A Stage-Structured Predator-Prey SI Model with Disease in the Prey and Impulsive Effects

This paper aims to develop a high-dimensional SI model with stage structure for both the prey (pest) and the predator, and then to investigate the dynamics of it. The model can be used for the study of Integrated Pest Management (IPM) which is a combination of constant pulse releasing of animal enemies and diseased pests at two different fixed moments. Firstly, we use analytical techniques for impulsive delay differential equations to obtain the conditions for global attractivity of the ‘pest-free’ periodic solution and permanence of the population model. It shows that the conditions strongly depend on time delay, impulsive release of animal enemies and infective pests. Secondly, we present a pest management strategy in which the pest population is kept under the economic threshold level (ETL) when the pest population is permanent. Finally, numerical analysis is presented to illustrate our main conclusion.


Introduction
Large-scale pest outbreak may bring serious ecological and economic problems to our society. For example, cotton bollworm outbreaks have caused severe losses of cotton in Xinjiang, Henan, Shandong, Hebei provinces of China in the past two decades; outbreaks of large-scale locusts, which feed on leaves, stems of crops, can cause serious ecological disaster. Organic chemicals (chemical insecticides) have been used to control bands and swarms for more than half a century. It turns out to be useful since they can quickly kill a significant portion of a pest population. Extensive use of chemical pesticides greatly enhances the human ability to control insects, however, it also brought well-known issues of environmental pollution, ecological balance, food safety, and so on. Pesticide caused environmental pollution is believed to be the number one killer to human health and other creatures. Therefore, how effectively and without compromise of environment to control insects has become an increasingly complex issue over the past two decades.
A pest control problem is a problem of population dynamics. It is proved to be more effective to control the pest by biological technologies, which include using predatory natural enemies, microbial, parasitic enemies, etc. It can be cultivated in biological laboratory or natural environment. This method is mostly welcomed by the people because it is harmless not only to environment and human health, but also the development of population. Nowadays, this method has been widely applied in the pest control of vegetables, fruits, some trees in agriculture and planting. In recent years, impulsive systems are found in many research areas of applied sciences [7,10,15,19,21]. Impulsive delay differential equations play a significant role in various branches of applied sciences including biology and population dynamics. Generally, the theory of impulsive delay differential equations is relatively mature [9,13] and the introduction of time delay and impulse to predator-prey models with stage-structure enriches the biological background greatly. But the consequence is obvious because the system becomes nonautonomous and quite complicated. Recently, twodimensional delayed stage-structured models with impulsive effect at one fixed impulsive moment have been investigated in [5,16]. The predator-prey models with stage structure for the predator have been extensively investigated, [4,20] to name but a few. In the real world, as immature prey takes τ units of time to mature, it is necessary to consider the death toll during the juvenile period. Hence time delay is endowed with more vital significance in stage-structured models, for example [1,6,17].
In this paper, we aim to propose a pest control SI model which is stagestructured for both the pest and the predator by introducing a constant periodic releasing animal enemies and infective pests at two different fixed moments. As the diseased and juvenile individuals of pests cause damage to crops very little, we assume in our model natural enemies prey on mature pests, but the diseased and juvenile pests, and we only need to control the adult pest (prey). Then we propose a high-dimensional delayed predator-prey model with two stage structures and two different fixed moment impulse effects. New technique is also developed to investigate the dynamics of the model.

Model formulation
The basic model considered is based on the following SI epidemic model where S(t) and I(t) denote the members of the population susceptible to the disease, and the infective members, respectively. Aiello and Freedman intro-duced the following famous model [1]: where they assumed for the mature population that the death rate is of a logistic nature, that is, proportional to the square of the population with proportionality constant e. Meng and Chen introduced a stage-structured SI ecoepidemiological model with time delay and impulsive controlling [14]. Cooke [3] formulated an SIR model with time delay effect by assuming that the force of infection at time t is given by βS(t)I(t − τ ), where β is the average number of contacts per infective per day and τ > 0 is a fixed time during which the infectious agents develop in the vector and it is only after that time that the infected vector can infect a susceptible human. Levin et al. adopted a incidence form like βS p (t)I q (t), βS p (t)I q (t)/N , p, q ∈ R + which depends on different infective diseases and environments [12]. We shall consider the case of p = 1, q = 2, N = 1, i.e. βS(t)I 2 (t). Motivated by recent work, we formulate the following delayed pest management SI model with stage structure for both the pests and the predators, and the effects of constant natural enemies and diseased pests impulsive input periodically at two different fixed moments: 1) where S j (t) and S(t) represent the density of the immature and mature pest, y j (t) and y(t), the density of the immature and mature natural enemy, respectively; I(t) represents the density of the diseased pest at time t, r is the growth rate of the mature pest in the absence of the predator, d 1 , µ, d 3 , d 4 and d 5 are the death rates of the immature pest, mature pest infective pest, immature natural enemy and mature natural enemy. d 2 is the mature pest death and overcrowding rate. β is the infection rate of infective pest, α is the predation rate of the predator, λ represents the conversion rate at which ingested prey in excess of what is needed for maintenance is translated into predator population increase, ω measures the psychological or inhibitory effect respectively; τ is the mean length of the juvenile period; T is the period of the impulsive effect; q is the released amount of natural enemies at every impulsive period (n + l − 1)T ; p represents the released amount of infective pests at every impulsive period nT . From the point of biology, we only consider system (1.1) in the following region: D = {(S j , S, I, y j , y | S j , S, I, y j , y ≥ 0}. Let For continuity of initial conditions, we require Since the variables S j (t) only appear in the first equation of system (1.1), we only need to consider the subsystem of system (1.1) as follows:

