The dynamics of an impulsive predator–prey model with communicable disease in the prey species only

Applied Mathematics and Computation 292 (2017) 320–335

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The dynamics of an impulsive predator–prey model with communicable disease in the prey species only Youxiang Xie a,b, Linjun Wang b,c,∗, Qicheng Deng c, Zhengjia Wu c a

College of Science Technology, China Three Gorges University, Yichang, Hubei 443002, PR China College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China c Hubei Key Laboratory of Hydroelectric Machinery Design and Maintenance, College of Mechanical and Power Engineering, China Three Gorges University, Yichang, Hubei 443002, PR China b

a r t i c l e

i n f o

Keywords: Impulsive differential equations Bifurcation Global asymptotical stability Floquet theorem Pest control

a b s t r a c t In this paper, we propose an impulsive predator–prey model with communicable disease in the prey species only and investigate its interesting biological dynamics. By the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we have deduced the sufficient conditions for the globally asymptotical stability of the semi-trivial periodic solution and the permanence of the proposed model. We also give the existences of the “infection-free” periodic solution and the “predator-free” solution. Finally, numerical results validate the effectiveness of theoretical analysis for the proposed model in this paper. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Biological control is an important component of integrated pest management programmes, and it relies on parasitism, predation, herbivore, or other natural mechanisms. However, it typically also contains active human management activities. Natural enemies play important roles in reducing the numbers of mites and pest insects. Moreover, biological control has a great benefit that is its safety for human health and the environment. There are some successful biological control examples which include uses of the predatory mites Phytoseiulus persimilis and Neoseiulus californicus against the red spider mite Tranychus urticae Koch in field-grown strawberries [1] and the use of the predatory arthropod Orius sauteri against the pest Thrips palmi Karny to protect eggplant crops in greenhouses [2]. Beauveria bassiana is often used to manage many kinds of insect pests including thrips, aphids, whiteflies and weevils. Because bacteria used for biological control infect insects by their digestive tracts, insects with sucking mouth parts like aphids and scale insects are very difficult to control with bacterial biological control [3–7]. In addition, bacillus thuringiensis is generally the most widely applied species of bacteria mainly used for biological control, with at least four sub-species used to control Coleoptera (beetles) , Diptera (true flies) and Lepidoptera (moths and butterflies) [8]. In recent years, many biologists tried their best to study the management of renewable natural resources, and extended impulsive differential equations to the models in the permanence of ecosystems [9–14]. Some researchers studied that the infected pests spread disease into the healthy wild population, and used the corresponding strategy in controlling pests [15].

∗ Corresponding author at: Hubei Key Laboratory of Hydroelectric Machinery Design and Maintenance, College of Mechanical and Power Engineering, China Three Gorges University, Yichang, Hubei 443002, PR China. E-mail address: [email protected] (L. Wang).

http://dx.doi.org/10.1016/j.amc.2016.07.042 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

Y. Xie et al. / Applied Mathematics and Computation 292 (2017) 320–335

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Phillips et al. researched that biocontrol can have disastrous consequences when it is poorly planned [16]. Lakshmikantham detailedly introduced the theory of impulsive differential equations and highlights the significance of Floquet theory [17]. Orr detailedly gave the history and informative review of biocontrol [18]. Li et al. proposed a SIS epidemic model incorporating media coverage and analyzed the dynamics of this disease model under constant and pulse vaccination, and they confirmed their theoretical results by numerical simulations [19]. Many experts investigated the impulsive differential systems such as population ecology [20], birth pulse [21] and chemotherapeutic treatment of disease [22], and they studied the properties of the stability and periodicity of solutions of the system. Tang and Chen studied the impulsive effect on the ecological situations by investigating a classical periodic Lotka–Volterra predator–prey system with impulsive effect [23]. Liu proposed a Lotka–Volterra predator–prey model with impulsive effect at fixed moment and investigated it according to the fact of integrated pest management [24]. Stone analyzed an SIR epidemic model with pulse vaccination strategy and thought that impulsive effect was important on epidemic models [25]. Donofrio studied the pulse vaccination strategy in the SIR epidemic model and stability properties of pulse vaccination strategy in SEIR epidemic model [26,27]. Hadeler investigated the first eco-epidemiological model of predator–prey population with parasite infection and obtained that a disease was spreading among interacting populations [28]. Panetta studied the bifurcation of nontrivial periodic solutions for an impulsively perturbed system of ordinary differential equations which models an integrated pest management strategy by means of a fixed point approach [29]. Lakmeeh studied the bifurcation of nontrivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment [30]. Liu et al. proposed a system of impulsive differential equations describing predator–prey dynamics with impulsive effect and analyzed the existences of “infection-free” periodic solution and the “predator-free” periodic solution via bifurcation [31]. Wang et al. proposed the pest management model with spraying microbial pesticide and releasing the infected pests, and investigated the dynamics of such a system by using the Floquet theory for impulsive differential equations [32]. Some experts have controlled the pests by exploiting viruses and simultaneously releasing the pest population [33,34]. First, a small amount of pathogens are introduced into a pest population with the expectation that it will generate an epidemic and that it will subsequently be endemic. The success of this method depends on the survival of the microbes which in turn depends on environmental factors. At the same time, we consider to release the pests infected in the laboratories to the pest population with periodic impulsive effect. The infected pests have little effect on the crops. The susceptible pests become infected through direct contact with the infective ones or through encountering the free-living infective stage in the environment. Thus it can infect the pest population and result in the death of them continuously. The main purpose of this paper therefore is to formulate and investigate an epidemiological model for the bio-control of a pest. In fact, the theoretical investigation and its application analysis can be found in almost every field [35–43]. This pest population is assumed to grow according to a logistic curve in the absence of disease [44,45]. Further, we consider the dynamics of an impulsive predator–prey model with communicable disease in the prey species only with periodic impulsive effect. The paper is organized as follows: In Section 2, we introduce the formulation of pest control model and some definitions. We give some preliminary lemmas and theorems and obtain the conditions for the globally asymptotical stability and the permanence of the semi-trivial periodic solution in Section 3. In Section 4, we analyze the existences of the “infection-free” periodic solution and the “predator-free” solution. We present numerical simulations to illustrate our results and the effects of impulse in Section 5.

