Abstract
We investigate the dynamics of a food-chain system with 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 is long enough or the harvesting rate 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.
1. 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 [2–4]. 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 time-delay 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 [5–13], and some of them [5–8] trend to focus on the impulsive model with distributed time-delay, in which a kernel function . 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] considered an ecological model with impulsive control strategy as follows: where , , and are the densities of one prey and two predators at time , respectively, and , , and . are the intrinsic growth rate, and measure the efficiency of the prey in evading a predator attack, and has similar meaning as that of . denote the efficiency with which resources are converted to new consumers, 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 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 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, is the intrinsic increasing rate of the prey population , is the death rate of the lower predator , and measure the efficiency of the prey in evading a predator attack. denote the efficiency that resources are converted to the new consumers, and is the mortality rates of the higher predator. 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 .
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.
2. Preliminaries
Let , be the set of all nonnegative integers. Denote as the map defined by the right-hand side of the first, second, and third equation of the system (2).
Let ; then is said to belong class if(1) is continuous in , and for each , , (2) is locally Lipschitzian for .
Definition 1. ; then for , the upper right derivative of with respect to the system (2) is defined as
Definition 2. System (2) is said to be permanent if there exist positive and , such that for any , each positive solution of system (2) satisfies with
The solution of system (2) is a piecewise continuous function. is continuous on , and exists, where . The smoothness properties of guarantee the global existence and the uniqueness of solution of system (2); more details can be seen in the books [15, 16].
Also, we will use the following comparison theorem of impulsive differential equation (see, [15]).
Lemma 3. Suppose , , and assume that
where is continuous in and for each , exists, is nondecreasing.
Let be the maximal (minimal) solution of the scalar impulsive differential equation
existing on . Then implies that , where is any solution of (2).
Lemma 4 (see [17]). Consider the following delay differential equation: where , , and are all positive constants and for all .(1)If , then .(2)If , then .
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 , and then we have the following results.(1)If , then the trivial periodic solution of system (10) is globally asymptotically stable.(2)If , then system (10) has a unique positive periodic solution which is globally asymptotically stable.
Proof. If , we can see that the stroboscopic map of (11) has a unique trivial fixed point , and by a direct calculation we can obtain
Hence, the trivial periodic solution is globally asymptotically stable.
If , the subsystem (10) has a trivial fixed point and a positive fixed point
Moreover,
So the trivial periodic solution is not stable.
Now we consider the stability of the positive fixed point .
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 for system (10), and then the following linear nonhomogeneous impulsive equation is obtained:
Thus, is the solution of system (10) with initial condition if is the solution of system (20) with initial condition .
Let
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
Thus,
That is, the positive periodic solution,
is globally asymptotically stable.
3. Main Results
Theorem 6. If , then for each solution of system (2), one has when is large enough.
Proof. Let be any solution of system (2) with initial condition (3), and we define
Then,
which yields
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.
Now we begin to study the global attractivity of predators-eradication periodic solution of system (2), which is the circumstance when both of the predator individuals are entirely absent from the system ultimately; that is, and .
Theorem 7. If system (2) satisfies and the following condition (H1): then the predators-eradication periodic solution of system (2) is globally attractive.
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 such that
when is large enough.
This yields
Hence, there exists a positive integer and arbitrarily small positive such that
for all .
On the other hand, since condition (H1) holds, then for above small enough.
At the moment, from the second equation of the system (2) we have
Then,
Then there exist and small enough, such that
and it follows from the last equation of system (2) that
when .
For above arbitrarily small positive , small enough, since condition (H1) holds, then
By Lemma 4, we have
Then for above small enough, there exists a such that
On the other hand, combining the first equation of system (2) with (43) and (47), we have
for , 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 as is large enough.
Let , and then
that is, .
At this time, it follows from (38) and (52) that
Thus, for large enough, we have
Combined with (42), (46), and (55), we have proved that the predators-eradication periodic solution of system (2) is globally attractive.
Corollary 8. If system (2) satisfies and then the predators-eradication periodic solution of system (2) is globally attractive.
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 of system (2) is globally attractive.
Remark 9. From Theorem 7 and its sufficient condition Corollary 8, if , , or , 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 , , and the following condition (H2): where , , , and , then system (2) is permanent.
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 and as , where is the unique and globally stable positive periodic solution of
Therefore, for sufficiently large , there exists a small enough such that
In the following, we will show that there exist two positive constants and , such that and for any large enough.
Step 1. We begin to find an such that for any large enough.
In order to achieve this goal, firstly we claim that the inequality cannot hold for all .
Otherwise, if for all , then from the first equation of (2),
where
Therefore, there exists a small enough and a , such that, for ,
where
When condition (H2) holds, we can choose small enough such that
Then at this time, from the second equation of system (2),
Let , and integrate (74) on , and we can get
Then as , which is contradicted with .
