# Permanence and Existence of Periodic Solutions for Nicholson’s Blowflies Model with Feedback Control and Delay on Time Scales

Authors: Lin-Lin Wang; Yong-Hong Fan
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

In 2008, we considered the following discrete Nicholson’s blowflies model with feedback control (see [1]):(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 inf⁡a>0, m is a nonnegative integer, and Δ is the first-order former difference operator and obtained the following.Theorem 1.
Assume that(2)infn∈0,+∞∩Z⁡pn-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 [2] 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 [3–6]). 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 [7]. For further study on equations with feedback control, we refer to [8–14] 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⁡-αtexp⁡xt-τt-ctyt-θt,yΔt=-atyt+btexp⁡xt-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 t∈T, (5)t-τt∈T,t-θt∈T,t-ηt∈T.We assume thatsup⁡T=∞, and, without loss of generality, suppose 0∈T. In view of the biological significance, we also assume that the coefficient functions δ, α, p, b, c, τ, θ, and η: T→R+ are all bounded rd-continuous and inft∈T⁡a>0 (the definition of rd-continuous function will be given in Section 2).WhenT=R, let N(t)=exp⁡{x(t)}, U(t)=y(t), and then (3) can be rewritten as(6)N′t=-δtNt+ptNtexp⁡-αtNt-τt-ctUt-θt,U′t=-atUt+btNt-ηt.WhenT=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=supt∈T⁡ft,fl=inft∈T⁡ft,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 t∈T, (10)μt≤1atu≔L.


## 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 subsetT of the real number R.Definition 3.
Fort∈T we define the forward jump operator σ:T→T and the backward jump operator ρ:T→T, by (11)σt≔inf⁡s∈T:s>t,ρt≔sup⁡s∈T:s<t,respectively.Definition 4.
Define the intervala,b in T by (12)a,b≔t∈T:a≤t≤b.Other types of intervals are defined similarly.Definition 5 (Definition 1.58 in [18], P22).
A functionf:T→R 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:T→R is denoted by (13)Crd=CrdT=CrdT,R.Definition 6.
Assumef:T→R and let t∈Tk, where (14)Tk=T∖ρsup⁡T,sup⁡T,ifsup⁡T<∞,T,ifsup⁡T=∞.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-s≤εσt-s,for all s∈U. 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 [18], P28).
IffΔ≥0, then f is increasing.Definition 8 (Definition 2.25 in [18], P58).
We say that a functionp:T→R is regressive provided that(16)1+μtpt≠0for  allt∈Tkholds. The set of all regressive and rd-continuous functions f:T→R will be denoted by (17)R=RT=RT,R.Definition 9.
Ifp∈R, we define the exponential function by(18)ept,s=exp⁡∫stξμτpτΔτ,fors,t∈T,where the cylinder transformation ξhz=1/hlog1+zh, for h>0.
Forh=0, we define ξ0z=z. In this case, T=R, μt=0, and ept,s=exp⁡∫stpτΔτ.
WhenT=hZ=hk:k∈Z, we know that μhk=hk+1-hk=h>0, and then for s<t, (19)ept,s=exp⁡∫st1hloge⁡1-haτΔτ=exp⁡∑k=s/ht/h-1loge⁡1-hahk=∏k=s/ht/h-11-hahk.Definition 10.
Ifp∈R, the function ⊖p is defined by (20)⊖pt=-pt1+μtpt,fort∈Tk.Lemma 11.
Supposep∈R; then(1)
⊖⊖p=p;(2)
epσt,s=1+μtptept,s;(3)
1/ept,s=e⊖pt,s;(4)
1/ep·,sΔ=-p·/epσ·,s;(5)
ep·,sΔ=p·ep·,s.Lemma 12 (Theorem 2.44 in [18], P66).
Assume thatp∈R and t0∈T. If 1+μ(t)p(t)>0 on Tk, then ep(t,t0)>0 for all t∈T.Lemma 13 (Theorem 2.77 in [18], P77).
