the prey, it is simultaneously reducing thepredator numbers with an even larger, positive, secondary influence on the prey. Asa result of this indirect effect, treatments can counter-intuitively lead to an effectiveincrease of the pest species, as has been documented (see Introduction) (Barclay 1982;Debach 1974; Roughgarden 1979; Ruberson et al. 1998). Furthermore, our theoreticalresults show that the effect of pesticide timing, the effectiveness of the natural enemies

123


and pesticide selectivity are all also crucial to pest depression and resurgence, infor-mation that could help field and glasshouse operators to decide on optimum timingsfor spray applications and optimum rates for natural enemy releases.

Repeated spraying of pesticides is often needed which accelerates the developmentof resistance. So, in cases when spraying is essential the following questions arise (1)what is the optimal timing for the pesticide applications? (2) How can adverse effectson the natural enemies be minimised? and (3) When there are short- or long-termresidual effects of the pesticides on pest and natural enemies, how should chemicaland biological control be applied with the aim of ensuring longer, effective, residualcontrol in accordance with Volterra’s principle?

To answer all these questions, and understand the ecological interactions amongstpest, natural enemy and pesticides which are important for successful pest control, wehave proposed detailed modelling methods involving residual effects of pesticides onboth pest and natural enemy. We employed both piecewise and step functions to modelthe killing efficiency rate and decay rate of pesticides for the pest or natural enemy,and addressed the following interesting pest control issues: (a) comparing differentintegrated pest control measures which result in different pest control outcomes; (b)investigating the residual effects of pesticides on both species in terms of thresholdconditions; (c) discussing the optimal number of pesticide applications in accordancewith Volterra’s principle; and (d) determining the most significant parameters forthe threshold conditions by using LHS/PRCC uncertainty and sensitivity analysistechniques.

Our results indicate that there exists an optimal number of pesticide applicationswhen pesticide applications are more frequent than the releases of natural enemies,which maximizes the threshold condition and results in longer, effective, residualcontrol. However, the optimal control tactics depend on a lot of factors, including thekilling efficiency rate and the decay rate of the pesticides, the duration of the resid-ual effects, the releasing period and the releasing constant. Our analyses clarify thatpest resurgence possibly occurs when a pesticide kills a large percentage of both thepest population and its natural enemies. Therefore, the most effective residual controlstrategy is to carefully choose pesticides such that they have longer effective residualeffects on the pest (i.e. longer residual effectiveness), while having only short-termeffects on the natural enemy (Raupp et al. 2001; Schmutterer 1988; Zacharda andHluchy 1991).

In the first case investigated in the present work we assumed that spraying pesti-cides is more frequent than releases of the natural enemy, i.e. over the releasing period,there are kp pesticide sprays. Thus, the likelihood of resistance develops faster withrepeated pesticide applications. Questions for future research include how to describethe resistance to pesticides in our present model and how to determine the optimal timewhen pesticides should be switched? According to the definition of IPM, a good pestcontrol programme should reduce pest populations to acceptable levels (here below theeconomic injury level) rather than eradication (Pedigo and Higley 1992; Van Lenterenand Woets 1988; van Lenteren 2000; Tang and Cheke 2005; Tang et al. 2008; Tangand Cheke 2008; Tang et al. 2009, 2010). How to model an IPM strategy with residualeffects of pesticides, taking account of the economic injury level, will also be studied

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20 S. Tang et al.

in future research to include analyses of the model’s dynamical behaviour and thebiological implications of the results.

Acknowledgments We would like to thank the referees and Prof. Sebastian Schreiber for their carefulreading of the original manuscript and many valuable comments that greatly improved the presentation ofthis paper.

Appendix A: the proof of Theorem 2.1

(i) Assume (P(t), N (t)) is any solution of system (5) with initial value (P(0), N (0)).The local asymptotical stability of periodic solution (9) can be determined by consid-ering the behaviour of small amplitude perturbations of the solution. Define P(t) =u(t), N (t) = N∗(t) + v(t), then there may be written

(u(t)v(t)

)

= Φ(t)

(u(0)

v(0)

)

, 0 < t ≤ T, (A.1)

where Φ(t) is the fundamental matrix of (5), which satisfies

dΦ(t)

dt=

(r(1 − b(t)) − βN∗(t) 0

λβN∗(t) −d

)

Φ(t),

and Φ(0) is the identity matrix.Therefore, the stability of the periodic solution (9) is determined by the eigenvalues

of

M =(

1 00 1

)

Φ(T ),

which are λ1 = exp(∫ T

0 r(1 − b(t)) − βN∗(t) dt), λ2 = e−dT < 1. According toFloquet theory, periodic solution (9) is asymptotically stable if and only if |λ1| < 1,i.e. R0 > 1.