2. The formulation of pest control model Let N(t) denote the density of an original insect pest population. Herein we give an assumption: it will grow in terms of the regulation of a logistic curve with the capacity r/a and a constant intrinsic birthrate r. The dynamics of N(t) is given as the following differential equation by establishing mathematical model and considering the practical value:

N (˙ t ) = N (t )(r − aN (t )). When a pest pathogen as biotic insecticide intrudes into the pest community, the pest species is divided into two classes: The first class is the susceptible pest whose density is represented by S(t) at the time t; the second class is the infected pest whose density is denoted by I(t) at the time t. So the total density of the population at any time t is

N (t ) = S(t ) + I (t ). We further assume that both the susceptible and infected pest individuals are capable of reproducing. The incidence is given by the simple mass action incidence with transmission coefficient λ > 0. The constant β > 0 acts as the mortality due to the illness. Thus, the insect-pathogen model yields



S˙ (t ) = [r − a(S(t ) + I (t ))]S(t ) − λS(t )I (t ), I˙(t ) = λS(t )I (t ) + (r − β − a(S(t ) + I (t )))I (t ).

If natural enemies of the pest are applied, these enemies only prey on the susceptible pest and the density of the natural enemies is represented by Y(t) at the time t, then the insect-pathogen model yields

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⎧. ⎨S(t ) = [r − a(S(t ) + I (t ))]S(t ) − λS(t )I (t ) − bS(t )Y (t ) · (t ) = λS(t )I (t ) + (r − β − a(S(t ) + I (t )))I (t ) ⎩I. Y (t ) = μbS(t )Y (t ) − dY (t ), where b is the coefficient of prey, μ is the coefficient of the conversion rate of susceptible prey into predators, and d is the mortality of natural enemy. We suppose that the pathogens are introduced at the pulse time, and the natural enemies of pest are introduced, then the biological control model can be obtained as follows:

⎫ ⎧. S(t ) = [r − a(S(t ) + I (t ) )]S(t ) − λS(t )I (t ) − bS(t )Y (t ) = f1 (t )⎬ ⎪ ⎪ · ⎪ ⎪ t = nT , ⎪ ⎨ I.(t ) = λS(t )I (t ) + (r − β − a(S(t ) + I (t )))I (t ) = f2 (t ) ⎭ Y (t ) = μbS(t )Y (t ) − dY (t ) = f3 (t ) ⎪ S(t ) = −uS(t ) ⎪ ⎪ ⎪ ⎪ ⎩ I (t ) = uS(t ) + α t = nT , Y (t ) = R

(2.1)

where

0 < u < 1, S(t ) = S(t + ) − S(t ), I (t ) = I (t + ) − I (t ), Y (t ) = Y (t + ) − Y (t ), u denotes the probability of the pest which is infected due to the pest pathogen at t = nT ; α > 0 represents the release amount of the infected pests at t = nT ; R represents the release amount of predators at t = nT ; T is the period of the impulsive effect and n ∈ N , N = {0, 1, 2 . . .} . In this paper, we assume that the birth rate is larger than the mortality rate, i.e., r > β , considering the illness. For the convenience of obtaining the results later, we need to introduce the following notations:

R+ = [0, +∞ ),





R3+ = X ∈ R3 X ≥ 0 , T

f = ( f1 , f2 , f3 ) . Let V : R+ × R3+ → R+ , then V is said to belong to class ν 0 if: (1) V is continuous in (nT , (n + 1 )T ] × R3+ (n ∈ N ). For each X ∈ R3+ , and there exists

lim

(t,Y )→(nτ + ,X )

V (t, Y ) = V (nτ + , X );

(2) V satisfies the local Lipschitz condition in X. Definition 1. Let V ∈ v0 , then for (t, X ) ∈ (nT , (n + 1 )T ] × R3+ , the upper right derivative of V(t, x) with respect to the system (2.1) is defined as

D+V (t, X ) = lim sup h→0

1 [V (t + h, X + h f (t, X )) − V (t, X )]. h

The solution of the system (2.1) is piecewise continuous for X : R+ → R3+ . X(t) is continuous and differentiable in

(nT , (n + 1 )T ](n ∈ N ) , and limt→nT + X (t ) = X (nT + ) exists. The smoothness characteristic of f ensures the global existence and uniqueness of solutions of the system [17].

Definition 2. System (2.1) is said to be permanent if there exists two positive constants M ≥ m > 0 that are independent of X0 . It also satisfies m ≤ S(t) ≤ M, m ≤ I(t) ≤ M, m ≤ Y(t) ≤ M when t is sufficiently large and S0+ > 0, I0+ > 0, Y0+ > 0. 3. The stability and persistence of semi-trivial periodic solutions Since variable cannot be negative, we hope that the solutions of the system can maintain non-negative when the initial condition is non-negative. Thus we need the following conclusions: Lemma 1. Let X (t ) = (S(t ), I (t ), Y (t )) be a solution of the system (2.1). If X0+ ≥ 0, then X(t) ≥ 0 for all t ≥ 0. Moreover, if X0+ > 0, then X(t) > 0 for all t ≥ 0. .

Proof. Assumes that there exists t∗ ∈ (0, T], then S(t) ≥ 0, I(t) ≥ 0, Y(t) ≥ 0, S(t .∗ ) = 0, S (t ∗ ) < 0, I(t∗ ) ≥ 0, and Y(t∗ ) ≥ 0 for all t ∈ (0, t∗ ). Exploiting the first equation of the system (2.1), we can obtain S (t ∗ ) = 0, which is a contradiction. From its first equation, we have

S(t ) = S(0+ )e

t 0

[r−a(S(ξ )+I (ξ )−λI (ξ )−bY (ξ ))]dξ

, t ∈ ( 0 , T ].