Supposep∈R and f:T→R is rd-continuous. Let t0∈T and y0∈R. 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 [19].Definition 14.
Letm be a constant and f:T→R;a positive semicycle relative to  m of f is defined as follows, as it consists of a “string” of terms:(23)ft,t∈s,t,s,t∈T,all greater than or equal to m, anda negative semicycle relative to  m of f is defined as follows, as it is a “string” of terms:(24)ft,t∈s,t,s,t∈T,all less than or equal to m.
The following two lemmas could be found in [20].Lemma 15.
Assume thatA,B>0 and x10=x0>0; furthermore suppose that
(1)(25)x1Δt≤B-Aexp⁡x1t,fort≥0,and then(26)limt→∞⁡sup⁡x1t≤BL+ln⁡BA,
(2)(27)x1Δt≥B-Aexp⁡x1t,fort≥0,and there exists a constant M>0, such that limt→∞⁡sup⁡x1t<M; then(28)limt→∞⁡inf⁡x1t≥B-Aexp⁡ML+ln⁡BA.Lemma 16.
Assume thatCt,Dt>0 are bounded rd-continuous functions and Cl>0; furthermore suppose that the Δ–differentiable function y1t satisfies
(1)(29)y1Δt+Cty1t≤Dt,fort≥T0,and Dt is Δ–differentiable; then there exists a constant T2>T0, such that for t>T2, we have(30)y1t≤y1T2e-Ct,T2+DtCl.Especially, if Dt is bounded above ultimately with respect to K1, then(31)limt→∞⁡sup⁡y1t≤K1Cl.
(2)(32)y1Δt+Cty1t≥Dt,fort≥T0,and there exists a constant T3>T0, such that, for t>T3, we have(33)y1t≥y1T3-DT3Cue-Ct,T3+DT3Cu.Especially, if Dt is bounded below ultimately with respect to k1, then(34)limt→∞⁡inf⁡y1t≥k1Cu.
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,ψtT∈Crd-λ,0,R×R+,λ=max⁡τu,ηu,θu,we have(36)λ1≤limt→∞⁡inf⁡xt≤limt→∞⁡sup⁡xt≤λ2,λ1≤limt→∞⁡inf⁡yt≤limt→∞⁡sup⁡yt≤λ2,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.
Letx(t),y(t) be any solution of system (3) with initial condition (35); then (37)exp⁡xt>0,yt>0for  allt∈T.Proof.
The exponential formexp⁡{x(t)} ensures that exp⁡{x(t)}>0 for all t∈T. Now we consider the second equation of system (3); by Lemma 13 and (4), we have(38)yt=e-at,0y0+∫0te-at,σsbsexp⁡xs-ηsΔs>0fort≥0.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 Theorem19. 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)λ1≤limt→∞⁡inf⁡exp⁡xt≤limt→∞⁡sup⁡exp⁡xt≤λ2,and(41)λ1≤limt→∞⁡inf⁡yt≤limt→∞⁡sup⁡yt≤λ2,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)limt→∞⁡sup⁡xt≤K1,where K1 can be chosen as (43)K1=exp⁡pu-δlτuαlln⁡puδl+pu-δll.
Proof. From the first equation of system (3) and the positivity of y(t), we have(44)xΔt≤-δt+ptexp⁡-αtexp⁡xt-τt≤-δl+puexp⁡-αlexp⁡xt-τt.From the first equation of (3), we have(45)xΔt≤pu-δl,for  allt∈T,thus(46)xt-xt-τt≤pu-δlτt≤pu-δlτu,which implies that(47)exp⁡xt-τt≥exp⁡-pu-δlτuexp⁡xt,and by (44), we have(48)xΔt≤-δl+puexp⁡-αlexp⁡-pu-δlτuexp⁡xt,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 someN1∈T such that x(t)≤1/A1ln⁡pu/δl for all t>N1, and then we have(50)limt→∞⁡sup⁡xt≤1A1ln⁡puδl<K1.
