# Permanence of a Single Species System with Distributed Time Delay and Feedback Control

Authors: Yali Shen; Fengqin Zhang; Kai Wang
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

We study the permanence of a classofsingle species system with distributed time delay and feedback controls. General criteria on permanence are established in this paper. A very important fact is found in our results; that is, the feedback control is harmless to the permanence of species.


---

## Body

## 1. Introduction

Ecosystems in the real world are continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. During the last decade, many scholars did works on the feedback control ecosystems. Motivated by those work especially [1], we consider the following single species system with distributed time delay and feedback control:(1.1)x′(t)=x(t){a(t)-b(t)x(t-τ(t))-c(t)∫-∞0H(s)x(t+s)ds-h(t)u(t)-r(t)u(t-λ(t))},u′(t)=-d(t)u(t)+e(t)∫-∞0H(s)x2(t+s)ds,
where x(t) is the density of the species at time t, u(t) is the control variable; a(t), b(t),c(t),d(t),e(t),h(t),r(t),τ(t) and λ(t) are defined on R+=[0,+∞) and are bounded and continuous functions; b(t),c(t),d(t),e(t),h(t),r(t), τ(t), and λ(t) are nonnegative for all t∈R+. Further, we assume that the delay kernels H(s) are nonnegative integral functions defined on R-=(-∞,0] such that ∫-∞0H(s)ds=1 and σ=∫-∞0sH(s)ds<+∞.We denote byBC+ the space of all bounded continuous functions ϕ:R-→R+ with norm |ϕ|=sups∈R-|ϕ(s)|. In this paper, we always assume that all solutions of system (1.1) satisfy the following initial condition:(1.2)x(s)=ϕ(s),u(s)=ψ(s)∀s∈R-,
where ϕ(s)∈BC+ and ψ(s)∈BC+.Let(x(t),u(t)) be the solution of system (1.1) satisfying initial condition (1.2). We easily prove x(t)>0 and u(t)>0 in maximal interval of the existence of the solution. For the sake of convenience, the solution of system (1.1) with initial condition (1.2) is said to be positive.In addition, for a functiong(t) defined on set I⊂R, we denote(1.3)g0=sup⁡t∈Ig(t),g̲=inf⁡t∈Ig(t).In the theory of mathematical biology, systems such as (1.1) are very important in a single species system in time-fluctuating environments, the effect of time delays and feedback controls. We see that there has been a series of articles which deal with the dynamical behaviors of the autonomous, periodic, and general nonautonomous population growth systems with feedback controls, for example [1–12] and reference cited therein. In [1] the authors proposed the following single species model with feedback regulation and distributed time delay of the form:(1.4)x′(t)=x(t){a(t)-b(t)∫0∞H(s)x(t-s)ds-c(t)u(t)},u′(t)=-d(t)u(t)+e(t)∫0∞H(s)x2(t-s)ds.
By using the continuation theorem of coincidence theory, a criterion which guarantees the existence of positive periodic solution of system (1.4) is obtained. In [2], the authors obtain sufficient condition which guarantees the global attractivity of the positive solution of system (1.4) by constructing a suitable Lyapunov functional. The aim of this paper is to establish new sufficient conditions on the permanence for all positive solutions of system (1.1) by improving the method given in [13–15].