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Apparently, S(t) ≥ 0 for S(0+ ) ≥ 0 and when S(0+ ) > 0, S(t) > 0 for t ∈ (0, T]. Similarly, we can prove it for I(t) and Y(t). Now we provide an important theorem about the following impulsive differential equations.  Lemma 2 [29]. Suppose V ∈ v0 satisfies the following differential inequalities:



D+V (t, X ) ≤ g(t , V (t , X )), t = nT V (t, X (t + )) ≤ ψn (V (t, X )), t = nT ,

(3.1)

where g : R+ × R+ → R is continuous in (nT , (n + 1 )T ] × R+ . For u ∈ R+ , n ∈ N, lim(t,v )→(nT + ,u ) g(t, v ) = g(nT + , u ) exists and ψn : R+ → R+ is non-decreasing. Let r(t) be the maximal solution of the scalar impulsive differential equation

.

u(t ) = g(t , u(t )), t = nT , u(t + ) = ψn (u(t )), t = nT , u ( 0+ ) = u0

existing on [0, ∞). Then we can obtain V(t, X(t)) ≤ r(t) for t ≥ t0 by the V (0+ , X0 ) ≤ u0 , where X(t) is a solution of system (2.1). We will provide some basic properties about the following impulsive differential equations. Solving the following impulsive differential equations:

.

I (t ) = (r − β − aI (t ))I (t ), t = nT , I (t + ) = I (t ) + α , t = nT , I (0+ ) = I0 , β > 0,

(3.2)

we have a unique positive periodic solution

I∗ (t ) =

r−β

where

I (0 ) = ∗

+

ω+

I∗ (0+ )e(r−β )(t−nT ) (r − β ) , t ∈ (nT , (n + 1 )T ], − aI∗ (0+ ) + aI∗ (0+ )e(r−β )(t−nT )



ω2 + 4aα (r − β )(e(r−β )T − 1 ) , 2a(e(r−β )T − 1 )

ω = (r − β + aα )(e(r−β )T − 1 ). For the equation

.

Y (t ) = −dY (t ), t = nT , Y (t + ) = Y (t ) + R, t = nT , Y (0+ ) = Y0 ,

(3.3)

there exists the only positive periodic solution

Y ∗ (t ) =

R · e−d (t−nT ) . 1 − e−dT

We can conclude the following results for the system (3.2) and (3.3). Lemma 3. Suppose aα > r − β , I0 ≥ 0, then |I (t ) − I∗ (t )| → 0, |Y (t ) − Y ∗ (t )| → 0 as t → ∞, where I(t) and Y(t) are a solution of the system (3.2) and (3.3), respectively. Proof. (Refer to [46]). In order to discuss the stability of semi-trivial periodic solution of the system (2.1), we need to present the Floquet theory for the linear T-periodic impulsive equation:

.

x(t ) = A(t )x(t ), t = Tk , t ∈ R, x(t + ) = x(t ) + Bk x(t ), t = Tk .

(3.4)

Firstly, we introduce the following assumptions: (H1) A(·) ∈ PC(R, Cn × n ) and A(t + T ) = A(t )(t ∈ R ), where PC(R, Cn × n ) is the set of all piecewise continuous matrix functions which are right continuous at t = Tk , and Cn × n is the set of all n × n matrices. (H2) Bk ∈ Cn × n , det(E + Bk ) = 0 , Tk < Tk+1 (k = 1, 2, 3 . . . ). (H3) There exists a q ∈ N, such that Bk+q = Bk and Tk+q = Tk + T (k = 1, 2, 3 . . . ). Let (t) be a fundamental matrix of Eq. (3.4), then there exists a unique non-singular matrix M ∈ Cn × n such that (t + T ) = (t )M (t ∈ R ). The matrix M is called the monodromy matrix of (3.4) corresponding to the fundamental matrix of (t). All the monodromy matrices of (3.4) are similar and have the same eigenvalues. The eigenvalues μ1, μ2 , . . . , μn of the monodromy matrices are called the Floquet multipliers of (3.4).  Lemma 4 (Lakshmikantham, Bainov and Simeonov [17], Floquet Theorem). Suppose that Hypothesis (H1)–(H3) hold, then the linear T-periodic impulsive (3.4) is

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(1) stable if and only if all the multipliers μ j ( j = 1, 2, . . . , n ) of (3.4) satisfy the inequality |μj | ≤ 1. Moreover, for those μj satisfying μ j = 1, there correspond simple elementary divisors; (2) asymptotically stable if and only if all the multipliers μj of (3.4) satisfy |μj | < 1; (3) unstable if |μj | > 1 for some j = 1, 2, . . . , n. Then we can obtain the following conclusions for the system (2.1): Theorem 1. Each solution of the system (2.1) are ultimately bounded, i.e., there exists a constant M > 0 such that S(t) ≤ M, I(t) ≤ M and Y(t) ≤ M for each solution X (t ) = (S(t ), I (t ), Y (t )) of system (2.1) with all t large enough. Proof. Let X (t ) = (S(t ), I (t ), Y (t )) is any solution of system (2.1) and V (t ) = S(t ) + I (t ) + Y (t ), then V (t ) ∈ v0 and

⎧ + D V = −aS2 (t ) + rS(t ) − bS(t )Y (t ) − aI2 (t ) − 2aS(t )I (t ) + (r − β )I (t ) ⎪ ⎪ ⎨ + μbS(t )Y (t ) − dY (t ), = −aS2 (t ) + (r + d )S(t ) − bS(t )Y (t ) − 2aS(t )I (t ) − aI2 (t ) + (r + d − β )I (t ) ⎪ ⎪ ⎩ + + μbS(t )Y (t ) − d (S(t ) + I (t ) + Y (t )), t = nT , V (t ) = V (t ) + α + R, t = nT .

Apparently,

−aS2 (t ) + (r + d )S(t ) − bS(t )Y (t ) − 2aS(t )I (t ) − aI2 (t ) + (r + d − β )I (t ) + μbS(t )Y (t ) is bounded, so we can conclude that there exists M0 > 0 such that



D+V ≤ −dV + M0 , t = nT , V (t + ) = V (t ) + α + R, t = nT .

According to Lemma 2, we conclude



V (t ) ≤ V (0+ ) −



M0 −dt (α + R )(1 − e−ndT )e−d(t−nT ) M0 e + + , t ∈ (nT , (n + 1 )T ]. d d 1 − e−dT

So Theorem 1 has been proved.



