DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2018/3403127 3403127 Research Article Permanence and Existence of Periodic Solutions for Nicholson’s Blowflies Model with Feedback Control and Delay on Time Scales http://orcid.org/0000-0002-6610-4406 Wang Lin-Lin 1 http://orcid.org/0000-0002-9763-1575 Fan Yong-Hong 1 Pancioni Luca School of Mathematics and Statistics Science Ludong University Yantai Shandong 264025 China ytnc.edu.cn 2018 782018 2018 21 04 2018 26 07 2018 782018 2018 Copyright © 2018 Lin-Lin Wang and Yong-Hong Fan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Sufficient conditions are established for the permanence in a delayed Nicholson’s blowflies model with feedback control on time scales. Our investigation confirms that the bounded feedback terms do not have any influence on the permanence of this system.

National Natural Science Foundation of China 11201213 11371183 Natural Science Foundation of Shandong Province ZR2015AM026 Project of Shandong Provincial Higher Educational Science and Technology J15LI07
1. Introduction

In 2008, we considered the following discrete Nicholson’s blowflies model with feedback control (see ):(1)xn+1=xnexp-δn+pnexp-αnxn-cnUn,ΔUn=-anUn+bnxn-m,where δ, α, p, b, and c: Z (integer number set) R+ (nonnegative real number set) are all bounded sequences, a:Z(0,1) is also a bounded sequence and infa>0, m is a nonnegative integer, and Δ is the first-order former difference operator and obtained the following.

Theorem 1.

Assume that(2)infn0,+Zpn-δn>0hold true; then system (1) is permanent.

The continuous and discrete systems always appear separately, until in 1988, the theory of time scales, which has recently received much attention, was initiated by Hilger  in his Ph.D. thesis to unify both difference and differential calculus in a consistent way. Since then many authors have investigated the dynamic equations on time scales (see ). This theory provides a powerful tool for applications to economics, population models, quantum physics, among others. In fact, the progressive field of dynamic equations on time scales contains, links, and extends the classical theory of differential and difference equations.

For the origin of mathematical model for Nicholson’s blowflies, one can see . For further study on equations with feedback control, we refer to  and references therein. And for investigation on delay differential equations, we refer to [15, 16] and so on.

In this paper, we will discuss the permanence of the following system:(3)xΔt=-δt+ptexp-αtexpxt-τt-ctyt-θt,yΔt=-atyt+btexpxt-ηt,on time scales T, where Δ stands for the delta-derivative and(4)1-μtat>0,in which μt=σt-t and σ(t) is the forward jump operator on T. To make (3) meaningful, we suppose that, for any tT, (5)t-τtT,t-θtT,t-ηtT.

We assume that supT=, and, without loss of generality, suppose 0T. In view of the biological significance, we also assume that the coefficient functions δ, α, p, b, c, τ, θ, and η: TR+ are all bounded rd-continuous and inftTa>0 (the definition of rd-continuous function will be given in Section 2).

When T=R, let N(t)=exp{x(t)}, U(t)=y(t), and then (3) can be rewritten as(6)Nt=-δtNt+ptNtexp-αtNt-τt-ctUt-θt,Ut=-atUt+btNt-ηt.

When T=Z, if we let N(t)=exp{x(t)}, then (3) can be rewritten as(7)Nn+1=Nnexp-δn+pnexp-αnxn-τn-cnUn-θn,ΔUn=-anUn+bnxn-ηn.Obviously, (7) includes (1).

In what follows we shall use the notations (8)fu=suptTft,fl=inftTft,where f is a bounded rd-continuous function in T. Throughout this paper, we assume that(9)al>0,δl>0.In this case, for any tT, (10)μt1atuL.

2. Preliminary

Before giving our main result, first we list some basic properties about time scales which could be found in ([2, 17, 18]).

Definition 2.

A time scale is an arbitrary nonempty closed subset T of the real number R.

Definition 3.