In the following, we prove the global attractivity of periodic solution (9). Choosean ε > 0 small enough such that

χ = exp

⎛

⎝T∫

0

r(1 − b(t)) − β(N∗(t) − ε) dt

⎞

⎠ < 1.

It follows from the second equation of system (5) that we have d N (t)dt > −d N (t), and

consider the following impulsive differential equation

⎧⎪⎨

⎪⎩

dy(t)dt = −dy(t), t �= nT,

y(t+) = y(t) + R, t = nT,

y(0+) = N (0).

(A.2)

123


According to Lemma 2.1 and the comparison theorem on impulsive differential equa-tions we get N (t) ≥ y(t) and y(t) → N∗(t) as t → ∞. Therefore

N (t) ≥ y(t) > N∗(t) − ε (A.3)

hold for ε(ε > 0) small enough and all t large enough. For simplicity, we may assumethat (A.3) holds for all t ≥ 0. Again it follows from the first equation of system (5)that we have

d P(t)

dt≤ r P(t)

[

1 − b(t) − β

r(N∗(t) − ε)

]

.

Now we consider the following scalar differential equation

{dx(t)

dt = r x(t)[1 − b(t) − β

r (N∗(t) − ε)]

x(0) = P(0).= P0.

(A.4)

According to the comparison theorem we have P(t) ≤ x(t). In general, for any func-tion b(t) the analytical solution of the model (A.4) with initial value x(0) = P0(P0 >

0) can be obtained, i.e.,

x(t) = P0 exp

⎡

⎣r

t∫

0

(1 − b(s) − β

r(N∗(s) − ε)) ds

⎤

⎦.

For any t , there exists a positive integer n such that t ∈ (nT, (n + 1)T ], so we have

x(t) = P0χn exp

⎡

⎣r

t∫

nT

(1 − b(s) − β

r(N∗(s) − ε)) ds

⎤

⎦.

This implies that x(t) → 0 as t → ∞. Consequently, we have P(t) → 0 as t → ∞.Next, we prove that N (t) → N∗(t) as t → ∞. For any 0 < ε < d

λβ, there must

exist a t1 > 0 such that 0 < P(t) < ε for t ≥ t1. Without loss of generality, we mayassume that 0 < P(t) < ε holds true for all t > 0, then we have

−d N (t) ≤ d N (t)

dt≤ (λβε − d)N (t).

For the left hand inequality, it follows from impulsive differential equation (A.2) thaty(t) ≤ N (t) and y(t) → N∗(t) as t → ∞. For the right hand inequality, we considerthe following impulsive differential equation

⎧⎪⎨

⎪⎩

dz(t)dt = (λβε − d)z(t), t �= nT,

z(t+) = z(t) + R, t = nT,

z(0+) = N (0).

(A.5)

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22 S. Tang et al.

By using the same methods as those for system (A.2) we can easily prove that system(A.5) has a globally stable periodic solution, denoted by z∗(t) and

z∗(t) = R exp((λβε − d)(t − nT ))

1 − exp((λβε − d)T ), nT < t ≤ (n + 1)T .

Therefore, for any ε1 > 0, there exists a t2 > 0 such that

N∗(t) − ε1 < N (t) < z∗(t) + ε1

for t > t2. Let ε → 0, then we have

N∗(t) − ε1 < N (t) < N∗(t) + ε1

for t > t2, which indicates that N (t) → N∗(t) as t → ∞. This completes the proofof first part.

Before the proof of permanence of system (5), we need the following Definitionand Lemma.

Definition A.1 The system (5) is said to be permanent if there exists a compact regionΩ ⊂ intR2+ such that every solution of system (5) starting from intR2+ will eventuallyenter and remain in region Ω .

Lemma A.1 There exists a constant M > 0 such that P(t) ≤ M and N (t) ≤ M foreach solution of (5) with all t large enough.