Boundedness
First we should point out the solution of (1.1), x(t) = (S j (t), S(t), I(t), y j (t), y(t)) is a piecewise continuous function on (nτ, (n + 1)τ ], and x(nτ + ) = lim t→nτ + x(t). For system (1.1) to be biologically meaningful, it is important to prove that all its state variables are non-negative for all time. In other word, solutions of system (1.1) with positive initial value remain positive for all time t > 0.
From the third equation of (1.1) we get for t = nT , that For t = nT , we have I(nT + ) = I(nT ) + p, p ≥ 0, it is easy to see that I(t) > 0 for all t > 0. From the fourth equation of (1.1) for t = (n + l − 1)T , we have that and for t = (n + l − 1)T , we have y j (nT + ) = y j (nT ) + q, q ≥ 0, thus we have y j (t) > 0 for all t > 0. Similarly from the fifth equation of (1.1), we get for t > 0 that Obviously we have y j (t) > 0 for all t > 0. Finally we consider the following equation: and comparing with (1.1), we note that if u(t) is the solution of (1.4) and if We now show by induction that S j (t) is positive on nτ < t < (n + l)τ , n = 0, 1, . . .. We have just shown that this is valid for n = 0. Assume it is valid for n = 0, 1, . . . , k − 1. Then S j (kτ ) > 0. Consider the equation given by (1.5) So we obtain u((k + 1)τ ) = 0 and so S j (τ ) > 0. Hence map defined by the right-hand side of the anterior five equations of system (1.1), and N the set of all non-negative integers. Let exist.
(ii) V is locally Lipschitzian in x.
Then we have the following definition: x) with respect to the impulsive differential system (1.1) is defined as Next, we will consider the boundedness of system (1.1). Let Then the upper right derivative of V (t) along a solution of system (1.1) with t = (n + l − 1)T , t = nT is given bẏ Since r, λ, d 1 , d 2 , d 3 , d 4 , d 5 > 0, one can deduce thaṫ We consider the following impulse differential inequalities.
According to impulse differential inequalities theory, we get we know that each positive solution of system is ultimately bounded.

Analysis of the Model
Then we have the following lemmas from previous work.

Global attractivity of the 'mature pest-extinction' periodic solution
Firstly, in this section, we investigate the pest-extinction solution of the system (1.3), in which the pest individual are entirely absent from the population permanently, i.e. S(t) = 0, t ≥ 0. In this case, system system (1.3) can be rewritten as follows: ∆y j (t) = q, ∆y(t) = 0, t = (n + l − 1)T.
Proof. Let (S(t), I(t), y j (t), y(t)) be any solution of system (1.3) with initial condition (1.2). From the second equation of system (1.3), it follows that dI(t)/dt ≥ −d 3 I(t), I(t + ) = I(t) + p, for nT < t ≤ (n + 1)T . By Lemma 3, impulse differential system has a globally asymptotically stable positive periodic solution, Then the comparison theorem implies, for any sufficiently small ε > 0, there exists an integer n 1 such that Similarly, for any sufficiently small ε 1 > 0, there exists an integer n 2 > n 1 such that y j (t) > y * j (t) − ε 1 , t ∈ (n + l − 1)T, (n + l)T , t > n 2 T, (2.7) from which and the fourth equation of system (1.3), we have Consider the system It is not difficult to verify that system (2.8) has a globally asymptotically stable positive periodic solution, where g(t) is continuous function on [(n + l − 1)T, (n + l)T ] and It has a unique stationary point at which