For tT we define the forward jump operator σ:TT and the backward jump operator ρ:TT, by (11)σtinfsT:s>t,ρtsupsT:s<t,respectively.

Definition 4.

Define the interval a,b in T by (12)a,btT:atb.Other types of intervals are defined similarly.

Definition 5 (Definition 1.58 in [<xref ref-type="bibr" rid="B18">18</xref>], P22).

A function f:TR is called rd-continuous provided that it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T. The set of rd-continuous functions f:TR is denoted by (13)Crd=CrdT=CrdT,R.

Definition 6.

Assume f:TR and let tTk, where (14)Tk=TρsupT,supT,if  supT<,T,if  supT=.Then we define fΔ(t) to be the number (provided it exists) with property that given any ε>0, there is a neighborhood U of t such that (15)fσt-fs-fΔtσt-sεσt-s,for all sU. We call fΔ(t) the delta (or Hilger) derivative of f(t) and it turns out that fΔ is the usual derivative if T=R and is the usual forward difference operator if T=Z.

Lemma 7 (Theorem 1.76 in [<xref ref-type="bibr" rid="B18">18</xref>], P28).

If fΔ0, then f is increasing.

Definition 8 (Definition 2.25 in [<xref ref-type="bibr" rid="B18">18</xref>], P58).

We say that a function p:TR is regressive provided that(16)1+μtpt0for  all  tTkholds. The set of all regressive and rd-continuous functions f:TR will be denoted by (17)R=RT=RT,R.

Definition 9.

If pR, we define the exponential function by(18)ept,s=expstξμτpτΔτ,for  s,tT,where the cylinder transformation ξhz=1/hlog1+zh, for h>0.

For h=0, we define ξ0z=z. In this case, T=R, μt=0, and ept,s=expstpτΔτ.

When T=hZ=hk:kZ, we know that μhk=hk+1-hk=h>0, and then for s<t, (19)ept,s=expst1hloge1-haτΔτ=expk=s/ht/h-1loge1-hahk=k=s/ht/h-11-hahk.

Definition 10.

If pR, the function p is defined by (20)pt=-pt1+μtpt,for  tTk.

Lemma 11.

Suppose pR; then

p=p;

epσt,s=1+μtptept,s;

1/ept,s=ept,s;

1/ep·,sΔ=-p·/epσ·,s;

ep·,sΔ=p·ep·,s.

Lemma 12 (Theorem 2.44 in [<xref ref-type="bibr" rid="B18">18</xref>], P66).

Assume that pR and t0T. If 1+μ(t)p(t)>0 on Tk, then ep(t,t0)>0 for all tT.

Lemma 13 (Theorem 2.77 in [<xref ref-type="bibr" rid="B18">18</xref>], P77).

Suppose pR and f:TR is rd-continuous. Let t0T and y0R. The unique solution of the initial value problem (21)yΔ=pty+ft,yt0=y0is given by(22)yt=ept,t0y0+t0tept,σsfsΔs.

In order to give our main result, we also need to establish the following definitions and lemmas. The first definition is the generalized version of the semicycle in discrete situation .

Definition 14.

Let m be a constant and f:TR; a positive semicycle relative to  m of f is defined as follows, as it consists of a “string” of terms:(23)ft,ts,t,s,tT,all greater than or equal to m, and a negative semicycle relative to  m of f is defined as follows, as it is a “string” of terms:(24)ft,ts,t,s,tT,all less than or equal to m.

The following two lemmas could be found in .

Lemma 15.

Assume that A,B>0 and x10=x0>0; furthermore suppose that

(1)(25)x1ΔtB-Aexpx1t,for  t0,and then(26)limtsupx1tBL+lnBA,

(2)(27)x1ΔtB-Aexpx1t,for  t0,and there exists a constant M>0, such that limtsupx1t<M; then(28)limtinfx1tB-AexpML+lnBA.

Lemma 16.

