- Email: [email protected]
Pest regulation by means of continuous and impulsive nonlinear controls
Mathematical and Computer Modelling 51 (2010) 810–822
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Pest regulation by means of continuous and impulsive nonlinear controls Pei Yongzhen a,∗ , Ji Xuehui a , Li Changguo b a
School of Science, Tianjin Polytechnic University, Tianjin, 300160, China
b
Department of Basic Science of Institute of Military Traffic, Tianjin, 300161, China
article
info
Article history: Received 21 March 2009 Received in revised form 17 October 2009 Accepted 19 October 2009 Keywords: Continuous control Impulsive control Susceptible pests Infective pests Natural enemy Permanence
abstract In this paper, two integrated pest management models are investigated, which rely on release of infective pest individuals and of natural enemies in a constant amount, together with spraying of pesticides. It is proved that the susceptible pests can be eradicated if the release amount of infected pests is above some threshold or the pesticide effect is above another threshold. Furthermore, permanent conditions are established when an impulsive control is used. Finally, numerical results show that (1) fewer infected pests or pesticides are needed as the impulsive strategy is taken, displaying its positive effect on the pest control; (2) our assumption that the natural enemies of the pests do not catch the infective pests would reduce the level of the susceptible pests. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Controlling insects and other arthropods has become an increasingly complex issue over the past two decades. Overuse of a single control tactic is discouraged to avoid or delay the development of resistance by the pest to the control tactic, to minimize damage to non-target organisms, and to preserve the quality of the environment [1]. Integrated pest management (IPM), which was introduced in the late 1950s [2] but more widely practiced during the 1970s and 1980s [3–6], promotes the use of several measures such as insecticide applications, crop rotation, biological control, harvest management, and the use of pest-resistant varieties of crops to reduce pest populations below economic levels [1]. Biological insecticides are used in IPM such as Bacillus thuringiensis or entomopathogenic fungi, disrupting the reproductive processes of the pest by releasing sterile pest individuals or spreading a disease in the pest population on the grounds that infective pests are usually less damaging to the environment [7]. And bacillus thuringiensis, which is available in commercial preparations, is used in the control of a large number of pests [8–11]. Another successful case of biological control in greenhouses is the use of the parasitoid Encarsia formosa against the greenhouse whitefly Trialeurodes vaporariorum on tomatoes and cucumbers [4,5]. Additionally, Baculoviruses have been found useful to control the diamondback moth Plutella xylostella in cabbage farms [12,13] after it had been noticed that the diamondback moth became resistant to chemical pesticides in many areas [14]. An advantage of using insect pathogens is that they are safe to man and are usually safe to beneficial insects [15]. Breeding natural enemies to control pest or regulate its density below the threshold for economical damage also is used by people. Natural biological control results when naturally occurring enemies maintain pests at a lower level than what would occur without them, and is generally characteristic of biodiverse systems [16]. Some examples of successful uses of biocontrol agents include the use of the predatory arthropod Orius sauteri against the pest Thrips palmi Karny to protect eggplant crops in greenhouses [17]. Other examples are the use of the predatory mites Phytoseiulus persimilis and Neoseiulus californicus against the red spider mite Tetranychus urticae Koch in field-grown strawberries [18] and welltimed inundant releases of Trichogramma egg wasps for codling moth control [16].
∗
Corresponding author. E-mail address: [email protected] (Y. Pei).
0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.10.013
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
811
Chemical treatments are used only in a crisis situation threatening rapid losses or when pests fail to succumb to more conservative methods. Pesticides have become less attractive because of a growing awareness that the chemicals in pesticides can pose health hazards to people [16]. But we do not argue for the abandonment of the use of pesticides, since they quickly kill a significant portion of a pest population and they sometimes provide the only feasible method for preventing economic loss [19]. Pesticides should be used only when other measures, such as biological controls, have failed to keep pest populations from approaching economically damaging levels [16]. Recently, many papers have been devoted to the analysis of mathematical models describing IPM strategies [4–7,19–21]. In [20] a predator–prey model with disease in the prey is constructed and investigated for the purpose of integrated pest management. Paul and Zhang [7] consider an integrated pest management model with disease in the pest and stage structure for its natural predator. But chemical treatment is not considered. Wang and Pang [21] investigate the dynamics of a predator–prey food chain with impulsive effect, periodic releasing natural enemies and spraying pesticide at different fixed times. Hui and Zhu [16] investigate the natural enemy–pest models with age structure for the predator, immature and mature natural enemies are released and pesticide is applied impulsively. However, releasing infective pest individuals is not investigated. As motivated by the above-mentioned brief literature survey, in this paper, we will construct two integrated pest management models which rely on release of infective pest individuals and of natural predators in a constant amount, together with spraying of pesticides in a more flexible manner. The infective pests can be cultivated in the laboratory and the natural enemy can be introduced from other regions. Once the susceptible pest meets with the infective pest, there is a chance to be infected. The infective pests have more possibility on death due to the disease and are less damaging to the crops and environment. We suppose the natural enemies catch the susceptible pests, but do not catch the infective pests. All species will be affected by pesticides. Because infected pests and predators are introduced in a uniform manner, our models may be quite appropriate for application to pest control in greenhouses rather than in normal fields where the space might have to be taken explicitly into account. The paper is organized as follows. In Section 2, two integrated pest management models are constructed and the main biological assumptions are formulated. In Section 3, the conditions of the local and global stability of the susceptible pest-eradication equilibrium are obtained. In Section 4, by using Floquet’s theory for impulsive differential equations, small-amplitude perturbation methods and comparison techniques, we investigate the global asymptotic stability of the susceptible pest-eradication periodic solution and the conditions for the permanence of the system. We demonstrate our results by numerical simulation in Section 5. Finally, we give a brief discussion by comparing the continuous model with the corresponding impulsive one and establish the feasible control strategies. 2. Model formulation In a natural ecosystem, there are two main types of predators—generalist and specialist. Generalist predators feed on many types of species. But specialist predators feed almost exclusively on one specific species of prey which may be possible for a parasitoid [22]. Based on this fact, we study the following two systems of differential equations for integrated pest management. (I) Continuous control
α SP θ1 S (t ) S+I 0 − λIS − − , S ( t ) = rS 1 − K 1 + aS T θ2 I (t ) u I 0 (t ) = λIS − µI − + T T P 0 (t ) = β SP − γ P − θ3 P (t ) + v . 1 + aS T T
(2.1)
(II) Impulsive control
S+I α SP 0 S ( t ) = rS 1 − − λ IS − , K 1 + aS 0 t 6= nT , I ( t ) = λ IS − µ I , β SP 0 P (t ) = − γ P, 1 + aS ) ∆S (t ) = −θ1 S (t ), t = nT , n = 1, 2, . . . . ∆I (t ) = −θ2 I (t ) + u, ∆P (t ) = −θ3 P (t ) + v,
(2.2)
Here functions S (t ) and I (t ) represent the densities of the susceptible prey (pest) population and the infective prey (pest) population, respectively. The function P (t ) is the density of predator (natural enemy) population. The model is derived with the following assumptions. (A.1) There is a disease among the prey population, and the prey is divided into two classes, susceptible and infective. The incidence rate is the classic bilinear λIS, and λ is the contact number per unit time for every infective prey with susceptible prey.
812
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
(A.2) The prey population grows according to a logistic fashion with the intrinsic growth rate r and the carrying capacity K . We suppose that the infective prey cannot produce offspring due to the disease; however, the infective prey still consume some crop. (A.3) The predators only catch the susceptible prey, and the predation functional response is Holling type II. The parameters α, a are positive constants, and β/α is the conversion rate from the prey to the predator. (A.4) Parameters µ, γ are the death rate for the infective prey and the predator, respectively. (A.5) We release infective pest population and predator population with releasing amount u(>0) and v(>0) respectively. Parameter T is the period of the impulse. (A.6) Parameters θi (i = 1, 2, 3) represent the portion of susceptible, infective pests and natural enemies, due to spraying of pesticides, respectively. 3. Continuous control In this section, we investigate the dynamics of the ordinary differential system (2.1) by means of stability analysis and apply the subsequently obtained stability results to the study of our control problem. For biological reasons, we restrict our discussion to the feasible region
{(S , I , P )|S ≥ 0, I ≥ 0, P ≥ 0}. The following Lemma implies that all solutions of system (2.1) are ultimately bounded. Lemma 3.1. For system (2.1), there exists a positive constant M1 such that S (t ) ≤ M1 , I (t ) ≤ M1 , P (t ) ≤ M1 for t large enough. Proof. Define function V (t ) = β S (t ) + β
r + 1 I (t ) + α P (t ), λK
and let δ = min{µ, γ }, then we have
h
r
h
K r
D+ V (t ) + δ V (t ) = β rS (t ) + δ S (t ) −
≤ β rS (t ) + δ S (t ) −
K
i
S 2 (t ) − β
r + 1 (µ − δ)I (t ) − α(γ − δ)P (t ) λK
i
S 2 (t ) ≤ M0 ,
(r +δ)2 K
where M0 = . 4r It follows from the methods of [16,23,24] that V (t ) is uniformly ultimately bounded. Thus, by the definition of V (t ), there exists a constant M1 > 0 such that S (t ) ≤ M1 , I (t ) ≤ M1 , P (t ) ≤ M1 for large enough t. The proof is completed. In what follows, the existence and stability of equilibria of the model (2.1) are investigated. There is one boundary equilibrium E1 (0, I1 , P1 ) =
0,
u
µT + θ 2
,
v γ T + θ3
,
representing the extinction of the susceptible pest population, the density of infected pest population and of predator population equilibrating at the carrying capacity. Below we will consider this equilibrium E1 . For the stability of the susceptible pest-eradication equilibrium we have one of the following main Theorems of this paper. The existence and stability of the positive equilibrium will be studied by means of numerical simulation. Theorem 3.1. The susceptible pest-eradication equilibrium E1 is locally asymptotically stable provided r <
u( r + λ K ) K (µT + θ2 )
+
αv . γ T + θ3
(3.1)
Otherwise, it is unstable. Proof. The Jacobian matrix of system (2.1) at E1 takes the form of
r− JE1 =
rI1
− λI1 − α P1 K λI1 β P1
0
−µ 0
0
. 0
−γ
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
813
It is easy to obtain that the eigenvalues for the matrix JE1 are
λ1 = −µ < 0, λ2 = −γ < 0, r + λK · λ3 = r −
αv u − . µT + θ2 γ T + θ3 If condition (3.1) holds, then λ3 < 0. Hence the susceptible pest-eradication equilibrium E1 is locally asymptotically K
stable. Otherwise, equilibrium E1 is unstable. This completes the proof.
Theorem 3.2. The susceptible pest-eradication equilibrium E1 is globally asymptotically stable provided r <
θ1 T
+
u(r + λK ) K (µT + θ2 )
.
(3.2)
Proof. For the globally asymptotical stability of the equilibrium E1 , we define the following Lyapunov function:
V1 = c1 S + c2
I − I1 − I1 ln
I
I1
P + c3 P − P1 − P1 ln , P1
where ci (i = 1, 2, 3) are positive constants and satisfy c1 =
λK r + λK
· c2 =
β · c3 = K . α
Along the solution of model (2.1), we have d dt
V1 = c1 S˙ + c2
(I − I1 ) ˙
I + c3
I
S+I
= c1 S r 1 −
1 + aS
= S c1 r − −
c1 θ1 T
−
P
−
αP θ1 c2 (µT + θ2 )(I − I1 )2 − λI − − + c2 λS (I − I1 ) − 1 + aS T IT 2 c3 (γ T + θ3 )(P − P1 ) PT
c2 λu
µT + θ2
c3 (γ T + θ3 )(P − P1 )2 PT
It follows from (3.2) that we have
P˙
K
c3 β S (P − P1 )
+
P − P1
d V dt 1
−
c3 βv S
(γ T + θ3 )(1 + aS )
−
c1 rS 2 K
−
c2 (µT + θ2 )(I − I1 )2 IT
.