## 2. Preliminaries

Throughout this paper, we will introduce the following assumptions:(H1)
there exists a constantω>0, such that
(2.1)liminf⁡t→∞∫tt+ωa(s)ds>0;(H2)
there exists a constantβ>0, such that
(2.2)liminf⁡t→∞∫tt+βb(s)ds>0;(H3)
there exists a constantγ>0 such that
(2.3)liminf⁡t→∞∫tt+γc(s)ds>0;(H4)
there exists a constantλ>0 such that
(2.4)liminf⁡t→∞∫tt+λd(s)ds>0;(H5)
there exists a constantζ>0 such that
(2.5)liminf⁡t→∞∫tt+ζe(s)ds>0.First, we consider the following nonautonomous logistic equation(2.6)x′(t)=x(t)(a(t)-b(t)x(t)),
where functions a(t),b(t) are bounded and continuous on R+. Furthermore, b(t)≥0 for all t≥0. We have the following result which is given in [15] by Teng and Li.Lemma 2.1.
Suppose that assumptions  (H1)-(H2) hold. Then, (a)
there exist positive constantsm and M such that
(2.7)m≤liminf⁡t→∞x(t)≤lim⁡⁡sup⁡t→∞x(t)≤M
for any positive solution x(t) of (2.6);(b)
limt→∞(x(1)(t)-x(2)(t))=0 for any two positive solutions x(1)(t) and x(2)(t) of (2.6).Further, we consider the following nonautonomous linear equation(2.8)u′(t)=r(t)-d(t)u(t),
where functions r(t) and d(t) are bounded continuous defined on R+, and r(t)≥0 for all t≥0. We have the following result.Lemma 2.2.
Suppose that assumptions  (H4) hold. Then,
(a)
there exists a positive constantU such that limsupt→∞u(t)≤U for any positive solution u(t) of (2.8);(b)
limt→∞(u(1)(t)-u(2)(t))=0 for any two positive solutions u(1)(t) and u(2)(t) of (2.8).The proof of Lemma2.2 is very simple, we hence omit it here.Lemma 2.3.
Suppose that assumption  (H4) holds. Then for any constants  ε>0 and  M>0 there exist constants  δ=δ(ε)>0 and  T=T(M)>0 such that for any  t0∈R+ and  u0∈R with  |u0|≤M, when  |r(t)|<δ for all  t≥t0, one has
(2.9)|u(t,t0,u0)|<ε∀t≥t0+T,
where u(t,t0,u0) is the solution of (2.8) with initial condition u(t0)=u0.The proof of Lemma2.3 can be found as Lemma  2.4 in [16] by Wang et al.Lemma 2.4.
Letx(t):R→R be a nonnegative and bounded continuous function, and let  H(s):R-→R+ be an integral function satisfying  ∫-∞0H(s)ds=1. Then one has
(2.10)lim⁡⁡inf⁡t→∞x(t)≤lim⁡⁡inf⁡t→∞∫-∞0H(s)x(t+s)ds≤lim⁡⁡sup⁡t→∞∫-∞0H(s)x(t+s)ds≤limsup⁡t→∞x(t).Lemma2.4 is given in [17] by Montes de Oca and Vivas.