Consider the third equation of system (1.3), we have
there exists a positive periodic solution, which is globally asymptotically stable. Hence, for a sufficiently small ε 4 > 0, when t is large enough, we have Combining system (2.7) with (2.18), we obtain which implies lim t→∞ y j (t) = y * j (t) as ε 1 , ε 3 and ε 4 are all sufficiently small constants. Since lim t→∞ y j (t) = y * j (t), by Lemma 4, we obtain lim t→∞ y(t) = y * (t). The proof is completed.
Let R 1 = 1, we can work out threshold value of parameter p, q and τ respectively. Denote .

Permanence and the pest control strategy
In Section 2.2, we will prove the pest-eradication solution (0, I * (t), y * j (t), y * (t)) of (1.3) is globally attractive when R 1 < 1, that is, the adult pest population is eradicated totally as time goes under the condition for the global attractivity.
Considering the principle of ecosystem balance, biological diversity of population and resources saving, we hope that the pest population can coexist with its natural enemy population while the pests do not bring immense economic loss, in other words, we only want to control the pest population under the economic threshold level (ETL). So in the following, we will give the sufficient condition for the permanence of (1.3), and discuss the strategy of regulating the pest. The following definition is necessary before stating our theorems. Denote , .

Then we have
Theorem 3. If R 2 > 1, there exists a positive constant σ such that each positive solution (S(t), I(t), y j (t), y(t)) of system (1.3) satisfies S(t) ≥ σ for t large enough.
Proof. The first equation of system (1.3) can be rewritten aṡ Let U (t) = S(t) + re −d1τ t t−τ S( ) d . Its derivative along the solution of system (1.3) is Next, we claim that the inequality S(t) < k * cannot hold for all t ≥ t 0 , here t 0 > 0 is arbitrary constant. Otherwise, there exists a positive constant t 0 such that S(t) < k * for all t ≥ t 0 . We shall prove that this can not happen. From the second equation of system (1.3), when t ≥ t 0 , we have (2.20) Thus there exists a T 1 > t 0 + τ such that (2.21) From the third equation of system (1.3), when t ≥ t 0 , we have Thus there exists a T 1 > t 0 + τ such that Let T 1 = max{T 1 , T 1 }. Then from the fourth equation of system (1.3), when t > T 1 , we have Similarly, there exits a T 2 > T 1 such that Then from (2.9), we have By the definition of k * , one can easily get k * > 0. Choose ε 1 , ε 2 , ε 3 > 0 to be small enough such that re −d1τ > d 2 k * + βCL + αD/(1 + ωD), (2.23) inequality (2.23) is further proved in Appendix A. Then we have We can show S(t) ≥ S l for all t ≥T . Actually, if there exists a nonnegative constant T 3 such that S(t) ≥ S l for t ∈ [T ,T + τ + T 3 ], S(T + τ + T 3 ) = S l andṠ(T + τ + T 3 ) ≤ 0. Then from the first equation of (1.3) and (2.21), we easily see thaṫ So we havė which is a contradiction. Hence S(t) ≥ S l > 0 for all t ≥T . From (2.24), we have which implies U (t) → +∞ as t → +∞. This is a contradiction to U (t) ≤ (1+rτ e −d1τ )L. Therefore, for any positive constant t 0 , the inequality S(t) < k * cannot hold for all t ≥ t 0 . If S(t) ≥ k * holds true for all t large enough, then we finish the proof of the theorem. Otherwise, S(t) is oscillatory about k * . We shall show our conclusion is also true in this case.
(2.27) From (2.27), we know that S(t) < k * fort < t <t + ρ, from the third equation of system (1.3), when t ≥t, we have Thus there exists a T 1 >t + τ such that Then from the fourth equation of system (1.3), when t > T 1 , we have Similarly, there exits a T 2 > T 1 such that Thus, we can eventually get y(t) < D fort + T 2 < t <t + ρ. Since S(t) is continuous and bounded, and is not effected by impulses, we conclude that S(t) is uniformly continuous. Hence there exists a constant T 4 with 0 < T 4 < τ and independent of the choice oft such that S(t) > 1 2 k * for allt ≤ t ≤t + T 4 . If ρ ≤ T 4 , our aim is obtained.
If ρ ≥ τ , then we have that S(t) ≥ k * e −(d2L+βL 2 + αL 1+ωL )τ fort < t ≤ t + τ . Next, we will show that S(t) ≥ k * e −(d2L+βL 2 + αL 1+ωL )τ fort + τ < t ≤ t + ρ. In fact, if this is not true, there exists a T 5 > 0 such that On the other hand, from the second equation of system (1.3) and (2.21), we easily seė thus we geṫ which is a contradiction toṠ(t + τ + T 5 ) ≤ 0. Hence we get that S(t) ≥ σ > 0 for all t ∈ [t,t + ρ]. Since interval [t,t + ρ] is arbitrarily chosen, we know S(t) ≥ σ for t large enough. Please notice the choice of σ is independent of the positive solution of (1.3) which satisfies that I(t) ≥ σ for sufficiently large t. This completes the proof. Proof. Suppose that (S(t), I(t), y j (t), y(t)) is any positive solution of system (1.3) with initial conditions (1.2). By Theorem 3, there exist positive constants σ and T * such that S(t) ≥ σ, for t ≥ T * . From the proof of Theorem 2, we can conclude that the following inequalities hold for t large enough. Noticing the boundedness of (1.1) gives S(t), I(t), y j (t), y(t) ≤ L for t large enough. Then system (1.3) is permanent, and the proof is completed.
Let R 2 = 1, we can work out threshold value of parameter p, q and τ respectively. Denote We have Corollary 2. If p < p * or q < q * or τ < τ * , then system (1.3) is permanent.
As any pest control professional can tell us, large-scale eradication is infeasible. Therefore we aim to keep pests under the economic threshold level (ETL) to protect the crop but to eradicate them. In the following, we consider pest control strategy and give conditions under which the pest population is under ETL. Then we have then the pest and its natural enemy may coexist. Furthermore, when t is large enough, we have S(t) < E, where the constant E is the economic threshold level (ETL).
Proof. Suppose (S(t), I(t), y j (t), y(t)) is a positive solution of (1.3) with initial conditions (1.2). Since we may choose three sufficiently small positive constants ε 1 , ε 2 , ε such that Furthermore, from the following inequality we can get and by (2.11), when t is large enough, we have which implies S(t) ≤ Z(t) for t large enough. Consider the comparison equation Using (2.28) and Lemma 2, we have This completes the proof.