< 0, which implies the equilibrium E1 is globally asymptotically stable.
Corollary 3.1. The susceptible pest-eradication equilibrium E1 is globally asymptotically stable provided u>
K (rT − θ1 )(µT + θ2 ) . = uˆ , T (r + λK )
(3.3)
or
θ1 > rT −
uT (r + λK ) . = θˆ1 .
K (µT + θ2 )
(3.4)
Remark 3.1. By Corollary 3.1, we know that the parameter uˆ is a releasing amount critical value of the infective pest population. Conditions (3.3) and (3.4) imply that releasing more infected pests or spraying more pesticides guarantees the extinction of the susceptible pests in a long term. 4. Pulse control In this section, we investigate the dynamics of system (2.2) by means of stability analysis and apply the subsequently obtained stability results to the study of our control problem. First, it is seen that the long-term survival of the infective pest population and of the predator populations is assured by the pulsed supply of individuals which occurs for t = nT , n ∈ N, while the susceptible pest population may be driven to extinction in certain circumstances, since a pulsed supply of susceptibles is not present. Also, at least when the susceptible pest population (the only one which is not impulsively controlled) tends to extinction, it is natural to expect that the solutions of (2.2) tend to a limiting periodic solution due to the forcing effects of the periodic impulsive perturbations. Thus we will study the existence and stability of the susceptible pest-eradication periodic solution (0, I ∗ , P ∗ ). To this purpose, it is seen
814
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
first that when S (t ) = 0, the system (2.2) reduces to
I (t ) = −µI , P 0 (t ) = −γ P ,
t 6= nT ,
I (t ) = (1 − θ2 )I (t ) + u, + P (t ) = (1 − θ3 )P (t ) + v, +
(4.1)
t = nT ,
which describes the dynamics of the system in the absence of the susceptible pest population. Obviously, system (4.1) has a positive periodic solution ue−µ(t −nT )
I ∗ ( t ) =
1−
P ∗ (t ) =
(1 − θ2 )e−µT −γ (t −nT )
,
I ∗ (0+ ) =
ve , 1 − (1 − θ3 )e−γ T
u 1 − (1 − θ2 )e−µT
P ∗ (0+ ) =
,
v , 1 − (1 − θ3 )e−γ T
t ∈ (nT , (n + 1)T ].
(4.2)
Since the solution of (4.1) is
u ∗ n ∗ + I (t ) = (1 − θ2 ) I (0 ) − 1 − (1 − θ ) exp(−µt ) exp(−µt ) + I (t ), 2 v P (t ) = (1 − θ3 )n P ∗ (0+ ) − exp(−γ t ) + P ∗ (t ), 1 − (1 − θ3 ) exp(−γ t )
t ∈ (nT , (n + 1)T ].
(4.3)
we obtain the following lemma. Lemma 4.1. System (4.1) has a positive periodic solution (I ∗ (t ), P ∗ (t )) and for every solution (I (t ), P (t )) of (4.1), I (t ) → I ∗ (t ), P (t ) → P ∗ (t ) as t → ∞. Therefore, system (2.2) has a pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )). Having proven the existence of the susceptible pest-eradication periodic solution, we are now ready to study its stability. Theorem 4.1. Let (S (t ), I (t ), P (t )) be any solution of (2.2). Then the susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )) is locally stable if rT −
u(r + λK )(1 − exp(−µT )) αv(1 − exp(−γ T )) 1 − < ln , µK [1 − (1 − θ2 ) exp(−µT )] γ [1 − (1 − θ3 ) exp(−γ T )] 1 − θ1
(4.4)
and unstable if the reverse inequality holds. Proof. The stability of periodic solution (0, I ∗ (t ), P ∗ (t )) may be determined by considering the behavior of small-amplitude perturbation of the solution. Denoted by S (t ) = x(t ),
I (t ) = y(t ) + I ∗ (t ),
P (t ) = z (t ) + P ∗ (t ),
the corresponding linear system of (2.2) at (0, I ∗ (t ), P ∗ (t )) reads as
I∗ ∗ ∗ 0 − λ I x − α P x , x ( t ) = rx 1 − K t 6= nT , y0 (t ) = λI ∗ x − µy, 0 ∗ z (t ) = β P x − γ z , ) ∆ x(t ) = −θ1 x(t ), t = nT , n = 1, 2, . . . . ∆y(t ) = −θ2 y(t ), ∆z (t ) = −θ3 z (t ),
(4.5)
So a fundamental matrix Φ (t ) of (4.5) satisfies
dΦ (t ) dt
rI ∗
r − K − λ I − α P = λI ∗ βP∗ ∗
∗
0
−µ 0
0 Φ (t ) 0
−γ
and Φ (0) = E3 , the identity matrix. Hence the fundamental solution matrix is
Φ (t ) =
Z t r−
exp 0
rI ∗ (s) K
− λI ∗ (s) − α P ∗ (s) ds ∆1 ∆2
0
0
exp{−µt } 0
0
.
exp{−γ t }
It is no need to give the exact form of ∆1 and ∆2 as they are not required in the analysis that follows.
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
815
If all eigenvalues of 1 − θ1 0 0
M =
0
0 0
1 − θ2 0
! Φ (T )
1 − θ3
have absolute values less than one, then the periodic solution (0, I ∗ (t ), P ∗ (t )) is locally stable. Since eigenvalues of M are
µ1 = (1 − θ1 ) exp
T
Z
r−
rI ∗ (s)
0
K
− λI (s) − α P (s) ds < 1, ∗
∗
µ2 = (1 − θ2 ) exp{−µt }, µ3 = (1 − θ3 ) exp{−γ t }, then |µ1 | < 1 if and only if (4.4) holds. According to Floquet’s theory of impulsive differential equations, the susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )) is locally stable. The proof is complete. Below we will observe the globally stability of the susceptible pest-eradication periodic solution. Theorem 4.2. The susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )) is globally asymptotically stable, provided that rT −
u(r + λK )(1 − exp(−µT )) 1 < ln . µK (1 − (1 − θ2 ) exp(−µT )) 1 − θ1
(4.6)
Proof. Choose ε > 0 such that
η = (1 − θ1 ) exp
Z t r−
r (I ∗ (s) − ε) K
0
− λ(I (s) − ε) ds < 1, ∗
(4.7)
Noting that I 0 (t ) ≥ −µI (t ) and P 0 (t ) ≥ −λP (t ), from Lemma 4.1 and the comparison theorem, we have I 0 (t ) ≥ I ∗ (t ) − ε, P 0 (t ) ≥ P ∗ (t ) − ε,
(4.8)
for all t large enough. For simplification, we may assume that (4.8) holds for t ≥ 0. From the first equation of (2.2) and (4.8), we get
S 0 (t ) ≤ rS (t ) 1 −
I ∗ (t ) − ε
S (t ) = (1 − θ1 )S (t ), +
K
− λ(I ∗ (t ) − ε)S (t ),
t 6= nT ,
(4.9)
t = nT .
Integrating (4.9) on (nT , (n + 1)T ], it is followed that S ((n + 1)T ) ≤ S (nT )(1 − θ1 ) exp
Z
(n+1)T
r−
nT
= S (nT )η = S (T )ηn .
rI ∗ (s) − r ε K
− λ(I (s) − ε) ds ∗
(4.10)
From (4.7), we have S ((n + 1)T ) → 0 as n → ∞. Therefore, S (t ) → 0 as t → ∞ since 0 < S (t ) ≤ S (nT )(1 − θ1 ) exp(rT ) for nT < t ≤ (n + 1)T . Then we prove that I (t ) → I ∗ (t ) as t → ∞. For 0 < ε1 < µ/λ, there must exist a t1 > 0, such that 0 < S (t ) < ε1 for t ≥ t1 . Without loss of generality, we assume that 0 < S (t ) < ε1 for all t ≥ 0, then from system (2.2) we have
(−λε1 − µ)I (t ) ≤ I 0 (t ) ≤ (λε1 − µ)I (t ). Again from Lemma 4.1 and comparison theorem, we have I1 (t ) ≤ I (t ) ≤ I2 (t ) and I1 (t ) → I1∗ (t ), I2 (t ) → I2∗ (t ) as t → ∞, where I1 (t ) and I2 (t ) are the solutions of
0 I1 (t ) = [−λε1 − µ]I1 (t ), I1 (t + ) = (1 − θ2 )I1 (t ) + u, I (0+ ) = I (0+ ), 1
t 6= nT , t = nT ,
0 I2 (t ) = [−λε1 − µ]I2 (t ), I2 (t + ) = (1 − θ2 )I2 (t ) + u, I (0+ ) = I (0+ ), 2
t 6= nT , t = nT ,
and
816
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
respectively, where I1∗ (t ) = I2∗ (t ) =
ue−(λε1 +µ)(t −nT ) 1 − (1 − θ2 )e−(λε1 +µ)T ue−(µ−λε1 )(t −nT ) 1 − (1 − θ2 )e−(µ−λε1 )T
,
t ∈ (nT , (n + 1)T ];
,
t ∈ (nT , (n + 1)T ].
Therefore, for any ε2 > 0, there exists a t2 > 0 such that I1∗ (t ) − ε2 ≤ I (t ) ≤ I2∗ (t ) + ε2 ,
for t > t2 .
(4.11)
Let ε1 → 0, we have I ∗ (t ) − ε2 ≤ I (t ) ≤ I ∗ (t ) + ε2 , for t large enough, which implies that I (t ) → I ∗ (t ) as t → ∞. Likewise, we prove that for any ε3 > 0, there exist t3 > 0, such that P ∗ (t ) − ε3 ≤ P (t ) ≤ P ∗ (t ) + ε3
(4.12)
for t > t3 , which implies that P (t ) → P (t ) as t → ∞. The proof is complete. ∗
Corollary 4.1. The susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )) is globally asymptotically stable, provided that u>
µK (1 − (1 − θ2 )e−µT )(rT + ln(1 − θ1 )) . = u˜ , (r + λK )(1 − e−µT )
or
θ1 > 1 − exp rT −
u(r + λK )(1 − exp(−µT ))
−1
µK [1 − (1 − θ2 ) exp(−µT )]
. = θ˜1 .
