# Permanence for a Discrete Ratio-Dependent Predator-Prey System with Holling Type III Functional Response and Feedback Controls

Authors: Jiangbin Chen; Shengbin Yu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

A new set of sufficient conditions for the permanence of a ratio-dependent predator-prey system with Holling type III functional response and feedback controls are
obtained. The result shows that feedback control variables have no influence on the
persistent property of the system, thus improving and supplementing the main result
of  Yang (2008).


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## Body

## 1. Introduction

The aim of this paper is to investigate the permanent property of the following discrete ratio-dependent predator-prey system with Holling type III and feedback controls:(1)x(n+1)=x(n)exp{c(n)x(n)y(n)x2(n)+m2(n)y2(n)a(n)-b(n)x(n)=x(n)eexpp-c(n)x(n)y(n)x2(n)+m2(n)y2(n)-e1(n)u1(n)},y(n+1)=y(n)exp{f(n)x2(n)x2(n)+m2(n)y2(n)-d(n)=x(n)eexpp+f(n)x2(n)x2(n)+m2(n)y2(n)-e2(n)u2(n)},Δu1(n)=-η1(n)u1(n)+q1(n)x(n),Δu2(n)=-η2(n)u2(n)+q2(n)y(n),
where x(t) and y(t) are the densities of the prey population and predator population at time t, respectively, for i=1,2,{a(n)},{b(n)},{c(n)},{d(n)},{f(n)},{m(n)},{ei(n)},{ηi(n)}, and qi(n) are all bounded nonnegative sequences such that
(2)0<aL≤a(n)≤aU,0<bL≤b(n)≤bU,0<cL≤c(n)≤cU,0<dL≤d(n)≤dU,0<fL≤f(n)≤fU,0<mL≤m(n)≤mU,0<eiL≤ei(n)≤eiU,0<ηiL≤ηi(n)≤ηiU<1,0<qiL≤qi(n)≤qiU.
Here, for any bounded sequence {a(n)}, aL=infn∈N{a(n)}, aU=supn∈N{a(n)}.By the biological meaning, we will focus our discussion on the positive solutions of system (1). So, we consider (1) together with the following initial conditions:
(3)x(0)>0,y(0)>0,ui(0)>0,i=1,2.
It is not difficult to see that the solutions of (1)–(3) are well defined and satisfy
(4)x(n)>0,y(n)>0,ui(n)>0,i=1,2,forn∈N.
Recently, Yang [1] proposed and studied the permanence of system (1). Set
(5)M1=exp{aU-1}bL,M2=fUM1exp{fU-dL}2dLmL,W1=q1UM1η1L,W2=q2UM2η2L.
Using the comparison theorem of difference equation, Yang obtained the following result.Theorem A (see [1]).
Assume that(A1)aL-cU2mL-e1UW1>0,(A2)fL-dU-e2UW2>0,
hold; then system (1) is permanent.Theorem A shows that feedback control variables play important roles in the persistent property of the system (1). But the question is whether or not the feedback control variables have influence on the permanence of the system. On the other hand, as was pointed out by Fan and Wang [2], “if we use the method of comparison theorem, then the additional condition, in some extent, is necessary. But for the system itself, this condition may not necessary.[sic]” In [2], by establishing a new difference inequality, Fan and Wang showed that feedback control has no influence on the permanence of a single species discrete model. Their success motivated us to consider the persistent property of system (1). Indeed, in this paper, we will apply the analysis technique of Fan and Wang [2] to establish sufficient conditions, which is independent of feedback control variables, to ensure the permanence of the system. We finally obtain the following main results.Theorem B.
Assume that(H1)aL-cU2mL>0,(H2)fL-dU>0,
hold; then system (1) is permanent.Comparing with Theorem A, it is easy to see that(Hi) in Theorem B are weaker than (Ai) in Theorem A (i=1,2) and feedback control variables have no influence on the permanent property of system (1), so our results improve the main results in [1]. For more works on this direction, one could refer to [3–10] and the references cited therein.The remaining part of this paper is organized as follows. In Section2, we will introduce several lemmas. The permanence of system (1) is then studied in Section 3. In Section 4, a suitable example together with its numerical simulations shows the feasibility of our results.


