Dynamical Behavior for a Food-Chain Model with Impulsive Harvest and Digest Delay

We investigate the dynamics of a food-chain systemwith digest delay and periodic harvesting for the prey. By using the comparison theorem, small amplitude skills in the impulsive differential equation, and a special qualitative analysis method in the delay differential equation, we prove that there exists a predators-eradication periodic solution which is globally attractive and show that the pest population can be controlled under the economic threshold level and the system can be uniformly permanent when the harvest period T is long enough or the harvesting rate δ is not too large. Furthermore, we perform a series of numerical simulations to display the effects of the digest delay and periodic harvesting on the dynamic behavior of the food-chain system.


Introduction
It is now widely believed that pest outbreaks often cause serious ecological and economic problems.As a result, ecologists and mathematics acknowledge the importance of controlling insect pests of agriculture and insect vectors of plant [1].Integrated pest management involves choosing appropriate tactics from a range of pest control techniques including biological, cultural, and chemical methods to suit individual cropping systems, pest complexes, and local environments [2][3][4].For example, as concerning the chemical control strategy, it seems to be quick and efficient to decrease the pests population by the chemical insecticides in a short time.But when we use excess of chemical insecticides to kill the pest population, not only is the environment polluted, but also the natural enemies (or beneficial species) will be killed at the same time, even leading to the adaptability of the pests and the ineffectiveness of the insecticides.And this will lead to the waste of the manpower and material resources and we cannot reach our expected results, even bringing negative effects.And as concerning the biological control strategy, that is, stocking the natural enemies periodically by artificial culture or immigration, we can avoid many human losses caused by environmental pollution in this way, while it will take us a long time and a complex process for the culture of the natural enemies.Therefore, it is important to establish mathematical models to provide valuable information about how to control pest outbreaks, especially to study the dynamical behavior of the pests and their natural enemies.
On the other hand, when the prey-predator system is referred, sometimes there is a digest and absorption time (which is the so-called digest delay) during the predation instead of translating the food into growth rate immediately.Hence, in order to model the relationship between the predator and the prey more accurately, it is more reasonable to introduce time delay into the model.Usually, there are two kinds of delays in the ecological model, that is, discrete timedelay and distributed time-delay (continuous time delay).Recently, it seems that much more attention is paid on the models with impulsive perturbations and time delay [5][6][7][8][9][10][11][12][13], and some of them [5][6][7][8] trend to focus on the impulsive model with distributed time-delay, in which a kernel function 2 Journal of Applied Mathematics () =  − ,  > 0. To the best of our knowledge, the study on the effect of the discrete time-delay on the impulsive system seems to be rare.
Recently, in an effort to seek more efficient pest management strategies, Yu et al. [14] where (), (), and () are the densities of one prey and two predators at time , respectively, and Δ() = ( + )−(), Δ() = ( + ) − (), and Δ() = ( + ) − ().  ( = 1, 2) are the intrinsic growth rate,   ( = 1, 2, 3) and   ( = 1, 2) measure the efficiency of the prey in evading a predator attack, and  3 has similar meaning as that of   .  ( = 1, 2, 3) denote the efficiency with which resources are converted to new consumers,   ( = 1, 2) are carrying capacity in the absence of predator,  is the mortality rates for the predator,  is the period of the impulsive effect,  ∈ ,  is the set of all nonnegative integers, and  > 0 is the release amount of predator at  = .
In [14], the authors studied the food-chain prey-predator model (1) with periodic release on the higher predator (enemy population) () and discussed some efficient biological control strategies for the system.But they had not considered the affection of the digest delay.
Based on the discussions above, we consider the following food-chain prey-predator model with periodic harvest on the prey (the pest population) (), but the lower predator () only lives on the prey.That is, if the prey () is extinct, the lower predator () has no other food resources, and it is inevitable to be extinct.Furthermore, we assume that there is a digest and absorption time  during the predation of the higher predator () instead of translating the food into growth rate immediately, and the final model we will study in this paper is as follows: where  1 ,  2 > 0 are the coefficients of density dependence of () and (); since the higher predator population always have stronger ability to migrate, then it is more possible for them to escape from the inner competition.Thus, the impact of density dependence is relatively small, so we do not consider the density dependence of higher predator () in the model.Further,  1 is the intrinsic increasing rate of the prey population (),  2 is the death rate of the lower predator (),   and   ( = 1, 2, 3) measure the efficiency of the prey in evading a predator attack.  ( = 1, 2, 3) denote the efficiency that resources are converted to the new consumers, and  is the mortality rates of the higher predator.0 <  < 1 is the harvesting rate at the periodic time  =  ( ∈  * ), and the initial condition for system (2) is From the viewpoint of ecological meanings, we only consider system (2) in the nonnegative region  = {(, , ) |  ≥ 0,  ≥ 0,  ≥ 0}.
The rest of this paper is organized as follows: in Section 2, we will give some basic definitions and several useful lemmas for the proof of our main results.In Section 3, we will state and prove our main results such as boundedness of the solution, global attractivity of the predators-eradication periodic solution, and sufficient conditions for the permanence of the system.In Section 4, we give some numerical examples to support our theoretical results.And in the last section, we provide a brief discussion and the summary of our main results.