## 3. Main Results

Theorem 3.1.
Suppose that assumption  (H1) holds and  (H2) or  (H3) holds. Then there exists a positive constant  M>0 such that
(3.1)lim⁡⁡sup⁡t→∞x(t)<M,limsup⁡t→∞u(t)<M,
for any positive solution ((x(t),u(t)) of system (1.1).Proof.
Letx(t) be a positive solution of the first equation system (1.1). Since
(3.2)dx(t)dt≤a(t)x(t)
for all t>0 as long as the solution exists and the function a(t) is bounded and continuous on R+. We can obtain that the solution exists for all t∈R+. For any t≥0,s≤0, and t+s≥0, by integrating (3.2) from t+s to t, we have
(3.3)x(t+s)≥x(t)exp⁡∫tt+sa(u)du.
For any t>τ0, where τ0=max{supt∈R+τ(t)≥0,supt∈R+λ(t)≥0}, by (3.3) we can directly from the first equation of (1.1)
(3.4)dx(t)dt≤x(t)(a(t)-b(t)x(t)exp⁡∫tt-τ(t)a(u)du-c(t)(∫-t0H(s)exp⁡∫tt+sa(u)duds)x(t))=x(t)(a(t)-θ(t)x(t)),
where θ(t)=b(t)exp∫tt-τ(t)a(u)du+c(t)∫-t0H(s)exp∫tt+sa(u)duds. Since for any t≥τ0 and s∈R-(3.5)∫tt-τ(t)a(u)du≥-τ0a0,∫tt+sa(u)du≥sa0.
We have
(3.6)d(t)≥b(t)exp⁡(-τ0a0)+c(t)∫-t0H(s)exp⁡(a0s)ds.
Let
(3.7)f(t)=b(t)exp⁡(-τ0a0)+c(t)∫-t0H(s)exp⁡(a0s)ds,g(t)=b(t)exp⁡(-τ0a0)+c(t)∫-∞0H(s)exp⁡(a0s)ds.
Since 0<a0, we have
(3.8)0<∫-∞0H(s)exp⁡(a0s)ds<∫-∞0H(s)ds<∞,lim⁡t→∞(g(t)-f(t))=lim⁡t→∞c(t)∫-∞-tH(s)exp⁡(a0s)ds=0.
Hence, assumption (H2) or (H3) implies
(3.9)lim⁡⁡inf⁡t→∞∫tt+σf(s)ds=liminf⁡t→∞[∫tt+σg(s)ds+∫tt+σ(f(s)-g(s))ds]=lim⁡⁡inf⁡t→∞∫tt+σg(s)ds>0,
where the constant σ=β or γ. Since θ(t)≥f(t) for all t≥τ0, we finally obtain
(3.10)lim⁡⁡inf⁡t→∞∫tt+δθ(s)ds≥lim⁡⁡inf⁡t→∞∫tt+δf(s)ds>0.
We consider the following auxiliary equation:
(3.11)dy(t)dt≤y(t)(a(t)-θ(t)y(t)),
then by conclusion (a) of Lemma 2.1 and inequality (3.10) we obtain that there exist a constant M1 such that limsupt→∞y(t)≤M1 for any solution of (3.11) with initial condition y(0)>0. Let y*(t) be the solution of (3.11) with initial condition y*(τ0)=x(τ0), then by the comparison theorem, we have from (3.4) that
(3.12)x(t)<y*(t),∀t>τ0.
Thus, we finally obtain that
(3.13)lim⁡⁡sup⁡t→∞x(t)<M1.
Moreover, by Lemma2.4, for any positive constant ε>0, there exists a constant T>0 such that
(3.14)∫-∞0H(s)x2(t+s)ds<M12+ε,∀t>T.
Considering the second equation of the system (1.1), we have
(3.15)du(t)dt≤-d(t)u(t)+e(t)(M12+ε).
We consider the following auxiliary equations:
(3.16)dv(t)dt=-d(t)v(t)+e(t)(M12+ε),
then, by assumption (H4) and conclusion (a) of the Lemma 2.2, we obtain that there is a constant M2 such that limsupt→∞v(t)≤M2 for any solution of (3.16) with initial condition v(0)>0. Let v*(t) be the solution of (3.16) with initial condition v*(T′)=u(T′) and T′>T, then, by comparison theorem, we have from (3.15) that