## 2. Preliminaries

In this section, we will introduce several useful lemmas.Lemma 1 (see [11]).
Assume that{x(n)} satisfies
(6)x(n+1)≥x(n)exp{a(n)-b(n)x(n)},n≥N0,limsupn→+∞x(n)≤x* and x(N0)>0, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants and N0∈N. Then
(7)liminfn→+∞x(n)≥min{aLbUexp{aL-bUx*},aLbU}.Lemma 2 (see [2]).
Assume thatA>0 and y(0)>0. Suppose that
(8)y(n+1)≤Ay(n)+B(n),n=1,2,….
Then, for any integer k≤n,
(9)y(n)≤Aky(n-k)+∑i=0k-1‍AiB(n-i-1).
In particular, if A<1 and B is bounded above with respect to M, then
(10)limsupn→+∞y(n)≤M1-A.Lemma 3 (see [2]).
Assume thatA>0 and y(0)>0. Suppose that
(11)y(n+1)≥Ay(n)+B(n),n=1,2,….
Then, for any integer k≤n,
(12)y(n)≥Aky(n-k)+∑i=0k-1‍AiB(n-i-1).
In particular, if A<1 and B is bounded above with respect to m*, then
(13)liminfn→+∞y(n)≥m*1-A.


## 3. Permanence

In this section, we detail the proof of our main result by several lemmas. The following lemma is a direct conclusion of [1].Lemma 4.
There exist two positive constantsM1 and W1 such that
(14)limsupn→+∞x(n)≤M1,limsupn→+∞y(n)≤M2,limsupn→+∞u1(n)≤W1,limsupn→+∞u2(n)≤W2,
where Mi and Wi(i=1,2) are defined in (5).Lemma 5.
Assume(H1′)aL-cU2mL>0,
holds then there exist two positive constants m1 and w1 such that
(15)liminfn→+∞x(n)≥m1,liminfn→+∞u1(n)≥w1,
where m1 and w1 are defined in the proof.Proof.
According to Lemma4, for any ε>0 small enough, there exists enough large N1>0, such that, for n≥N1,
(16)x(n)≤M1+ε,u1(n)≤W1+ε.
Thus, it follows from (16) and the first equation of system (1) that
(17)x(n+1)≥x(n)×exp{aL-bU(M1+ε)-cU2mL-e1U(W1+ε)}≥x(n)exp{-bU(M1+ε)-cU2mL-e1U(W1+ε)}=Δx(n)exp{D1ε},
for n≥N1, where D1ε=-bU(M1+ε)-(cU/2mL)-e1U(W1+ε)<0. For any integer k≤n, it follows from (17) that
(18)∏j=n-kn-1‍x(j+1)x(j)≥∏j=n-kn-1‍exp{D1ε}=exp{D1εk}.
Thus
(19)x(n-k)≤x(n)exp{-D1εk}.
From the third equation of system (1), we have
(20)u1(n+1)=(1-η1(n))u1(n)+q1(n)x(n)≤(1-η1L)u1(n)+q1Ux(n)=ΔA1u1(n)+B1(n),
where A1=1-η1L and B1(n)=q1Ux(n). Then, for any k≤n, according to Lemma 2, (19), and (20)
(21)u1(n)≤A1ku1(n-k)+∑i=0k-1‍A1iB1(n-i-1)=A1ku1(n-k)+∑i=0k-1‍A1iq1Ux(n-i-1)≤A1ku1(n-k)+q1Ux(n)∑i=0k-1‍A1iexp{-D1ε(i+1)}.
Note that 0<η1L<1; hence 0<A1<1. Therefore,
(22)0≤A1ku1(n-k)≤A1k(W1+ε)⟶0,ask⟶∞.
Then, there exists a positive integer N2>N1 such that, for any positive solution of system (1), e1UA1N2(W1+ε)<(1/2)(aL-cU/2mL) for all n≥N2. In fact, we could choose N2=lnP1/lnA1+1, where P1=(2mLaL-cU)/4e1UmL(W1+ε). Fixing N2, for n≥N1+N2, we get
(23)u1(n)≤A1N2u1(n-N2)+q1Ux(n)∑i=0N2-1‍A1iexp{-D1ε(i+1)}≤A1N2(W1+ε)+q1Ux(n)∑i=0N2-1‍A1iexp{-D1ε(i+1)}=ΔA1N2(W1+ε)+G1εx(n),
where G1ε=q1U∑i=0N2-1A1iexp{-D1ε(i+1)}.
Substituting (23) into the first equation of system (1), we can get
(24)x(n+1)≥x(n)exp{aL-bUx(n)-cU2mL-e1Uu1(n)}≥x(n)exp{aL-bUx(n)-cU2mL-e1U(A1N2(W1+ε)+G1εx(n))cU2mL}=x(n)exp{aL-cU2mL-e1UA1N2(W1+ε)-(bU+e1UG1ε)x(n)cU2mL}≥x(n)exp{12(aL-cU2mL)-(bU+e1UG1ε)x(n)}=Δx(n)exp{E1-E2εx(n)},
where E1=(1/2)(aL-cU/2mL) and E2ε=bU+e1UG1ε.
By applying Lemma1 to (24), it immediately follows that
(25)liminfn→+∞x(n)≥min{E1E2εexp{E1-E2εM1},E1E2ε}.
Setting ε→0 in the above inequality, we obtain
