We study a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls. We establish a series of criteria under which a part of n-species of the systems is driven to extinction while the remaining part of the species is persistent. Particularly, as a special case, a series of new sufficient conditions on the persistence for all species of system are obtained. Several examples together with their numerical simulations show the feasibility of our main results.

1. Introduction

In this paper, we consider the following nonautonomous n-species Lotka-Volterra competitive system with infinite delay and feedback controls:
(1)x˙i(t)=xi(t)[ri(t)-∑j=1naij(t)∫0+∞Kij(s)xj(t-s)ds00000000i0000-bi(t)∫0+∞Hi(s)ui(t-s)ds∑j=1n],u˙i(t)=-ci(t)ui(t)+di(t)∫0+∞Ri(s)xi(t-s)ds,000000000000000000000000000000i=1,2,…,n,
where xi(t)(i=1,2,…,n) is the density of the ith species at time t and ui(t)(i=1,2,…,n) is the indirect control variable.

In particular, when the coefficients bi(t)≡0, ci(t)≡0, and di(t)≡0 for all t∈R and i=1,2,…,n, the system (1) will degenerate into the following pure delay type system:
(2)x˙i(t)=xi(t)[ri(t)-∑j=1naij(t)∫0+∞Kij(s)xj(t-s)ds],000000000000000000000000000000000000i=1,2,…,n.

As is well known, systems such as (2) without feedback controls are very important mathematical models of multispecies populations dynamics. This is a generalization from Ahmad [1] about two-species system without delays to n-species system of infinite delay. Systems without delays such as [1] have attracted the interest of many researchers (see, e.g., [2–5]), and systems with delays have been studied extensively in the past twenty years, and some good results on the permanence, extinction and persistence or uniform persistence, global stability, and almost periodic solution have been developed (see [6–18]). In [19], Montes de Oca and Pérez provided for us a very interesting work for system (2), who showed that if the coefficients are bounded and continuous and satisfy certain inequalities, then any solution with initial function of system (2) in an appropriate space will have n-1 of its components tenting to zero, while the remaining one will stabilize at a certain solution of a logistic differential equation. And for more works about single species dynamic behaviors of infinite delay, one could refer to [20, 21].

On the other hand, as was pointed out by Fan and Wang [22], feedback control is the basic mechanism by which systems, whether mechanical, electrical, or biological, maintain their equilibrium or homeostasis. Many scholars have done works on the ecosystem with feedback controls (see, e.g., [23–29] and the references cited therein). In [23], Shi et al. proposed the feedback control system (1). By using the method of multiple Lyapunov functionals and by developing a new analysis technique, Shi et al. established the sufficient conditions which guarantee part species xr+1,xr+2,…,xr+n of the n-species driven to extinction. But in the paper [23], they did not discuss the survival problems for the remaining species. The main aim of this paper is to study the persistence of the remaining species x1,x2,…,xr of system (1). By the new method motivated by work [11, 27, 28], we will establish new sufficient conditions for which surplus species x1,x2,…,xr of system (1) remain persistent.

The organization of the paper is as follows. In the next section, some assumptions and lemmas are introduced. In Section 3, we state and prove our main results. Finally, several examples with their numerical simulations are presented to show the feasibility of the main results.

2. Preliminaries

Throughout this paper, for system (1), we introduce the following hypotheses.

ri(t), aij(t), bi(t), ci(t), and di(t)(i,j=1,2,…,n) are bounded and continuous, defined on [0,∞). Furthermore, aij(t)(i≠j), bi(t), ci(t), and di(t) are nonnegative on [0,∞), and 0<aiil≤aii(t)≤aiiu<∞. Here, we denote fl=inft≥0f(t) and fu=supt≥0f(t).

Kij:[0,∞)→[0,∞), Hi:[0,∞)→[0,∞), and Ri:[0,∞)→[0,∞), i, j=1,2,…,n, are piecewise continuous and satisfy
(3)∫0+∞Kij(s)ds=1,Kij=∫0+∞sKij(s)ds<∞,∫0+∞Hi(s)ds=1,Hi=∫0+∞sHi(s)ds<∞,∫0+∞Ri(s)ds=1,Ri=∫0+∞sRi(s)ds<∞.

