Time Delayed Stage-Structured Predator-Prey Model with Birth Pulse and Pest Control Tactics

and Applied Analysis 3 which implies that corresponding periodic solution of (3) on (nT, (n + 1)T] is ?̃? (t) = a (1 − p) + b − ae d3T 1 − (1 − p) e −d3T e −d3(t−nT) . (9) Thus (1) has a mature prey-extinction periodic solution (x 1 (t), 0, y(t)), where x 1 (t) = (μe −ω(t−nT) )/(1 − (1 − p 0 )e −ωT ), y(t) = (a(1 − p) + b − ae d3T )/(1 − (1 − p)e −d3T )e 3. Theorem 3. The mature prey-extinction periodic solution (x 1 (t), 0, y(t)) is globally asymptotically stable if R 1 < 1 (10) holds, where R 1 = (re −ωτ − d 1 )[1 − (1 − p)e −d3T ]/β[a(1 − p)e −d3T + be −d3T − a]. Proof. Let (x 1 (t), x 2 (t), y(t)) be any solution of system (1). Following from the third equation of system (1), we notice that ̇ y(t) ≥ −d 3 y(t). Consider the following impulsive differential system: ̇ y 1 (t) = −d 3 y 1 (t) , t ̸ = nT, y 1 (t + ) = (1 − p) y 1 (t) + by 1 (t) a + y 1 (t) , t = nT. (11) According to Lemma 2, we obtain the fact that y 1 (t) = ((a(1− p)+b−ae d3T )/(1−(1−p)e −d3T ))e 3, for nT < t ≤ (n+1)T. By using the comparison differential theorem of impulsive equation, we havey(t) ≥ y 1 (t) andy 1 (t) → y 1 (t) as t → ∞. Therefore, for an arbitrarily small positive constant ε and all t large enough: y (t) ≥ y 1 (t) ≥ y 1 (t) − ε (12) holds true. For simplicity, we assume that (12) holds for all t > 0. From (12), we have


Introduction
The outbreak of pest often triggers serious ecological and economic problems.In recent years, the management of pest has increasingly become the focus of attention.How to effectively control pest is one primarily concern problem.In practice, lots of factors can affect the efficiency of pest control, for instance, the time of impulsive effects, the number of prey stocked or naturally released, and the proportion of killing or catching pests.Mathematical modelling is one of the main ways for estimating and predicting the range of ecological interactions between pest and predator.Lately, many papers have been devoted to propose and analyze the predator-prey systems [1][2][3][4].
In one aspect, many species have the life history that goes through two stages, immature and mature, which has significant morphological and behavioral differences between them.Therefore, it is necessary to account for these differences, and the dynamics of mathematical models with stage-structured prey-predator model has been widely studied [5][6][7][8][9][10][11] in recent years.In other aspects, in the natural world, the immature predator always undergo a certain time  (which is called maturation time delay) to be mature, so time delays play an important role in biological meanings in age-structured models.Time delay and stage-structured systems are introduced into predator-prey models, which greatly enriches biologic background.Many age-structured models with time delay were extensively studied [12][13][14].A single species model with stage structure and time delay was invested by Aiello and Freedman [15].
As far as the population dynamics is concerned, most models often considered that the population reproduces throughout all year.However, many species give birth seasonally or in regular pules.In this regard, the continuous reproduction of mature species should be removed from the model and termed this growth form as birth pulse.For instance, Tang and Chen study an age-structured model with density-dependent birth pulse in [16] and Xiang et al. [17] have considered a delayed Lotka-Volterra model with birth pulse and impulsive effect at different moment on the prey.Theories of impulsive differential equations have been studied lately [18][19][20].Impulsive systems are found in many domains of applied sciences.The application of impulsive differential equations to population dynamics is an interesting topic since it is reasonable and corrects in modeling the evolution of populations.Virtually, the pulses occur at different moments

Predator-Prey Model with Impulsive Effects at the Same Time
The purpose of this paper is to address how the time of impulsive effects influences the pest control.On the basis of the above discussion and motivated by [8,16], we first will extend the following impulsive model which introduces the impulsive control tactics at the same time: where  1 () and  2 () are densities of the immature prey and mature prey, respectively.() denotes the density of predator. represents a constant time to maturity and ,  1 ,  2 ,  3 , , , , ,  are positive constants. is the birth rate of immature prey,  ( >  3 ),  1 are the mortality rates of the immature prey and mature prey, and  3 ( 3 >  1 ) is the death coefficient of (). 2 is the intraspecific competition rate of the mature prey. is the transmission coefficient.The term  −  2 ( − ) represents the immatures who were born at time  −  and survive at time  (with the immature death rate ) and therefore represents the transformation of immatures to matures ( − >  1 ). is the rate of conversing prey into predator. 0 ,  (0 ≤  0 ,  < 1) represent the killing (or poisoning) rate of the prey and the predator, respectively. ≥ 0 is the stocking amount of the immature prey at  = .()/(+()) denotes the birth effort of predator population at  = , where  means the maximum birth effort per unit of time and  the predator size at which 50% saturation (()/( + ())| ()= = /2) measures how soon saturation occurs.
Proof.Integrating the first equation of (3) on  <  < ( + 1), we have With the successive pulse, we can obtain the following stroboscopic map of system (3): and unstable if  < 1 + (/) −   3  , since If  < 1 + (/) −   3  , (5) has a unique positive fixed point: By the same way, we can obtain the fact that the fixed point ) is globally asymptotically stable if  < 1 + (/) −   3  .This completes the proof.