In what follows, we will discuss and study the stability of the susceptible pest-eradication periodic solution. It is easy to check that system (2.1) has a semi-trivial periodic solution X (t ) = (0, I∗ (t ), Y ∗ (t )), where I∗ (t) and Y∗ (t) are the positive periodic solution of the system (3.2) and (3.3), respectively. Theorem 2. If aα > r − β λ ≥ a and (1 − u )e of system (2.1) is globally asymptotical stable.

T 0

(r−(a+λ )I∗ (t )−bY ∗ (t ))dt < 1, then the semi-trivial periodic solution (0, I∗ (t), Y∗ (t))

Proof. We first prove its local stability. Let H (t ) = S(t ), P (t ) = I (t ) − I∗ (t ), and Z (t ) = Y (t ) − Y ∗ (t ), so the linear approximation system about the semi-trivial periodic solution (0, I∗ (t), Y∗ (t)) of the system is as follows:

⎛·

⎞



H (t )

⎜· ⎟ ⎝ P (t ) ⎠ = · Z (t )

r − (a + λ )I∗ (t ) − bY ∗ (t ) 0 0 (λ − a )I∗ (t ) r − β − 2aI∗ (t ) 0 μbY ∗ (t ) 0 −d

Thus the fundamental matrix of system (3.5) is

⎛



t ∗ ∗ e 0 (r−(a+λ)I (s )−bY (s ))ds ∗ ∗∗

(t ) = ⎝

t

0

∗ e 0 (r−β −2aI (s ))ds ∗∗∗

0 0





H (t ) P (t ) . Z (t )

(3.5)

⎞ ⎠.

e−dt

As it is not required in the following discussion, so it is not necessary to give the specific forms of ∗, ∗∗, ∗∗∗ , and it can be discarded. The linear part of the fourth, fifth and sixth equation of the system (2.1) can be expressed as





H (nT + ) P (nT + ) Z (nT + )



=

1−u 0 u 1 0 0



0 0 1



H (nT ) P (nT ) . Z (nT )

According to Floquent theory, if all the eigenvalues of



M=



1−u 0 0 u 1 0 (T ) 0 0 1

satisfy |λ1 | < 1, |λ2 | < 1 and |λ3 | < 1, then the periodic solution (0, I∗ (t), Y∗ (t)) is locally asymptotically stable. In fact, λ2 = e

T 0

(r−β −2aI∗ (s ))ds = · · · < 1 ( Refer to [38]);

T

λ3 = e−dt < 1; λ1 = (1 − u )e 0

(r−(a+λ )I∗ (s )−bY ∗ (s ))ds .

Y. Xie et al. / Applied Mathematics and Computation 292 (2017) 320–335 T

∗

325

∗

Hence while |λ1 | < 1, i.e., (1 − u )e 0 (r−(a+λ )I (t )−bY (t ))dt < 1 , we can assert that the solution (0, I∗ (t), Y∗ (t)) is locally asymptotically stable. In the following part, we will prove the global attractivity of (0, I∗ (t), Y∗ (t)). Choose ε 1 > 0 such that δ = (1 − u )e that

T 0

[r−(a+λ )(I∗ (t )−ε1 )−b(Y ∗ (t )−ε1 )]dt

< 1. Exploiting Lemmas 2 and 3, we can conclude

I (t ) ≥ I∗ (t ) − ε1 , Y (t ) ≥ Y ∗ (t ) − ε1

(3.6)

for sufficiently large t. So we may assume that (3.6) holds for all t ≥ 0, and thus we can conclude from system (2.1) and Eq. (3.6) that

S((n + 1 )T ) ≤ S(nT )(1 − u )e

T 0

[r−(a+λ )(I∗ (t )−ε1 )−b(Y ∗ (t )−ε2 )]dt

≤ S ( 0+ )δ n .

Therefore, we have that S(nT) → 0 as n → ∞ and it follows that S(t) → 0, limt→∞ |I (t ) − I∗ (t )| = 0 and

lim |Y (t ) − Y ∗ (t )| = 0

t→∞

as n → ∞ due to 0 < S(t ) ≤ S(nT )(1 − u )erT . So we can obtain that (0, I∗ (t), Y∗ (t)) is globally asymptotically stable. We will consider the permanence of system (2.1) in the next part.  Theorem 3. If aα > r − β , λ ≥ a and T

(1 − u )e 0 [r−(a+λ)I

∗

(t )−bY ∗ (t )]dt

>1

(3.7)

then system (2.1) is permanent. Proof. Let (S(t), I(t), Y(t)) is any solution of system (2.1) with S0 > 0, I0 > 0, Y0 > 0. From Theorem 1, we have S(t) ≤ M, I(t) ≤ M, Y(t) ≤ M for t ≥ 0. From Lemma 3, we know I (t ) ≥ I∗ (t ) − ε1 and Y (t ) ≥ Y ∗ (t ) − ε1 for all sufficiently large t and some ε 1 > 0. From the continuity and periodicity of the function I∗ (t) and Y∗ (t), we can obtain that for sufficiently large t, there exists m2 > 0 such that I (t ) ≥ I∗ (t ) − ε1 ≥ m2 , Y (t ) ≥ Y ∗ (t ) − ε1 ≥ m2 . Thus we only need to find a m1 > 0 such that S(t) ≥ m1 for sufficiently large t in the next part. We will prove it in the following two steps. Step 1: By the inequality (3.7), we can select m3 > 0 and ε 0 > 0 small enough such that δ = (1 − 

T ∗ ∗ u )e 0 (r−(a+λ )I (t )−bY (t ))dt−am3 T −(a+λ+b)ε0 T > 1 . Next we prove that S(t) < m3 cannot permanently hold for all t ≥ 0. Otherwise,

.

I (t ) ≤ (λm3 + r − β )I (t ) − aI2 (t ), t = nT , I (t ) ≤ α + um3 , t = nT .