There exists a positive constant ω such that for each i=1,2,…,n(4)liminft→∞∫tt+ωri(s)ds>0.

There exist positive constants λ and γ such that for each i=1,2,…,n(5)liminft→∞∫tt+λci(s)ds>0,liminft→∞∫tt+γdi(s)ds>0.

We will consider system (1) together with the initial conditions
(6)xi(θ)=ϕi(θ),ui(θ)=ψi(θ),θ≤0,i=1,2,…,n,
where ϕi,ψi∈BC+, i=1,2,…,n, and
(7)BC+={φ∈C[(-∞,0],[0,+∞)]:0000000φ(0)>0,φisbounded}.
It is easy to verify that solutions of (1) satisfying the initial condition (6) are well defined for all t≥0 and satisfy
(8)xi(t)>0,ui(t)>0,∀t≥0.

We now introduce several lemmas which will be useful in the proofs of the main results.

We consider the following nonautonomous linear equation:
(9)x˙(t)=a(t)-b(t)x(t),
where nonnegative functions a(t) and b(t) are bounded and continuous, defined on [0,+∞). We have the following results.

Lemma 1 (see [<xref ref-type="bibr" rid="B30">30</xref>]).

Suppose that there exist positive constants η1 and η2 such that
(10)liminft→∞∫tt+η1a(s)ds>0,liminft→∞∫tt+η2b(s)ds>0.
Then, there exist positive constants M≥m such that
(11)m≤liminft→∞x(t)≤limsupt→∞x(t)≤M,
for any positive solution x(t) of (9).

Lemma 2 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Suppose that assumptions (H1)–(H4) hold; then there exist constants x¯i>0 and u¯i>0 such that
(12)limsupt→∞xi(t)<x¯i,limsupt→∞ui(t)<u¯i,i=1,2,…,n,
for any positive solution (x1(t),x2(t),…,xn(t),u1(t), u2(t),…,un(t)) of system (1).

Remark 3.

If all parameters ri(t), aij(t), bi(t), ci(t), and di(t)(i,j=1,2,…,n) of system (1) have the positive lower bound on [0,+∞), then, from Lemma 2.2 in [23], we can choose
(13)x¯i=riuaiil∫0+∞kii(s)exp(-rius)ds,u¯i=diucilx¯i.

Lemma 4 (see [<xref ref-type="bibr" rid="B1">6</xref>]).

Let x(t):R→R be a nonnegative and bounded continuous function, and let k(s):[0,+∞)→[0,+∞) be an integral function satisfying ∫0+∞k(s)ds=1. Then
(14)liminft→∞x(t)≤liminft→∞∫0+∞k(s)x(t-s)ds≤limsupt→∞∫0+∞k(s)x(t-s)ds≤limsupt→∞x(t).

3. Main Results

In this section, we discuss the persistence of part species xi(1≤i≤r) of system (1), where integer r∈{1,2,…,n}. Let functions
(15)Aij(t)=∫0+∞aij(t+s)Kij(s)ds,Bi(t)=∫0+∞bi(t+s)Hi(s)ds,Di(t)=∫0+∞di(t+s)Ri(s)ds,0000000000000i,j=1,2,…,n.

Lemma 5.

Suppose that assumptions (H1)–(H4) hold and there exists an integer 1≤r<n such that for any k>r there exists an integer ik<k such that
(16)limsupt→∞∫tt+wrk(s)ds∫tt+wrik(s)ds<liminft→∞Akj(t)Aikj(t)∀j≤k,liminft→∞Bk(t)ck(t)>limsupt→∞(Aikk(t)Dk(t)limsupt→∞∫tt+wrk(s)ds∫tt+wrik(s)ds-Akk(t)Dk(t)),limsupt→∞Bik(t)cik(t)<liminft→∞(Akik(t)Dik(t)liminft→∞∫tt+wrik(s)ds∫tt+wrk(s)ds-Aikik(t)Dik(t)).
Then for each i=r+1,…,n we have
(17)limt→∞xi(t)=0,limt→∞ui(t)=0,∫0∞xi(t)dt<∞,
for any positive solution (x1(t),x2(t),…,xn(t),u1(t), u2(t),…,un(t)) of system (1).

