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

Authors: Baodan Tian; Yanhong Qiu; Yucai Ding
Journal: Journal of Control Science and Engineering
 (2016)

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## Abstract

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.


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## Body

## 1. 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 [2–4]) 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 [5–8]), 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 three-species food-chain system with multiple delays and impulsive perturbation in this paper, and the model is described as the following impulsive differential equations: (1)dxdt=r1txt1-xt-τ1k1t-a1txtytb1t+xt-a2txtztb2t+xt,dydt=r2tyt1-yt-τ2k2t+e1ta1txt-τ4ytb1t+xt-τ4-a3tytztb3t+yt,dzdt=zte2ta2txt-τ5b2t+xt-τ5+e3ta3tyt-τ6b3t+yt-τ6-mtzt-τ3,t≠tk,xt+=1+q1kxt,yt+=1+q2kyt,zt+=1+q3kzt,t=tk,k∈N∗,where x(t), y(t), and z(t) denote the population of the prey and the lower and higher predator at time t, respectively. r1(t) and r2(t) are the intrinsic growth rate of the prey x(t) and the lower predator y(t); k1(t) and k2(t) denote the environmental carrying capacity of the prey and the lower predator. ai(t) and bi(t) are the coefficient of the functional response, m(t) is the density dependent coefficient of the higher predator z(t), and ei(t)(i=1,2,3) represents the transform coefficients during the predation. More details of the background of this model can be found in [5].LetI={tk}∈R:tk<tk+1,k∈N∗,limk→±∞⁡tk=±∞, and the set of all sequences are bounded and strictly increasing. Let Ω⊂R,Ω≠⌀,τ=max1≤i≤6⁡{τi}, ξ0∈R. PC(ξ0) is the space of all function ϕ:[ξ0-τ,ξ0]→Ω having points of discontinuity at μ1,μ2,…,∈[ξ0-τ,ξ0] of the first kind and left continuous at these points.ForJ⊂R,PC(J,R) is the space of all piecewise continuous functions from J to R with points of discontinuity of the first kind tk at which it is left continuous.Letϕi∈PC(0)(i=1,2,3), denote x(t)=x(t;0,ϕ1), y(t)=y(t;0,ϕ2), z(t)=z(t;0,ϕ3), and xt,yt,ztT is the solution of system (1) satisfying the following initial conditions: (2)0≤xt;0,ϕ1=ϕ1s≤∞,s∈-τ,0,x0+0;0,ϕ1=ϕ10>0;0≤yt;0,ϕ2=ϕ2s≤∞,s∈-τ,0,y0+0;0,ϕ2=ϕ20>0;0≤zt;0,ϕ3=ϕ3s≤∞,s∈-τ,0,z0+0;0,ϕ3=ϕ30>0.It is easy to verify that solutions of system (1) with above initial conditions are positive for all t≥0.Furthermore, throughout the present paper, we assume(H1)
0<t1<t2⋯<tk<⋯ are fixed impulsive points with limk→∞⁡tk=∞;(H2)
qik are real sequences with qik>-1;(H3)
there exist some positive constantsMi and mi such that(3)mi≤∏0<tk<t1+qik≤Mi,i=1,2,3,where k∈N∗; N∗ denotes set of the positive integer.For a real positive and continuous functionf(t), we denote (4)fM=maxt∈0,∞⁡ft,fL=mint∈0,∞⁡ft.


## 2. Preliminaries

In this section, we will consider the following impulsive system (5) with time delay and give two important lemmas: (5)u′t=utA-But-τ,t≠tk,ut+=1+qikut,t=tk,k∈N∗.Under the hypotheses (H1)–(H3), the corresponding nonimpulsive system of system reads(6)v′t=vtA-B∏0<tk<t1+qikvt-τ.Lemma 1 (see [9]).
Assume that (H1)–(H3) hold:(1)
Ifv(t) is a solution of system (6) on [-τ,∞), then(7)ut=∏0<tk<t1+qikvtis a solution of system (5) on [-τ,∞).(2)
Ifu(t) is a solution of system (5) on [-τ,∞), then (8)vt=∏0<tk<t1+qik-1utis a solution of system (6) on [-τ,∞).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).For a delay logistic equation of the form(9)N′t=Ntb-a1Nt-τ1-a2Nt-τ2,with initial conditions, (10)Ns=φs≥0,s∈-τ,0,τ=max⁡τ1,τ2,φ∈C-τ,0,R+,φ0>0,where b,a1,a2,τ1,τ2∈(0,+∞).Lemma 2.
For the delay logistic equation (9) with initial conditions (10), if condition(11)bτ1+τ2exp⁡bτ1+τ2<1holds, then(12)limt→∞⁡Nt=ba1+a2.By the above lemma, we have the following corollary.Corollary 3.
For any solutionu(t) of the functional differential equation(13)u′t=utA-But-τ,with initial condition(14)us=φs∈C-τ,0,R+,φ0>0,if condition(H4)
(15)AτeAτ<1holds, then (16)limt→∞⁡ut=AB.


