Impulsive Control Strategy for a Nonautonomous Food-Chain System with Multiple Delays

A nonautonomous food-chain system with Holling II functional response is studied, in which multiple delays of digestion are also considered. By applying techniques in differential inequalities, comparison theorem in ordinary differential equations, impulsive differential equations, and functional differential equations, some effective control strategies are obtained for the permanence of the system. Furthermore, effects of some important coefficients and delays on the permanence of the system are intuitively and clearly shown by series of numerical examples.


Introduction
From the beginning of the early 21st century, people began to study the complexity of biology by using a variety of mathematical models and methods; then the development of mathematical biology entered a new period, especially in the field of population dynamics (see [1]) and epidemic dynamics.Moreover, some emerging theoretical tools such as the complex network were also introduced to study mathematical models in biology (see [2][3][4]) and so forth.
Meanwhile, it is known that many evolution processes are characterized by the fact at certain moments when they experience a change of state abruptly.These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process.Thus, models involving impulsive effects seem to be a hot research field (see [5][6][7][8]), which could describe the real relationship among the species more accurately.On the other hand, when prey-predator system is referred, sometimes there is a digest and absorption time (which is always called the digest delay) during the predation instead of transforming 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.
Enlightened by above ecological backgrounds and based on model (1.1) in [5], we consider a nonautonomous threespecies food-chain system with multiple delays and impulsive perturbation in this paper, and the model is described as the following impulsive differential equations: where (), (), and () denote the population of the prey and the lower and higher predator at time , respectively. 1 () and  2 () are the intrinsic growth rate of the prey () and the lower predator ();  1 () and  2 () denote the environmental carrying capacity of the prey and the lower predator.  () and   () are the coefficient of the functional response, () is the density dependent coefficient of the higher predator (), and   () ( = 1, 2, 3) represents the transform coefficients during the predation.More details of the background of this model can be found in [5].
For  ⊂ , PC(, ) is the space of all piecewise continuous functions from  to  with points of discontinuity of the first kind   at which it is left continuous.
For a real positive and continuous function (), we denote
The proof of this lemma can be found in [10], and we omit it here.
In the following, we will give the other important lemma, which was introduced by Gopalsamy in [11] (page 57: Theorem 1.4.3).

Lemma 2.
For the delay logistic equation ( 9) with initial conditions (10), if condition holds, then By the above lemma, we have the following corollary.

Corollary 3. For any solution 𝑢(𝑡) of the functional differential equation
with initial condition if condition (H4) holds, then

Main Results
Theorem 4. Assume that (H1)-(H4) hold; then for any solution of system ( 1) there exist  0 > 0 and Proof.From the first equation of system (1), we have whose corresponding comparison impulsive system is According to Lemma 1, the corresponding nonimpulsive equation of (18) reads By hypothesis (H3), it follows from (19) that whose corresponding comparison functional differential system is By the corollary, it follows from (21) that lim then, for any small positive  1 > 0, there exists  1 > 0, such that  1 () < Δ 1 +  1 for  >  1 .
On the other hand, by hypothesis (H3), the comparison theorem on impulsive differential equation (see [12]), and the comparison theorem on functional differential equation (see [13]), we have Also, from the second and the third equations of system (1), when  ≥  1 we have whose comparison system is And the corresponding nonimpulsive system of (25) is whose comparison system is By the corollary again, it follows from (27) that lim then, for any small positive  2 > 0, there exists  2 > 0, such that  2 () < Δ 2 +  2 for  >  2 .Thus, by hypothesis (H3) and the comparison theorem we have Finally, when  ≥  2 it follows from the third equation of system (1) that we have Repeating the above process, we can derive that, for any small positive  2 > 0, there exists  0 > 0, such that where Theorem 5.If (H1)-(H4) hold, further assume that (H5) (H6) hold; then for any solution of system (1) there exist  >  0 and   > 0 ( = 1, 2, 3) such that Proof.From the first equation of system (1), when  >  0 , whose corresponding comparison impulsive system is By Lemma 1, the corresponding nonimpulsive equation of system (37) reads For the functional differential equation (39), it follows from the corollary that lim →∞  3 () =  1 , where Then, for any small positive  4 > 0, there exists  3 >  0 , such that  3 () >  1 −  4 for  >  0 .
On the other hand, by hypothesis (H3) and the comparison theorem again we have Consider the second equation of system (1); when  >  3 we have Repeating the above process, we can derive that, for any small positive  5 > 0, there exists  4 >  3 >  0 , such that for  >  4 , where Finally, considering the third equation of system (1), when  >  4 we have In the same way, we have that, for any small positive  6 > 0, there exists  >  4 , such that where Theorem 6.If conditions (H1)-(H6) hold, then system (1) is permanent.
That is to say, system (1) is permanent.
Remark 7. From the above theorems, we know that if we choose suitable control conditions (H1)-(H6), then the system will be controlled to be permanent.These results may provide some reasonable control strategies for relevant ecological departments.

Numerical Simulations and Discussions
In the above section, we focused our attention on the permanence of the food-chain system with multiple delays in theory; some control conditions have been obtained to guarantee the permanence of system (1).In this section we will give some numerical examples and simulations and then make some discussions.We denote According to Theorem 5, we can see that the signs of  and  are very important factors for controlling the permanence of system (1).In order to verify this point, we only change one or two parametric values based on Case 1 as follows.
Case 1.We consider the following choice of parametric values: The delays are given as On the one hand, one can verify that conditions (H1)-(H6) are satisfied.According to Theorem 5, system (1) is permanent.On the other hand, system (1) is numerically solved for the above choice of parameters and initial conditions.It is obvious that the system is permanent and has quasiperiodic solutions; it is clear to see the time-series and phase portrait intuitively in Figure 1.
Case 2. If we increase the value of   1 , for example, if we let  1 () = 3.5 while other parametric values are the same as Case 1, then system (1) is numerically solved in Figure 2. One can find that the highest predator  will be extinct finally, and the density of the prey will decrease to zero periodically, while the middle predator can be permanent.
Case 3. When we go on increasing the value of   1 such that  1 () = 3.9, one can see that both the highest predator  and the prey  will be extinct finally, while the middle predator  can be permanent at the moment (see Figure 3).
Case 4. If we increase the value of   3 , for example, we let  3 () = 2.3, while other parametric values are the same as Case 1.It is very strange and interesting that both the highest predator  and the prey  can be permanent, while the middle predator  will be extinct gradually (see Figure 4).Case 5.In the following, we will study effects of the delays on the dynamical behavior of the system.To this end, we enlarge the value of all of the delays and choose while the other parameters and initial conditions are the same as Case 1.It is easy to verify that condition (H4) does not hold any more.By the numerical simulations, we can see that the prey  and the higher predator  will be permanent while the density of the lower predator () oscillates, and it shows a strange dynamic characteristic of intermittent extinction (see Figure 5).
By the above theoretical analysis and the numerical experiments, it is shown that we can seek some reasonable control strategies so that the system can be controlled to be permanent, such as changing values of some important parameters of the system.On the other hand, by the numerical simulations in Case 5, when the delays are too long, the density of the species will oscillate and leads to strange phenomena of intermittent extinction, which can explain the complexity of biological systems.In addition, we can also seek some efficient measures to guarantee that some of the species (beneficial insects) in the system can be permanent while the other species (pest insects) will be extinct finally.And this method can be extended to the study of the other systems with variable coefficients, such as epidemic systems and neural network systems.