Preliminaries
+ ,  be the set of all nonnegative integers.Denote as  = ( 1 ,  2 ,  3 ) the map defined by the right-hand side of the first, second, and third equation of the system (2).
Lemma 4 (see [17]).Consider the following delay differential equation: where  1 ,  2 , and  are all positive constants and () > 0 for all  ∈ [−, 0]. ( ( In order to discuss the predators-eradication periodic solution of system (2) in the next section, we will give some basic properties about the following subsystem of system (2) at first: It is easy to solve above system (10) between pulses, yielding Then we can obtain the stroboscopic map of ( 11) as follows: If we denote  = ( + ), then which has two fixed points: Then we have Lemma 5 for the subsystem (10) by the method in [9].Lemma 5. Suppose  * = 1 −  − 1  , and then we have the following results.
Proof.If  >  * , we can see that the stroboscopic map of (11) has a unique trivial fixed point  * 1 = 0, and by a direct calculation we can obtain Hence, the trivial periodic solution  * 1 = 0 is globally asymptotically stable.

Journal of Applied Mathematics
If  <  * , the subsystem (10) has a trivial fixed point  * 1 = 0 and a positive fixed point Moreover, So the trivial periodic solution  * 1 = 0 is not stable.Now we consider the stability of the positive fixed point  * 2 .
In fact, if we substitute into ( 11), then we can get which is a positive periodic solution of system (10).
In the following we will show that the positive periodic solution is globally asymptotically stable.
In order to do this, we take the transformation () = 1/() for system (10), and then the following linear nonhomogeneous impulsive equation is obtained: Thus, () = (,  0 ) is the solution of system (10) with initial condition (0 By the Cauchy matrix of the respective homogeneous equation, we have that is the solution of system (20).Thus, On the other hand, when  <  * , which leads to lim Thus, That is, the positive periodic solution, is globally asymptotically stable.
Proof.Let () = ((), (), ()) be any solution of system (2) with initial condition (3), and we define Then, On the other hand, by a simple calculation Therefore, By Lemma 2.2 in [14] we have which leads to This completes the proof of this theorem.
Proof.By the first equation and the impulsive effect, we have whose comparison system is (10).Then, by comparison theorem (Lemma 3) of impulsive differential equations, there exists an arbitrarily small positive  > 0 such that when  is large enough.This yields lim Hence, there exists a positive integer  1 ∈  and arbitrarily small positive  1 > 0 such that for all  ≥  1 .
At the moment, from the second equation of the system (2) we have Then, lim Then there exist  1 > 0 and  2 > 0 small enough, such that and it follows from the last equation of system (2) that when  > max{ 1 ,  1 } + .
For above arbitrarily small positive  1 ,  2 small enough, since condition (H1) holds, then By Lemma 4, we have lim Then for above  2 > 0 small enough, there exists a  2 >  1 such that On the other hand, combining the first equation of system (2) with ( 43) and (47), we have for  >  2 , where Note that the corresponding comparison system of (48) is By Lemma 5, if  <  * , system (50) also has the following positive periodic solution: which is globally asymptotically stable.Thus, by Lemma 3 again we have for above arbitrarily small  > 0 as  is large enough.Let  2 → 0, and then that is,  * () →  * ().
In fact, if the conditions of Corollary 8 hold, then It follows from (57) that which yields Note that and then That is, In the same way, from (58), we have another inequality: Therefore, all the conditions of Theorem 7 hold, and then the predators-eradication periodic solution ( * (), 0, 0) of system ( 2) is globally attractive.Remark 9. From Theorem 7 and its sufficient condition Corollary 8, if  <  * ,  <  * 1 , or Δ * <  <  * , then the natural enemies (both of the predators' population) in the model are extinct while the pest population is still not controlled when the pest population is poisoned exclusively.From the viewpoint of ecosystem and protecting the variety of the rare species, we only need to control the pest population under a certain threshold level and should not eradicate the enemy population.That is, the pest population and the enemy population can coexist when the pest cannot cause immense economic losses, so it is more important to consider the uniform persistence for the system.Theorem 10.If system (2) satisfies  2 ≤  1  3 ,  < , and the following condition (H2): Proof.From Theorem 6, we have obtained the upper bound of each solution () = ((), (), ()) of the system (2) with  large enough.Thus, we only need to search for the lower bound of the solution in the following.