Proof Let V0 = {V : R+ × R2+ → R+, continuous on (nT, (n + 1)T ] × R2+, andlim(t,y)→(nT,x) V (t, y) = V (nT +, x) exists}. Suppose (P(t), N (t)) is any solution of(5). Let V = λP(t) + N (t), and then V ∈ V0 with

D+V (t) + γ V (t) = λ[r(1 − b(t)) + γ ]P(t) − λrηP2(t)

−[d − γ ]N (t), t �= nT, (A.6)

V (nT +) = V (nT ) + R,

where D+ denotes the right upper derivative. Obviously, if parameter γ is small enoughsuch as 0 < γ < d, then the right hand of the first equation of (A.6) is upper bounded.So selecting such a γ0 and letting K be the bound, yields

D+V (t) ≤ −γ0V (t) + K , t �= nT,

V (nT +N ) = V (nT ) + R.

According to the comparison theorem of impulsive differential equations we have

V (t) ≤(

V (0) − K

γ0

)

e−γ0t + R(1 − e−nγ0T )

1 − e−γ0Te−γ0(t−nT ) + K

γ0

123


for t ∈ (nTN , (n + 1)TN ]. Therefore, V (t) is ultimately bounded by a constant, andconsequently for t large enough there exists a constant M > 0 such that P(t) ≤M, N (t) ≤ M for each solution of (5).

(ii) Now we prove the permanence of (5) if R0 < 1. Let q2 = N∗(T ) − ε2 >

0, ε2 > 0, where N∗(t) is given by (8). We can easily prove that N (t) > q2 for all tlarge enough. In fact, according to (A.2) we have

N (t) > N∗(t) − ε2 ≥ N∗(T ) − ε2 = q2

for t large enough. Next we claim that there exists a q1 > 0 such that P(t) ≥ q1 for tlarge enough. We do it in the following two steps. We first note that for the followingsystem

{du(t)

dt = u(t)[ω − d], t �= nT,

u(nT +) = u(nT ) + R, t = nT,(A.7)

there exists a unique T −periodic solution (denoted by u(t)) which is globally stable,where ω is defined later in Step 1 and

u(t) = R exp((ω − d)(t − nT ))

1 − exp((ω − d)T ), t ∈ (nT, (n + 1)T ].

Step 1 Since R0 < 1, we can select q3 > 0, ε1 > 0 small enough such that ω =λβq3 < d, σ = ∫ T

0 [r − rb(t) − rηq3 − β(u(t) + ε1)] dt > 0, where u(t) = u(t) forn = 0.

We claim that P(t) < q3 cannot hold true for all t ≥ 0. Otherwise,

d N (t)

dt≤ N (t)(ω − d),

then N (t) ≤ u(t) and u(t) → u(t) as t → ∞. Therefore, there exists a T1 > 0 suchthat

N (t) ≤ u(t) < u(t) + ε1

for t ≥ T1. Thus

d P(t)

dt≥ r P(t)[1 − b(t) − ηq3 − β

r(u(t) + ε1)] (A.8)

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24 S. Tang et al.

for t ≥ T1. So there exists a n1 ∈ N , such that n1T > T1. Integrating (A.8) on(nT, (n + 1)T ] with n ≥ n1 yields

P((n + 1)T ) ≥ P(nT ) exp

⎡

⎣

(n+1)T∫

nT

(r − rb(t) − rηq3 − βu(t) − βε1) dt

⎤

⎦

= P(nT )eσ .

Then P((n + k)T ) ≥ P(nT )ekσ → ∞ as k → ∞, which is a contradiction. There-fore, there exists a t1 > 0 such that P(t1) ≥ q3.

Step 2 If P(t1) ≥ q3 for all t ≥ t1, then the permanence of the system (5) follows.Otherwise, there exists a t ′ ≥ t1 such that P(t ′) < q3, let t∗ = inf t≥t1{P(t) < q3}.Then P(t) ≥ q3 for all t ∈ [t1, t∗) and P(t∗) = q3. Assuming t∗ ∈ (n2T, (n2 +1)T ], n2 ∈ N and selecting n3, n4 ∈ N such that

n3 >ln

(ε1

M+R

)

(ω − d)T, e(n3+1)σ1T +n4σ > 1,

and σ1 = −rηq3 − βM < 0. We confirm that there must be a t2 ∈ [(n2 + 1)T, (n2 +1 + n3 + n4)T ] such that P(t2) ≥ q3. Otherwise P(t) < q3, t ∈ [(n2 + 1)T, (n2 +1 + n3 + n4)T ]. It follows from (A.7) with u((n2 + 1)T +) = N ((n2 + 1)T +) that wehave

u(t) =[

u((n2 + 1)T +) − R

1 − exp((ω − d)T )