(3.17)u(t)≤v*(t),∀t>T′.
Thus, we finally obtain that
(3.18)lim⁡⁡sup⁡t→∞v(t)<M2.
Choose the constant M=max{M1,M2}, then we finally obtain
(3.19)lim⁡⁡sup⁡t→∞x(t)<M,lim⁡⁡sup⁡t→∞u(t)<M.
This completes the proof.Theorem 3.2.
Suppose that assumptions  (H1)–(H5) hold. Then there exists a constant  ηx>0 which is independent of the solution of system (1.1) such that
(3.20)lim⁡⁡inf⁡t→∞x(t)≥ηx,
for any positive solutions (x(t),u(t)) of system (1.1).Proof.
Let(x(t),u(t)) be a solution of system (1.1); from Theorem 3.1 there exists a T0 such that for all t>T0 we have x(t)≤M,u(t)≤M. According to the assumption (H1) we can choose positive constants ε>0 and T0>0 such that, for all t≥T0, we have
(3.21)∫tt+δ(a(s)-b(s)ε-2c(s)ε-h(s)ε-r(s)ε)ds≥ε.
Consider the following equation:
(3.22)dv(t)dt=-d(t)v(t)+e(t)(α0+α02),
where α0>0 is a parameter. By Lemma 2.3, for given in above ε>0 and positive constant M (Theorem 3.1), there exist constants δ0=δ0(ε) and T̃0=T̃0(M)>0 such that for any t0∈R+ and 0≤v0≤M, when e(t)(α0+α02)<δ0, for all t≥t0, we have
(3.23)0≤v(t,t0,u0)<ε∀t≥t0+T̃0,
where v(t,t0,u0) is the solution of (3.22) with initial condition v(t0)=v0. Hence, we can choose a positive constant α0≤min{ε,δ0/2(e0+1)} such that, for all t≥T0,
(3.24)∫tt+δ(a(s)-b(s)α0-2c(s)α0-h(s)ε-r(s)ε)ds≥α0.
We first prove that(3.25)lim⁡⁡sup⁡t→∞x(t)≥α0.
In fact, if (3.25) is not true, then there exists a constant T1≥T0 such that x(t)<α0 for all t≥T1. Choose constants τ1>τ0,τ2>τ1, such that
(3.26)∫-∞-τ1H(s)ds<α0M02,∫-∞-τ2H(s)ds<α0M0,
whereM0=sup{x(t+s):t∈R+,S∈R-}, then for t≥T1+τ1 we have
(3.27)u′(t)=-d(t)u(t)+e(t)(∫-∞-τ1+∫-τ10)H(s)x2(t+s)ds≤-d(t)u(t)+e(t)(α0M02M02+α02)≤-d(t)u(t)+e(t)(α0+α02).
Letv(t) be the solution of (3.22) with initial condition v(T1+τ1)=u(T1+τ1), then by the comparison theorem we have
(3.28)u(t)≤v(t),∀t≥T1+τ1.
In (3.23), we choose t0=T1+τ1 and v0=u(T1+τ1), since (α0+α02)e(t)<δ0 for all t≥T1+τ1, we have
(3.29)v(t)=v(t,T1+τ1,v(T1+τ1))<ε∀t≥T1+τ1+T̃0.
Hence, we further have
(3.30)u(t)<ε∀t≥T1+τ1+T̃0.
For any t≥T1+τ1+T̃0+τ2, from the first equation of (1.1) we have
(3.31)x′(t)=x(t){a(t)-b(t)x(t-τ(t))-c(t)(∫-∞-τ2+∫-τ20)H(s)x(t+s)ds-h(t)u(t)-r(t)u(t-λ(t))}≥x(t)(a(s)-b(s)α0-2c(s)α0-h(s)ε-r(s)ε).
Integrating (3.31) from T1+τ1+T̃0+τ2 to t≥T1+τ1+T̃0+τ2 we obtain
(3.32)x(t)≥x(T1+τ1+T̃0+τ2)exp⁡{∫T1+τ1+T̃0+τ2t(a(s)-b(s)α0-2c(s)α0-h(s)ε-r(s)ε)ds}.