Assume that Ct,Dt>0 are bounded rd-continuous functions and Cl>0; furthermore suppose that the Δ–differentiable function y1t satisfies

(1)(29)y1Δt+Cty1tDt,for  tT0,and Dt is Δ–differentiable; then there exists a constant T2>T0, such that for t>T2, we have(30)y1ty1T2e-Ct,T2+DtCl.Especially, if Dt is bounded above ultimately with respect to K1, then(31)limtsupy1tK1Cl.

(2)(32)y1Δt+Cty1tDt,for  tT0,and there exists a constant T3>T0, such that, for t>T3, we have(33)y1ty1T3-DT3Cue-Ct,T3+DT3Cu.Especially, if Dt is bounded below ultimately with respect to k1, then(34)limtinfy1tk1Cu.

Before giving our main result, we list the definition of uniform ultimate boundedness.

Definition 17.

Solutions for system (3) are said to be uniformly ultimate bounded if there exist two constants λ1 and λ2 such that, for any initial condition(35)ϕt,ψtTCrd-λ,0,R×R+,λ=maxτu,ηu,θu,we have(36)λ1limtinfxtlimtsupxtλ2,λ1limtinfytlimtsupytλ2,where λ1 and λ2 are independent of ϕ(t),ψtT.

3. Main Results

First, we give a lemma which will be useful for our further discussion.

Lemma 18.

Let x(t),y(t) be any solution of system (3) with initial condition (35); then (37)expxt>0,yt>0for  all  tT.

Proof.

The exponential form exp{x(t)} ensures that exp{x(t)}>0 for all tT. Now we consider the second equation of system (3); by Lemma 13 and (4), we have(38)yt=e-at,0y0+0te-at,σsbsexpxs-ηsΔs>0for  t0.The proof is complete.

In the sequel, we assume that(39)pn-δnl>0.

Theorem 19.

Assume that (39) holds true; then system (3) is uniformly ultimate bounded.

We now prove the following result before proving Theorem 19. In fact, the two theorems are equivalent to each other.

Theorem 20.

Assume that (39) holds true; then there exist two positive constants λ1 and λ2 such that (40)λ1limtinfexpxtlimtsupexpxtλ2,and(41)λ1limtinfytlimtsupytλ2,for any solutions (x(t),y(t)) of (3) with initial condition (35).

Proof.

We divided the proof into four claims.

Claim 1. There exists a positive constant K1 such that(42)limtsupxtK1,where K1 can be chosen as (43)K1=exppu-δlτuαllnpuδl+pu-δll.

Proof. From the first equation of system (3) and the positivity of y(t), we have(44)xΔt-δt+ptexp-αtexpxt-τt-δl+puexp-αlexpxt-τt.From the first equation of (3), we have(45)xΔtpu-δl,for  all  tT,thus(46)xt-xt-τtpu-δlτtpu-δlτu,which implies that(47)expxt-τtexp-pu-δlτuexpxt,and by (44), we have(48)xΔt-δl+puexp-αlexp-pu-δlτuexpxt,and for simplicity, we set(49)A1=αlexp-pu-δlτu,obviously, A1>0. In the following, we divided the proof into three cases:

(1) There exists some N1T such that x(t)1/A1lnpu/δl for all t>N1, and then we have(50)limtsupxt1A1lnpuδl<K1.

(2) There exists some N2T such that x(t)>1/A1lnpu/δl for all t>N2, and then from (48), we have (51)xΔt0for  t>N2+τu,which implies that limt+x(t) exists and if we denote limt+x(t)=λ, and then(52)λ1A1lnpuδl,and thus (48) shows that(53)-δl+puexp-A1λ0,that is,(54)λ1A1lnpuδl,while from (52) and (54), we get (55)λ=1A1lnpuδl;therefore(56)limtsupxt=1A1lnpuδl<K1.