]

e(ω−d)(t−(n2+1)T ) + u(t),

t ∈ (nT, (n + 1)T ], n2 + 1 ≤ n ≤ n2 + 1 + n3 + n4. Therefore,

|u(t) − u(t)| < (M + R)en3(ω−d)T < ε1,

this gives

N (t) ≤ u(t) ≤ u(t) + ε1,

(n2 + 1 + n3)T ≤ t ≤ (n2 + 1 + n3 + n4)T, which implies that (A.8) holds for(n2 + 1 + n3)T ≤ t ≤ (n2 + 1 + n3 + n4)T. From Step 1, we get

P((n2 + 1 + n3 + n4)T ) ≥ P((n2 + 1 + n3)T )en4σ .

It follows from R0 = 1T

∫ T0 [b(s)+ β

r N∗(s)] ds > 1 that∫ (n2+1)T

n2T (1−b(t))dt > 0.

Consequently, we have∫ (n2+1)T

t∗ (1 − b(t))dt > 0 due to function 1 − b(t) = 1 −me−δ(t−n2T ) is increasing in [n2T, (n2 + 1)T ]. Integrating

d P(t)

dt≥ r P(t)

[

1 − b(t) − ηq3 − β

rM

]

123


on [t∗, (n2 + 1 + n3)T ], one gives

P((n2 + 1 + n3)T ) ≥ q3 exp

⎛

⎝

(n2+1+n3)T∫

t∗r

[

1 − b(t) − ηq3 − β

rM

]

dt

⎞

⎠

= q3 exp

⎛

⎝

(n2+1+n3)T∫

t∗r(1 − b(t)) dt

⎞

⎠

exp

⎛

⎝

(n2+1+n3)T∫

t∗r [−ηq3 − β

rM] dt

⎞

⎠

≥ q3 exp

⎛

⎝

(n2+1+n3)T∫

t∗σ1 dt

⎞

⎠

≥ q3eσ1(n3+1)T.

Thus P((n2 + 1 + n3 + n4)T ) ≥ q3eσ1(n3+1)T en4σ > q3, which is a contradic-tion. Therefore, there exists a t2 ∈ [(n2 + 1)T, (n2 + 1 + n3 + n4)T ] such thatP(t2) ≥ q3. Let t = inf t≥t∗{P(t) ≥ q3}, thus P(t) ≥ q3. For t ∈ [t∗, t), we haveP(t) ≥ P(t∗)eσ1(t−t∗) ≥ q3eσ1(1+n3+n4)T .= q1. For t > t, the same argument can becontinued since P(t) ≥ q3. Hence P(t) ≥ q1 for all t ≥ t1. The proof is complete.

Appendix B: analyzing the dynamical behaviour of Case 1 and calculating thethreshold value RTN

0

Firstly, the basic properties of the following subsystem

{d N (t)

dt = −[d + b2(t)]N (t), t �= hTN ,

N (hT +N ) = N (hTN ) + R, t = hTN

(B.1)

play a key role in analyzing the pest control, where h is a positive integer. In any giventime interval (hTN , (h + 1)TN ], we investigate the dynamical behaviour of model(B.1). Integrating the first equation of model (B.1) from hTN to (h + 1)TN yields

N ((h + 1)TN ) = N (hT +N ) exp

⎡

⎢⎣

(h+1)TN∫

hTN

−(d + b2(s)) ds

⎤

⎥⎦,

and at time (h + 1)TN , release of natural enemies occurs once and

123

26 S. Tang et al.

N ((h + 1)T +N ) = N (hT +

N ) exp

⎡

⎢⎣

(h+1)TN∫

hTN

−(d + b2(s)) ds

⎤

⎥⎦ + R.