The proof of the extinction of part species xr+1,xr+2,…,xr+n of system (1) could be found in [23] and we hence omit it here.

On the persistence of part species xi(1≤i≤r) of system (1), we state and prove the following results.

Theorem 6.

Suppose that all assumptions of Lemma 5 hold and there exists a positive constant η>0 such that
(18)liminft→∞∫tt+η[ri(s)-∑j≠irAij(s)x¯j-Bi(s)u¯i]ds>0,00000000000000000000000000000000000000∀i≤r.
Then, for each i=1,2,…,r, there exist positive constants m and M, with m<M, such that
(19)m≤liminft→∞xi(t)≤limsupt→∞xi(t)≤M,m≤liminft→∞ui(t)≤limsupt→∞ui(t)≤M,
for any positive solution (x(t),u(t))=(x1(t),x2(t),…,xn(t),u1(t),u2(t),…,un(t)) of system (1).

Proof.

Let (x(t),u(t))=(x1(t),x2(t),…,xn(t), u1(t),u2(t),…,un(t)) be any positive solution of system (1). By Lemma 2, let M=max1≤i≤r{x¯i,u¯i}; for each i=1,2,…,r, we have limsupt→∞xi(t)≤M and limsupt→∞ui(t)≤M. So, we only need to prove that there exists a positive constant m such that liminft→∞xi(t)≥m and liminft→∞ui(t)≥m for all i≤r.

First of all, assumption (18) implies that there are positive constants α0 and T0 such that
(20)∫tt+η[ri(s)-∑j≠irAij(s)x¯j-Bi(s)u¯i]ds≥α0,
for all t≥T0 and i≤r.

By Lemmas 2 and 5, we obtain that, for any constant ɛ>0, there is a T(ɛ)>T0 such that, for all t≥T(ɛ),
(21)xi(t)≤x¯i+ɛ,ui(t)≤u¯i+ɛ,∀i≤r,xi(t)≤ɛ,ui(t)≤ɛ,∀i>r.

