# Two Positive Periodic Solutions for a Neutral Delay Model of Single-Species Population Growth with Harvesting

Authors: Hui Fang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

By coincidence degree theory fork-set-contractive mapping, this paper establishes a new criterion for the existence of at least two positive periodic solutions for a neutral delay model of single-species population growth with harvesting. An example is given to illustrate the effectiveness of the result.


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## Body

## 1. Introduction

In 1993, Kuang [1] proposed the following open problem (Open Problem 9.2): obtain sufficient conditions for the existence of positive periodic solutions for
(1.1)N′(t)=N(t)[a(t)-β(t)N(t)-b(t)N(t-τ(t))-c(t)N′(t-τ(t))],
where all parameters are nonnegative continuous T-periodic functions. Fang and Li [2] gave an answer to the above open problem. In recent years, many papers have been published on the existence of multiple positive periodic solutions for some population systems with periodic harvesting terms by using Mawhin's coincidence degree theory (see, e.g., [3–7]). However, to our knowledge, few papers deal with the existence of multiple positive periodic solutions for neutral delay population models with harvesting. The main difficulty is that Mawhin's coincidence degree theory is generally not available to neutral delay models. Moreover, it is also hard to obtain a priori bounds on solutions for neutral delay models.In this paper, we consider the following neutral delay model of single-species population growth with harvesting(1.2)N′(t)=N(t)[a(t)-β(t)N(t)-b(t)N(t-τ(t))-c(t)N′(t-τ(t))]-h(t),
where a(t),β(t),b(t),τ(t),c(t),andh(t) are nonnegative continuous T-periodic functions, and  h(t) denotes the harvesting rate.The purpose of this paper is to establish the existence of at least two positive periodic solutions for neutral delay model (1.2). To show the existence of solutions to the considered problems, we will use the coincidence degree theory for k-set contractions [8–10] and a priori bounds on solutions.


## 2. Preliminaries

We now briefly state the part of coincidence degree theory fork-set-contractive mapping (see [8–10]).LetZ be a Banach space. For a bounded subset A⊂Z, let ΓZ(A) denote the (Kuratowski) measure of noncompactness defined by
(2.1)ΓZ(A)=inf⁡{δ>0:∃a finite number of subsetsAi⊂A,A=⋃i‍Ai,diam⁡(Ai)≤δ}.
Here, diam (Ai) denotes the maximum distance between the points in the set Ai.LetX and Z be Banach spaces with norms ∥·∥x and ∥·∥z respectively, and Ω a bounded open subset of X· A continuous and bounded mapping N:Ω-→Z is called k-set-contractive if, for any bounded A⊂Ω-, we have
(2.2)ΓZ(N(A))≤kΓX(A).Also, for a continuous and bounded mapT:X→Y, we define
(2.3)l(T)=sup⁡{r≥0:∀bounded subsetA⊂X,rΓX(A)≤ΓY(T(A))}.LetL:domL⊂X→Z be a linear mapping and N:X→Z a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim KerL=codimImL<+∞ and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors P:X→X and Q:Z→Z such that ImP=KerL,ImL=KerQ=Im(I-Q). If we define LP:domL∩KerP→ImL as the restriction L|domL∩KerP of L to domL∩KerP, then LP is invertible. We denote the inverse of that map by KP. If Ω is an open bounded subset of X, the mapping N will be called L-k-set-contractive on Ω- if QN(Ω-) is bounded and KP(I-Q)N:Ω-→X is k-set contractive. Since ImQ is isomorphic to KerL, there exists isomorphism J:ImQ→KerL.Lemma 2.1 (see[8], Proposition XI.2.).
LetL be a closed Fredholm mapping of index zero, and let N:Ω-→Z be k′-set contractive with
(2.4)0≤k′<l(L).
Then N:Ω-→Z is a L-k-set contraction with constant k=k′/l(L)<1.The following lemma (see [8], page 213) will play a key role in this paper.Lemma 2.2.
LetL be a Fredholm mapping of index zero, and let N:Ω-→Z be L-k-set contractive on Ω-,k<1. Suppose that(i)
Lx≠λNx for every x∈domL∩∂Ω and every λ∈(0,1);(ii)
QNx≠0 for every x∈∂Ω∩KerL;(iii)
Brouwer degreedegB(JQN,Ω∩KerL,0)≠0.Then Lx=Nx has at least one solution in domL∩Ω-.