According to Lemmas 2 and 3, we know I(t) ≤ Z(t) and Z (t ) − Z ∗ (t ) → 0 as t → ∞, where z(T) is a solution of the following equations:

.

Z (t ) = (λm3 + r − β )Z (t ) − aZ 2 (t ), t = nT , Z (t ) = α + um3 , t = nT , Z ( 0+ ) = Z0 > 0

(3.8)

and

Z ∗ (t ) =

r−β

Z ∗ (0+ )e(r−β +λm3 )(t−nT ) (r − β + λm3 ) , t ∈ (nT , (n + 1 )T ], n ∈ N. + λm3 − aZ ∗ (0+ ) + aZ ∗ (0+ )e(r−β +λm3 )(t−nT )

So there exists n0 > 0 such that I (t ) ≤ Z (t ) ≤ Z ∗ (t ) + ε0 for t ≥ n0 T = T1 . Similarly, we can conclude that

Y (t ) ≤ and

Re(ubm3 −d )(t−nT ) + ε0 1 − e(ubm3 −d )T

⎧. 3 −d )(t−nT ) ⎨S(t ) ≥ S(t )[r − a(m3 + Z ∗ (t ) + ε0 ) − λ(Z ∗ (t ) + ε0 ) − b( Re(ubm(ubm + ε0 )], t = nT , 3 −d )T 1−e S(t ) = −uS(t ), t = nT . ⎩ + S ( 0 ) = S0

(3.9)

So it follows from (3.9) that

S((n + 1 )T ) ≥ S(nT )(1 − u )e ≥ δ S(nT )

 (n+1)T nT

[r−a(m3 +Z ∗ (t )+ε0 )−λ(Z ∗ (t )+ε0 )−b( Re

(ubm3 −d )(t−nT ) (ubm3 −d )T

1−e

+ε0 )]dt

for n ≥ n0 . This indicates that S((n0 + k )T ) ≥ S(n0 T )δ k → ∞ as k → ∞, which contradicts the bounded S(t). So there exists t1 > 0 such that S(t1 ) > m3 .

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Step 2: If S(t) > m3 for all t ≥ t1 , then we have obtained the aim. If not, we may let t ∗ = inf{t ≥ t1 : S(t ) < m3 }. Then we have that S(t) ≥ m3 for all t ∈ [t1 , t∗ ]. We also have S(t ∗ ) = m3 by the continuity of S(t). In the next step, we consider this in two possible cases. Case 1: t∗ = nT, n ∈ N. Then we have S(t) ≥ m3 , S(t ∗ ) = m3 for all t ∈ [t1 , t∗ ). Suppose t ∗ ∈ [n1 T , (n + 1 )T ) for some n1 ∈ N. Choosing n2 , n3 ∈ N such that

n2 T > T2 =

ε1 ln M+ Z∗ 0

,

λm3 + r − β (1 − u )n2 +1 e(n2 +1)δ1 T δ n3 > 1, α + um3 ∗

Z0 =

1 − eλm3 +r−β

,

where δ1 = r − am3 − (a + λ + b)M < 0. Let T ∗ = (n2 + n3 )T . We can know that there must be a t ∈ [(n1 + 1 )T , (n1 + 1 )T + T ∗ ) such that S(t2 ) ≥ m3 . Otherwise, we can assume that S(t) < m3 for any t ∈ [(n1 + 1 )T , (n1 + 1 )T + T ∗ ). Considering (3.8) with Z ((n1 + 1 )T + ) = I ((n1 + 1 )T + ), we have

Z (t ) = (Z ((n1 + 1 )T + ) − Z0∗ )e(λm3 +r−β )(t−(n1 +1)T ) + Z ∗ (t ), t ∈ [nT , (n + 1 )T ) for n1 + 1 ≤ n ≤ n1 + 1 + n2 + n3 . Then

|Z (t ) − Z ∗ (t )| ≤ (M + Z0∗ )e(λm3 +r−β )n2 T < ε0 and

I (t ) ≤ Z (t ) ≤ Z ∗ (t ) + ε0 , (n1 + 1 + n2 )T ≤ t ≤ (n1 + 2 )T + T ∗ . Thus, for (n1 + 1 + n2 )T ≤ t ≤ (n1 + 2 )T + T ∗ , it follows from (3.9) that

S((n1 + 1 + n2 + n3 )T ) ≥ S((n1 + 1 + n2 )T )δ n3 and

.

S(t ) ≥ S(t )[r − am3 − (a + λ + b)M], t = nT , S(t ) = −uS(t ), t = nT , S ( 0+ ) = S0 .

Integrating it from t∗ to (n1 + 1 + n2 )T , we have

S((n1 + 1 + n2 )T ) ≥ m3 (1 − u )n2 +1 e(n2 +1)δ1 T . Thus

S((n1 + 1 + n2 + n3 )T ) ≥ m3 (1 − u )n2 +1 e(n2 +1)δ1 T δ n3 > m3 , which contradicts

S(t ) < m3 , t ∈ [(n1 + 1 )T , (n1 + 1 )T + T ∗ ). Let t = inf{t ≥ t ∗ : S(t ) ≥ m3 }. Then S(t ) ≥ m3 . For t ∈ [t ∗ , t ], we have ∗ S(t ) ≥ S(t ∗ )(1 − u )1+n2 +n3 e(t −t )δ1 ≥ m3 (1 − u )1+n2 +n3 e(1+n2 +n3 )δ1 T = m1 .

For t > t , we can obtain the same result S(t) ≥ m1 while S(t ) ≥ m3 . Hence S(t) ≥ m1 for all t ≥ t1 . Case 2: t ∗ = nT for some n ∈ N. Then S(t) ≥ m3 for t ∈ [t1 , t∗ ] and

(1 − u )m3 ≤ S(t ∗+ ) = (1 − u )S(t ∗ ) < m3 . The proof is similar to Case 1. Hence S(t) ≥ m1 for all t ≥ t1 .