(3) For simplicity, set β=1/A1lnpu/δl. Assume that x(t) oscillates about β. By (48), we know that u(t)β implies that xΔ(t)0. Thus, by Lemma 7, if we let x(tl) be the first element of a lth positive semicycle relative to β of x(t), then(57)limtsupxt=limtsupxtl.Now we divided the proof into two cases: tl is left-dense and tl is left-scattered. If the former holds, then(58)xtl=β.If the latter holds, then x(ρ(tl))β, by (45), we have(59)ρtltlxΔsΔsρtltlpu-δlΔs=pu-δltl-ρtlpu-δlL,and therefore(60)xtlxρtl+pu-δllβ+pu-δlL,while from (58) and (60), we have(61)limtsupxtlβ+pu-δlL=K1.By (50), (56), and (61), we complete the proof of Claim 1.

Claim 2. There exists a positive constant K2 such that(62)limtsupytK2.

Proof. From (42), we know that, for any arbitrary positive number ε, there exists an N3 such that x(t)K1+ε for all t>N3, and then from the second equation of system (3), we have (63)yΔt-atyt+buexpK1+ε,for  t>N3+ηu,by Lemma 16 (1),(64)limtsupytbuexpK1+εal.Let ε0, and then we can obtain(65)limtsupytbuexpK1alK2.The proof of Claim 2 is complete.

Claim 3. There exists a constant k1 such that(66)limtinfxtk1.

By Claims 1 and 2 and the first equation of system (3), we have(67)xΔt-δu+plexp-αuexpK1-cuK2-A2,for t sufficiently large, where A2>0 is a constant. Then (68)xt-xt-ηt-A2ηt-A2ηu,xt-xt-τt-A2τt-A2τu.From the second equation of system (3), we have (69)yΔt-atyt+buexpA2τuexpxt,and then Lemma 16 (1) implies that, for any t>T2,(70)ytyT2e-at,T2+buexpA2τuexpxtal,noting that(71)yT2e-at,T20,as  t+,and hence there exists a positive integer K>T2 such that, for any solution (x(t),y(t)) of system (3), yT2c(t)e-at,T2<γ/2, as t>K, where γ=inftT(p(t)-δ(t)). Fixing K, we get (72)ytyT2e-at,T2+buexpA2τuexpxtalfor  t>K.

Set(73)M1=buexpA2τual,and then(74)ytyT2e-at,T2+M1expxt.Notice that e-x1-x, for x>0, and then from the first equation of system (3), we have (75)xΔt-δt+pt1-αtexpxt-τt-ctyt-θtpt-δt-ptαtuexpA2τuexpxt-ctyT2e-at,T2-M1ctexpxt,for t>K. According to the choosing of K, we have(76)xΔtγ2-ptαtuexpA2τu+M1cuexpxtfor  t>K.By Lemma 15 (2), we have(77)limt+infxtB-AexpK1L+lnBA,where(78)B=ptαtuexpA2τu+M1cu,A=γ2,and choosing(79)k1=B-AexpK1L+lnBA,this shows that the conclusion holds true.

Claim 4. There exists a positive constant k2 such that(80)limt+infytk2.Notice that (66) implies that there exists an N4 such that x(t)k1/2 for all t>N4, and then from the second equation of system (3), we have (81)yΔt-atyt+blexpk12,for  t>N4+ηu,by Lemma 16 (2),(82)limt+infytblexpk1/2auk2.The proof of Claim 4 is complete.

Choose(83)λ1=minexpk1,k2,λ2=maxexpK1,K2,obviously, (84)λi>0,i=1,2.And(85)λ1limtinfexpxtlimtsupexpxtλ2,λ1limtinfyklimtsupytλ2.Thus we complete the proof of Theorem 20.

Theorem 21.

If the coefficient functions δ, α, p, b, c, τ, θ, and η:TR+ are all ω–periodic rd-continuous functions and also satisfy p¯>δ¯ and (9), then system (3) has at least an ω–periodic solution with strictly positive for its second component.

Corollary 22.

If all the conditions in Theorem 21 hold, then (6) or (7) has at least a positive ω–periodic solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by NSF of China (11201213, 11371183), NSF of Shandong Province (ZR2015AM026), and the Project of Shandong Provincial Higher Educational Science and Technology (J15LI07).

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