Now, for any i≤r, we define the Lyapunov function as follows:
(22)Wi(t)=xi(t)exp[-∑j=1n∫0+∞Kij(s)∫t-staij(θ+s)xj(θ)dθds=xi(t)exp11-∫0+∞Hi(s)∫t-stbi(θ+s)ui(θ)dθds∑j=1n].
By assumptions (H1) and (H2) and Lemma 2, we have
(23)∑j=1n∫0+∞Kij(s)∫t-staij(θ+s)xj(θ)dθds≤∑j=1naiju∫0+∞sKij(s)dssupt∈Rxj(t)<∞,∫0+∞Hi(s)∫t-stbi(θ+s)ui(θ)dθds≤biu∫0+∞sHi(s)dssupt∈Rui(t)<∞.
So we see that Wi(t) has definition for all t≥0. From (23), we can obtain that for any i≤r there is a positive constant di<1, and di may be dependent on the positive solution of system (1) such that
(24)dixi(t)≤Wi(t)≤xi(t),∀t≥0.
Calculating the derivative of Wi(t) with respect to t, we have
(25)W˙i(t)=Wi(t)[ri(t)-∑j=1naij(t)∫0+∞Kij(s)xj(t-s)ds=Wi(t)-bi(t)∫0+∞Hi(s)ui(t-s)ds=Wi(t)-∑j=1n∫0+∞Kij(s)aij(t+s)dsxj(t)=Wi(t)+∑j=1n∫0+∞Kij(s)aij(t)xj(t-s)ds=Wi(t)-∫0+∞Hi(s)bi(t+s)dsui(t)=Wi(t)+∫0+∞Hi(s)bi(t)ui(t-s)ds∑j=1n]=Wi(t)[ri(t)-∑j=1n∫0+∞Kij(s)aij(t+s)dsxj(t)=Wi(t)-∫0+∞Hi(s)bi(t+s)dsui(t)∑j=1n]=Wi(t)[ri(t)-∑j=1nAij(t)xj(t)-Bi(t)ui(t)],0000000000000000000000000000∀i≤r,t≥0.
Let βi(t,ɛ)=ri(t)-∑j≠inAij(t)ɛ-∑j≠irAij(t)x¯j-Bi(t)(u¯i+ɛ). From (21), for all t≥T(ɛ)>0, we have
(26)W˙i(t)≥Wi(t)[βi(t,ɛ)-Aii(t)xi(t)]≥Wi(t)[βi(t,ɛ)-Aii(t)di-1Wi(t)].
Obviously, from inequality (20), we can find enough small positive constants δi and ɛ0 such that
(27)∫tt+η[βi(s,ɛ0)-Aii(s)di-1δi]ds>12α0,
for all t≥T1=T(ɛ0). So for the above ɛ0, when t≥T1=T(ɛ0),
(28)W˙i(t)≥Wi(t)[βi(t,ɛ0)-Aii(t)di-1Wi(t)].
Consider the auxiliary equation
(29)W˙i(t)=Wi(t)[βi(t,ɛ0)-Aii(t)di-1Wi(t)];
then by (28), we obtain that
(30)Wi(t)≥Wi*(t),∀t≥T1,
where Wi*(t) is the solution of (29) with the initial condition Wi(T1)=Wi*(T1). If Wi*(t)<δi for all t≥T1, then Wi*(t) is defined on [T1,+∞). Integrating inequality (29) from T1 to t, we obtain
(31)Wi*(t)=Wi*(T1)exp∫T1t[βi(s,ɛ0)-Aii(s)di-1Wi*(s)]ds≥Wi*(T1)exp∫T1t[βi(s,ɛ0)-Aii(s)di-1δi]ds,
for all t≥T1. Putting t=T1+mη, m=1,2,…, then, from (27) and (31), we have
(32)Wi*(T1+mη)≥Wi*(T1)exp(12mα0),m=1,2,….
Letting m→+∞, we have Wi*(T1+mη)→+∞, a contradiction. Hence, there is a ti≥T1 such that Wi*(ti)>δi. Now, we prove that
(33)Wi*(t)≥δiexp(-βi(δi,ɛ0)η),∀t≥ti,
where βi(δi,ɛ0)=supt≥0{|βi(t,ɛ0)|+Aii(t)di-1δi}, and the definition of βi(t,ɛ0) implies 0<βi(δi,ɛ0)<∞. In fact, if (33) is not true, then there are t1 and t2, t1<t2, such that
(34)Wi*(t2)<δiexp(-βi(δi,ɛ0)η),Wi*(t1)=δi,Wi*(t)<δi,∀t∈(t1,t2].
Choosing the integer m≥0 such that t2∈(t1+mη,t1+(m+1)η], then, by (27) and (29), it follows that
(35)δiexp(-βi(δi,ɛ0)η)>Wi*(t2)=Wi*(t1)exp∫t1t2[βi(t,ɛ0)-Aii(t)di-1Wi*(t)]dt≥δiexp{∫t1t1+mη+∫t1+mηt2[βi(t,ɛ0)-Aii(t)di-1δi]dt}≥δiexp{∫t1+mηt2[βi(t,ɛ0)-Aii(t)di-1δi]dt}≥δiexp(-βi(δi,ɛ0)η),
which is a contradiction.

From (24), (30), and (33), we can obtain that
(36)xi(t)≥δiexp(-βi(δi,ɛ0)η)∀t≥ti.
Finally, we define the constants mi=δiexp(-βi(δi,ɛ0)η) and T=maxi≤r{ti}; then we have
(37)xi(t)≥mi∀t≥T,i≤r.
Letting mi*={inft∈[0,T]xi(t)>0} and m*=min1≤i≤r{mi,mi*}, we have
(38)liminft→∞xi(t)≥m*,
for all i≤r.