Denote Yh = N (hT +N ), then we have the following difference equation

Yh+1 = Yh exp

⎡

⎢⎣

(h+1)TN∫

hTN

−(d + b2(s)) ds

⎤

⎥⎦ + R,

which has a unique steady state

Y ∗ = R

1 − exp[∫ (h+1)TN

hTN−(d + b2(s)) ds

] . (B.2)

Due to the periodicity of function b2, the exponential term in Y ∗ can be calculated asfollows

exp

⎡

⎢⎣

(h+1)TN∫

hTN

−(d + b2(s)) ds

⎤

⎥⎦ = exp

⎡

⎢⎣

kp∑

i=1

τi∫

τi−1

−(d + b2(s)) ds+TN∫

τk p

−(d+b2(s)) ds

⎤

⎥⎦

= exp

⎡

⎢⎣−dTN −

kp∑

i=2

τi∫

τi−1

b2(s) ds −TN +τ1∫

τk p

b2(s) ds

⎤

⎥⎦

= exp

⎡

⎣−dTN − m2

δ2

kp+1∑

i=2

(1 − e−δ2(τi −τi−1))

⎤

⎦

with τ0 = 0 and∫ TN +τ1τk p

b2(s) ds = ∫ τ1τ0

b2(s)ds + ∫ TNτk p

b2(s) ds due to the residual

effects of pesticide on the interval (0, τ1) and (τkp , TN ). Therefore, we obtain the ana-lytical formula of the TN -periodic solution (denoted by N TN (t)) of the model (B.1),i.e.


⎡

⎢⎣

t∫

hTN

−(d + b2(s)) ds

⎤

⎥⎦, t ∈ (hTN , (h + 1)TN ].

Using the same methods as in the proof of Lemma 2.1, we can prove that thesubsystem (B.1) has the following main property.

Lemma B.1 The sub-system (B.1) has a positive periodic solution N TN (t) and forevery solution N (t) of (B.1) we have |N (t) − N TN (t)| → 0 as t → ∞, where

123



⎡

⎢⎣

t∫

hTN

−(d + b2(s)) ds

⎤

⎥⎦ , t ∈ (hTN , (h + 1)TN ] (B.3)

where b2(t) is given by (13) and Y ∗ is given by (B.2).Furthermore, we obtain the complete expression for the pest-free periodic solution,

(0, N TN (t)), of system (12) over the h-th time interval hTN < t ≤ (h + 1)TN . Thus inthe following, we will discuss the threshold condition which guarantees the stabilityof (0, N TN (t)).

Let (P(t), N (t)) be any solution of system (12) with positive initial value (P(0),

N (0)). By using the comparison theorem of impulsive differential equations and thesame methods as in the proof of Theorem 2.1, we can choose ε(ε > 0) small enoughsuch that N (t) > N TN (t) − ε holds for t large enough. Without loss of generality, wemay assume that N (t) > N TN (t) − ε holds for all t ≥ 0. Consequently, we have

d P(t)

dt≤ r P(t)[(1 − b1(t)) − ηP(t)] − β P(t)[N TN (t) − ε]

= r P(t)

[(

1 − b1(t) − β

r(N TN (t) − ε)

)

− ηP(t)

]

(B.4)

.= r P(t)[(1 − b(t)) − ηP(t)].

Considering the following scale differential equation

dy(t)

dt= r y(t)(t)[(1 − b(t)) − ηy(t)]

with initial value y(0) = P(0). It follows from the discussion of Sect. 1 that if

〈b(t)〉 = 1

TN

(h+1)TN∫

hTN

b(s) ds > 1,

then we have y(t) → 0 as t → ∞. Consequently, we have 0 < P(t) < y(t) → 0 ast → ∞, which indicates that P(t) → 0 as t → ∞.

Thus, it follows from the analytical solution of N TN , that the threshold condi-tion, denoted by RTN

0 , which guarantees the stability of the pest-free periodic solution(0, N TN (t)) is given as follows

RTN0 = 1

TN

(h+1)TN∫

hTN

[

b1(s) + β

rN TN (s)

]

ds

= 1

TN

(h+1)TN∫

hTN

b1(s) ds + β

rTN

(h+1)TN∫

hTN

N TN (s) ds

123

28 S. Tang et al.

= 1

TN

(h+1)TN∫

hTN

b1(s) ds + βY ∗

rTN

(h+1)TN∫

hTN

exp

⎡

⎢⎣

s∫

hTN

−(d + b2(ξ)) dξ

⎤

⎥⎦ ds

= m1

δ1TN

kp+1∑

i=2

(1−e−δ1(τi −τi−1))+ βY ∗

rTN

(h+1)TN∫

hTN

exp

⎡

⎢⎣

s∫

hTN

−(d+b2(ξ)) dξ

⎤

⎥⎦ ds.