## 3. Main Result

LetCT0 denote the linear space of real-valued continuous T-periodic functions on R. The linear space CT0 is a Banach space with the usual norm for x∈CT0 given by |x|0=maxt∈R|x(t)|. Let CT1 denote the linear space of T-periodic functions with the first-order continuous derivative. CT1 is a Banach space with norm |x|1=max{|x|0,|x′|0}.LetX=CT1 and Y=CT0, and let L:X→Y be given by Lx=dx/dt. Since |Lx|0=|x′|0≤|x|1, we see that L is a bounded (with bound =1) linear map.Since we are concerning with positive solutions of (1.2), we make the change of variables as follows:
(3.1)N(t)=ex(t).
Then (1.2) is rewritten as
(3.2)x′(t)=a(t)-β(t)ex(t)-b(t)ex(t-τ(t))-c(t)x′(t-τ(t))ex(t-τ(t))-h(t)ex(t).
Next define a nonlinear map N:X→Y by
(3.3)N(x)(t)=a(t)-β(t)ex(t)-b(t)ex(t-τ(t))-c(t)x′(t-τ(t))ex(t-τ(t))-h(t)ex(t).
Now, if Lx=Nx for some x∈X, then the problem (3.2) has a periodic solution x(t).In the following, we denote that(3.4)g-=1T∫0T‍g(t)dt,gl=min⁡t∈[0,T]g(t),gu=max⁡t∈[0,T]g(t),
where g(t) is a continuous nonnegative T-periodic solution.From now on, we always assume that(H1)
a(t)∈C(R,(0,+∞)),β(t),b(t)∈C(R,R+),c(t),τ(t)∈C1(R,R+),τ′<1;(H2)
c0′(t)<b(t),  where c0(t)=c(t)/(1-τ′(t));(H3)
al>cuM0+2[βu+bu]hu,cueR1<1, where
(3.5)M0=aueR1+(βu+bu)e2R1+hu1-cueR1,R1=ln⁡a-βl+c0ua-(b-c0′)l+2a-T.For further convenience, we introduce 6 positive numbers as below(3.6)l±=au+cuM0±[au+cuM0]2-4βlhl2βl,x±=a-±(a-)2-4[β-+b-]h-2[β-+b-],u±=al-cuM0±[al-cuM0]2-4[βu+bu]hu2[βu+bu].
Set the following:
(3.7)g-(x,y,z)=y-y2-4xz2x=2zy+y2-4xz(x>0,y>0,z>0),g+(x,y,z)=y+y2-4xz2x(x>0,y>0,z>0),
where y2>4xz.By the monotonicity of the functionsg-(x,y,z),g+(x,y,z) on x,y,z, it is not difficult to see that
(3.8)l-<x-<u-<u+<x+<l+.Theorem 3.1.
In addition to(H1), (H2),   (H3), assume further that the following condition holds:(H4)
k*=cumax{eR1,l+}<1.Then (1.2) has at least two positive T-periodic solutions.Before proving Theorem3.1, we need the following lemmas.Lemma 3.2 (see [11]).
L is a Fredholm map of index 0 and satisfies
(3.9)l(L)≥1.Lemma 3.3.
Under the assumptions of Theorem3.1, let