(2) There exists someN2∈T such that x(t)>1/A1ln⁡pu/δl for all t>N2, and then from (48), we have (51)xΔt≤0fort>N2+τu,which implies that limt→+∞⁡x(t) exists and if we denote limt→+∞⁡x(t)=λ, and then(52)λ≥1A1ln⁡puδl,and thus (48) shows that(53)-δl+puexp⁡-A1λ≥0,that is,(54)λ≤1A1ln⁡puδl,while from (52) and (54), we get (55)λ=1A1ln⁡puδl;therefore(56)limt→∞⁡sup⁡xt=1A1ln⁡puδl<K1.
(3) For simplicity, setβ=1/A1ln⁡pu/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)limt→∞⁡sup⁡xt=limt→∞⁡sup⁡xtl.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)∫ρtltlxΔsΔs≤∫ρtltlpu-δlΔs=pu-δltl-ρtl≤pu-δlL,and therefore(60)xtl≤xρtl+pu-δll≤β+pu-δlL,while from (58) and (60), we have(61)limt→∞⁡sup⁡xtl≤β+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)limt→∞⁡sup⁡yt≤K2.
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+buexp⁡K1+ε,fort>N3+ηu,by Lemma 16 (1),(64)limt→∞⁡sup⁡yt≤buexp⁡K1+εal.Let ε→0, and then we can obtain(65)limt→∞⁡sup⁡yt≤buexp⁡K1al≔K2.The proof of Claim 2 is complete.
Claim 3. There exists a constant k1 such that(66)limt→∞⁡inf⁡xt≥k1.
By Claims 1 and 2 and the first equation of system (3), we have(67)xΔt≥-δu+plexp⁡-αuexp⁡K1-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+buexp⁡A2τuexp⁡xt,and then Lemma 16 (1) implies that, for any t>T2,(70)yt≤yT2e-at,T2+buexp⁡A2τuexp⁡xtal,noting that(71)yT2e-at,T2→0,ast→+∞,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=inft∈T⁡(p(t)-δ(t)). Fixing K, we get (72)yt≤yT2e-at,T2+buexp⁡A2τuexp⁡xtalfort>K.
Set(73)M1=buexp⁡A2τual,and then(74)yt≤yT2e-at,T2+M1exp⁡xt.Notice that e-x≥1-x, for x>0, and then from the first equation of system (3), we have (75)xΔt≥-δt+pt1-αtexp⁡xt-τt-ctyt-θt≥pt-δt-ptαtuexp⁡A2τuexp⁡xt-ctyT2e-at,T2-M1ctexp⁡xt,for t>K. According to the choosing of K, we have(76)xΔt≥γ2-ptαtuexp⁡A2τu+M1cuexp⁡xtfort>K.By Lemma 15 (2), we have(77)limt→+∞⁡inf⁡xt≥B-Aexp⁡K1L+ln⁡BA,where(78)B=ptαtuexp⁡A2τu+M1cu,A=γ2,and choosing(79)k1=B-Aexp⁡K1L+ln⁡BA,this shows that the conclusion holds true.
Claim 4. There exists a positive constant k2 such that(80)limt→+∞⁡inf⁡yt≥k2.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+blexp⁡k12,fort>N4+ηu,by Lemma 16 (2),(82)limt→+∞⁡inf⁡yt≥blexp⁡k1/2au≔k2.The proof of Claim 4 is complete.
Choose(83)λ1=min⁡exp⁡k1,k2,λ2=max⁡expK1,K2,obviously, (84)λi>0,i=1,2.And(85)λ1≤limt→∞⁡inf⁡exp⁡xt≤limt→∞⁡sup⁡exp⁡xt≤λ2,λ1≤limt→∞⁡inf⁡yk≤limt→∞⁡sup⁡yt≤λ2.Thus we complete the proof of Theorem 20.Theorem 21.
If the coefficient functionsδ, α, p, b, c, τ, θ, and η:T→R+ 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 Theorem21 hold, then (6) or (7) has at least a positive ω–periodic solution.



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