Obviously, inequality (3.24) implies that x(t)→∞ as t→∞, which leads to a contradiction.
Now, we prove the conclusion of Theorem3.2 to hold. Assume that it is not true, then there exists a sequence {Z(m)}={(φ(m),ψ(m))} of initial functions of system (1.1) such that
(3.33)lim⁡⁡inf⁡t→∞x(t,Z(m))<α0(m+1)2∀m=1,2,…,
where (x(t,Z(m)),u(t,Z(m))) is the solution of system (1.1) with initial condition
(3.34)x(s)=φ(m)(s),u(s)=ψ(m)(s)∀s∈R-.
From (3.25) and (3.33), we obtain that for every m there are two time sequences {sq(m)} and {tq(m)}, satisfying
(3.35)0<s1(m)<t1(m)<s2(m)<t2(m)<⋯<sq(m)<tq(m)<⋯,(3.36)sq(m)⟶∞,tq(m)⟶∞asq⟶∞,(3.37)x(sq(m),ϕ(m),ψ(m))=α0m+1,x(tq(m),ϕ(m),ψ(m))=α0(m+1)2,(3.38)α0(m+1)2≤x(t,ϕ(m),ψ(m))≤α0m+1∀t∈(sq(m),tq(m)).
Let Mm=sup{x(t+s,ϕ(m),ψ(m)):t∈R+,S∈R-}. For each m=1,2,… we can choose a constant τ1(m)>τ0 such that
(3.39)∫-∞-τ1(m)H(s)x(t+s,ϕ(m),ψ(m))ds≤Mm∫-∞-τ1(m)H(s)ds<α0.
By Theorem 3.1, there exists a constant M such that for each m=1,2,… there exists a T1(m)>T0 such that
(3.40)x(t,ϕ(m),ψ(m))≤M,u(t,ϕ(m),ψ(m))≤M∀t>T1(m).
Further, from (3.36) there is an integer K1(m)>0 such that sq(m)>T1(m)+τ1(m) for all q≥K1(m) for all m=1,2,…. Hence, for any t∈(sq(m),tq(m)) and q≥K1(m) by (3.39) and (3.40) we have
(3.41)dx(t,ϕ(m),ψ(m))dt≥x(t,ϕ(m),ψ(m)){a(t)-b(t)M-c(t)(∫-∞-τ1(m)+∫-τ1(m)0)H(s)x(t+s)ds-h(t)M-r(t)M}≥x(t,ϕ(m),ψ(m))(a(t)-b(t)M-c(t)(α0+M)-h(t)M-r(s)M)≥-γ1x(t,ϕ(m),ψ(m)),
where γ1=supt∈R+{|a(t)|+b(t)M+c(t)(α0+M0)+r(t)M+h(t)M}>0. Therefore, for any q≥K1(m) and m=1,2,… integrating the above inequality from sq(m) to tq(m), we further have
(3.42)α0(m+1)2=x(tq(m),ϕ(m),ψ(m))≥x(sq(m),ϕ(m),ψ(m))exp⁡{-γ1(tq(m)-sq(m))}=α0m+1exp⁡{-γ1(tq(m)-sq(m))}.
Consequently,
(3.43)tq(m)-sq(m)≥ln⁡(m+1)γ1∀q≥K1(m),m=1,2,….
For any q≥K1(m) and t>sq(m) we can obtain
(3.44)∫-∞T1(m)H(θ-t)x(θ,ϕ(m),ψ(m))dθ≤Mm∫-∞T1(m)-tH(s)ds,∫-∞T1(m)H(θ-t)x2(θ,ϕ(m),ψ(m))dθ≤Mm2∫-∞T1(m)-tH(s)ds,∫T1(m)sq(m)H(θ-t)x(θ,ϕ(m),ψ(m))dθ≤M∫-∞sq(m)-tH(s)ds,∫T1(m)sq(m)H(θ-t)x2(θ,ϕ(m),ψ(m))dθ≤M2∫-∞sq(m)-tH(s)ds.
(3.20)For each m=1,2,… by (3.36) there exists a K2(m)≥K1(m) and constant L>τ0 such that
(3.45)Mm∫-∞T1(m)-sq(m)H(s)ds<Mm2∫-∞T1(m)-sq(m)H(s)ds≤α02,∀q≥K2(m),M∫-∞-LH(s)ds<M2∫-∞-LH(s)ds≤α02.
By (3.43) there exists a large enough m̂1 such that
(3.46)tq(m)-sq(m)≥L+δ+T̃0,∀m≥m̂1,q≥K2(m).
Hence, for any m≥m̂1,q≥K2(m) and t∈[sq(m)+L,tq(m)], by (3.38), (3.20), and (3.45) we have
(3.47)du(t,Z(m))dt=-d(t)u(t,Z(m))+e(t)(∫-∞T1(m)+∫T1(m)sq(m)+∫sq(m)t)H(θ-t)x2(θ)dθ≤-d(t)u(t,Z(m))+e(t)Mm2∫-∞T1(m)-tH(θ)dθ+e(t)M∫-∞sq(m)-tH(θ)dθ+e(t)α02(m+1)2∫sq(m)tH(θ-t)dθ≤-d(t)u(t,Z(m))+e(t)(α02+α02+α02(m+1)2)≤-d(t)u(t,Z(m))+e(t)(α0+α02).