An important concept in IPM is that of the economic threshold (ET). An ET is usually defined as the number of insect pests in the field when control actions must be taken to prevent the economic injury level (EIL) from being reached and exceeded, where EIL is the lowest pest population density that will cause economic damage. From a pest control point of view, our aim is to keep pests at acceptably low levels; not to eradicate them. So we need to investigate the permanence of system (2.2). Before starting our theorems, we give the following definition. Definition 1. System (2.2) is said to be permanent if there are three positive constants m, M and T0 such that every positive solution (S (t ), I (t ), P (t )) of system (2.2) satisfies m ≤ S (t ) ≤ M ,
m ≤ I (t ) ≤ M ,
m ≤ P (t ) ≤ M .
for all t ≥ T0 . Theorem 4.3. There exists a constant M > 0, such that S (t ) ≤ M , I (t ) ≤ M and P (t ) ≤ M for each solution (S (t ), I (t ), P (t )) of system (2.2) with all t large enough. Proof. Define a function V (t ) as r V (t ) = β S (t ) + β + 1 I (t ) + α P (t ). λK For t 6= nT , denoted by δ = min{µ, γ }, we have
h
r
h
K r
D+ V (t ) + δ V (t ) = β rS (t ) + δ S (t ) −
≤ β rS (t ) + δ S (t ) −
K
i
S 2 (t ) − β
r + 1 (µ − δ)I (t ) − α(γ − δ)P (t ) λK
i
S 2 (t ) ≤ M0 ,
where M0 = (r + δ)2 K /4r. When t = nT , V (nT + ) ≤ V (nT ) + β( λrK + 1)u + αv . By comparison theorem, for t ≥ 0 we have
r e−δ(t −T ) r eδ T (1 − e−δt ) + β + 1 u + αv + β + 1 u + αv δ λK 1 − eδ T λK eδ T − 1 δT M0 r e → + β + 1 u + αv δT (as t → ∞). δ λK e −1 Then V (t ) is uniformly upper bounded. Hence, by the definition of V (t ), there exists a constant M > 0 such that S (t ) ≤ M , I (t ) ≤ M and P (t ) ≤ M for all t large enough. The proof is completed. V (t ) ≤ V (0)e−δ t +
M0
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
817
Theorem 4.4. System (2.2) is permanent provided that rT −
u(r + λK )(1 − exp(−µT )) αv(1 − exp(−γ T )) 1 − > ln . µK [1 − (1 − θ2 ) exp(−µT )] γ [1 − (1 − θ3 ) exp(−γ T )] 1 − θ1
(4.13)
Proof. Suppose that (S (t ), I (t ), P (t )) is a solution of (2.2) with S (0) > 0, I (0) > 0, P (0) > 0. From Theorem 4.3, we may assume I (t ) ≤ M,P (t ) ≤ Mand M > for t > 0. From (4.11) and (4.12) we know I (t ) > I ∗ (t ) − ε2 and P (t ) > P ∗ (t ) − ε3 for all t large enough, then I (t ) ≥
ue−µT 1 − (1 − θ2 )e−µT
− ε2 = m 2
and P (t ) ≥
v e−γ T − ε3 = m 3 1 − (1 − θ3 )e−γ T
for t large enough. Thus we only need to find m1 > 0 such that S (t ) ≥ m1 for t large enough. We will do it in the following two steps.
¯ 1 > 0 and ε1 > 0 small enough such that Step 1. By condition (4.13), one can select m ¯ 1 < min m
µ γ , λ β − aγ
,
δ1 = λM < µ,
δ2 =
βM <γ 1 + aM
and
σ = rT −
¯ 1T rm K
−
u(r + λK )(1 − exp((−µ + δ1 )T )) K (µ − δ1 )[1 − (1 − θ2 ) exp(−µT )]
−
αv(1 − exp((−γ + δ2 )T )) > 0. (γ − δ2 )[1 − (1 − θ3 exp(−γ T ))]
¯ 1 cannot hold for all t ≥ 0. Otherwise, note that if S (t ) ≥ 0, then We will prove S (t ) < m
I 0 (t ) ≤ I (t )(−µ + δ1 ), I (t + ) = (1 − θ2 )I (t ) + u,
t 6= nT , t = nT ,
P 0 (t ) ≤ P (t )(−γ + δ2 ), P (t + ) = (1 − θ3 )I (t ) + v,
t 6= nT , t = nT .
and
By Lemma 4.1 and Theorem 4.3, we have I (t ) ≤ z1 (t ), P (t ) ≤ z2 (t ) and z1 (t ) → z1∗ (t ), z2 (t ) → z2∗ (t ) as t → ∞, where z1∗ (t ) = z2∗ (t ) =
ue(−µ+δ1 )(t −nT )
,
t ∈ (nT , (n + 1)T ],
v e(−γ +δ2 )(t −nT ) , 1 − (1 − θ3 )e(−γ +δ2 )T
t ∈ (nT , (n + 1)T ],
1 − (1 − θ2 )e(−µ+δ1 )T
and z1 (t ) and z2 (t ) are the solutions of the following equations
0 z1 (t ) = (−µ + δ1 )z1 (t ), z1 (t + ) = (1 − θ2 )z1 (t ) + u, z (0+ ) = I (0+ ), 1
t 6= nT , t = nT ,
(4.14)
0 z2 (t ) = (−γ + δ1 )z2 (t ), z2 (t + ) = (1 − θ3 )z1 (t ) + v, z (0+ ) = P (0+ ), 2
t 6= nT , t = nT ,
(4.15)
and
respectively. Therefore, there exists a T1 > 0 such that for t ≥ T1 , we have I (t ) ≤ z1 (t ) ≤ z1∗ (t ) + ε1 , and P (t ) ≤ z2 (t ) ≤ z2∗ (t ) + ε1 .
818
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
Then
S (t ) ≥ S (t ) r
S (t ) = (1 − θ1 )S (t ),
0
1−
m¯1
+
K
−
z1∗ (t ) + ε1
K
− λ(z1 (t ) + ε1 ) − α(z2 (t ) + ε1 ) , ∗
∗
t 6= nT ,
(4.16)
t = nT .
Let N1 ∈ N and N1 T ≥ T1 , integrating (4.16) on (nT , (n + 1)T ](n ≥ N1 ), we have S ((n + 1)T ) ≥ S (nT )(1 − θ1 ) exp
(n+1)T
Z
r
1−
m¯1 + z1∗ (s) + ε1
nT
σ
K
− λ(z1 (s) + ε1 ) − α(z2 (t ) + ε1 ) ds ∗
∗
= (1 − θ1 )S (nT )e . Then S ((N1 + k)T ) ≥ (1 − θ1 )k S (nT )ekσ → ∞ as k → ∞, which is a contradiction to the boundedness of S (t ). Hence there ¯ 1. exists a t1 > 0 such that S (t ) ≥ m ¯ 1 for all t ≥ t1 , then our aim is attained. Hence, we only need to consider those solutions which leave Step 2. If S (t ) ≥ m ¯ 1 }, there are two possible ¯ 1 } and reenter it again. Let t ∗ = inft >t1 {S (t ) < m region R = {(S (t ), I (t ), P (t )) ∈ R3+ : S (t ) < m cases for t ∗ . ¯ 1 for t ∈ [t1 , t ∗ ) and Case I. If t ∗ = nT , n ∈ N, then S (t ) ≥ m
¯ 1 ≤ S (t ∗+ ) = (1 − θ1 )S (t ∗ ) < m ¯ 1. (1 − θ1 )m Choose n2 , n3 ∈ N, such that
ε1 ε1 n2 T > max ln /(−µ + δ1 ), ln /(−γ + δ2 ) , M +u M +v
(1 − θ1 )n2 en3 σ en2 σ1 T > (1 − θ1 )n2 en3 σ e(n2 +1)σ1 T > 1, ¯ +M ) r (m
1 − (λ + α)M < 0. Put T 0 = n2 T + n3 T , we claim that there must exist a t10 ∈ (t ∗ , t ∗ + T 0 ], such that where σ1 = r − K ¯ 1 . Otherwise, consider the solutions of (4.14) and (4.15) with z1 (t ∗+ ) = I (t ∗+ ) and z2 (t ∗+ ) = P (t ∗+ ), we have S1 (t10 ) > m
z1 (t ) =
z1 (t ∗+ ) −
u 1 − (1 − θ2
)e(−µ+δ1 )T
∗ e(−µ+δ1 )(t −t ) + z1∗ (t )
and z2 (t ) =
z2 (t
∗+
v )− 1 − (1 − θ3 )e(−γ +δ2 )T
∗ e(−γ +δ2 )(t −t ) + z2∗ (t )
for t ∈ ((n − 1)T , nT ], n1 ≤ n ≤ n1 + n2 + n3 . Therefore,
|z1 (t ) − z1∗ (t )| < (M + u) exp((−µ + δ1 )(t − (n1 + 1)T )) < ε1 , and
|z2 (t ) − z2∗ (t )| < (M + v) exp((−γ + δ2 )(t − (n1 + 1)T )) < ε1 . Hence we have I (t ) ≤ z1 (t ) ≤ z1∗ (t ) + ε1 and P (t ) ≤ z2 (t ) ≤ z2∗ (t ) + ε1 for t ∗ + n2 T ≤ t ≤ t ∗ + T 0 , which implies (4.16) holds on [t ∗ + n2 T , t ∗ + T 0 ]. Similar to that in step one, we have S (t ∗ + T 0 ) ≥ S (t ∗ + n2 T )en3 σ . If t ∈ [t ∗ , t ∗ + n2 T ], it follows from (2.4) that
S 0 (t ) ≥ S (t ) r −
¯ 1 + M) r (m
S (t + ) = (1 − θ1 )S (t ),
K
− (λ + α)M
= σ1 S (t ),
t 6= nT , t = nT .
Integrating (4.17) on [t ∗ , t ∗ + n2 T ], we have
¯ 1 (1 − θ1 )n2 eσ1 n2 T . S (t ∗ + n2 T ) ≥ m Thus we have
¯ 1 (1 − θ1 )n2 eσ1 n2 T en3 σ > m ¯ 1, S (t ∗ + T ) ≥ m which is a contradiction. ¯ 1 }, then S (t¯) > m ¯ 1 for t ∈ [t ∗ , t˜) and S (t ∗ ) = m, ¯ we have Let t˜ = inft >t ∗ {x1 (t ) > m ∗ ¯ 1 eσ1 (n2 +n3 ) = m1 S (t ) ≥ (1 − θ1 )n2 +n3 S (t ∗ )eσ1 (t −t ) ≥ (1 − θ1 )n2 +n3 m
¯ 1. for t ≥ t˜. So we have S (t ) ≥ m1 . The same arguments can be continued since S (t¯) ≥ m
(4.17)
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
819
Table 1 Threshold values for the stability and permanence of the systems, with regard to u and θ1 . Parameter
GAS of (0, I1 , P1 )
GAS of (0, I ∗ (t ), P ∗ (t ))
Permanence
θ1 = 0.3 u = 0.3
u > uˆ = 0.5325 θ1 > θˆ1 = 0.867606
u > u˜ = 0.507 θ1 > θ˜1 = 0.578695
u < u¯ = 0.264757 θ1 < θ¯1 = 0.236819
¯ 1 for t ≥ t 0 , and S (t ∗ ) = m. ¯ Let t ∗ = inft ≥t 0 {S (t ) < m ¯ }, then S (t ) ≥ m ¯ for t ∈ (t 0 , t ∗ ) Case II. If t ∗ 6= nT , n ∈ N, then S (t ) ≥ m 0 0 0 0 0 ∗ ∗ 00 ¯ For t ∈ [n1 T , (n1 + 1)T ), n1 ∈ N, we claim that must exist a t ∈ [(n1 + 1)T , (n1 + 1)T + T 0 ] such that and S (t ) = m. ¯ 1. S (t 00 ) ≥ m Considering (4.14) and (4.15) z1 ((n01 + 1)t + ) = I ((n01 + 1)t + ) and z2 ((n01 + 1)t + ) = P ((n01 + 1)t + ), we have z1 (t ) =
z1 ((n1 + 1)t ) − +
u
e(−µ+δ1 )(t −(n1 +1)T ) + z1∗ (t ),
e(−γ +δ2 )(t −(n1 +1)T ) + z2∗ (t ),
1 − (1 − θ2 )e(−µ+δ1 )T
0
and z2 (t ) =
z2 ((n1 + 1)t + ) −
v 1 − (1 − θ3 )e(−µ+δ2 )T
0
for t ∈ (nT , (n + 1)T ], n01 + 1 ≤ n ≤ n01 + 1 + n2 + n3 , then
|z1 (t ) − z1∗ (t )| < (M + u) exp((−µ + δ1 )(t − (n01 + 1)T )) < ε1 , and
|z2 (t ) − z2∗ (t )| < (M + v) exp((−γ + δ2 )(t − (n01 + 1)T )) < ε1 . Hence I (t ) ≤ z1 (t ) ≤ z1∗ (t ) + ε1 and P (t ) ≤ z2 (t ) ≤ z2∗ (t ) + ε1 for (n01 + 1 + n2 )T ≤ t ≤ (n01 + 1 + T 0 )T , which implies (4.16) holds for (n01 + 1 + n2 )T ≤ t ≤ (n01 + 1 + T 0 )T . As in step 1, we have S ((n01 + 1 + n2 + n3 )T ) ≥ (1 − θ1 )S ((n01 + 1 + n2 )T )en3 σ . Again integrating (4.17) on [t ∗ , (n01 + 1 + n + 2)T ], we have
¯ 1 (1 − θ1 )n2 +1 eσ1 (n2 +1)T . S ((n01 + 1 + n2 )T ) ≥ m Thus we have
¯ 1 (1 − θ1 )n2 +1 eσ1 (n2 +1)T en3 σ > m ¯ 1, S ((n01 + 1 + n2 + n3 )T ) ≥ m ¯ 1 }, thus S (t¯) ≥ m ¯ 1 for t ∈ [t ∗ , t¯], so we have which is a contradiction. Let t¯ = inft ≥t ∗ {S (t ) ≥ m ∗ ¯ σ1 (1+n2 +n3 ) = m1 S (t ) ≥ S (t ∗ )eσ1 (t −t ) ≥ me
The proof is completed.
for t ≥ t1 .
