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.
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: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 by BC+ 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: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 function g(t) defined on set I⊂R, we denoteg0=supt∈Ig(t),g̲=inft∈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: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:
there exists a constant ω>0, such that
liminft→∞∫tt+ωa(s)ds>0;
there exists a constant β>0, such that
liminft→∞∫tt+βb(s)ds>0;
there exists a constant γ>0 such that
liminft→∞∫tt+γc(s)ds>0;
there exists a constant λ>0 such that
liminft→∞∫tt+λd(s)ds>0;
there exists a constant ζ>0 such that
liminft→∞∫tt+ζe(s)ds>0.
First, we consider the following nonautonomous logistic equationx′(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,
there exist positive constants m and M such that
m≤liminft→∞x(t)≤limsupt→∞x(t)≤M
for any positive solution x(t) of (2.6);
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 equationu′(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,
there exists a positive constant U such that limsupt→∞u(t)≤U for any positive solution u(t) of (2.8);
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 Lemma 2.2 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
|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 Lemma 2.3 can be found as Lemma 2.4 in [16] by Wang et al.
Lemma 2.4.
Let x(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
liminft→∞x(t)≤liminft→∞∫-∞0H(s)x(t+s)ds≤limsupt→∞∫-∞0H(s)x(t+s)ds≤limsupt→∞x(t).
Lemma 2.4 is given in [17] by Montes de Oca and Vivas.
3. Main ResultsTheorem 3.1.
Suppose that assumption (H1) holds and (H2) or (H3) holds. Then there exists a positive constant M>0 such that
limsupt→∞x(t)<M,limsupt→∞u(t)<M,
for any positive solution ((x(t),u(t)) of system (1.1).
Proof.
Let x(t) be a positive solution of the first equation system (1.1). Since
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
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)
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-∫tt-τ(t)a(u)du≥-τ0a0,∫tt+sa(u)du≥sa0.
We have
d(t)≥b(t)exp(-τ0a0)+c(t)∫-t0H(s)exp(a0s)ds.
Let
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
0<∫-∞0H(s)exp(a0s)ds<∫-∞0H(s)ds<∞,limt→∞(g(t)-f(t))=limt→∞c(t)∫-∞-tH(s)exp(a0s)ds=0.
Hence, assumption (H2) or (H3) implies
liminft→∞∫tt+σf(s)ds=liminft→∞[∫tt+σg(s)ds+∫tt+σ(f(s)-g(s))ds]=liminft→∞∫tt+σg(s)ds>0,
where the constant σ=β or γ. Since θ(t)≥f(t) for all t≥τ0, we finally obtain
liminft→∞∫tt+δθ(s)ds≥liminft→∞∫tt+δf(s)ds>0.
We consider the following auxiliary equation:
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
x(t)<y*(t),∀t>τ0.
Thus, we finally obtain that
limsupt→∞x(t)<M1.
Moreover, by Lemma 2.4, for any positive constant ε>0, there exists a constant T>0 such that
∫-∞0H(s)x2(t+s)ds<M12+ε,∀t>T.
Considering the second equation of the system (1.1), we have
du(t)dt≤-d(t)u(t)+e(t)(M12+ε).
We consider the following auxiliary equations:
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
u(t)≤v*(t),∀t>T′.
Thus, we finally obtain that
limsupt→∞v(t)<M2.
Choose the constant M=max{M1,M2}, then we finally obtain
limsupt→∞x(t)<M,limsupt→∞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
liminft→∞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
∫tt+δ(a(s)-b(s)ε-2c(s)ε-h(s)ε-r(s)ε)ds≥ε.
Consider the following equation:
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
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,
∫tt+δ(a(s)-b(s)α0-2c(s)α0-h(s)ε-r(s)ε)ds≥α0.
We first prove that
limsupt→∞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
∫-∞-τ1H(s)ds<α0M02,∫-∞-τ2H(s)ds<α0M0,
whereM0=sup{x(t+s):t∈R+,S∈R-}, then for t≥T1+τ1 we have
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).
Let v(t) be the solution of (3.22) with initial condition v(T1+τ1)=u(T1+τ1), then by the comparison theorem we have
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
v(t)=v(t,T1+τ1,v(T1+τ1))<ε∀t≥T1+τ1+T̃0.
Hence, we further have
u(t)<ε∀t≥T1+τ1+T̃0.
For any t≥T1+τ1+T̃0+τ2, from the first equation of (1.1) we have
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
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 Theorem 3.2 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
liminft→∞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
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
0<s1(m)<t1(m)<s2(m)<t2(m)<⋯<sq(m)<tq(m)<⋯,sq(m)⟶∞,tq(m)⟶∞asq⟶∞,x(sq(m),ϕ(m),ψ(m))=α0m+1,x(tq(m),ϕ(m),ψ(m))=α0(m+1)2,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
∫-∞-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
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
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
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,
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
∫-∞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
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
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
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
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
v(t)=v(t,sq(m)+L,v(sq(m)+L))<ε∀t∈[sq(m)+L+T̃0,tq(m)].
Hence, we further have
u(t,Z(m))<ε
for all t∈[sq(m)+L+T̃0,tq(m)],q≥K2(m) and m≥m̂1.
For any q≥K2(m),m≥m̂1, and t∈[sq(m)+L+T̃0,tq(m)], from (3.38)–(3.40), we have
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
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
liminft→∞x(t)≥ηx,
for any positive solutions (x(t),u(t)) of system (1.1). This completes the proof.
Applying Theorem 3.2 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 Theorem 3.2 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.
Acknowledgments
This work is supported by the National Sciences Foundation of China (11071283), the Sciences Foundation of Shanxi (2009011005-3), the Young Foundation of Shanxi Province (no. 2011021001-1), Research Project supported by Shanxi Scholarship Council of China (2011-093) and the Major Subject Foundation of Shanxi and Doctoral Scientific Research Fund of Xinjiang Medical University.
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