Assume that v(t) is the solution of (3.22) satisfying initial condition v(sq(m)+L)=u(sq(m)+L), then we have
(3.48)u(t,Z(m))≤v(t),∀t∈[sq(m)+L,tq(m)],m≥m̂1,q≥K2(m).
In (3.23), we choose t0=sq(m)+L and v0=u(sq(m)+L), since (α0+α02)e(t)<δ0 for all t≥sq(m)+L, we have
(3.49)v(t)=v(t,sq(m)+L,v(sq(m)+L))<ε∀t∈[sq(m)+L+T̃0,tq(m)].
Hence, we further have
(3.50)u(t,Z(m))<ε
for all t∈[sq(m)+L+T̃0,tq(m)],q≥K2(m) and m≥m̂1.
For anyq≥K2(m),m≥m̂1, and t∈[sq(m)+L+T̃0,tq(m)], from (3.38)–(3.40), we have
(3.51)dx(t,ϕ(m),ψ(m))dt≥x(t,ϕ(m),ψ(m)){∫-∞T1(m)a(t)-b(t)x(t-T(t))-c(t)(∫-∞T1(m)+∫T1(m)sq(m)+∫sq(m)t)H(s)x(t+s)ds-h(t)α0-r(t)α0∫-∞T1(m)}>x(t)(a(t)-b(t)α0m+1-c(t)Mm∫-∞T1(m)-tH(θ)ds-c(t)M∫T1(m)sq(m)-tH(θ)ds-c(t)α0m+1∫sq(m)tH(θ-t)ds-h(t)ε-r(t)ε∫T1(m)sq(m)-t)≥x(t)(a(t)-b(t)α0-2c(t)α0-h(t)ε-r(t)ε).
Integrating the above inequality from tq(m)-δ to tq(m), then by (3.24), (3.37) and (3.38) we obtain
(3.52)α0(m+1)2=x(tq(m),Z(m))≥x(tq(m)-δ,Z(m))exp⁡∫tq(m)-δtq(m)(a(s)-b(s)α0-2c(s)α0-h(s)ε-r(s)ε)ds≥α0(m+1)2exp⁡{α0}>α0(m+1)2
which leads to a contradiction. Therefore, this contradiction shows that there exists constant ηx>0 such that
(3.53)lim⁡⁡inf⁡t→∞x(t)≥ηx,
for any positive solutions (x(t),u(t)) of system (1.1). This completes the proof.Applying Theorem3.2 to system (1.4), we have the following corollary.Corollary 3.3.
Suppose that assumptions  (H1)–(H5) hold, then system (1.4) is permanent.Obviously, we will first consider system (1.1) which is more general than system (1.4). Moreover, Corollary 3.3 is a very good improvement of Theorem  2.1 in [2]. From Corollary 3.3, we find that it is established that a very weak sufficient condition for the permanent of system (1.4).Remark 3.4.
From Theorem3.2 we directly see that for system (1.1) the feedback control is harmless to the permanence of system (1.1).Remark 3.5.
In this paper, the biological model about a single species is considered. However, complex networks have attracted increasing attention from various fields of science and engineering in recent years. Meanwhile, in some sense, the coupled species systems can be treated as a typical complex networks. The real world biological systems have more complex structures and relationships. Motivated by above work [18–21], our future work is that how to apply the current complex networks theory to improve the current work.



---

*Source: 867364-2012-06-19.xml*