Corollary 4.2. System (2.2) is permanent provided that u<
µK [1 − (1 − θ2 ) exp(−µT )] αv(1 − exp(−γ T )) 1 . rT − − ln = u¯ , (r + λK )(1 − exp(−µT )) γ [1 − (1 − θ3 ) exp(−γ T )] 1 − θ1
v<
γ [1 − (1 − θ3 ) exp(−γ T )] u(r + λK )(1 − exp(−µT )) 1 . rT − − ln = v¯ . α(1 − exp(−γ T )) µK [1 − (1 − θ2 ) exp(−µT )] 1 − θ1
or
Corollary 4.3. System (2.2) is permanent provided that u(r + λK )(1 − exp(−µT ))
αv(1 − exp(−γ T )) θ1 < 1 − exp rT − − µK [1 − (1 − θ2 ) exp(−µT )] γ [1 − (1 − θ3 exp(−γ T ))]
−1
. = θ¯1 .
5. Numerical simulations First, to confirm our mathematical findings and facilitate their interpretation, we proceed to investigate them by using numerical simulations (Table 1). Let parameters r = 0.8, λ = 0.6, α = 0.35, µ = 0.35, a = 0.5, v = 0.6, γ = 0.35, β = 0.25, K = 3, T = 2, θ1 = 0.3, θ2 = 0.01, θ3 = 0.01 and S (0) = 0.6, I (0) = 0, P (0) = 0.4. It is seen from Corollary 3.3 that for u > uˆ or θ1 > θˆ1
820
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
a
b
Fig. 1. (a) The stability of susceptible pest-eradication equilibrium (0, I1 , P1 ) of system (2.1). (b) The stability of the positive equilibrium.
a
b
c
Fig. 2. The susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )). Here (a), (b) and (c) are the time series of S (t ), I (t ) and P (t ), respectively.
a
b
c
Fig. 3. The permanence of system (2.2), where (a), (b) and (c) are the time series of S (t ), I (t ) and P (t ), respectively.
the susceptible pest-eradication equilibrium (0, I1 , P1 ) of system (2.1) is globally asymptotically stable. Indeed, in Fig. 1(a), we capture a stable behavior of the susceptible pest-eradication equilibrium of system (2.1) for θ1 = 0.3, u = 0.7 > uˆ = 0.5175, and a stable behavior of the positive equilibrium of system (2.1) for θ1 = 0.3, u = 0.1 < uˆ = 0.5325 in Fig. 1 (b). According to Corollary 4.1, a suitable increase in the value of u has the potential to stabilize the behavior of the system (2.2) globally and make all solutions tend to the susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )). A typical example of stable behavior is captured in Fig. 2 for θ1 = 0.3, u = 0.7 > u˜ = 0.507. For u = 0.23 < u¯ = 0.264757, it follows from Corollary 4.2 that in this case system (2.2) is permanent. The behavior of the trajectory is depicted in Fig. 3. Next, we examine the behaviors of the susceptible pest-eradication solution when the natural enemies have the same effect on infected as on susceptible pests. Suppose that the predation effect on infected pests is ξ IP /(1 + bI ) and the conversion rate is η/ξ . Then the behaviors of the trajectories are captured in Fig. 4 (a) for T = 2, θ1 = 0.25, θ2 = 0.01, θ3 = 0.01, u = 0.35, v = 0.6, ξ = 0.35, η = 0.25, b = 0.5. and in Fig. 4(b) for T = 2, θ1 = 0.25, θ2 = 0.01, θ3 = 0.01, u =
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
a
821
b
Fig. 4. The behavior difference of the susceptible pest when the natural enemies have the same effect on infected as on susceptible pest. The solid line denotes the level of the susceptible when the natural enemies catch the infected pests. The dotted line denotes the level of the susceptible when the natural enemies do not feed on the infected pests.
0.1, v = 0.6, ξ = 0.35, η = 0.25, b = 0.5. Fig. 4 indicates that it has great effect on the level of the susceptible pests whether the natural enemies catch the infected pests or not. From a biological point of view, our assumption that the natural enemies of the pests do not catch the infective pests would tend to increase the spread of the infections. Hence it results in the lower level of the susceptible pests. Finally, from Table 1 together with Figs. 1 and 2, it is preferable to control the target pest population by using an impulsive control since u˜ < uˆ and θ˜1 < θˆ1 , that is there needs fewer infected pests or pesticides under the impulsive control than the continuous control. 6. Conclusion In this paper, we construct two integrated pest management models which rely on releasing of infective pest individuals and of natural predators in a constant amount, together with spraying of pesticides in a more flexible manner. In the case in which a continuous control is used, we obtain the condition which guarantees that the stability of susceptible pest-eradication equilibrium. In the case in which an impulsive control is used, the stability of the susceptible pesteradication periodic solution is guaranteed. Consequently, the condition for the permanence of the system is obtained. From Corollaries 3.1 and 4.1, we know that the releasing amount of infected pest or the pesticide effect on the susceptible pest plays an important role in the system which determines the extinction of the susceptible pest. From our models, we know that the natural enemy could not in itself bring about extinction of the population of susceptible pests since it would hold the susceptible population to a steady state level γ /(β − αγ ) without the release effect and the pesticide effect. If this is sufficiently low (below the economic threshold) there will be no need for the other pest control methods. Otherwise, integrated pest managements would be used to attain this objective. Although, these managements generally are quite costly, but they are useful during the fastigium of the oviposition. From pest control point of view, our results indicate that (1) the susceptible pests can be eradicated if the release amount of infected pests is above some threshold or the amount of spraying of pesticides is above another threshold; (2) the system is permanent under some conditions; (3) fewer infected pests or pesticides are needed as the impulsive strategy is taken, displaying its positive effect on the pest control; (4) our assumption that the natural enemies of the pests do not catch the infective pests would reduce the level of the susceptible pests. Acknowledgements The authors would like to thank the editor and the referee for constructive comments which significantly improve this paper. Funding: The National Natural Science Foundation of China (10971037). References [1] S. Tang, R.A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol. 50 (2005) 257–292. [2] V.M. Stern, Economic thresholds, Ann. Rev. Entomol. (1973) 259–280. [3] J.C. Van Lenteren, Integrated pest management in protected crops, in: D. Dent (Ed.), Integrated Pest Management, Chapman & Hall, London, 1995, pp. 311–320. [4] J.C. Van Lenteren, Measures of success in biological control of arthropods by augmentation of natural enemies, in: S. Wratten, G. Gurr (Eds.), Measures of Success in Biological Control, Kluwer Academic Publishers, Dordrecht, 2000, pp. 77–89. [5] J.C. Van Lenteren, J. Woets, Biological and integrated pest control in greenhouses, Ann. Rev. Entmol. 33 (1988) 239–250.
822
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
[6] M.L. Flint, Integrated pest management for walnuts, University of California Statewide Integrated Pest Management Project, Division of Agriculture and Natural Resources, Second Edition, University of California, Oakland, CA, publication 3270, 1987, pp. 3641. [7] P. Georgescu, H. Zhang, An impulsively controlled predator–pest model with disease in the pest, Nonlinear Anal. Real World Appl. (2008) doi:10.1016/j.nonrwa.2008.10.060. [8] L.A. Falcon, Use of Bacteria for Microbial Control of Insects, Academic Press, New York, NY, 1971. [9] L.A. Falcon, Problems associated with the use of arthropod viruses in pest control, Ann. Rev. Entomol. 21 (1976) 305–324. [10] N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Griffin, London, 1975. [11] H.D. Burges, N.W. Hussey, Microbial Control of Insects and Mites, Academic Press, New York, NY, 1971. [12] M. Sarfraz, A. Keddie, L. Dosdall, Biological control of the diamondback moth, Plutella xylostella: A review, Biocontrol Sci. Tech. 15 (2005) 763–789. [13] D. Grzywacz, M. Parnell, G. Kibata, G. Oduor, W. Ogutu, D. Miano, D. Winstanley, The development of endemic baculoviruses of Plutella xylostella (diamondback moth, DBM) for control of DBM in East Africa, in: N. Endersby, P. Ridland (Eds.), The Management of Diamondback Moth and other Cruciferous Pests (Proceedings of the Fourth International Workshop on Diamondback Moth, Melbourne University), 2001, pp. 179–183. [14] R. Roush, Insecticide resistance management in diamondback moth: quo vadis? in: A. Sivapragasam, W.H. Loke, A.K. Hussan, G.S. Lim (Eds.), The Management of Diamondback Moth and Other Crucifer Pests, (Proceedings of the Third International Workshop, Kuala Lumpur, Malaysia, Malaysian Agricultural Research and Development Institute (MARDI)), 1997, pp. 21–24. [15] J. Jiao, L. Chen, A pest management SI model with periodic biological and chemical control concern, Appl. Math. Comput. 183 (2006) 1018–1026. [16] J. Hui, D. Zhu, Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos Solitons Fractals 29 (2006) 233–251. [17] K. Nagai, E Yano, Predation by Orius sauteri (Poppius) (Heteroptera: Anthocoridae) on Thrips palmi Karny (Thysanoptera: Thripidae). Functional response and selective predation, Appl. Entomol. Zool. 35 (2000) 565–574. [18] C.M. Port, N.E.A. Scopes, Biological control by predatory mites (Phytoseiulus persimilis Athias–Henriot) of red spider mite (Tetranychus urticae Koch) infesting strawberries grown in walk-in plastic tunnels, Plant Pathol. 30 (1981) 95–99. [19] B. Liu, Y. Zhang, The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management, Nonlinear Anal. Real World Appl. 6 (2005) 227–243. [20] R. Shi, L. Chen, A predator–prey model with disease in the prey and two impulses for integrated pest management, Appl. Math. Mode. 33 (2009) 2248–2256. [21] F. Wang, Bifurcation and chaos of a pest-control food chain model with impulsive effects, Chaos Solitons Fractals 39 (2009) 1903–1914. [22] M. Martcheva, Evolutionary consequences of predation for pathogens in prey, B. Math. Biol. 71 (2009) 819–844. [23] Z. Liu, Positive periodic solution for neutral delay competitive system, J. Math. Anal. Appl. 293 (2004) 181–189. [24] Y. Kuang, Delay Differential Equations with Appilcations in Population Dynamics, Academiac Press. INC. Harcourt Brace Iovanovich. Publishers, Boston, 1993.