(B.5)

In the following, we show how to calculate the second term of RTN0 . Denote bc

.= m2δ2

and the second integration term is given by

(h+1)TN∫

hTN

exp

⎡

⎢⎣

s∫

hTN

−(d + b2(ξ)) dξ

⎤

⎥⎦ ds =

hTN +τ1∫

hTN

exp

⎡

⎢⎣

s∫

hTN

−(d + b2(ξ)) dξ

⎤

⎥⎦ ds

+kp−1∑

i=1

hTN +τi+1∫

hTN +τi

exp

⎡

⎢⎣

s∫

hTN

−(d + b2(ξ)) dξ

⎤

⎥⎦ ds

+(h+1)TN∫

hTN +τk p

exp

⎡

⎢⎣

s∫

hTN

−(d + b2(ξ)) dξ

⎤

⎥⎦ ds.

Since

hTN +τ1∫

hTN

b2(s) ds = bc[e−δ2(TN −τk p ) − e−δ2(TN +τ1−τk p )],

i−1∑

j=1

hTN +τ j+1∫

hTN +τ j

b2(ξ) dξ =i−1∑

j=1

bc[1 − e−δ2(τ j+1−τ j )]

and

s∫

hTN +τi

b2(ξ) dξ = bc[1 − e−δ2(s−hTN −τi )].

123


So we have

hTN +τ1∫

hTN

e[∫ s

hTN−(d+b2(ξ)) dξ ]

ds =hTN +τ1∫

hTN

e(−d(s−hTN ))e− ∫ s

hTNm2e

[−δ2(ξ−(h−1)TN −τk p )] dξ

ds

=hTN +τ1∫

hTN

e[−d(s−hTN )−bce−δ2(TN −τk p )

(1−e−δ2(s−hTN ))] ds,

hTN +τi+1∫

hTN +τi

e[∫ s


ds

=hTN +τi+1∫

hTN +τi

e−d(s−hTN )e[−(

∫ hTN +τ1hTN

+i−1∑

j=1

∫ hTN +τ j+1hTN +τ j

+ ∫ shTN +τi

)b2(ξ) dξ ]ds

=hTN +τi+1∫

hTN +τi

exp[−d(s − hTN ) − bc((e−δ2(TN −τk p ) − e−δ2(TN +τ1−τk p )

)

+i−1∑

j=1

(1 − e−δ2(τ j+1−τ j )) + (1 − e−δ2(s−hTN −τi )))]

and

(h+1)TN∫

hTN +τk p

e[∫ s


ds

=(h+1)TN∫

hTN +τk p

e−d(s−hTN )e[−(

∫ hTN +τ1hTN

+k p−1∑

j=1

∫ hTN +τ j+1hTN +τ j

+ ∫ shTN +τk p

)b2(ξ) dξ ]ds

=(h+1)TN∫

hTN +τk p

exp[−d(s − hTN ) − bc((e−δ2(TN −τk p ) − e−δ2(TN +τ1−τk p )

)

+kp−1∑

j=1

(1 − e−δ2(τ j+1−τ j )) + (1 − e−δ2(s−hTN −τk p )))].

123

30 S. Tang et al.

For simplification, we denote

Ci = (e−δ2(TN −τk p ) − e−δ2(TN +τ1−τk p )) +

i−1∑

j=1

(1 − e−δ2(τ j+1−τ j )), i = 1, 2, . . . , kp.

and

B(s) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

exp [−ds − bce−δ2(TN −τk p )(1 − e−δ2s)], s ∈ [0, τ1)

exp [−ds − bc(Ci + (1 − e−δ2(s−τi )))], s ∈ [τi , τi+1),

i = 1, 2, . . . , kp − 1,

exp [−ds − bc(Ckp + (1 − e−δ2(s−τk p )))], s ∈ [τkp , TN ).

Therefore, the formula of RTN0 becomes

RTN0 = m1

δ1TN

kp+1∑

i=2

(1 − e−δ1(τi −τi−1)) + βY ∗

rTN

TN∫

0

B(s) ds (B.6)

In particular, if we assume that the impulsive points τ1, τ2, . . . , τkp divide the periodTN into kp + 1 equal small intervals, denote the length of these small intervals as ι,then the constant Ci becomes

Ci = (e−δ2ι − e−2δ2ι) +i−1∑

j=1

(1 − e−δ2ι) = (1 − e−δ2ι)(i − 1 + e−δ2ι),

i = 1, 2, . . . , kp

and

RTN0 = m1

δ1TN[(kp − 1)(1 − e−δ1ι) + (1 − e−2δ1ι)] + βY ∗

rTN

TN∫

0

B(s) ds. (B.7)

Therefore, by using the same methods as in the proof of Theorem 2.1 we concludethat if RTN

0 > 1 then the pest will die out eventually. Consequently, P(t) → 0 ast → ∞. Similarly, we can prove N (t) → N TN (t) as t → ∞. This indicates that ifRTN

0 > 1 then the pest eradication periodic solution (0, N TN (t)) is globally attractive.