Further, by Lemma 4 and (38), we can choose constants ϵ>0 and T*>0 such that for all i≤r and t≥T*(39)∫0+∞Ri(s)xi(t-s)ds≥m*-ϵ>0.
Considering the second equation of system (1), from (39), for any t≥T*, we obtain
(40)u˙i(t)=-ci(t)ui(t)+di(t)∫0+∞Ri(s)xi(t-s)ds≥-ci(t)ui(t)+di(t)(m*-ϵ).
We consider the following auxiliary equation:
(41)v˙i(t)=-ci(t)vi(t)+di(t)(m*-ϵ).
Then by assumption (H4) and applying Lemma 1 there exists a constant u_i>0 such that
(42)liminft→∞vi(t)>u_i,
for any positive solution vi(t) of (41). Let vi*(t) be the solution of (41) with the initial condition vi*(T*)=ui(T*); then by the comparison theorem we have
(43)ui(t)≥vi*(t)∀t≥T*.
Thus, we finally obtain
(44)liminft→∞ui(t)≥u_i.
Let m=min1≤i≤r{m*,u_i}; from (38) and (44), we obtain that liminft→∞xi(t)≥m and liminft→∞ui(t)≥m. This completes the proof of Theorem 6.

As consequences of Theorem 6 we have the following corollaries.

Corollary 7.

If, in system (1), bi(t)=ci(t)=di(t)=0(i=1,2,…,n) for all t≥0, then system (1) will be reduced to the following n-species competitive system with infinite delay:
(45)x˙i(t)=xi(t)[ri(t)-∑j=1naij(t)000000000000×∫0+∞Kij(s)xj(t-s)ds∑j=1n],i=1,2,…,n.
Suppose that assumptions (H1)–(H3) hold and there exists an integer 1≤r<n such that for any k>r there exists an integer ik<k such that
(46)limsupt→∞∫tt+wrk(s)ds∫tt+wrik(s)ds<liminft→∞Akj(t)Aikj(t),∀j≤k.
Furthermore, there exists a positive constant η>0 such that
(47)liminft→∞∫tt+η[ri(s)-∑j≠irAij(s)x-j]ds>0,∀i≤r.
Then, for each i=1,2,…,r, there exist positive constants m≤M such that
(48)m≤liminft→∞xi(t)≤limsupt→∞xi(t)≤M,
and for each i=r+1,…,n we have
(49)limt→∞xi(t)=0,∫0∞xi(t)dt<∞,
for any positive solution (x1(t),x2(t),…,xn(t)) of system (45).

Proof.

From the condition,
(50)limsupt→∞∫tt+wrk(s)ds∫tt+wrik(s)ds<liminft→∞Akj(t)Aikj(t),∀j≤k.
And the assumptions (H1)–(H3) hold; from Corollary 7 in [23], for each i=r+1,…,n we have
(51)limt→∞xi(t)=0,∫0∞xi(t)dt<∞.
Further condition
(52)liminft→∞∫tt+η[ri(s)-∑j≠irAij(s)x¯j]ds>0,∀i≤r
holds, so we see, from Theorem 6, for each i=1,2,…,r, that there exist positive constants m≤M such that
(53)m≤liminft→∞xi(t)≤limsupt→∞xi(t)≤M.

Remark 8.

When r=1, the conditions of Corollary 7 will reduce to the assumptions that (H1)–(H3) hold and for any k>1 such that
(54)limsupt→∞∫tt+wrk(s)ds∫tt+wr1(s)ds<liminft→∞Akj(t)A1j(t),∀j≤k.
We have that there exist positive constants m≤M such that
(55)m≤liminft→∞x1(t)≤limsupt→∞x1(t)≤M,
and for each i=2,…,n we have
(56)limt→∞xi(t)=0,∫0∞xi(t)dt<∞.
In comparison with the assumptions (1.5) together with Proposition 2.2 given by Montes de Oca and Pérez [19], we can see that our assumptions in Corollary 7 are weaker.