In fact, from the first equation of system (7), we have By the comparison theorem (Lemma 3) we have () ≥ V 1 () and V 1 () → Ṽ1 () as  → ∞, where V 1 () is the unique and globally stable positive periodic solution of Therefore, for sufficiently large , there exists a  3 > 0 small enough such that In the following, we will show that there exist two positive constants  2 and  3 , such that () ≥  2 and () ≥  3 for any  large enough.
Step 1.We begin to find an  2 > 0 such that () ≥  2 for any  large enough.
In order to achieve this goal, firstly we claim that the inequality () <  2 cannot hold for all  ≥  1 .
Step 2. Now we try to find an  3 > 0 such that () ≥  3 for all  is large enough.
In the same method, we claim that the inequality () <  3 cannot hold for all  >  3 .
Otherwise, if there exists a  3 > 0 such that () <  3 for all  ≥  3 + , then by the first equation of (2), where Therefore, there exists a  5 > 0 small enough and a  4 ≥  3 + , such that for  ≥  4 , where Now we define a Liapunov functional and then When the condition (H2) holds, we can choose  5 > 0 small enough such that Therefore, () <  3 cannot hold for all  ≥  3 , and there are two cases as follows.
If () ≥  3 for all  ≥  3 , then our aim is obtained.Otherwise, if () is oscillatory around  3 , when  is sufficiently large, let then we can show that () ≥  3 as  is large enough.In fact, suppose there exist two positive constants  2 > 0,  2 > 0, such that (95) Since () is continuous, bounded, and not affected by impulses, we conclude that () is uniformly continuous; then exists a  5 > 0 (with 0 <  5 <  and  5 is independent of the choice of  2 ) such that () ≥  3 /2 for all  2 <  <  2 +  5 .
In a similar way to the discussion of Corollary 8, we can obtain the following two sufficient conditions for the permanence.
where  1 ,  2 and  * ,  is the same as Theorem 10, then system (2) is permanent.

Numerical Simulations and Discussions
In this paper, we consider a food-chain prey-predator system with digest delay and impulsive harvest on the prey.Our main aim is to investigate how the impulsive harvest and digest delay affect the dynamical behavior of the system.Especially, we focus on the suitable impulsive period so that we could guarantee that the predators will not be extinct before the prey.Furthermore, we are also concerned when the system will be permanent and how to control the population of the prey (pests) under a certain economic threshold level (ETL).
In the following, we will verify our main results by numerical simulation.
On the other hand, the selection of the economic threshold level (ETL) is closely related to the dynamical behavior of the pest population, especially to the maximum population after a period.Therefore, when the conditions of the permanent theorem (Theorem 10) hold, from the first equation of system (2), we can change some of the parameters of system (2) to decrease the economic threshold level, such as decreasing the value of parameter  1 or increasing the value of parameter  1 .To verify this point, we consider the following Case 3.
Case 3. If we choose  1 = 1.86 < 1.96,  1 = 0.36 > 0.3 while the other values keep the same as Case 2, and plot the time-series of the pest population (see Figure 3(a)), it is obvious to see that the pest population can be controlled under a new ETL = 1.05, which is lower than the previous ETL = 1.2.Furthermore, multiple periodic solutions or periodic oscillations appear from the phase portrait at the moment (see Figure 3

Conclusions and Discussions
In this paper, we investigate the dynamics of a threedimensional food-chain system incorporating digest delay and periodic harvesting for the prey.The value of our study