Numerical Analysis and Discussion
In this paper, we further developed impulsive delayed models with staged structure, and investigated a high-dimensional delayed pest management SI model with impulsive natural enemies and diseased pest transmission at different fixed moments. Our main purpose is to study dynamics of the model such as attractivity of periodic solution, permanence of the system and to give pest control strategies for Integrated Pest Management (IPM). Using the theory for impulsive delay differential equation, we obtained some interesting results. In   Section 2, we analyzed extinction of pests and coexistence of pests and natural enemies. Section 2.1, discussed the conditions for the global asymptotical attractivity of the 'pest-extinction' periodic solution, and in Section 2.2, we got the conditions for the permanence of the system and also considered the pest control strategy. From Theorem 2, we can see that a large amount of infective prey input, p or a large amount of natural enemy input, q or a long juvenile period of the predator, τ is a sufficient condition for the global attractivity of the 'pest-extinction' periodic solution. From Theorems 3 and 4, we can see that a small amount of infective pest or a small amount of natural enemy or a short juvenile period of the predator (with τ ) is a sufficient condition for the permanence of the system. Theorems 2, 3 and 4 show that R 1 and R 2 depend on the time delay τ , so, we call it "profitless", and we obtained critical values of time delay τ * and τ * . To verify the theoretical results obtained in this paper, we will give some numerical simulations by Maple and Matlab, which also show some new phenomena different to previous work. For this purpose, parameters have been   (i) From Theorem 3.2, we know that the mature pest tend to die out when T = 0.3, p = 0.3, q = 0.3, τ = 0 (R 1 = 0.3608 < 1, see Fig. 1(a)).
(iii) Also we can fix T = 3, p = 0.3, q = 0.3 and increase maturation time delay to τ = 5.8 (R 1 = 0.9787 < 1), then we find that the pest tends to be extinct (see Fig. 4), which implies the great effect of the maturation time delay on dynamics of the system.
(iv) Fig. 2(b), Fig. 2(c) and Fig. 2(d) show the system has a global stable   positive periodic solution when it is permanent.
(v) From the standpoint of ecological balance and saving resources, we maintain the pest population under the economic threshold level (ETL = 0.12) only, but to eradicate the pests totally (see Fig. 5).
All these results show that dynamical behaviors of system (1.3) become more complex under periodically impulsive effects.