## 3. Main Results

Theorem 4.
Assume that (H1)–(H4) hold; then for any solution of system (1) there exist T0>0 and ηi>0 such that x(t)≤η1,y(t)≤η2,z(t)≤η3  for t>T0.Proof.
From the first equation of system (1), we have (17)x′t≤r1Mxt1-xt-τ1k1M,t≠tk,xt+=1+q1kxt,t=tk,k∈N∗,whose corresponding comparison impulsive system is (18)u1′t=r1Mu1t1-u1t-τ1k1M,t≠tk,u1t+=1+q1ku1t,t=tk,k∈N∗.
According to Lemma1, the corresponding nonimpulsive equation of (18) reads(19)v1′t=r1Mv1t1-1k1M∏0<tk<t1+q1kv1t-τ1.
By hypothesis (H3), it follows from (19) that (20)v1′t≤r1Mv1t1-m1k1Mv1t-τ1,whose corresponding comparison functional differential system is (21)w1′t=r1Mw1t1-m1k1Mw1t-τ1.
By the corollary, it follows from (21) that (22)limt→∞⁡w1t=k1Mm1=Δ1;then, for any small positive ε1>0, there exists T1>0, such that w1(t)<Δ1+ε1 for t>T1.
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(23)xt≤u1t=∏0<tk<t1+q1kv1t≤M1v1t≤M1w1t≤M1Δ1+ε1=η1.
Also, from the second and the third equations of system (1), when t≥T1 we have (24)y′t≤ytr2M+e1Ma1Mη1b1L+η1-r2Myt-τ2k2M,t≠tk,yt+=1+q2kyt,t=tk,k∈N∗,whose comparison system is (25)u2′t=u2tr2M+e1Ma1Mη1b1L+η1-r2Mu2t-τ2k2M,t≠tk,u2t+=1+q2ku2t,t=tk,k∈N∗.
And the corresponding nonimpulsive system of (25) is (26)v2′t=v2tr2M+e1Ma1Mη1b1L+η1-r2Mk2M∏0<tk<t1+q2kv2t-τ2≤v2tr2M+e1Ma1Mη1b1L+η1-m2r2Mk2Mv2t-τ2,whose comparison system is(27)w2′t=w2tr2M+e1Ma1Mη1b1L+η1-m2r2Mk2Mw2t-τ2.
By the corollary again, it follows from (27) that(28)limt→∞⁡w2t=r2M+e1Ma1Mη1/b1L+η1m2r2M/k2M=Δ2;then, for any small positive ε2>0, there exists T2>0, such that w2(t)<Δ2+ε2 for t>T2.
Thus, by hypothesis (H3) and the comparison theorem we have(29)yt≤u2t=∏0<tk<t1+q2kv2t≤M2v2t≤M2w2t≤M2Δ2+ε2=η2.
Finally, whent≥T2 it follows from the third equation of system (1) that we have(30)z′t≤zte2Ma2Mη1b2L+η1+e3Ma3Mη2b3L+η2-mLzt-τ3,t≠tk,zt+=1+q3kzt,t=tk,k∈N∗.
Repeating the above process, we can derive that, for any small positiveε2>0, there exists T0>0, such that (31)zt≤M3Δ3+ε3=η3fort>T0,where(32)Δ3=1mLm3e2Ma2Mη1b2L+η1+e3Ma3Mη2b3L+η2.Theorem 5.
If (H1)–(H4) hold, further assume that(H5)
(33)r1L>a1Mη1η2b2L+η1+a2Mη1η3b2L+η1,(H6)
(34)r2L+a2Le2Lξ1b1M+ξ1>a3Mη3b3Lhold; then for any solution of system (1) there exist T>T0 and ξi>0(i=1,2,3) such that (35)xt≥ξ1,yt≥ξ2,zt≥ξ3,fort>T.Proof.
From the first equation of system (1), when t>T0, (36)x′t≥xtr1L-a1Mη1η2b1L+η1-a2Mη1η3b2L+η1-r1Mk1Lxt-τ1,t≠tk,xt+=1+q1kxt,t=tk,k∈N∗,whose corresponding comparison impulsive system is (37)u3′t=u3tr1L-a1Mη1η2b1L+η1-a2Mη1η3b2L+η1-r1Mk1Lu3t-τ1,t≠tk,u3t+=1+q1ku3t,t=tk,k∈N∗.
By Lemma1, the corresponding nonimpulsive equation of system (37) reads(38)v3′t=v3tr1L-a1Mη1η2b1L+η1-a2Mη1η3b2L+η1-r1Mk1L∏0<tk<t1+q1kv3t-τ1≥v3tr1L-a1Mη1η2b1L+η1-a2Mη1η3b2L+η1-r1MM1k1Lv3t-τ1,and the comparison equation of (38) is (39)w3′t=w3tr1L-a1Mη1η2b1L+η1-a2Mη1η3b2L+η1-r1MM1k1Lw3t-τ1.
For the functional differential equation (39), it follows from the corollary that limt→∞w3(t)=δ1, where (40)δ1=k1Lr1MM1r1L-a1Mη1η2b1L+η1-a2Mη1η3b2L+η1.
Then, for any small positiveε4>0, there exists T3>T0, such that w3(t)>δ1-ε4 for t>T0.
On the other hand, by hypothesis (H3) and the comparison theorem again we have(41)xt≥u3t=∏0<tk<t1+q1kv3t≥m1v3t≤m1w3t≥m1δ1-ε4=ξ1.
Consider the second equation of system (1); when t>T3 we have (42)y′t≥ytr2L+a1Le1Lξ1b1M+ξ1-a3Mη3b3L+η3-r2Mk2Lyt-τ2,t≠tk,yt+=1+q2kyt,t=tk,k∈N∗.
Repeating the above process, we can derive that, for any small positiveε5>0, there exists T4>T3>T0, such that (43)yt≥m2δ2-ε5=ξ2,for t>T4, where(44)δ2=k2Lr2MM2r2L+a1Le1Lξ1b1M+ξ1-a3Mη3b3L+η3.
Finally, considering the third equation of system (1), when t>T4 we have (45)z′t≥zta2Le2Lξ1b2M+ξ1+a3Le3Lξ2b3M+ξ2-mMzt-τ3,t≠tk,zt+=1+q3kzt,t=tk,k∈N∗.
In the same way, we have that, for any small positiveε6>0, there exists T>T4, such that (46)zt≥m3δ3-ε6=ξ3,fort>T,where(47)δ3=1mMM3r2L+a2Le2Lξ1b2M+ξ1+a3Le3Lξ2b3M+ξ2.Theorem 6.
If conditions (H1)–(H6) hold, then system (1) is permanent.Proof.
If we denoteξ=min⁡{ξ1,ξ2,ξ3}, η=max⁡{η1,η2,η3}. Further, if conditions (H1)–(H6) hold, then any solution xt,yt,ztT of system (1) will eventually enter and remain in the compact set K=ξ≤x,y,z≤η.
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.