Predator-Prey Model with Impulsive Effects at the Different Time
In particular, in order to avoid the adverse effects of pesticides on the newly released natural enemies, we consider the following method implemented in practice to avoid such antagonism.That is, we assume that the pulse occurs at  = ( + ) (0 <  < 1) and  = ( + 1).The detailed changes are shown as follows: If the mature prey  2 () is absent, then system ( 22) reduces to We can easily obtain the analytic solution of system (23) at the interval [, ( + 1)): Lemma 5.
Proof.Consider system (24), which yields the following stroboscopic map: and unstable if  < 1 −   3  /( + ), since has a positive fixed point denoted by  * , where Further, we can show that  * is globally asymptotically stable, if the following statements are satisfied: This yields () > .So we have  * > () >  under the assumption  * >  > 0; otherwise  > () >  * if  >  * .Thus the statements (i) and (ii) are satisfied.This completes the proof.
Now we can deduce that the positive equilibrium  * of system ( 25) is globally asymptotically stable.So we have the corresponding positive periodic solution ( x1 (), ỹ()) in the following, which is globally stable: Theorem 6. Assume that holds, where Then the mature prey-extinction periodic solution ( x1 (), 0, ỹ()) is globally asymptotically stable.
The proof is completed.The biological significance of Corollary 7 is the same as Corollary 4, so we omit it.
In biological terms, the permanence of ( 22) implies that prey (both immature and mature) and predators will coexist, none of them facing extinction or growing indefinitely.Denote holds, where Proof.It is seen that the second equation of system ( 22) can be rewritten as Let us consider any positive solution ( 1 (),  2 (), ()) of system (22).By (51), () is defined as Taking the first derivative of () with respect to , we have As seen in Lemma 8, (53) can be written as From the definition of  * 2 , we have and then  > 0; it is easy to know that there exists a sufficiently small  such that We claim that, for any  0 > 0, it is impossible that  2 () <  for all  >  0 .Otherwise, there is a  0 > 0 such that  for all  >  0 .It follows from the third equation of ( 22) that for all  >  0 .Consider the following comparison impulsive system of (57): and we have In view of the comparison differential theorem for equation, there exists a  1 > 0, such that for  >  1 , which implies that () ≤ z9 () +  1 , where Thus From (54), we get By ( 54) and ( 63), we have Let In the following we show that  2 () ≥   2 for all  ≥  1 .Otherwise, there is a it is a contradiction.Thus,  2 ≥   2 for all  ≥  1 .Hence, for  ≥  1 , we get which means () → ∞ as  → ∞.This is contrary to the boundedness of ().The claim is proved.By the claim, consider the following two possibilities.
Case 2.  2 oscillates about  * 2 for  large enough.Set We will show that  2 () ≥  for all  large enough.The conclusion is evident in Case 1.For Case 2, suppose that there exists the positive constant ,  satisfying  2 () =  2 ( + ) =  * 2 and  2 () <  * 2 for all  <  <  + , where  is enough large such that since  2 () is continuous and bounded and is not affected by impulses.Thus  2 () is uniformly continuous.Consequently, there exists a constant  3 (0 <  3 <  and  3 is independent of the choice of ) such that  2 () > ( * 2 /2) for  ≤  ≤  +  3 .If  ≤  3 , our aim is obtained.Then we consider the case  3 <  < .From the second equation of (22) and the above assumption, we have that ẋ It is clear that  2 () ≥  for  <  ≤  + .If  ≥ , then we have  2 () ≥  for  <  ≤  + .The same arguments can be continued; we can obtain  2 () ≥  for + <  ≤ +.Since the kind of interval [, +] is chosen in arbitrary way, we have  2 () ≥  for  large enough.In view of the above discussion, the selection of  is independent of the positive solution of ( 22) which satisfies  2 () ≥  for all  large enough.
Next, from the first, the fourth, and the seventh equation of system (22), we have Therefore for any  4 > 0 small enough such that  1 () ≥ z10 () −  4 for  sufficiently large, In view of the comparison theorem of impulsive differential equation and the Theorem 6, we get From ( 34) and (68), set  ≜  3 and  ≜  2 .Define  = min{ 1 ,  2 ,  3 }, and thus we have  1 () ≥ ,  2 () ≥ , () ≥ .By Lemma 8 and the above discussion, the system is permanent.The proof is completed.