(3.10)Ω={x∈X|max⁡t∈[0,T]⁡x(t)∈(ln⁡(l--δ),ln⁡⁡(max⁡⁡{eR1,l+}+δ)),min⁡t∈[0,T]⁡x(t)∈(ln⁡(l--δ),ln⁡⁡(max⁡⁡{eR1,l+}+δ)),max⁡t∈[0,T]⁡|x′(t)|<M1.},
where
(3.11)M1=au+(βu+bu)eR1+(hu/l-)1-cueR1,
and 0<δ<l- such that
(3.12)k0=cu[max⁡⁡{eR1,l+}+δ]<1.
Then  N:Ω-→Y is a k0-set-contractive map.Proof.
The proof is similar to that of lemma  3.3 in [9], but for the sake of completeness we give the proof here. Let A⊂Ω- be a bounded subset and let η=ΓX(A). Then for any ε>0, there is a finite family of subsets {Ai} with A=⋃i‍Ai and diam1(Ai)≤η+ε.
Set the following:(3.13)g(t,x,x1,x2)=a(t)-β(t)ex-b(t)ex1-c(t)x2ex1-h(t)ex.
Now it follows from the fact that g(t,x,x1,x2) is uniformly continuous on any compact subset of R×R3, and from the fact A and Ai are precompact in CT0 with norm |·|0, that there is a finite family of subsets {Aij} of Ai such that Ai=⋃j‍Aij with
(3.14)|g(t,x(t),x(t-τ(t)),u′(t-τ(t)))-g(t,u(t),u(t-τ(t)),u′(t-τ(t)))|<ε
for any x,u∈Aij. Therefore, for x,u∈Aij we have
(3.15)|Nx-Nu|0=sup⁡0≤t≤T|g(t,x(t),x(t-τ(t)),x′(t-τ(t)))-g(t,u(t),u(t-τ(t)),u′(t-τ(t)))|≤sup⁡0≤t≤T|g(t,x(t),x(t-τ(t)),x′(t-τ(t)))-g(t,x(t),x(t-τ(t)),u′(t-τ(t)))|+sup⁡0≤t≤T|g(t,x(t),x(t-τ(t)),u′(t-τ(t)))-g(t,u(t),u(t-τ(t)),u′(t-τ(t)))|≤cu[max⁡⁡{eR1,l+}+δ]sup⁡0≤t≤T|x′(t-τ(t))-u′(t-τ(t))|+ε≤k0|x′-u′|0+ε≤k0η+(k0+1)ε.
That is ΓY(N(A))≤k0ΓX(A). The proof is complete.Lemma 3.4.
If the assumptions of Theorem3.1 hold, then every solution x∈X of the problem
(3.16)Lx=λNx,λ∈(0,1)
satisfies
(3.17)max⁡t∈[0,T]⁡x(t)∈[ln⁡⁡l-,ln⁡⁡u-]∪[ln⁡⁡u+,ln⁡⁡l+],min⁡t∈[0,T]⁡x(t)∈[ln⁡l-,ln⁡l+],max⁡t∈[0,T]⁡|x′(t)|<M1.Proof.
LetLx=λNx for x∈X, that is,
(3.18)x′(t)=λ[a(t)-β(t)ex(t)-b(t)ex(t-τ(t))-c(t)x′(t-τ(t))ex(t-τ(t))-h(t)ex(t)],λ∈(0,1).
Therefore, we have
(3.19)x′(t)=λ[a(t)-β(t)ex(t)-b(t)ex(t-τ(t))-c0(t)[ex(t-τ(t))]′-h(t)ex(t)],λ∈(0,1),
where c0(t)=c(t)/(1-τ′(t)).
By (3.19), we have
(3.20)[x(t)+λc0(t)ex(t-τ(t))]′=λ[a(t)-β(t)ex(t)-(b(t)-c0′(t))ex(t-τ(t))-h(t)ex(t)].
Integrating this identity leads to
(3.21)∫0T[β(t)ex(t)+(b(t)-c0′(t))ex(t-τ(t))+h(t)ex(t)]dt=∫0T‍a(t)dt.
From (3.20),(3.21), we have