Appendix C: analyzing the dynamical behaviour of Case 2 and calculating thethreshold value RTP

0

For Case 2, there are kn natural enemy releases during period TP . Firstly, the basicproperties of the following subsystem

123


{d N (t)

dt = −[d + b2(t)]N (t), t �= λm,

N (λ+m) = N (λm) + R, t = λm

(C.1)

play a key role in analyzing the pest control for Case 2. In any given time interval(hTP , (h +1)TP ] (where h is a positive integer), we first integrate Eq. (C.1) at intervalt ∈ (hTP , hTP + λ1], to yield

N (t) = N (hT +P ) exp

⎛

⎜⎝−

t∫

hTP

(d + b2(s))

⎞

⎟⎠ ds

= N (hT +P ) exp [−d(t − hTP ) − bc(1 − e−δ2(t−hTP ))].

For t ∈ (hTP + λi , hTP + λi+1], i = 1, 2, . . . , kn − 1, we have

N (t) = N [(hTP + λi )+] exp

⎛

⎜⎝−

t∫

hTP+λi

(d + b2(s))

⎞

⎟⎠ ds

= N [(hTP + λi )+] exp [−d(t − λi − hTP ) − bc(e

−δ2λi − e−δ2(t−hTP ))],and for t ∈ (hTP + λkn , (h + 1)TP ], we have

N (t) = N [(hTP + λkn )+] exp

⎛

⎜⎝−

t∫

hTP+λkn

(d + b2(s))

⎞

⎟⎠ ds

= N [(hTP + λkn )+] exp [−d(t − λkn − hTP ) − bc(e

−δ2λkn − e−δ2(t−hTP ))].Due to pulse releasing of the natural enemies at time points hTP +λi , i = 1, 2, . . . , kn ,so we have

N [(hTP + λ1)+] = N (hT +

P ) exp [−dλ1 − bc(1 − e−δ2λ1)] + R

N [(hTP + λi )+] = N [(hTP + λi−1)

+] exp[−d(λi − λi−1)

−bc(e−δ2λi−1 − e−δ2λi )] + R = N (hT +

P ) exp [−dλi − bc(1 − e−δ2λi )]

+Ri−1∑

j=1

exp [−d(λi − λ j ) − bc(e−δ2λ j − e−δ2λi )] + R, i = 2, . . . , kn

and

N [((h + 1)TP )+] = N [(hTP + λkn )+] exp [−d(TP − λkn ) − bc(e

−δ2λkn − e−δ2TP )]= N (hT +

P ) exp [−dTP − bc(1 − e−δ2TP )]

+Rkn∑

i=1

exp [−d(TP − λi ) − bc(e−δ2λi − e−δ2TP )].

123

32 S. Tang et al.

Denote Yh = N (hT +N ), then we have the following difference equation

Yh+1 = Yh exp [−dTP − bc(1 − e−δ2TP )]

+Rkn∑

i=1

exp [−d(TP − λi ) − bc(e−δ2λi − e−δ2TP )],

which has a unique steady state

Y ∗ =R

kn∑

i=1exp [−d(TP − λi ) − bc(e−δ2λi − e−δ2TP )]1 − exp [−dTP − bc(1 − e−δ2TP )] . (C.2)

Therefore, the model (C.1) has a TP periodic solution (denoted by N TP (t)), and

N TP (t)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Y ∗ exp [−d(t − hTP ) − bc(1 − e−δ2(t−hTP ))], t ∈ (hTP , hTP + λ1],Y ∗ exp [−d(t − hTP ) − bc(1 − e−δ2(t−hTP ))]

+R∑i

j=1 exp [−d(t − λ j − hTP ) − bc(e−δ2λ j − e−δ2(t−hTP ))],t ∈ (hTP + λi , hTP + λi+1], i = 1, 2, . . . , kn − 1,

Y ∗ exp [−d(t − hTP ) − bc(1 − e−δ2(t−hTP ))]+R

∑knj=1 exp [−d(t − λ j − hTP ) − bc(e−δ2λ j − e−δ2(t−hTP ))],

t ∈ (hTP + λkn , (h + 1)TP ].