Remark 9.

When r=n, from Corollary 7 we can easily obtain a criterion on the persistence of all species (x1(t),x2(t),…,xn(t)) of system (45).

Remark 10.

The conclusion of Corollary 7 improves that of Proposition 2.2 given by Montes de Oca and Pérez [19].

Corollary 11.

Suppose that (H1)–(H4) hold and there exists a positive constant η>0 such that
(57)liminft→∞∫tt+η[ri(s)-∑j≠inAij(s)x¯j-Bi(s)u¯i]ds>0,000000000000000000000000000i000∀i=1,2,…,n.
Then, for each i=1,2,…,n, there exist positive constants m and M, with m<M, such that
(58)m≤liminft→∞xi(t)≤limsupt→∞xi(t)<M,m≤liminft→∞ui(t)≤limsupt→∞ui(t)<M,
for any positive solution (x1(t),x2(t),…,xn(t),u1(t), u2(t),…,un(t)) of system (1).

Remark 12.

From Corollary 11 we can easily obtain a criterion on the persistence of all species (x1(t),x2(t),…,xn(t),u1(t),u2(t),…,un(t)) of system (1).

4. Examples

In this section, we will give several examples to illustrate the conclusions of Corollary 7, Theorem 6, and Corollary 11. In the first part we will illustrate the conclusions of Corollary 7, in the second we will illustrate the conclusions of Theorem 6, and in the last we will illustrate the conclusions of Corollary 11.

Example 1.

Consider the system
(59)x˙i(t)=xi(t)[ri(t)-∑j=13aij(t)∫0+∞Kij(s)xj(t-s)ds],000000000000000000000000000000000000000i=1,2,3,
where
(60)r1(t)=5+3sint,r2(t)=3+3cost,r3(t)=2+2sint,a11(t)=4,a12(t)=5,a13(t)=6,a21(t)=3,a22(t)=4,a23(t)=5,a31(t)=2,a32(t)=3,a33(t)=4,K11(t)=e-t,K12(t)=2e-2t,K13(t)=3e-3t,K21(t)=e-t,K22(t)=2e-2t,K23(t)=3e-3t,K31(t)=e-t,K32(t)=2e-2t,K33(t)=3e-3t.
Obviously, we have that the period of system (59) is ω=2π. By calculating, we obtain
(61)limsupt→∞∫tt+wr2(s)ds∫tt+wr1(s)ds=35<liminft→∞A21(t)A11(t)=34,limsupt→∞∫tt+wr2(s)ds∫tt+wr1(s)ds=35<liminft→∞A22(t)A12(t)=45,limsupt→∞∫tt+wr3(s)ds∫tt+wr2(s)ds=25<liminft→∞A31(t)A21(t)=23,limsupt→∞∫tt+wr3(s)ds∫tt+wr2(s)ds=25<liminft→∞A32(t)A22(t)=34,limsupt→∞∫tt+wr3(s)ds∫tt+wr2(s)ds=25<liminft→∞A33(t)A23(t)=45.
We can choose r=1, since all conditions of Corollary 7 hold; therefore, species x2 and x3 in system (59) are extinct, and only species x1 is persistent (see Figure 1). However, conditions (1.5) of Proposition 2.2 given by Montes de Oca and Pérez [19] do not apply in this example.

Dynamic behaviors of system (59). Here, we take the initial conditions x1(θ)≡x2(θ)≡x3(θ)=0.3 for all θ∈(-∞,0].

Example 2.