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Pest regulation by means of continuous and impulsive nonlinear controls Pei Yongzhen a,∗ , Ji Xuehui a , Li Changguo b a
School of Science, Tianjin Polytechnic University, Tianjin, 300160, China
b
Department of Basic Science of Institute of Military Traffic, Tianjin, 300161, China
article
info
Article history: Received 21 March 2009 Received in revised form 17 October 2009 Accepted 19 October 2009 Keywords: Continuous control Impulsive control Susceptible pests Infective pests Natural enemy Permanence
abstract In this paper, two integrated pest management models are investigated, which rely on release of infective pest individuals and of natural enemies in a constant amount, together with spraying of pesticides. It is proved that the susceptible pests can be eradicated if the release amount of infected pests is above some threshold or the pesticide effect is above another threshold. Furthermore, permanent conditions are established when an impulsive control is used. Finally, numerical results show that (1) fewer infected pests or pesticides are needed as the impulsive strategy is taken, displaying its positive effect on the pest control; (2) our assumption that the natural enemies of the pests do not catch the infective pests would reduce the level of the susceptible pests. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Controlling insects and other arthropods has become an increasingly complex issue over the past two decades. Overuse of a single control tactic is discouraged to avoid or delay the development of resistance by the pest to the control tactic, to minimize damage to non-target organisms, and to preserve the quality of the environment [1]. Integrated pest management (IPM), which was introduced in the late 1950s [2] but more widely practiced during the 1970s and 1980s [3–6], promotes the use of several measures such as insecticide applications, crop rotation, biological control, harvest management, and the use of pest-resistant varieties of crops to reduce pest populations below economic levels [1]. Biological insecticides are used in IPM such as Bacillus thuringiensis or entomopathogenic fungi, disrupting the reproductive processes of the pest by releasing sterile pest individuals or spreading a disease in the pest population on the grounds that infective pests are usually less damaging to the environment [7]. And bacillus thuringiensis, which is available in commercial preparations, is used in the control of a large number of pests [8–11]. Another successful case of biological control in greenhouses is the use of the parasitoid Encarsia formosa against the greenhouse whitefly Trialeurodes vaporariorum on tomatoes and cucumbers [4,5]. Additionally, Baculoviruses have been found useful to control the diamondback moth Plutella xylostella in cabbage farms [12,13] after it had been noticed that the diamondback moth became resistant to chemical pesticides in many areas [14]. An advantage of using insect pathogens is that they are safe to man and are usually safe to beneficial insects [15]. Breeding natural enemies to control pest or regulate its density below the threshold for economical damage also is used by people. Natural biological control results when naturally occurring enemies maintain pests at a lower level than what would occur without them, and is generally characteristic of biodiverse systems [16]. Some examples of successful uses of biocontrol agents include the use of the predatory arthropod Orius sauteri against the pest Thrips palmi Karny to protect eggplant crops in greenhouses [17]. Other examples are the use of the predatory mites Phytoseiulus persimilis and Neoseiulus californicus against the red spider mite Tetranychus urticae Koch in field-grown strawberries [18] and welltimed inundant releases of Trichogramma egg wasps for codling moth control [16].
∗
Corresponding author. E-mail address: [email protected] (Y. Pei).
0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.10.013
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
811
Chemical treatments are used only in a crisis situation threatening rapid losses or when pests fail to succumb to more conservative methods. Pesticides have become less attractive because of a growing awareness that the chemicals in pesticides can pose health hazards to people [16]. But we do not argue for the abandonment of the use of pesticides, since they quickly kill a significant portion of a pest population and they sometimes provide the only feasible method for preventing economic loss [19]. Pesticides should be used only when other measures, such as biological controls, have failed to keep pest populations from approaching economically damaging levels [16]. Recently, many papers have been devoted to the analysis of mathematical models describing IPM strategies [4–7,19–21]. In [20] a predator–prey model with disease in the prey is constructed and investigated for the purpose of integrated pest management. Paul and Zhang [7] consider an integrated pest management model with disease in the pest and stage structure for its natural predator. But chemical treatment is not considered. Wang and Pang [21] investigate the dynamics of a predator–prey food chain with impulsive effect, periodic releasing natural enemies and spraying pesticide at different fixed times. Hui and Zhu [16] investigate the natural enemy–pest models with age structure for the predator, immature and mature natural enemies are released and pesticide is applied impulsively. However, releasing infective pest individuals is not investigated. As motivated by the above-mentioned brief literature survey, in this paper, we will construct two integrated pest management models which rely on release of infective pest individuals and of natural predators in a constant amount, together with spraying of pesticides in a more flexible manner. The infective pests can be cultivated in the laboratory and the natural enemy can be introduced from other regions. Once the susceptible pest meets with the infective pest, there is a chance to be infected. The infective pests have more possibility on death due to the disease and are less damaging to the crops and environment. We suppose the natural enemies catch the susceptible pests, but do not catch the infective pests. All species will be affected by pesticides. Because infected pests and predators are introduced in a uniform manner, our models may be quite appropriate for application to pest control in greenhouses rather than in normal fields where the space might have to be taken explicitly into account. The paper is organized as follows. In Section 2, two integrated pest management models are constructed and the main biological assumptions are formulated. In Section 3, the conditions of the local and global stability of the susceptible pest-eradication equilibrium are obtained. In Section 4, by using Floquet’s theory for impulsive differential equations, small-amplitude perturbation methods and comparison techniques, we investigate the global asymptotic stability of the susceptible pest-eradication periodic solution and the conditions for the permanence of the system. We demonstrate our results by numerical simulation in Section 5. Finally, we give a brief discussion by comparing the continuous model with the corresponding impulsive one and establish the feasible control strategies. 2. Model formulation In a natural ecosystem, there are two main types of predators—generalist and specialist. Generalist predators feed on many types of species. But specialist predators feed almost exclusively on one specific species of prey which may be possible for a parasitoid [22]. Based on this fact, we study the following two systems of differential equations for integrated pest management. (I) Continuous control
α SP θ1 S (t ) S+I 0 − λIS − − , S ( t ) = rS 1 − K 1 + aS T θ2 I (t ) u I 0 (t ) = λIS − µI − + T T P 0 (t ) = β SP − γ P − θ3 P (t ) + v . 1 + aS T T
(2.1)
(II) Impulsive control
S+I α SP 0 S ( t ) = rS 1 − − λ IS − , K 1 + aS 0 t 6= nT , I ( t ) = λ IS − µ I , β SP 0 P (t ) = − γ P, 1 + aS ) ∆S (t ) = −θ1 S (t ), t = nT , n = 1, 2, . . . . ∆I (t ) = −θ2 I (t ) + u, ∆P (t ) = −θ3 P (t ) + v,
(2.2)
Here functions S (t ) and I (t ) represent the densities of the susceptible prey (pest) population and the infective prey (pest) population, respectively. The function P (t ) is the density of predator (natural enemy) population. The model is derived with the following assumptions. (A.1) There is a disease among the prey population, and the prey is divided into two classes, susceptible and infective. The incidence rate is the classic bilinear λIS, and λ is the contact number per unit time for every infective prey with susceptible prey.
812
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
(A.2) The prey population grows according to a logistic fashion with the intrinsic growth rate r and the carrying capacity K . We suppose that the infective prey cannot produce offspring due to the disease; however, the infective prey still consume some crop. (A.3) The predators only catch the susceptible prey, and the predation functional response is Holling type II. The parameters α, a are positive constants, and β/α is the conversion rate from the prey to the predator. (A.4) Parameters µ, γ are the death rate for the infective prey and the predator, respectively. (A.5) We release infective pest population and predator population with releasing amount u(>0) and v(>0) respectively. Parameter T is the period of the impulse. (A.6) Parameters θi (i = 1, 2, 3) represent the portion of susceptible, infective pests and natural enemies, due to spraying of pesticides, respectively. 3. Continuous control In this section, we investigate the dynamics of the ordinary differential system (2.1) by means of stability analysis and apply the subsequently obtained stability results to the study of our control problem. For biological reasons, we restrict our discussion to the feasible region
{(S , I , P )|S ≥ 0, I ≥ 0, P ≥ 0}. The following Lemma implies that all solutions of system (2.1) are ultimately bounded. Lemma 3.1. For system (2.1), there exists a positive constant M1 such that S (t ) ≤ M1 , I (t ) ≤ M1 , P (t ) ≤ M1 for t large enough. Proof. Define function V (t ) = β S (t ) + β
r + 1 I (t ) + α P (t ), λK
and let δ = min{µ, γ }, then we have
h
r
h
K r
D+ V (t ) + δ V (t ) = β rS (t ) + δ S (t ) −
≤ β rS (t ) + δ S (t ) −
K
i
S 2 (t ) − β
r + 1 (µ − δ)I (t ) − α(γ − δ)P (t ) λK
i
S 2 (t ) ≤ M0 ,
(r +δ)2 K
where M0 = . 4r It follows from the methods of [16,23,24] that V (t ) is uniformly ultimately bounded. Thus, by the definition of V (t ), there exists a constant M1 > 0 such that S (t ) ≤ M1 , I (t ) ≤ M1 , P (t ) ≤ M1 for large enough t. The proof is completed. In what follows, the existence and stability of equilibria of the model (2.1) are investigated. There is one boundary equilibrium E1 (0, I1 , P1 ) =
0,
u
µT + θ 2
,
v γ T + θ3
,
representing the extinction of the susceptible pest population, the density of infected pest population and of predator population equilibrating at the carrying capacity. Below we will consider this equilibrium E1 . For the stability of the susceptible pest-eradication equilibrium we have one of the following main Theorems of this paper. The existence and stability of the positive equilibrium will be studied by means of numerical simulation. Theorem 3.1. The susceptible pest-eradication equilibrium E1 is locally asymptotically stable provided r <
u( r + λ K ) K (µT + θ2 )
+
αv . γ T + θ3
(3.1)
Otherwise, it is unstable. Proof. The Jacobian matrix of system (2.1) at E1 takes the form of
r− JE1 =
rI1
− λI1 − α P1 K λI1 β P1
0
−µ 0
0
. 0
−γ
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
813
It is easy to obtain that the eigenvalues for the matrix JE1 are
λ1 = −µ < 0, λ2 = −γ < 0, r + λK · λ3 = r −
αv u − . µT + θ2 γ T + θ3 If condition (3.1) holds, then λ3 < 0. Hence the susceptible pest-eradication equilibrium E1 is locally asymptotically K
stable. Otherwise, equilibrium E1 is unstable. This completes the proof.
Theorem 3.2. The susceptible pest-eradication equilibrium E1 is globally asymptotically stable provided r <
θ1 T
+
u(r + λK ) K (µT + θ2 )
.
(3.2)
Proof. For the globally asymptotical stability of the equilibrium E1 , we define the following Lyapunov function:
V1 = c1 S + c2
I − I1 − I1 ln
I
I1
P + c3 P − P1 − P1 ln , P1
where ci (i = 1, 2, 3) are positive constants and satisfy c1 =
λK r + λK
· c2 =
β · c3 = K . α
Along the solution of model (2.1), we have d dt
V1 = c1 S˙ + c2
(I − I1 ) ˙
I + c3
I
S+I
= c1 S r 1 −
1 + aS
= S c1 r − −
c1 θ1 T
−
P
−
αP θ1 c2 (µT + θ2 )(I − I1 )2 − λI − − + c2 λS (I − I1 ) − 1 + aS T IT 2 c3 (γ T + θ3 )(P − P1 ) PT
c2 λu
µT + θ2
c3 (γ T + θ3 )(P − P1 )2 PT
It follows from (3.2) that we have
P˙
K
c3 β S (P − P1 )
+
P − P1
d V dt 1
−
c3 βv S
(γ T + θ3 )(1 + aS )
−
c1 rS 2 K
−
c2 (µT + θ2 )(I − I1 )2 IT
.