## 4. 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 (48)f≔r1L-a1Mη1η2b2L+η1+a2Mη1η3b2L+η1;g≔r2L+a2Le2Lξ1b1M+ξ1-a3Mη3b3L.According to Theorem5, we can see that the signs of f and g 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:(49)r1t=4+sin⁡t,r2t=b+cos⁡t,k1t=2+0.1sin⁡t,k2t=3,b1t=3+sint,b2t=4-0.1cos⁡t,b3t=4,mt=0.2+0.1cos⁡2t,a1t=1,a2t=0.2,a3t=2,e1t=18,e2t=3,e3t=6,q1k=0.1,q2k=0.2,q3k=0.2,k=1,2,…with initial conditions (50)ϕ1s=x0=0.5,ϕ2s=y0=0.8,ϕ3s=z0=0.6,s∈-τ,0.The delays are given as (51)τ1=τ2=τ3=0.01,τ4=τ5=τ6=0.02.
On the one hand, one can verify that conditions (H1)–(H6) are satisfied. According to Theorem5, 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.Figure 1
Permanence of system (1) with the impulsive control strategy with ϕ1(s)=x(0)=0.5,  ϕ2(s)=y(0)=0.8,  ϕ3(s)=z(0)=0.6,  s∈[-τ,0].Case 2.
If we increase the value ofa1M, for example, if we let a1(t)=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 z will be extinct finally, and the density of the prey will decrease to zero periodically, while the middle predator can be permanent.Figure 2
Extinction of the higher predator of system (1) with ϕ1(s)=x(0)=0.5,  ϕ2(s)=y(0)=0.8,  ϕ3(s)=z(0)=0.6,  s∈[-τ,0].Case 3.
When we go on increasing the value ofa1M such that a1(t)=3.9, one can see that both the highest predator z and the prey x will be extinct finally, while the middle predator y can be permanent at the moment (see Figure 3).Figure 3
Extinction of the prey of system (1) with ϕ1(s)=x(0)=0.5,  ϕ2(s)=y(0)=0.8,  ϕ3(s)=z(0)=0.6,  s∈[-τ,0].Case 4.
If we increase the value ofa3M, for example, we let a3(t)=2.3, while other parametric values are the same as Case 1. It is very strange and interesting that both the highest predator z and the prey x can be permanent, while the middle predator y will be extinct gradually (see Figure 4).Figure 4
Extinction of the lower predator of system (1) with ϕ1(s)=x(0)=0.5,  ϕ2(s)=y(0)=0.8,  ϕ3(s)=z(0)=0.6,  s∈[-τ,0].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(52)τ1=τ2=τ3=τ4=τ5=τ6=0.2,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 x and the higher predator z will be permanent while the density of the lower predator y(t) oscillates, and it shows a strange dynamic characteristic of intermittent extinction (see Figure 5).Figure 5
Effect of delays on the dynamical behavior of system (1) with ϕ1(s)=x(0)=0.5,  ϕ2(s)=y(0)=0.8,  ϕ3(s)=z(0)=0.6,  s∈[-τ,0].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 Case5, 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.



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