Numerical Analysis
In the previous sections, we introduced the analytical tools and used them for a qualitative analysis of the system obtaining some results about the dynamics of the system.In this section, we perform a numerical analysis of the model Those inequalities demonstrate that threshold value  1 is a monotonic increasing function with respect to  and  and is a monotonic decreasing function with .These results indicate that the smaller killing (or poisoning) rate  or the shorter pulse period , the smaller the threshold value  1 which follows, so, the result shows that it is more effective for pest control.Similarly, when pests have very short maturity , the bigger the threshold value  1 which follows, and hence make it more difficult for pest control.Therefore, we must carefully select the impulsive catching (or poisoning) rate or the time of pesticide applications or reduce the dosages of the pesticide.
So far, we have considered the global attractive mature prey-eradication solution of systems ( 1) and (22).We first focus on the system (1); when threshold value  1 < 1, the mature prey is eradicated totally and the immature prey and the predator population will tend to a stable level; see Figure 1, where  = 1.2,  = 0.5,  = 0.2,  = 2,  = 2,  = 0.2,  1 = 0.1,  2 = 0.2,  3 = 0.15,  0 = 0.1,  = 0.5,  = 5,  = 6,  = 3, and  1 ≈ 0.7997 < 1, while if we chose the parameters set as those in Figure 2, then the all populations oscillate periodically, which indicates that the system (1) is permanent.Furthermore, we can carry out two-parameter bifurcation analysis for the threshold value  1 , as those shown in Figures 3(a)-3(d).In each subplot, we let two key parameters vary simultaneously and see how those parameters affect the threshold value  1 .All simulation results shown in Figure 3 indicate that the  1 appears to be quite sensitive to small changes in parameters, , , , and .However, we note that the dynamical behaviour of system (1) is dominated by several types of periodic solutions and their coexistence, even with very high intrinsic growth rates.In Figure 4, we studied the influence of killing rate  on the complexity of system (1); numerical results show that when killing rate  varies from 0.45 to 0.95, the dynamical behaviour of system (1) is very complex.A typical chaos oscillation is captured when  = 0.7 (see Figure 5).Figure 6 shows the bifurcation diagrams of system (1); the system analyzed here could take on many forms of complexity, including period doubling bifurcation, symmetry-breaking pitchfork bifurcation, and chaotic solutions.Meanwhile, system (1) also has a certain type of coexistence if the intrinsic growth rate of the pest population is large enough.
We are more interested in how different patterns of insecticide applications affect the two threshold values ( 1 ,  * 1 ) and which strategy is more conducive to pest control.To do this, we investigate the superiority of the two strategies.Assume that the  1 and  * 1 are with the same parameters; Figure 9 provides the details of how different control strategies affect the threshold values  1 and  * 1 .If  = 0, that is, do not spray insecticide,  1 =  * 1 , the effect of two control strategies is; if  ̸ = 0, then  1 >  * 1 ; this result indicates that if natural enemies were born at the same time as spraying insecticide, the side-effects of pesticides on the natural enemy population are very strong.Obviously, part of the newborn natural enemies will be killed and the number of natural enemies will be sharply reduced; the contribution of chemical control for the threshold value  1 is stronger than  * 1 .From the practical point of view, we should avoid to spray insecticide when the predators are born.

Discussion
When using integrated pest management as an approach to control insect pests, one must be committed to a long-term strategy.It is well known that pesticides usually act not only on the pest species but also, with even stronger impact, on their natural enemies; then, despite the fact that pesticides kill the target pest, they are simultaneously reducing the population density of natural enemies; the influence of pesticides on natural enemies is even greater than on insect pests.As a result of this indirect effect, treatments can counterintuitively lead to an effective increase of the pest species, even pest breakout again.
In this paper, our idea is to contribute to pest control programs with pesticide applications and with birth pulse in natural enemies and to provide some strategies; that is, spraying pesticides and natural enemies were born at the same time and at a different time.The threshold conditions which guarantee the existence and stability of the mature prey-extinction periodic solution are provided.If the integrated control methods cannot completely eradicate the mature prey, the pest population can have outbreaks at different scales.By numerical bifurcation investigations, we found that, when choosing different parameter spaces, two attractors from which the pest population oscillates with different amplitudes can coexist for a wide range of parameters.Moreover, the effects of times of spraying pesticides (or the enemy was born) and control tactics on the threshold conditions were carefully investigated.In particular, the effects of the killing rates of pesticides for pest and natural enemy populations and spraying period on the stability of the mature prey-extinction periodic solutions were discussed.The results imply that the modeling methods described can help the pest control specialist to decide appropriate control strategies and assist management decision-making.

Figure 5 :
Figure 5: Time-series of the system (1) with pulse, where killing rate  = 0.7; other parameters are the same as of those in Figure 4.