< 0, which implies the equilibrium E1 is globally asymptotically stable.
Corollary 3.1. The susceptible pest-eradication equilibrium E1 is globally asymptotically stable provided u>
K (rT − θ1 )(µT + θ2 ) . = uˆ , T (r + λK )
(3.3)
or
θ1 > rT −
uT (r + λK ) . = θˆ1 .
K (µT + θ2 )
(3.4)
Remark 3.1. By Corollary 3.1, we know that the parameter uˆ is a releasing amount critical value of the infective pest population. Conditions (3.3) and (3.4) imply that releasing more infected pests or spraying more pesticides guarantees the extinction of the susceptible pests in a long term. 4. Pulse control In this section, we investigate the dynamics of system (2.2) by means of stability analysis and apply the subsequently obtained stability results to the study of our control problem. First, it is seen that the long-term survival of the infective pest population and of the predator populations is assured by the pulsed supply of individuals which occurs for t = nT , n ∈ N, while the susceptible pest population may be driven to extinction in certain circumstances, since a pulsed supply of susceptibles is not present. Also, at least when the susceptible pest population (the only one which is not impulsively controlled) tends to extinction, it is natural to expect that the solutions of (2.2) tend to a limiting periodic solution due to the forcing effects of the periodic impulsive perturbations. Thus we will study the existence and stability of the susceptible pest-eradication periodic solution (0, I ∗ , P ∗ ). To this purpose, it is seen
814
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
first that when S (t ) = 0, the system (2.2) reduces to
I (t ) = −µI , P 0 (t ) = −γ P ,
t 6= nT ,
I (t ) = (1 − θ2 )I (t ) + u, + P (t ) = (1 − θ3 )P (t ) + v, +
(4.1)
t = nT ,
which describes the dynamics of the system in the absence of the susceptible pest population. Obviously, system (4.1) has a positive periodic solution ue−µ(t −nT )
I ∗ ( t ) =
1−
P ∗ (t ) =
(1 − θ2 )e−µT −γ (t −nT )
,
I ∗ (0+ ) =
ve , 1 − (1 − θ3 )e−γ T
u 1 − (1 − θ2 )e−µT
P ∗ (0+ ) =
,
v , 1 − (1 − θ3 )e−γ T
t ∈ (nT , (n + 1)T ].
(4.2)
Since the solution of (4.1) is
u ∗ n ∗ + I (t ) = (1 − θ2 ) I (0 ) − 1 − (1 − θ ) exp(−µt ) exp(−µt ) + I (t ), 2 v P (t ) = (1 − θ3 )n P ∗ (0+ ) − exp(−γ t ) + P ∗ (t ), 1 − (1 − θ3 ) exp(−γ t )
t ∈ (nT , (n + 1)T ].
(4.3)
we obtain the following lemma. Lemma 4.1. System (4.1) has a positive periodic solution (I ∗ (t ), P ∗ (t )) and for every solution (I (t ), P (t )) of (4.1), I (t ) → I ∗ (t ), P (t ) → P ∗ (t ) as t → ∞. Therefore, system (2.2) has a pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )). Having proven the existence of the susceptible pest-eradication periodic solution, we are now ready to study its stability. Theorem 4.1. Let (S (t ), I (t ), P (t )) be any solution of (2.2). Then the susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )) is locally stable if rT −
u(r + λK )(1 − exp(−µT )) αv(1 − exp(−γ T )) 1 − < ln , µK [1 − (1 − θ2 ) exp(−µT )] γ [1 − (1 − θ3 ) exp(−γ T )] 1 − θ1
(4.4)
and unstable if the reverse inequality holds. Proof. The stability of periodic solution (0, I ∗ (t ), P ∗ (t )) may be determined by considering the behavior of small-amplitude perturbation of the solution. Denoted by S (t ) = x(t ),
I (t ) = y(t ) + I ∗ (t ),
P (t ) = z (t ) + P ∗ (t ),
the corresponding linear system of (2.2) at (0, I ∗ (t ), P ∗ (t )) reads as
I∗ ∗ ∗ 0 − λ I x − α P x , x ( t ) = rx 1 − K t 6= nT , y0 (t ) = λI ∗ x − µy, 0 ∗ z (t ) = β P x − γ z , ) ∆ x(t ) = −θ1 x(t ), t = nT , n = 1, 2, . . . . ∆y(t ) = −θ2 y(t ), ∆z (t ) = −θ3 z (t ),
(4.5)
So a fundamental matrix Φ (t ) of (4.5) satisfies
dΦ (t ) dt
rI ∗
r − K − λ I − α P = λI ∗ βP∗ ∗
∗
0
−µ 0
0 Φ (t ) 0
−γ
and Φ (0) = E3 , the identity matrix. Hence the fundamental solution matrix is
Φ (t ) =
Z t r−
exp 0
rI ∗ (s) K
− λI ∗ (s) − α P ∗ (s) ds ∆1 ∆2
0
0
exp{−µt } 0
0
.
exp{−γ t }
It is no need to give the exact form of ∆1 and ∆2 as they are not required in the analysis that follows.
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
815
If all eigenvalues of 1 − θ1 0 0
M =
0
0 0
1 − θ2 0
! Φ (T )
1 − θ3
have absolute values less than one, then the periodic solution (0, I ∗ (t ), P ∗ (t )) is locally stable. Since eigenvalues of M are
µ1 = (1 − θ1 ) exp
T
Z
r−
rI ∗ (s)
0
K
− λI (s) − α P (s) ds < 1, ∗
∗
µ2 = (1 − θ2 ) exp{−µt }, µ3 = (1 − θ3 ) exp{−γ t }, then |µ1 | < 1 if and only if (4.4) holds. According to Floquet’s theory of impulsive differential equations, the susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )) is locally stable. The proof is complete. Below we will observe the globally stability of the susceptible pest-eradication periodic solution. Theorem 4.2. The susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )) is globally asymptotically stable, provided that rT −
u(r + λK )(1 − exp(−µT )) 1 < ln . µK (1 − (1 − θ2 ) exp(−µT )) 1 − θ1
(4.6)
Proof. Choose ε > 0 such that
η = (1 − θ1 ) exp
Z t r−
r (I ∗ (s) − ε) K
0
− λ(I (s) − ε) ds < 1, ∗
(4.7)
Noting that I 0 (t ) ≥ −µI (t ) and P 0 (t ) ≥ −λP (t ), from Lemma 4.1 and the comparison theorem, we have I 0 (t ) ≥ I ∗ (t ) − ε, P 0 (t ) ≥ P ∗ (t ) − ε,
(4.8)
for all t large enough. For simplification, we may assume that (4.8) holds for t ≥ 0. From the first equation of (2.2) and (4.8), we get
S 0 (t ) ≤ rS (t ) 1 −
I ∗ (t ) − ε
S (t ) = (1 − θ1 )S (t ), +
K
− λ(I ∗ (t ) − ε)S (t ),
t 6= nT ,
(4.9)
t = nT .
Integrating (4.9) on (nT , (n + 1)T ], it is followed that S ((n + 1)T ) ≤ S (nT )(1 − θ1 ) exp
Z
(n+1)T
r−
nT
= S (nT )η = S (T )ηn .
rI ∗ (s) − r ε K
− λ(I (s) − ε) ds ∗
(4.10)
From (4.7), we have S ((n + 1)T ) → 0 as n → ∞. Therefore, S (t ) → 0 as t → ∞ since 0 < S (t ) ≤ S (nT )(1 − θ1 ) exp(rT ) for nT < t ≤ (n + 1)T . Then we prove that I (t ) → I ∗ (t ) as t → ∞. For 0 < ε1 < µ/λ, there must exist a t1 > 0, such that 0 < S (t ) < ε1 for t ≥ t1 . Without loss of generality, we assume that 0 < S (t ) < ε1 for all t ≥ 0, then from system (2.2) we have
(−λε1 − µ)I (t ) ≤ I 0 (t ) ≤ (λε1 − µ)I (t ). Again from Lemma 4.1 and comparison theorem, we have I1 (t ) ≤ I (t ) ≤ I2 (t ) and I1 (t ) → I1∗ (t ), I2 (t ) → I2∗ (t ) as t → ∞, where I1 (t ) and I2 (t ) are the solutions of
0 I1 (t ) = [−λε1 − µ]I1 (t ), I1 (t + ) = (1 − θ2 )I1 (t ) + u, I (0+ ) = I (0+ ), 1
t 6= nT , t = nT ,
0 I2 (t ) = [−λε1 − µ]I2 (t ), I2 (t + ) = (1 − θ2 )I2 (t ) + u, I (0+ ) = I (0+ ), 2
t 6= nT , t = nT ,
and
816
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
respectively, where I1∗ (t ) = I2∗ (t ) =
ue−(λε1 +µ)(t −nT ) 1 − (1 − θ2 )e−(λε1 +µ)T ue−(µ−λε1 )(t −nT ) 1 − (1 − θ2 )e−(µ−λε1 )T
,
t ∈ (nT , (n + 1)T ];
,
t ∈ (nT , (n + 1)T ].
Therefore, for any ε2 > 0, there exists a t2 > 0 such that I1∗ (t ) − ε2 ≤ I (t ) ≤ I2∗ (t ) + ε2 ,
for t > t2 .
(4.11)
Let ε1 → 0, we have I ∗ (t ) − ε2 ≤ I (t ) ≤ I ∗ (t ) + ε2 , for t large enough, which implies that I (t ) → I ∗ (t ) as t → ∞. Likewise, we prove that for any ε3 > 0, there exist t3 > 0, such that P ∗ (t ) − ε3 ≤ P (t ) ≤ P ∗ (t ) + ε3
(4.12)
for t > t3 , which implies that P (t ) → P (t ) as t → ∞. The proof is complete. ∗
Corollary 4.1. The susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )) is globally asymptotically stable, provided that u>
µK (1 − (1 − θ2 )e−µT )(rT + ln(1 − θ1 )) . = u˜ , (r + λK )(1 − e−µT )
or
θ1 > 1 − exp rT −
u(r + λK )(1 − exp(−µT ))
−1
µK [1 − (1 − θ2 ) exp(−µT )]
. = θ˜1 .