(3.22)∫0T‍|[x(t)+λc0(t)ex(t-τ(t))]′|dt≤λ(∫0T‍a(t)dt+∫0T‍[β(t)ex(t)+(b(t)-c0′(t))ex(t-τ(t))+h(t)ex(t)]dt)<2∫0T‍a(t)dt=2Ta-.
By (3.21), we have
(3.23)∫0T‍a(t)dt≥∫0T‍[β(t)ex(t)+(b(t)-c0′(t))ex(t-τ(t))]dt.
It follows that
(3.24)∫0T‍a(t)dt≥Tβlex(ξ)+T(b-c0′)lex(ξ-τ(ξ)),
for some  ξ∈[0,T].
Therefore, we have(3.25)x(ξ)≤ln⁡a-βl,ex(ξ-τ(ξ))≤a-(b-c0′)l.
From (3.22) and (3.25), we see that
(3.26)x(t)+λc0(t)ex(t-τ(t))≤x(ξ)+λc0(ξ)ex(ξ-τ(ξ))+∫0T‍|[x(t)+λc0(t)ex(t-τ(t))]′|dt<ln⁡a-βl+c0ua-(b-c0′)l+2a-T=R1.
Hence, we have x(t)<R1.
Lets=t-τ(t). It follows from (3.19) that
(3.27)x′(t)=λ[a(t)-β(t)ex(t)-b(t)ex(t-τ(t))-c(t)dex(s)ds-h(t)ex(t)],λ∈(0,1).
Then from (3.27) and the inequality x(t)<R1, we obtain that
(3.28)|[ex(t)]′|≤λ[a(t)ex(t)+β(t)e2x(t)+b(t)ex(t)+x(t-τ(t))+c(t)ex(t)|dex(s)ds|+h(t)]<aueR1+(βu+bu)e2R1+cueR1|dex(s)ds|+hu,∀t∈R.
So that
(3.29)|[ex(t)]′|<aueR1+[βu+bu]e2R1+cueR1|[ex]′|0+hu,∀t∈R.
Since cueR1<1, we have
(3.30)|[ex]′|0<aueR1+(βu+bu)e2R1+hu1-cueR1=M0.
Choose tM,tm∈[0,T], such that
(3.31)x(tM)=max⁡t∈[0,T]x(t),x(tm)=min⁡t∈[0,T]x(t).
Then, it is clear that
(3.32)x′(tM)=0,x′(tm)=0.
From this and (3.27), we obtain that
(3.33)a(tM)ex(tM)=β(tM)e2x(tM)+b(tM)ex(tM)+x(tM-τ(tM))+c(tM)ex(tM)[dex(s)ds]s=tM-τ(tM)+h(tM),(3.34)a(tm)ex(tm)=β(tm)e2x(tm)+b(tm)ex(tm)+x(tm-τ(tm))+c(tm)ex(tm)[dex(s)ds]s=tm-τ(tm)+h(tm).
It follows from (3.33) that
(3.35)[βu+bu]e2x(tM)-[al-cuM0]ex(tM)+hu≥0.
By (H3), we have
(3.36)x(tM)≤ln⁡u-orx(tM)≥ln⁡u+.
It also follows from (3.33) that
(3.37)βle2x(tM)-[au+cuM0]ex(tM)+hl≤0.
By (H3), we have
(3.38)ln⁡l-≤x(tM)≤ln⁡l+.
Similarly, it follows from (3.34) that
(3.39)βle2x(tm)-[au+cuM0]ex(tm)+hl≤0.
By (H3), we have
(3.40)ln⁡⁡l-≤x(tm)≤ln⁡⁡l+.
Hence, it follows from (3.36), (3.38), and (3.40) that
(3.41)x(tM)∈[ln⁡l-,ln⁡u-]∪[ln⁡u+,ln⁡l+],x(tm)∈[ln⁡l-,ln⁡l+].
From the above inequality and (3.18), we obtain that
(3.42)|x′(t)|≤λ[a(t)+β(t)ex(t)+b(t)ex(t-τ(t))+c(t)|x′(t-τ(t))|ex(t-τ(t))+h(t)ex(t)]<au+(βu+bu)eR1+cueR1|x′|0+huex(tm).
So that
(3.43)|x′|0<au+[βu+bu]eR1+cueR1|x′|0+hul-.
Since cueR1<1, we have
(3.44)|x′|0<au+(βu+bu)eR1+hu/l-1-cueR1=M1.
The proof is complete.The Proof of Theorem3.1