4. Existences of the “infection-free” periodic solution and the “predator-free” solution 4.1. Preliminaries In this section, we follow the technique used by Lakmeche and Arino in [29], and herein we give the existences of the “infection-free” and the “predator-free” periodic solutions. We introduce some notations and state some preliminary results. Let us consider the following impulsive system:

⎧.  ⎪ S.(t ) = F1 (S(t ), Y (t )) ⎪ t = nT , ⎨ Y (t ) = F2 (S(t ), Y (t ))  S(t + ) = θ (S(t ), Y (t )) ⎪ ⎪ ⎩ Y (t + ) = θ1 (S(t ), Y (t )) t = nT . 2

(4.1)

Y. Xie et al. / Applied Mathematics and Computation 292 (2017) 320–335

.

S (t )=F1 (S (t ),0 ), t=nT, S (t + )=θ1 (S (t ),0 ), t =nT,

Suppose that

327

which is obtained by taking Y (t ) = 0 in (4.1), has a periodic solution denoted by V(t). We

denote a trivial periodic solution of system (4.1) by

π = (0, V )T Let  be the flow associated to system (4.1), we have V (t ) = (t, S0 , Y0 )(0 < t ≤ T ), where V0 = (S0 , Y0 ). The flow  applies to time T. So V (T ) = (T , V0 ). The following notations of [47] will be used:



∂θ2 ∂ 2 d0 = 1 − ∂Y ∂Y



∂θ1 ∂ 1 a0 = 1 − ∂S ∂S





b0 = −

∂θ1 ∂ 1 ∂ S ∂Y



, (T0 ,V0 )

 , (T0 ,V0 )

 , (T0 ,V0 )

  ∂ 2 θ2 ∂ 1 (T0 , V0 ) ∂ 1 (T0 , V0 ) 1 ∂θ1 ∂ 1 (T0 , V0 ) ∂ 2 (T0 , V0 ) B=− + ∂ S∂ Y ∂t ∂S ∂t ∂Y a0 ∂ S  2  ∂θ2 ∂ 2 (T0 , V0 ) 1 ∂θ1 ∂ 1 (T0 , V0 ) ∂ 2 2 (T0 , V0 ) − + , ∂Y ∂ S∂ Y ∂t ∂ t ∂Y a0 ∂ S    2 b0 ∂ 1 (T0 , V0 ) ∂ 1 (T0 , V0 ) ∂ 2 (T0 , V0 ) ∂ 2 θ2 ∂ 2 (T0 , V0 ) ∂ 2 θ2 C = −2 − − ∂ S∂ Y ∂S ∂Y ∂Y ∂Y ∂Y 2 a0

+2

∂θ2 b0 ∂ 2 (T0 , V0 ) ∂θ2 ∂ 2 2 (T0 , V0 ) − , ∂ Y a 0 ∂ S∂ Y ∂Y ∂Y 2

where T0 is the root of d0 = 0, and

∂ 1 (t, V0 ) ∂S ∂ 2 (t, V0 ) ∂Y ∂ 1 (t, V0 ) ∂Y ∂ 2 2 (t, V0 ) ∂Y ∂ S 2 ∂ (t, V0 ) ∂Y 2

=e

 t ∂ F1 (π (ζ )) 0

∂S

dζ

,

 t ∂ F2 (π (ζ ))

= e 0 ∂ Y dζ ,  t  t ∂ F1 (π (ζ )) ∂ F1 (π (v )) 0v ∂ F2 (∂πY(ζ )) dζ = d v, e v ∂ S dζ e ∂Y 0  t  t ∂ F2 (π (ζ )) ∂ 2 F2 (π (v )) 0v ∂ F2 (∂πY(ζ )) dζ = d v, e v ∂ Y dζ e ∂Y ∂ S 0  t   t  t ∂ F2 (π (ζ )) t ∂ 2 F2 (π (v )) 0v ∂ F2 (∂πY(ζ )) dζ = e v ∂ Y dζ e d v + ev 2 ∂Y 0 0   v  v ∂ F1 (π (ζ ))  ∂ F ( π ( ζ )) θ ∂ F1 (π (v )) 0 2 ∂Y dζ × e 0 ∂ Y dζ e d θ d v, ∂Y 0

t ∂ 2 2 (t, V0 ) ∂ F2 (π (v )) 0t ∂ F2 (∂πY(ζ )) dζ ˜ = e = (ubS˜ (t ) − d )e 0 (ubS ∂Y ∂ t ∂Y ∂ 1 (T0 , V0 ) ˜ = S (T0 ), ∂t

∂ F2 (π (ζ )) dζ ∂Y

(ζ )−d )dζ

∂ 2 F (π (v )) ∂Y ∂ S



,

then we get the following lemma.



Lemma 5. If |1 − a0 | < 1 and d0 = 0, then we get: (a) If BC = 0, then we have a bifurcation. Moreover, we have a bifurcation of nontrivial periodic solution of system (4.1) if BC < 0, and a subcritical case if BC > 0. (b) If BC = 0, then we have an undetermined case.

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4.2. Existence of the “infection-free” periodic solution Let us consider the following set of differential equations.

 ⎧.  ( t ) = (r − aS(t ))S(t ) − μbS(t )Y (t ) = F1 (S, Y ) S ⎪ ⎪ t = nT , ⎨.  Y (t ) = μbS(t )Y (t ) − dY (t ) = F2 (S, Y )  + ⎪ ⎪ ⎩ S(t ) = (1 − u )S(t) = θ1 (S, Y ) t = nT . Y (t + ) = Y (t ) + R = θ2 (S, Y ) In order to apply Lemma 5, we compute the following:

d0 = 1 − e

 T0 ∂ F2 (π (ζ )) 0

∂y

dζ

=1−e

 T0 0

(μbS∗ −d )dζ

= 1 − e−dT0 e

 T0 0

μbS∗ dζ ,



if d0 = 0, we can get

T0 =

−μb ar ln(1 − u ) −μbr ln(1 − u ) = . r (μb ar − d ) r (μbr − ad )

Fig. 1. Numerical solution of system (2.1) with r = 3, λ = 1, a = 1, u = 0.1, β = 5, α = 5, T = 1.35, b = 0.2, μ = 0.1, R = 4, d = 0.5, and all the parameters T ∗ ∗ satisfy (1 − u )e 0 (r−(a+λ)I (t )−bY (t ))dt < 1. (a) Phase (S(t), I(t), Y(t)) (b) Time-series of the susceptible pest species S(t). (c) Time-series of the corresponding infected pest species I(t). (d) Time-series of natural enemies of the pest Y(t).