An important concept in IPM is that of the economic threshold (ET). An ET is usually defined as the number of insect pests in the field when control actions must be taken to prevent the economic injury level (EIL) from being reached and exceeded, where EIL is the lowest pest population density that will cause economic damage. From a pest control point of view, our aim is to keep pests at acceptably low levels; not to eradicate them. So we need to investigate the permanence of system (2.2). Before starting our theorems, we give the following definition. Definition 1. System (2.2) is said to be permanent if there are three positive constants m, M and T0 such that every positive solution (S (t ), I (t ), P (t )) of system (2.2) satisfies m ≤ S (t ) ≤ M ,
m ≤ I (t ) ≤ M ,
m ≤ P (t ) ≤ M .
for all t ≥ T0 . Theorem 4.3. There exists a constant M > 0, such that S (t ) ≤ M , I (t ) ≤ M and P (t ) ≤ M for each solution (S (t ), I (t ), P (t )) of system (2.2) with all t large enough. Proof. Define a function V (t ) as r V (t ) = β S (t ) + β + 1 I (t ) + α P (t ). λK For t 6= nT , denoted by δ = min{µ, γ }, we have
h
r
h
K r
D+ V (t ) + δ V (t ) = β rS (t ) + δ S (t ) −
≤ β rS (t ) + δ S (t ) −
K
i
S 2 (t ) − β
r + 1 (µ − δ)I (t ) − α(γ − δ)P (t ) λK
i
S 2 (t ) ≤ M0 ,
where M0 = (r + δ)2 K /4r. When t = nT , V (nT + ) ≤ V (nT ) + β( λrK + 1)u + αv . By comparison theorem, for t ≥ 0 we have
r e−δ(t −T ) r eδ T (1 − e−δt ) + β + 1 u + αv + β + 1 u + αv δ λK 1 − eδ T λK eδ T − 1 δT M0 r e → + β + 1 u + αv δT (as t → ∞). δ λK e −1 Then V (t ) is uniformly upper bounded. Hence, by the definition of V (t ), there exists a constant M > 0 such that S (t ) ≤ M , I (t ) ≤ M and P (t ) ≤ M for all t large enough. The proof is completed. V (t ) ≤ V (0)e−δ t +
M0
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
817
Theorem 4.4. System (2.2) is permanent provided that rT −
u(r + λK )(1 − exp(−µT )) αv(1 − exp(−γ T )) 1 − > ln . µK [1 − (1 − θ2 ) exp(−µT )] γ [1 − (1 − θ3 ) exp(−γ T )] 1 − θ1
(4.13)
Proof. Suppose that (S (t ), I (t ), P (t )) is a solution of (2.2) with S (0) > 0, I (0) > 0, P (0) > 0. From Theorem 4.3, we may assume I (t ) ≤ M,P (t ) ≤ Mand M > for t > 0. From (4.11) and (4.12) we know I (t ) > I ∗ (t ) − ε2 and P (t ) > P ∗ (t ) − ε3 for all t large enough, then I (t ) ≥
ue−µT 1 − (1 − θ2 )e−µT
− ε2 = m 2
and P (t ) ≥
v e−γ T − ε3 = m 3 1 − (1 − θ3 )e−γ T
for t large enough. Thus we only need to find m1 > 0 such that S (t ) ≥ m1 for t large enough. We will do it in the following two steps.
¯ 1 > 0 and ε1 > 0 small enough such that Step 1. By condition (4.13), one can select m ¯ 1 < min m
µ γ , λ β − aγ
,
δ1 = λM < µ,
δ2 =
βM <γ 1 + aM
and
σ = rT −
¯ 1T rm K
−
u(r + λK )(1 − exp((−µ + δ1 )T )) K (µ − δ1 )[1 − (1 − θ2 ) exp(−µT )]
−
αv(1 − exp((−γ + δ2 )T )) > 0. (γ − δ2 )[1 − (1 − θ3 exp(−γ T ))]
¯ 1 cannot hold for all t ≥ 0. Otherwise, note that if S (t ) ≥ 0, then We will prove S (t ) < m
I 0 (t ) ≤ I (t )(−µ + δ1 ), I (t + ) = (1 − θ2 )I (t ) + u,
t 6= nT , t = nT ,
P 0 (t ) ≤ P (t )(−γ + δ2 ), P (t + ) = (1 − θ3 )I (t ) + v,
t 6= nT , t = nT .
and
By Lemma 4.1 and Theorem 4.3, we have I (t ) ≤ z1 (t ), P (t ) ≤ z2 (t ) and z1 (t ) → z1∗ (t ), z2 (t ) → z2∗ (t ) as t → ∞, where z1∗ (t ) = z2∗ (t ) =
ue(−µ+δ1 )(t −nT )
,
t ∈ (nT , (n + 1)T ],
v e(−γ +δ2 )(t −nT ) , 1 − (1 − θ3 )e(−γ +δ2 )T
t ∈ (nT , (n + 1)T ],
1 − (1 − θ2 )e(−µ+δ1 )T
and z1 (t ) and z2 (t ) are the solutions of the following equations
0 z1 (t ) = (−µ + δ1 )z1 (t ), z1 (t + ) = (1 − θ2 )z1 (t ) + u, z (0+ ) = I (0+ ), 1
t 6= nT , t = nT ,
(4.14)
0 z2 (t ) = (−γ + δ1 )z2 (t ), z2 (t + ) = (1 − θ3 )z1 (t ) + v, z (0+ ) = P (0+ ), 2
t 6= nT , t = nT ,
(4.15)
and
respectively. Therefore, there exists a T1 > 0 such that for t ≥ T1 , we have I (t ) ≤ z1 (t ) ≤ z1∗ (t ) + ε1 , and P (t ) ≤ z2 (t ) ≤ z2∗ (t ) + ε1 .
818
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
Then
S (t ) ≥ S (t ) r
S (t ) = (1 − θ1 )S (t ),
0
1−
m¯1
+
K
−
z1∗ (t ) + ε1
K
− λ(z1 (t ) + ε1 ) − α(z2 (t ) + ε1 ) , ∗
∗
t 6= nT ,
(4.16)
t = nT .
Let N1 ∈ N and N1 T ≥ T1 , integrating (4.16) on (nT , (n + 1)T ](n ≥ N1 ), we have S ((n + 1)T ) ≥ S (nT )(1 − θ1 ) exp
(n+1)T
Z
r
1−
m¯1 + z1∗ (s) + ε1
nT
σ
K
− λ(z1 (s) + ε1 ) − α(z2 (t ) + ε1 ) ds ∗
∗
= (1 − θ1 )S (nT )e . Then S ((N1 + k)T ) ≥ (1 − θ1 )k S (nT )ekσ → ∞ as k → ∞, which is a contradiction to the boundedness of S (t ). Hence there ¯ 1. exists a t1 > 0 such that S (t ) ≥ m ¯ 1 for all t ≥ t1 , then our aim is attained. Hence, we only need to consider those solutions which leave Step 2. If S (t ) ≥ m ¯ 1 }, there are two possible ¯ 1 } and reenter it again. Let t ∗ = inft >t1 {S (t ) < m region R = {(S (t ), I (t ), P (t )) ∈ R3+ : S (t ) < m cases for t ∗ . ¯ 1 for t ∈ [t1 , t ∗ ) and Case I. If t ∗ = nT , n ∈ N, then S (t ) ≥ m
¯ 1 ≤ S (t ∗+ ) = (1 − θ1 )S (t ∗ ) < m ¯ 1. (1 − θ1 )m Choose n2 , n3 ∈ N, such that
ε1 ε1 n2 T > max ln /(−µ + δ1 ), ln /(−γ + δ2 ) , M +u M +v
(1 − θ1 )n2 en3 σ en2 σ1 T > (1 − θ1 )n2 en3 σ e(n2 +1)σ1 T > 1, ¯ +M ) r (m
1 − (λ + α)M < 0. Put T 0 = n2 T + n3 T , we claim that there must exist a t10 ∈ (t ∗ , t ∗ + T 0 ], such that where σ1 = r − K ¯ 1 . Otherwise, consider the solutions of (4.14) and (4.15) with z1 (t ∗+ ) = I (t ∗+ ) and z2 (t ∗+ ) = P (t ∗+ ), we have S1 (t10 ) > m
z1 (t ) =
z1 (t ∗+ ) −
u 1 − (1 − θ2
)e(−µ+δ1 )T
∗ e(−µ+δ1 )(t −t ) + z1∗ (t )
and z2 (t ) =
z2 (t
∗+
v )− 1 − (1 − θ3 )e(−γ +δ2 )T
∗ e(−γ +δ2 )(t −t ) + z2∗ (t )
for t ∈ ((n − 1)T , nT ], n1 ≤ n ≤ n1 + n2 + n3 . Therefore,
|z1 (t ) − z1∗ (t )| < (M + u) exp((−µ + δ1 )(t − (n1 + 1)T )) < ε1 , and
|z2 (t ) − z2∗ (t )| < (M + v) exp((−γ + δ2 )(t − (n1 + 1)T )) < ε1 . Hence we have I (t ) ≤ z1 (t ) ≤ z1∗ (t ) + ε1 and P (t ) ≤ z2 (t ) ≤ z2∗ (t ) + ε1 for t ∗ + n2 T ≤ t ≤ t ∗ + T 0 , which implies (4.16) holds on [t ∗ + n2 T , t ∗ + T 0 ]. Similar to that in step one, we have S (t ∗ + T 0 ) ≥ S (t ∗ + n2 T )en3 σ . If t ∈ [t ∗ , t ∗ + n2 T ], it follows from (2.4) that
S 0 (t ) ≥ S (t ) r −
¯ 1 + M) r (m
S (t + ) = (1 − θ1 )S (t ),
K
− (λ + α)M
= σ1 S (t ),
t 6= nT , t = nT .
Integrating (4.17) on [t ∗ , t ∗ + n2 T ], we have
¯ 1 (1 − θ1 )n2 eσ1 n2 T . S (t ∗ + n2 T ) ≥ m Thus we have
¯ 1 (1 − θ1 )n2 eσ1 n2 T en3 σ > m ¯ 1, S (t ∗ + T ) ≥ m which is a contradiction. ¯ 1 }, then S (t¯) > m ¯ 1 for t ∈ [t ∗ , t˜) and S (t ∗ ) = m, ¯ we have Let t˜ = inft >t ∗ {x1 (t ) > m ∗ ¯ 1 eσ1 (n2 +n3 ) = m1 S (t ) ≥ (1 − θ1 )n2 +n3 S (t ∗ )eσ1 (t −t ) ≥ (1 − θ1 )n2 +n3 m
¯ 1. for t ≥ t˜. So we have S (t ) ≥ m1 . The same arguments can be continued since S (t¯) ≥ m
(4.17)
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
819
Table 1 Threshold values for the stability and permanence of the systems, with regard to u and θ1 . Parameter
GAS of (0, I1 , P1 )
GAS of (0, I ∗ (t ), P ∗ (t ))
Permanence
θ1 = 0.3 u = 0.3
u > uˆ = 0.5325 θ1 > θˆ1 = 0.867606
u > u˜ = 0.507 θ1 > θ˜1 = 0.578695
u < u¯ = 0.264757 θ1 < θ¯1 = 0.236819
¯ 1 for t ≥ t 0 , and S (t ∗ ) = m. ¯ Let t ∗ = inft ≥t 0 {S (t ) < m ¯ }, then S (t ) ≥ m ¯ for t ∈ (t 0 , t ∗ ) Case II. If t ∗ 6= nT , n ∈ N, then S (t ) ≥ m 0 0 0 0 0 ∗ ∗ 00 ¯ For t ∈ [n1 T , (n1 + 1)T ), n1 ∈ N, we claim that must exist a t ∈ [(n1 + 1)T , (n1 + 1)T + T 0 ] such that and S (t ) = m. ¯ 1. S (t 00 ) ≥ m Considering (4.14) and (4.15) z1 ((n01 + 1)t + ) = I ((n01 + 1)t + ) and z2 ((n01 + 1)t + ) = P ((n01 + 1)t + ), we have z1 (t ) =
z1 ((n1 + 1)t ) − +
u
e(−µ+δ1 )(t −(n1 +1)T ) + z1∗ (t ),
e(−γ +δ2 )(t −(n1 +1)T ) + z2∗ (t ),
1 − (1 − θ2 )e(−µ+δ1 )T
0
and z2 (t ) =
z2 ((n1 + 1)t + ) −
v 1 − (1 − θ3 )e(−µ+δ2 )T
0
for t ∈ (nT , (n + 1)T ], n01 + 1 ≤ n ≤ n01 + 1 + n2 + n3 , then
|z1 (t ) − z1∗ (t )| < (M + u) exp((−µ + δ1 )(t − (n01 + 1)T )) < ε1 , and
|z2 (t ) − z2∗ (t )| < (M + v) exp((−γ + δ2 )(t − (n01 + 1)T )) < ε1 . Hence I (t ) ≤ z1 (t ) ≤ z1∗ (t ) + ε1 and P (t ) ≤ z2 (t ) ≤ z2∗ (t ) + ε1 for (n01 + 1 + n2 )T ≤ t ≤ (n01 + 1 + T 0 )T , which implies (4.16) holds for (n01 + 1 + n2 )T ≤ t ≤ (n01 + 1 + T 0 )T . As in step 1, we have S ((n01 + 1 + n2 + n3 )T ) ≥ (1 − θ1 )S ((n01 + 1 + n2 )T )en3 σ . Again integrating (4.17) on [t ∗ , (n01 + 1 + n + 2)T ], we have
¯ 1 (1 − θ1 )n2 +1 eσ1 (n2 +1)T . S ((n01 + 1 + n2 )T ) ≥ m Thus we have
¯ 1 (1 − θ1 )n2 +1 eσ1 (n2 +1)T en3 σ > m ¯ 1, S ((n01 + 1 + n2 + n3 )T ) ≥ m ¯ 1 }, thus S (t¯) ≥ m ¯ 1 for t ∈ [t ∗ , t¯], so we have which is a contradiction. Let t¯ = inft ≥t ∗ {S (t ) ≥ m ∗ ¯ σ1 (1+n2 +n3 ) = m1 S (t ) ≥ S (t ∗ )eσ1 (t −t ) ≥ me
The proof is completed.
for t ≥ t1 .