Hence, there exists a , such that .
If for all , then our aim is obtained.
Otherwise, if is oscillatory around , let
And we assume that there exists two positive constants and such that
Since is continuous, bounded, and not affected by impulses, we conclude that is uniformly continuous; then exists a (with and is independent of the choice of ) such that for all .
If , then for all .
If , then, from the second equation of (2),
Integrate (78) on , and we have
If , from the second equation of (2), we can also obtain
Proceeding exactly as above analysis, we can conclude that , for .
Thus, no matter which case we have for all , since the interval is arbitrarily chosen, then there exist , such that for is large enough.
Step 2. Now we try to find an such that for all is large enough.
In the same method, we claim that the inequality cannot hold for all .
Otherwise, if there exists a such that for all , then by the first equation of (2),
where
Therefore, there exists a small enough and a , such that for ,
where
Now we define a Liapunov functional
and then
When the condition (H2) holds, we can choose small enough such that
Let , and we claim that
Otherwise, if there exists a nonnegative constant such that
When , from the last equation of (2), we have
Thus,
which is a contradiction.
Therefore,
which implies , as , and this is contradicted with
Therefore, cannot hold for all , and there are two cases as follows.
If for all , then our aim is obtained.
Otherwise, if is oscillatory around , when is sufficiently large, let
then we can show that as is large enough.
In fact, suppose there exist two positive constants , , such that
Since is continuous, bounded, and not affected by impulses, we conclude that is uniformly continuous; then exists a (with and is independent of the choice of ) such that for all .
If , then for all .
If , then from the last equation of (2),
Integrate (96) on , and we have
If , from the second equation of system (2), we can also obtain
Proceeding exactly as the proof for above claim (88), we can obtain for all , then for .
Thus, no matter which case we have for all , since the interval is arbitrarily chosen, then there exist , such that for is large enough.
Set . From above proof, we know that is the global attractor, and each solution of system (2) will eventually enter and remain in region . According to Definition 2, system (2) is permanent.
In a similar way to the discussion of Corollary 8, we can obtain the following two sufficient conditions for the permanence.
Corollary 11. If system (2) satisfies and where, , , and , is the same as Theorem 10, then system (2) is permanent.
Corollary 12. If system (2) satisfies where , and , is the same as Theorem 10, then system (2) is permanent.
4. 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.
Case 1. If we choose , , , , , , , , , , , , , , , , and with initial conditions , , and , it is easy to calculate , , and , , which satisfies the condition of Theorem 7. From the time-series diagram of , and (see Figures 1(a), 1(b), and 1(c)), we can see that the predators and become extinct while the prey population (pests population) is much more than the initial , and Figure 1(d) is the phase portrait of this circumstance. This means that when we capture or poison the pests more frequently, natural enemies will become extinct before the pests while the number of pests may increase than before.
(a)
(b)
(c)
(d)
Case 2. If we choose , , , , , , , , , , , , , , , , and with initial conditions , , and , which satisfies the condition of Theorem 10, then from the time-series diagram of , , and we can see that the system is permanent, and all of the population can coexist in this case. From the phase portrait (Figure 2(d)) we can see there is a periodic solution. Moreover, if we set the economic threshold level (which is much less than the initial value ), and from Figure 2(a), we can see that the pests population is less than ETL ultimately. That is to say, our control strategy is effective.
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 or increasing the value of parameter . To verify this point, we consider the following Case 3.
(a)
(b)
(c)
(d)
Case 3. If we choose , 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 , which is lower than the previous . Furthermore, multiple periodic solutions or periodic oscillations appear from the phase portrait at the moment (see Figure 3(b)).
(a)
(b)
5. Conclusions and Discussions
In this paper, we investigate the dynamics of a three-dimensional food-chain system incorporating digest delay and periodic harvesting for the prey. The value of our study lies in two aspects: mathematically, we prove the existence of a predators-eradication periodic solution which is globally attractive and show that the pest population can be controlled under the economic threshold level (ETL) and the system can be uniformly permanent when the harvest period is long enough or the harvesting rate is not too large. Biologically, we succeed to find some strategies to control the population of the pests under a certain economic threshold level (ETL) and provide some reasonable suggestions for relevant ecological departments by these conclusions.
However, these control conditions are sufficient and tedious; then how to obtain some simpler and more extensive control conditions is desirable in future studies.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the anonymous referees for very helpful suggestions and comments which led to improvement of their original paper. This work is supported by the National Natural Science Foundation of China (41072235, 51349011) and Scientific Research Fund of Sichuan Provincial Education Department (11ZB192, 14ZB0115).