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329

Fig. 2. Numerical solution of system (2.1) with r = 3, λ = 1, a = 1, u = 0.1, β = 5, α = 5, T = 1.5, b = 0.2, μ = 0.1, R = 4, d = 0.5, and all the parameters satT ∗ ∗ isfy (1 − u )e 0 [r−(a+λ)I (t )−bY (t )]dt > 1. (a) Phase (S(t), I(t), Y(t)). (b) Time-series of the susceptible pest species S(t). (c) Time-series of the corresponding infected pest species I(t). (d) Time-series of natural enemies of the pest Y(t).

In addition,

a0 = 1 − (1 − u )erT0 −2a

b0 = 1 − ( 1 − u ) ∂θ

∂θ



T0

e

 T0 0

T v

0

S ( ζ )d ζ

> 0,

(r−2a S (ζ ))dζ

0

∂θ

(−b S(v ))e−dT0 +μb ∂θ

∂2θ

 T0 0

S ( ζ )d ζ d

v > 0.

∂2θ

Obviously, ∂ Y1 = ∂ S2 = 0, ∂ S1 = 1 − u, ∂ Y2 = 1, ∂ Y 22 = ∂ S∂ Y2 = 0. It is easy to verify that C > 0, and if

μb



T0

0

then

S(ζ )dζ − dT0 > 0,

 B = −t

 T0 1−u ∂ 2 2 (T0 , V0 ) S˜ (T0 ) + [μbS˜(T0 ) − d]e 0 ∂ S ∂ t a0

According to Lemma 5, we can obtain the following result.

∂ F2 (π (ζ )) dζ ∂Y

 < 0.

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Fig. 3. Bifurcation diagrams of system (2.1) with respect to period T in range [0, 16]. The system exhibits stable solution, periodic doubling bifurcation and chaotic phenomenon corresponding with different T. (a) The bifurcation diagram of population S(t) with respect to period T. (b) The bifurcation diagram of population I(t) with respect to period T.

ln(1−u ) Theorem 4. System (2.1) has a positive periodic “infection” solution if T > T0 and is close to T0 , where T0 = −rμ(br μbr−ad ) . This theorem illustrates that the “semi-trivial” solution becomes unstable if the periodic T is more than T0 and close to T0 , and becomes the “infection-free” solution.

4.3. Existence of the “predator-free” periodic solution Let us consider the following set of differential equations.

 ⎧.  S(t ) = [r − a(S(t ) + I (t ))]S(t ) − λS(t )I (t ) = F1 (S, I ) ⎪ ⎪ t = nT , ⎨.  + I (t ))]I (t ) = F2 (S, I ) I (t ) = λS(t )I (t ) + [r − β − a(S(t )   + ⎪ ⎪ ⎩ S(t ) = (1 − u )S(t ) = θ1(S, I ) t = nT . I (t + ) = I (t ) + uS(t ) + α = θ2 (S, I )

We compute the following:

d0 = 1 − e

 T0 ∂ F2 (π (ζ )) 0

∂I



dζ



if d0 = 0, we can get T0 =

a0 = 1 − ( 1 − u )e

b0 = − ( 1 − u ) ∂θ



 T0 0



T0

e



= 1 − e(r−β )T0 +

−(λ−a ) ar

ln(1−u ) . r ((λ−a ) ar +r−β )

(r−2aS˜(t ))dt

 T0 v

∂θ

0

(λS˜(ζ )−aS˜(ζ ))dζ

,

In addition,

> 0,

(r−2aS˜(ζ ))dζ

0

 T0

 v ∂ F2

(−(a + λ )S˜(v ))e 0

∂θ

∂θ

∂2θ

∂ I dζ

d v > 0.

∂2θ

Apparently, ∂ I1 = 0, ∂ S2 = u, ∂ S1 = 1 − u, ∂ I2 = 1, ∂ I22 = ∂ S∂2I = 0. It is easy to verify that C > 0, and if

 0



T0

[(λ − a )S˜(ζ ) + r − β ]dζ < 0,

then

B = −(λS˜(T0 ) + r − β )e

 T0 0

(r−β +λS˜(ζ ))dζ

−

1 ˜ (1 − u )S (T0 ) a0

According to Lemma 4, we get the following result.

 0



T0

e

 T ∂ F2 0 v

∂ I dS

 V0 ∂ F2

( λ − a )e 0

∂ I dS

d v < 0.

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331

Fig. 4. Numerical solution of system (2.1) with r = 3, λ = 1, a = 1, u = 0.1, β = 5, α = 5, T = 1, b = 0.2, μ = 0.1, R = 4, d = 0.5, and all the parameters satisfy T ∗ ∗ (1 − u )e 0 (r−(a+λ)I (t )−bY (t ))dt < 1. (a) Phase (S(t), I(t), Y(t)) (b) Time-series of the susceptible pest species S(t). (c) Time-series of the corresponding infected pest species I(t)(t ∈ [0, 35]). (d) Time-series of natural enemies of the pest Y(t)(t ∈ [0, 35]).









Theorem 5. System (2.1) has a positive periodic “predator-free” solution if T > T0 and is close to T0 , where T0 satisfies T0 = −(λ−a ) ar ln(1−u ) . r ((λ−a ) ar +r−β )





This theorem illustrates that the “semi-trivial” solution becomes unstable if the period T is more than T0 and close to T0 . The variable I(t) begins to oscillate with an amplitude, and becomes the “predator-free” solution.