Corollary 4.2. System (2.2) is permanent provided that u<
µK [1 − (1 − θ2 ) exp(−µT )] αv(1 − exp(−γ T )) 1 . rT − − ln = u¯ , (r + λK )(1 − exp(−µT )) γ [1 − (1 − θ3 ) exp(−γ T )] 1 − θ1
v<
γ [1 − (1 − θ3 ) exp(−γ T )] u(r + λK )(1 − exp(−µT )) 1 . rT − − ln = v¯ . α(1 − exp(−γ T )) µK [1 − (1 − θ2 ) exp(−µT )] 1 − θ1
or
Corollary 4.3. System (2.2) is permanent provided that u(r + λK )(1 − exp(−µT ))
αv(1 − exp(−γ T )) θ1 < 1 − exp rT − − µK [1 − (1 − θ2 ) exp(−µT )] γ [1 − (1 − θ3 exp(−γ T ))]
−1
. = θ¯1 .
5. Numerical simulations First, to confirm our mathematical findings and facilitate their interpretation, we proceed to investigate them by using numerical simulations (Table 1). Let parameters r = 0.8, λ = 0.6, α = 0.35, µ = 0.35, a = 0.5, v = 0.6, γ = 0.35, β = 0.25, K = 3, T = 2, θ1 = 0.3, θ2 = 0.01, θ3 = 0.01 and S (0) = 0.6, I (0) = 0, P (0) = 0.4. It is seen from Corollary 3.3 that for u > uˆ or θ1 > θˆ1
820
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
a
b
Fig. 1. (a) The stability of susceptible pest-eradication equilibrium (0, I1 , P1 ) of system (2.1). (b) The stability of the positive equilibrium.
a
b
c
Fig. 2. The susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )). Here (a), (b) and (c) are the time series of S (t ), I (t ) and P (t ), respectively.
a
b
c
Fig. 3. The permanence of system (2.2), where (a), (b) and (c) are the time series of S (t ), I (t ) and P (t ), respectively.
the susceptible pest-eradication equilibrium (0, I1 , P1 ) of system (2.1) is globally asymptotically stable. Indeed, in Fig. 1(a), we capture a stable behavior of the susceptible pest-eradication equilibrium of system (2.1) for θ1 = 0.3, u = 0.7 > uˆ = 0.5175, and a stable behavior of the positive equilibrium of system (2.1) for θ1 = 0.3, u = 0.1 < uˆ = 0.5325 in Fig. 1 (b). According to Corollary 4.1, a suitable increase in the value of u has the potential to stabilize the behavior of the system (2.2) globally and make all solutions tend to the susceptible pest-eradication periodic solution (0, I ∗ (t ), P ∗ (t )). A typical example of stable behavior is captured in Fig. 2 for θ1 = 0.3, u = 0.7 > u˜ = 0.507. For u = 0.23 < u¯ = 0.264757, it follows from Corollary 4.2 that in this case system (2.2) is permanent. The behavior of the trajectory is depicted in Fig. 3. Next, we examine the behaviors of the susceptible pest-eradication solution when the natural enemies have the same effect on infected as on susceptible pests. Suppose that the predation effect on infected pests is ξ IP /(1 + bI ) and the conversion rate is η/ξ . Then the behaviors of the trajectories are captured in Fig. 4 (a) for T = 2, θ1 = 0.25, θ2 = 0.01, θ3 = 0.01, u = 0.35, v = 0.6, ξ = 0.35, η = 0.25, b = 0.5. and in Fig. 4(b) for T = 2, θ1 = 0.25, θ2 = 0.01, θ3 = 0.01, u =
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
a
821
b
Fig. 4. The behavior difference of the susceptible pest when the natural enemies have the same effect on infected as on susceptible pest. The solid line denotes the level of the susceptible when the natural enemies catch the infected pests. The dotted line denotes the level of the susceptible when the natural enemies do not feed on the infected pests.
0.1, v = 0.6, ξ = 0.35, η = 0.25, b = 0.5. Fig. 4 indicates that it has great effect on the level of the susceptible pests whether the natural enemies catch the infected pests or not. From a biological point of view, our assumption that the natural enemies of the pests do not catch the infective pests would tend to increase the spread of the infections. Hence it results in the lower level of the susceptible pests. Finally, from Table 1 together with Figs. 1 and 2, it is preferable to control the target pest population by using an impulsive control since u˜ < uˆ and θ˜1 < θˆ1 , that is there needs fewer infected pests or pesticides under the impulsive control than the continuous control. 6. Conclusion In this paper, we construct two integrated pest management models which rely on releasing of infective pest individuals and of natural predators in a constant amount, together with spraying of pesticides in a more flexible manner. In the case in which a continuous control is used, we obtain the condition which guarantees that the stability of susceptible pest-eradication equilibrium. In the case in which an impulsive control is used, the stability of the susceptible pesteradication periodic solution is guaranteed. Consequently, the condition for the permanence of the system is obtained. From Corollaries 3.1 and 4.1, we know that the releasing amount of infected pest or the pesticide effect on the susceptible pest plays an important role in the system which determines the extinction of the susceptible pest. From our models, we know that the natural enemy could not in itself bring about extinction of the population of susceptible pests since it would hold the susceptible population to a steady state level γ /(β − αγ ) without the release effect and the pesticide effect. If this is sufficiently low (below the economic threshold) there will be no need for the other pest control methods. Otherwise, integrated pest managements would be used to attain this objective. Although, these managements generally are quite costly, but they are useful during the fastigium of the oviposition. From pest control point of view, our results indicate that (1) the susceptible pests can be eradicated if the release amount of infected pests is above some threshold or the amount of spraying of pesticides is above another threshold; (2) the system is permanent under some conditions; (3) fewer infected pests or pesticides are needed as the impulsive strategy is taken, displaying its positive effect on the pest control; (4) our assumption that the natural enemies of the pests do not catch the infective pests would reduce the level of the susceptible pests. Acknowledgements The authors would like to thank the editor and the referee for constructive comments which significantly improve this paper. Funding: The National Natural Science Foundation of China (10971037). References [1] S. Tang, R.A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol. 50 (2005) 257–292. [2] V.M. Stern, Economic thresholds, Ann. Rev. Entomol. (1973) 259–280. [3] J.C. Van Lenteren, Integrated pest management in protected crops, in: D. Dent (Ed.), Integrated Pest Management, Chapman & Hall, London, 1995, pp. 311–320. [4] J.C. Van Lenteren, Measures of success in biological control of arthropods by augmentation of natural enemies, in: S. Wratten, G. Gurr (Eds.), Measures of Success in Biological Control, Kluwer Academic Publishers, Dordrecht, 2000, pp. 77–89. [5] J.C. Van Lenteren, J. Woets, Biological and integrated pest control in greenhouses, Ann. Rev. Entmol. 33 (1988) 239–250.
822
Y. Pei et al. / Mathematical and Computer Modelling 51 (2010) 810–822
[6] M.L. Flint, Integrated pest management for walnuts, University of California Statewide Integrated Pest Management Project, Division of Agriculture and Natural Resources, Second Edition, University of California, Oakland, CA, publication 3270, 1987, pp. 3641. [7] P. Georgescu, H. Zhang, An impulsively controlled predator–pest model with disease in the pest, Nonlinear Anal. Real World Appl. (2008) doi:10.1016/j.nonrwa.2008.10.060. [8] L.A. Falcon, Use of Bacteria for Microbial Control of Insects, Academic Press, New York, NY, 1971. [9] L.A. Falcon, Problems associated with the use of arthropod viruses in pest control, Ann. Rev. Entomol. 21 (1976) 305–324. [10] N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Griffin, London, 1975. [11] H.D. Burges, N.W. Hussey, Microbial Control of Insects and Mites, Academic Press, New York, NY, 1971. [12] M. Sarfraz, A. Keddie, L. Dosdall, Biological control of the diamondback moth, Plutella xylostella: A review, Biocontrol Sci. Tech. 15 (2005) 763–789. [13] D. Grzywacz, M. Parnell, G. Kibata, G. Oduor, W. Ogutu, D. Miano, D. Winstanley, The development of endemic baculoviruses of Plutella xylostella (diamondback moth, DBM) for control of DBM in East Africa, in: N. Endersby, P. Ridland (Eds.), The Management of Diamondback Moth and other Cruciferous Pests (Proceedings of the Fourth International Workshop on Diamondback Moth, Melbourne University), 2001, pp. 179–183. [14] R. Roush, Insecticide resistance management in diamondback moth: quo vadis? in: A. Sivapragasam, W.H. Loke, A.K. Hussan, G.S. Lim (Eds.), The Management of Diamondback Moth and Other Crucifer Pests, (Proceedings of the Third International Workshop, Kuala Lumpur, Malaysia, Malaysian Agricultural Research and Development Institute (MARDI)), 1997, pp. 21–24. [15] J. Jiao, L. Chen, A pest management SI model with periodic biological and chemical control concern, Appl. Math. Comput. 183 (2006) 1018–1026. [16] J. Hui, D. Zhu, Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos Solitons Fractals 29 (2006) 233–251. [17] K. Nagai, E Yano, Predation by Orius sauteri (Poppius) (Heteroptera: Anthocoridae) on Thrips palmi Karny (Thysanoptera: Thripidae). Functional response and selective predation, Appl. Entomol. Zool. 35 (2000) 565–574. [18] C.M. Port, N.E.A. Scopes, Biological control by predatory mites (Phytoseiulus persimilis Athias–Henriot) of red spider mite (Tetranychus urticae Koch) infesting strawberries grown in walk-in plastic tunnels, Plant Pathol. 30 (1981) 95–99. [19] B. Liu, Y. Zhang, The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management, Nonlinear Anal. Real World Appl. 6 (2005) 227–243. [20] R. Shi, L. Chen, A predator–prey model with disease in the prey and two impulses for integrated pest management, Appl. Math. Mode. 33 (2009) 2248–2256. [21] F. Wang, Bifurcation and chaos of a pest-control food chain model with impulsive effects, Chaos Solitons Fractals 39 (2009) 1903–1914. [22] M. Martcheva, Evolutionary consequences of predation for pathogens in prey, B. Math. Biol. 71 (2009) 819–844. [23] Z. Liu, Positive periodic solution for neutral delay competitive system, J. Math. Anal. Appl. 293 (2004) 181–189. [24] Y. Kuang, Delay Differential Equations with Appilcations in Population Dynamics, Academiac Press. INC. Harcourt Brace Iovanovich. Publishers, Boston, 1993.