(26)liminfn→+∞x(n)≥min{E1E2exp{E1-E2M1},E1E2}=Δm1.
It follows from (26) that there exists large enough N3≥N1+N2 such that
(27)x(n)≥m12,∀n≥N3.
This together with the third equation of system (1) leads to
(28)Δu1(n)≥-η1(n)u1(n)+m1q1(n)2,∀n≥N3.
Hence,
(29)u1(n+1)≥(1-η1U)u1(n)+m1q1L2,∀n≥N3.
By applying Lemma 3, it follows from (29) that
(30)liminfn→+∞u1(n)≥m1q1L2η1U=Δw1.
This completes the proof of Lemma 5.Lemma 6.
Assume that(H2′)fL-dU>0,
holds; then there exist two positive constants m2 and w2 such that
(31)liminfn→+∞y(n)≥m2,liminfn→+∞u2(n)≥w2,
where m2 and w2 are defined in the proof.Proof.
According to Lemmas4 and 5, for any ε>0 small enough, there exists enough large N4>0, such that, for n≥N4,
(32)y(n)≤M1+ε,x(n)≥m1-ε,u2(n)≤W2+ε.
Thus, it follows from (32) and the second equation of system (1) that
(33)y(n+1)=y(n)exp{-d(n)+f(n)-f(n)m2(n)y2(n)x2(n)+m2(n)y2(n)-e2(n)u2(n)f(n)m2(n)y2(n)x2(n)+m2(n)y2(n)}≥y(n)exp{-dU+fL-f(n)m(n)y(n)2x(n)-e2U(W2+ε)}≥y(n)exp{-dU+fL-fUmU(M1+ε)2(m1-ε)-e2U(W2+ε)}≥y(n)exp{-dU-fUmU(M1+ε)2(m1-ε)-e2U(W2+ε)}=Δy(n)exp{D2ε},
for n≥N4, where D2ε=-dU-fUmU(M1+ε)/2(m1-ε)-e2U(W2+ε)<0. For any integer k≤n, it follows from (33) that
(34)∏j=n-kn-1‍y(j+1)y(j)≥∏j=n-kn-1‍exp{D2ε}=exp{D2εk}.
Thus
(35)y(n-k)≤y(n)exp{-D2εk}.
From the fourth equation of system (1), we can get
(36)u2(n+1)=(1-η2(n))u2(n)+q2(n)y(n)≤(1-η2L)u2(n)+q2Uy(n)=ΔA2u2(n)+B2(n),
where A2=1-η2L and B2(n)=q2Uy(n). Then, for any k≤n, according to Lemma 2, (35) and (36), that
(37)u2(n)≤A2ku2(n-k)+∑i=0k-1‍A2iB2(n-i-1)=A2ku2(n-k)+∑i=0k-1‍A2iq2Uy(n-i-1)≤A2ku2(n-k)+q2Uy(n)∑i=0k-1‍A2iexp{-D2ε(i+1)}.
Note that 0<η2L<1; hence 0<A2<1. Therefore,
(38)0≤A2ku2(n-k)≤A2k(W2+ε)⟶0,ask⟶∞.
Then, there exists a positive integer N5>N4 such that, for any positive solution of system (1), e2UA2N5(W2+ε)<(1/2)(fL-dU) for all n≥N5. In fact, we could choose N5=lnP2/lnA2+1, where P2=(fL-dU)/2e2U(W2+ε). Fixing N5, for n≥N4+N5, we get
(39)u2(n)≤A2N5u2(n-N5)+q2Uy(n)∑i=0N5-1‍A2iexp{-D2ε(i+1)}≤A2N5(W2+ε)+q2Uy(n)∑i=0N5-1‍A2iexp{-D2ε(i+1)}=ΔA2N5(W2+ε)+G2εy(n),
where G2ε=q2U∑i=0N5-1A2iexp{-D2ε(i+1)}.
Substituting (39) into the second equation of system (1), we can get
(40)y(n+1)≥y(n)×exp{-dU+fL-f(n)m(n)y(n)2x(n)-e2Uu2(n)}≥y(n)exp{-dU+fL-fUmUy(n)2(m1-ε)-e2U(A2N5(W2+ε)+G2εy(n))fUmUy(n)2(m1-ε)}=y(n)exp{(fUmU2(m1-ε)+e2UG2ε)-dU+fL-e2UA2N5(W2+ε)-(fUmU2(m1-ε)+e2UG2ε)y(n)}≥y(n)exp{(fUmU2(m1-ε)+e2UG2ε)12(-dU+fL)-(fUmU2(m1-ε)+e2UG2ε)y(n)}=Δy(n)exp{E3-E4εy(n)},
for all n≥N5, where E3=(1/2)(-dU+fL) and E4ε=fUmU/2(m1-ε)+e2UG2ε.
By applying Lemma1 to (40), it immediately follows that
(41)liminfn→+∞y(n)≥min{E3E4εexp{E3-E4εM2},E3E4ε}.
Settingε→0 in the above inequality, we obtain
(42)liminfn→+∞y(n)≥min{E3E4exp{E3-E4M1},E3E4}=Δm2.
It follows from (42) that there exists large enough N6≥N4+N5 such that
(43)y(n)≥m22,∀n≥N6.
This together with the fourth equation of system (1) leads to
(44)Δu2(n)≥-η2(n)u2(n)+m2q2(n)2,∀n≥N6.
Hence,
(45)u2(n+1)≥(1-η2U)u2(n)+m2q2L2,∀n≥N6.
By applying Lemma 3, it follows from (45) that
(46)liminfn→+∞u2(n)≥m2q2L2η2U=Δw2.
This completes the proof of Lemma 6.Lemmas4–6 show that the conclusion of Theorem B holds.