5. Numerical simulations In this section, we provide the results of numerical simulations to validate our theoretical results and further prove the complex and rich properties of system (2.1). Nowadays, lots of management technology is applied to agriculture and ecosystem. By exploiting some advanced technology, we can quickly and conveniently control or even eliminate the pest, which can protect crops from harm at a large extent. Firstly, we assume that r = 3, λ = 1, a = 1, u = 0.1, β = 5, α = 5, T = 1.35, b = 0.2, μ = 0.1, R = 4, d = 0.5, and by verification, we know that these parameters satisfy the conditions of Theorem 2, then the mature predator-extinction periodic solution is globally attractive, which can be shown in Fig. 1. Fig. 1a is the phase portraits of S(t), I(t) and Y(t). Fig. 1b is the phase portraits of S(t) and t. S(t) goes to extinction. Fig. 1c is the phase portraits of I(t) and t. I(t) goes oscillatory. Fig. 1d

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Fig. 5. Numerical solution of system (2.1) with r = 3, λ = 1, a = 1, u = 0.1, β = 5, α = 5, T = 2, b = 0.2, μ = 0.1, R = 4, d = 0.5, and all the parameters satisfy T ∗ ∗ (1 − u )e 0 [r−(a+λ)I (t )−bY (t )]dt > 1. (a) Phase (S(t), I(t), Y(t)). (b) Time-series of the susceptible pest species S(t). (c) Time-series of the corresponding infected pest species I(t)(t ∈ [0, 35]). (d) Time-series of natural enemies of the pest Y(t) (t ∈ [0, 35]).

is the phase portraits of Y(t) and t . Y(t) goes oscillatory. From Fig. 1, we can find that the periodic solution is globally attractive. Secondly, we investigate the permanence of system (2.1). Choosing that r = 3, λ = 1, a = 1, u = 0.1, β = 5, α = 5, T = 1.5, b = 0.2, μ = 0.1, R = 4, d = 0.5, we can verify that these parameters satisfy the conditions of Theorem 3. Hence system (2.1) is permanent. Fig. 2b and c are the positive time series of prey species in the interval [0, 35]. Fig. 2d shows the positive time series of predator species in the interval [0, 35]. Fig. 2 illustrates that time series of three species become stable when t is large enough. Therefore, the permanence of system (1.3) can be fully demonstrated by these figures. Thirdly, by numerical analysis, we aim to investigate the effect of impulsive period T. The bifurcation diagrams of S(t) and I(t) with respect to parameter T in range [0, 16] are shown in Fig. 3. The parameter values of Fig. 3 are S(0 ) = 1, I (0 ) = 1, Y (0 ) = 1, r = 3, λ = 1, a = 0.01, u = 0.1, β = 3.4, α = 5, b = 0.2, μ = 0.1, R = 4, d = 0.5. From Fig. 3, we can find that there are bifurcations when T = 5.61, 6.1, 6.9, 8.2, 12.8. It implies that when T < 5.61, even if the stocking of susceptible pest and the releasing of infected pest are frequent, the solution of this system analyzed here is stable, and hence we can harvest predator Y(t) continuously. However, when T ≥ 5.61, the system exhibits periodic behavior for both susceptible pest and predator. If we give a moderate pulse, then the system appears chaotic phenomenon, that is, the evolution of this system is unpredictable. Furthermore, it is easy to find that there are more than one periodic solution in the interval T ∈ [6, 16.4]. In a word, the system analyzed here can take on many forms of complexity, including stable solutions, cycles, cascade, chaos and so on, which are shown in Figs. 1–6. From Figs. 1, 4 and 6, we can find that system (2.1) is globally asymptotical stable and

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333

Fig. 6. Numerical solution of system (2.1) with r = 3, λ = 1, a = 1, u = 0.1, β = 5, α = 5, T = 0.8, b = 0.2, μ = 0.1, R = 4, d = 0.5, and all the parameters satT ∗ ∗ isfy (1 − u )e 0 (r−(a+λ)I (t )−bY (t ))dt < 1. (a) Phase (S(t), I(t), Y(t)) (b) Time-series of the susceptible pest species S(t). (c) Time-series of the corresponding infected pest species I(t)(t ∈ [0, 35]). (d) Time-series of natural enemies of the pest Y(t)(t ∈ [0, 35]).

susceptible pests go to extinction when impulsive period T < 1.36. Moreover, the smaller impulsive period T is, the faster the susceptible pests perish. This is an important result from ecological view point since in this case the infected pests have little effect on the crops, but the pests that have huge effects or harm on the crops will tend to extinction. Thus we can obtain the prospective results in scientific and effective control of pests. In addition, the purpose of controlling the pests is not to eradicate the pests, but to control the pests under the level of economic harm. 6. Conclusions In this paper, we have studied an impulsive predator–prey model with communicable disease in the prey species only. Exploiting the Floquet theories and small amplitude perturbation technique, we have proved that If aα > r − β λ ≥ a and T

(r−(a+λ )I∗ (t )−bY ∗ (t ))dt < 1, then the semi-trivial periodic solution (0, I∗ (t), Y∗ (t)) of system (2.1) is globally asymptotT ∗ ∗ ical stable; If aα > r − β , λ ≥ a and (1 − u )e 0 [r−(a+λ )I (t )−bY (t )]dt > 1, then system (2.1) is permanent. We have also proved

( 1 − u )e

0

the existences of the “infection-free” periodic solution and the “predator-free” solution. Therefore, we can use the impulsive control strategy method to drive the susceptible pest to extinction by the effect of the viruses on the environment and cost of the releasing pest infected in a laboratory such that T < T0 ≈ 1.36. Numerical results show that some attractors can coexist for a wide range of parameters if parameter spaces are differently chosen, and the pest population oscillates with different amplitudes from them. These results can be helpful in determining key parameters including the transmission co-

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efficient, the mortality, the coefficient of prey, the coefficient of the conversion rate of susceptible prey into predators, the mortality of natural enemy, the birth rate and the release amount of the infected pests and play important part in designing appropriate control strategies and assisting decision-making management. We can be sure that impulse plays an important part in managing the agricultural resources, which makes the ecosystem exhibit more unpredictable and complex behaviors. Moreover, the method used in this study can be extended to more complex and realistic epidemiological models. Acknowledgment This work is supported by the National Natural Science Foundation of China (11202116), the Open Fund of Hubei key Laboratory of Hydroelectric Machinery Design and Maintenance (2016KJX01) and the Foundation of China Three Gorges University (KJ2011B033). 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