Clearly,l±,u± are independent of λ. Now, let us consider QN(x) with x∈R. Note that
(3.45)QN(x)=a--[β-+b-]ex-h-ex.
It is easy to see that QN(x)=0 has two distinct solutions:
(3.46)u~1=ln⁡x-,u~2=ln⁡x+.
By (3.8), one can take v-,v+>0 such that
(3.47)u-<v-<v+<u+.
Let
(3.48)Ω1={x∈X|max⁡t∈[0,T]⁡x(t)∈(ln⁡(l--δ),ln⁡v-),min⁡t∈[0,T]⁡x(t)∈(ln⁡(l--δ),ln⁡(l++δ)),max⁡t∈[0,T]|x′(t)|<M1.},Ω2={x∈X|max⁡t∈[0,T]⁡x(t)∈(ln⁡v+,ln⁡(l++δ)),min⁡t∈[0,T]⁡x(t)∈(ln⁡⁡(l--δ),ln⁡⁡(l++δ)),max⁡t∈[0,T]⁡|x′(t)|<M1.}.
Then Ω1andΩ2 are bounded open subsets of X. Clearly, Ωi⊂Ω(i=1,2). It follows from Lemma 3.3 that N:Ω-i→Y is a k0-set-contractive map (i=1,2). Therefore, it follows from Lemmas 2.1 and 3.2 that N:Ω-i→Z is L-k-set contractive on Ω-i(i=1,2) with k=k0/l(L)≤k0<1.
It follows from (3.8) and (3.46) that u~i∈Ωi(i=1,2). From (3.8), (3.47) and Lemma 3.4, it is easy to see that Ω-1∩Ω-2=∅ and Ωi satisfies (i) in Lemma 2.2 for i=1,2. Moreover, QN(x)≠0 for x∈∂Ωi∩KerL(i=1,2).
A direct computation gives the following:(3.49)deg{JQN,Ω1∩Ker⁡L,0}=1,deg{JQN,Ω2∩Ker⁡L,0}=-1.
Here, J is taken as the identity mapping since ImQ=KerL. So far we have proved that Ωi satisfies all the assumptions in Lemma 2.2(i=1,2). Hence, (3.2) has at least two T-periodic solutions: xi*(t) and xi*∈domL∩Ω-i(i=1,2). Obviously, xi*(i=1,2) are different. Let Ni*(t)=exi*(t)(i=1,2). Then Ni*(t)(i=1,2) are two different positive T-periodic solutions of (1.2). The proof is complete.Example 3.5.
Take the following:(3.50)τ(t)=1+0.5sin⁡⁡t,a(t)=2-sin⁡⁡t,β(t)=b(t)=1+0.5sin⁡⁡t,c(t)=ϵ(1-0.5cos⁡⁡t),h(t)=ϵ(2+sin⁡⁡t),
where the constant ϵ>0.
Clearly, we have(3.51)al=1,βl=0.5,βu=bu=1.5,cu=1.5ϵ,hu=3ϵ,c0(t)=ϵ.
Therefore, we have c0′(t)<b(t). Moreover, it is easy to see that R1,M0,andl+ are bounded with respect to ϵ. Hence, for some sufficiently small ϵ>0, we have
(3.52)al>cuM0+2[βu+bu]hu,k*=cumax⁡⁡{eR1,l+}=1.5ϵmax⁡⁡{eR1,l+}<1.
In this case, all necessary conditions of Theorem 3.1 hold. By Theorem 3.1, (1.2) has at least two positive 2π-periodic solutions.



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