## 4. Example and Numerical Simulation

Consider the following system:(47)x(n+1)=x(n)exp{(0.75+0.25sin(7n))x(n)y(n)x2(n)+(0.8+0.2cos(11n))2y2(n)1.32+0.32sin(17n)-(5.3+0.2sin(n))x(n)-(0.75+0.25sin(7n))x(n)y(n)x2(n)+(0.8+0.2cos(11n))2y2(n)-(2+0.1cos(13n))u1(n)(0.75+.25sin(7n))x(n)y(n)x2(n)+(0.8+0.2cos(11n))2y2(n)},y(n+1)=y(n)exp{(0.25+0.05sin(3n)))x2(n)x2(n)+(0.8+0.2cos(11n))2y2(n)-(0.125+0.025cos(5n))+(0.25+0.05sin(3n))x2(n)x2(n)+(0.8+0.2cos(11n))2y2(n)-(1.5+0.5sin(n))u2(n)(0.25+0.05sin(3n)))x2(n)x2(n)+(0.8+0.2cos(11n))2y2(n)}.Δu1(n)=-(0.09+0.01sin(7n))u1(n)+(0.3+0.2cos(n))x(n),Δu2(n)=-(0.06+0.04cos(11n))u2(n)+(0.1+0.05sin(n))y(n).
In this case, we have
(48)aL-cU2mL≈0.1667>0,fL-dU=0.05>0.
Equation (48) means that all conditions of Theorem B are satisfied in system (47). Thus, the system (47) is permanent. Our numerical simulation supports our result (see Figure 1). However,
(49)aL-cU2mL-e1UW1≈-4.7139<0,
that is to say, (A1) does not hold and we could not obtain the result of the permanence from Theorem A. Thus our results improve the main results in [1].Figure 1
Dynamic behavior of the system (47) with the initial condition (x(0),y(0),u1(0),u2(0))=(0.2,0.4,0.1,0.14)T and (0.4,0.5,0.21,0.3)T, respectively.



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