Consider the system
(62)x˙i(t)=xi(t)[ri(t)-∑j=13aij(t)∫0+∞Kij(s)xj(t-s)ds000000000000-bi(t)∫0+∞Hi(s)ui(t-s)ds∑j=13],u˙i(t)=-ci(t)ui(t)+di(t)∫0+∞Ri(s)xi(t-s)ds,0000000000000000000000000000000000i=1,2,3,
where
(63)r1(t)=2+sint,a11(t)=4+cost,a12(t)=15,a13(t)=16,b1(t)=110,r2(t)=3+cost,a21(t)=14,a22(t)=6+sint,a23(t)=27,b2(t)=13,r3(t)=1+14cost,a31(t)=52,a32(t)=52,a33(t)=2+sint,b3(t)=25,c1(t)=4+2sint,c2(t)=6+2sint,c3(t)=5+2sint,d1(t)=1+12cost,d2(t)=1+13cost,d3(t)=1+14cost,Kij(t)=Hi(t)=Ri(t)=e-t,i,j=1,2,3.
Obviously, we have that the period of system (62) is ω=2π. By calculating, we obtain
(64)limsupt→∞∫tt+wr3(s)ds∫tt+wr1(s)ds=12<liminft→∞A31(t)A11(t)=58+2,limsupt→∞∫tt+wr3(s)ds∫tt+wr1(s)ds=12<liminft→∞A32(t)A12(t)=252,limsupt→∞∫tt+wr3(s)ds∫tt+wr1(s)ds=12<liminft→∞A33(t)A13(t)=12-32,limsupt→∞∫tt+wr3(s)ds∫tt+wr2(s)ds=13<liminft→∞A31(t)A21(t)=10,limsupt→∞∫tt+wr3(s)ds∫tt+wr2(s)ds=13<liminft→∞A32(t)A22(t)=512+2,limsupt→∞∫tt+wr3(s)ds∫tt+wr2(s)ds=13<liminft→∞A33(t)A23(t)=7-742.
From Remark 3, we can choose r=2, x¯1=4, x¯2=4, x¯3=45/16, u¯1=3, u¯2=4/3, u¯3=75/64, and η=2π such that
(65)liminft→∞∫tt+2π[r1(s)-A12(s)x¯2-A13(s)x¯3-B1(s)u¯1]ds=0.8625π>0,liminft→∞∫tt+2π[r2(s)-A21(s)x¯1-A23(s)x¯3-B2(s)u¯2]ds≈1.504π>0.
All the conditions of Theorem 6 hold; therefore, species x1 and x2 coexist, and species x3 in system (62) is extinct (see Figure 2).

Dynamic behaviors of system (62). Here, we take the initial conditions x1(θ)≡x2(θ)≡x3(θ)≡u1(θ)≡u2(θ)≡u2(θ)=0.3 for all θ∈(-∞,0].

Example 3.

Consider the system
(66)x˙i(t)=xi(t)[ri(t)-∑j=13aij(t)∫0+∞Kij(s)xj(t-s)ds000000000000000-bi(t)∫0+∞Hi(s)ui(t-s)ds∑j=13],u˙i(t)=-ci(t)ui(t)+di(t)∫0+∞Ri(s)xi(t-s)ds,000000000000000000000000000000000i=1,2,3,
where
(67)r3(t)=4+cost,a31(t)=12,a32(t)=27,a33(t)=7+sint,
and the coefficients and the other kernels are as Example 2. In this case, we can choose x¯1=4, x¯2=4, x¯3=5, u¯1=3, u¯2=4/3, u¯3=25/12, and η=2π such that
(68)liminft→∞∫tt+2π[r1(s)-A12(s)x¯2-A13(s)x¯3-B1(s)u¯1]ds=215π>,liminft→∞∫tt+2π[r2(s)-A21(s)x¯1-A23(s)x¯3-B2(s)u¯2]ds=1663π>0,liminft→∞∫tt+2π[r3(s)-A31(s)x¯1-A32(s)x¯2-B3(s)u¯3]ds=121π>0.
All conditions of Corollary 11 hold, so all the species x1, x2, and x3 are persistent (see Figure 3).

Dynamic behaviors of system (66). Here, we take the initial conditions x1(θ)≡x2(θ)≡x3(θ)≡u1(θ)≡u2(θ)≡u3(θ)=0.3 for all θ∈(-∞,0].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Foundation of Fujian Education Bureau (nos. JA12369 and JA13361).

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