(C.3)

Using the same methods as in the proof of Lemma 2.1, we can prove that thesubsystem (C.1) has the following main property.

Lemma C.1 The sub-system (C.1) has a positive periodic solution N TP (t) and forevery solution N (t) of (C.1) we have |N (t) − N TP (t)| → 0 as t → ∞ with

N TP (t) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Y ∗C(t), t ∈ (hTP , hTP + λ1],Y ∗C(t) + R

∑i−1j=1 D j (t), t ∈ (hTP + λi , hTP + λi+1],

i = 1, 2, . . . , kn − 1,

Y ∗C(t) + R∑kn

j=1 D j (t), t ∈ (hTP + λkn , (h + 1)TP ](C.4)

where we denote

C(t).= exp [−d(t − hTP ) − bc(1 − e−δ2(t−hTP ))]

and

D j (t).= exp [−d(t − λ j − hTP) − bc(e

−δ2λ j − e−δ2(t−hTP ))].

Furthermore, for Case 2 we obtain the complete expression for the pest-free periodicsolution, (0, N TP (t)), of system (12) over the h-th time interval hTP < t ≤ (h +1)TP ,and in the following we will discuss its stability.

123


Let (P(t), N (t)) be any solution of system (12) with positive initial value (P(0), N (0))

and choosing ε(ε > 0) small enough, then it follows from the discussion in AppendixB that we have N (t) > N TP (t) − ε for t large enough. For simplification we mayassume N (t) > N TP (t) − ε holds for all t ≥ 0. Thus we have

d P(t)

dt≤ r P(t)[(1 − b1(t)) − ηP(t)] − β P(t)[N TP (t) − ε]

= r P(t)

[(

1 − b1(t) − β

r(N TP (t) − ε)

)

− ηP(t)

]

(C.5)

.= r P(t)[(

1 − b(t))

− ηP(t)],

and if

〈b(t)〉 = 1

TP

(h+1)TP∫

hTP

[

b1(s) + β

r(N TP (s) − ε)

]

ds > 1, (C.6)

then we can show that P(t) → 0 as t → ∞.Therefore, it follows from the analytical solution of N TP , that the threshold condi-

tion, denoted by RTP0 , which guarantees the stability of the pest-free periodic solution

(0, N TP (t)) is given as follows

RTP0 = 1

TP

(h+1)TP∫

hTP

[

b1(s) + β

rN TP (s)

]

ds

= 1

TP

(h+1)TP∫

hTP

b1(s) ds + β

rTP

(h+1)TP∫

hTP

N TP (s) ds

= m1

δ1TP(1 − e−δ1TP ) + β

rTP

TP∫

0

B(s) ds,

where for i = 1, 2, . . . , kn − 1

B(s) =

⎧⎪⎪⎨

⎪⎪⎩

Y ∗e[−ds−bc(1−e−δ2s )], s ∈ (0, λ1],Y ∗e[−ds−bc(1−e−δ2s )] + R

∑ij=1 e[−d(s−λ j )−bc(e

−δ2λ j −e−δ2s )], s ∈ (λi , λi+1],Y ∗e[−ds−bc(1−e−δ2s )] + R

∑knj=1 e[−d(s−λ j )−bc(e

−δ2λ j −e−δ2s )], s ∈ (λkn , TP ].

For simplification, if we denote that

E(s) = e[−ds+bce−δ2s )], D(i) =i∑

j=1

e[dλ j −bce−δ2λ j ],

123

34 S. Tang et al.

then the function B(s) becomes

B(s) =

⎧⎪⎨

⎪⎩

E(s)e−bc Y ∗, s ∈ (0, λ1],E(s)[e−bc Y ∗ + RD(i)], s ∈ (λi , λi+1], i = 1, 2, . . . , kn − 1,

E(s)[e−bc Y ∗ + RD(kn)], s ∈ (λkn , TP ].

Therefore, if RTP0 > 1 then the pest will die out eventually. Consequently, P(t) → 0

as t → ∞. Similarly, we can prove that N (t) → N TP (t) as t → ∞. This indicatesthat if RTP

0 > 1 then the pest eradication periodic solution (0, N TP (t)) is globallyattractive.

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Threshold conditions for integrated pest management models with pesticides that have residual effects