# Double Periodic Solutions for a Ratio-Dependent Predator-Prey System with Harvesting Terms on Time Scales

Authors: Zhenjie Liu
Journal: Discrete Dynamics in Nature and Society
 (2009)

---

## Abstract

We investigate a nonautonomous ratio-dependent predator-prey model with Beddington-DeAngelis functional response and multiple harvesting (or exploited) terms on time scales. By means of using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of at least two periodic solutions for the system. Moreover, when the time scale𝕋 is chosen as ℝ or ℤ, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.


---

## Body

## 1. Introduction

In recent years, the existence of periodic solutions for the predator-prey model has been widely studied. Models with harvesting (or exploited) terms are often considered (see, e.g., [1–4]). Generally, the model with harvesting (or exploited) terms is described as follows:
(1.1)ẋ=xf(x,y)-h,ẏ=yg(x,y)-k,
where x and y are functions of time representing densities of prey and predator, respectively; h and k are harvesting (or exploited) terms standing for the harvests (see [5]). Considering the inclusion of the effect of changing environment, Zhang and Hou [6] considered the following model of ordinary differential equations with Holling-type II functional response and harvesting (or exploited) terms:
(1.2)ẋ(t)=x(t)(a(t)-b(t)x(t)-c(t)y(t)m(t)y(t)+x(t))-h1(t),ẏ(t)=y(t)(-d(t)+f(t)x(t)m(t)y(t)+x(t))-h2(t),
where the parameters in system (1.2) are continuous positive ω-periodic functions. Authors discussed the existence of positive periodic solutions of system (1.2) in the region D={(x,y)∣x>0,y>0}.On the other hand, the theory of calculus on time scales unifies continuous and discrete analysis, many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on general time scales can reveal such discrepancies and help avoid proving results twice-once for differential equations and once again for difference equations. The two main features of the calculus on time scales are unification and extension. To prove a result for a dynamic equation on a time scale is not only related to the set of real numbers or set of integers but those pertaining to more general time scales.The principle aim of this paper is to systematically unify the existence of multiple periodic solutions of population models modelled by ordinary differential equations and their discrete analogues in form of difference equations and to extend these results to more general time scales. The approach is based on a continuation theorem in coincidence degree, which has been widely applied to deal with the existence of periodic solutions of differential equations and difference equations. Therefore, we consider the following ratio-dependent predator-prey system with Beddington-DeAngelis functional response and harvesting terms on time scales𝕋:
(1.3)z1Δ(t)=a(t)-b(t)ez1(t)-c1(t)ez2(t)ez2(t)+α(t)ez1(t)+β(t)e2z2(t)-h1(t)e-z1(t),z2Δ(t)=-d(t)+c2(t)ez1(t)ez2(t)+α(t)ez1(t)+β(t)e2z2(t)-h2(t)e-z2(t),
where z1(t) and z2(t) represent the prey and the predator population, respectively; a(t),b(t),ci(t)(i=1,2),d(t),hi(t)(i=1,2), and α(t),β(t) are all rd-continuous positive ω-periodic functions denoting the prey intrinsic growth rate, death rate, capture rate, conversion rate of predator, death rate of predator, harvesting rate, and Beddington-DeAngelis functional response parameters, respectively.Remark 1.1.
In (1.3), set xi(t)=exp{zi(t)},i=1,2. If 𝕋=ℝ, then (1.3) reduces to the ratio-dependent predator-prey system with Beddington-DeAngelis functional response and harvesting terms governed by the ordinary differential equations
(1.4)x1′(t)=x1(t)(a(t)-b(t)x1(t)-c1(t)x2(t)x2(t)+α(t)x1(t)+β(t)x22(t))-h1(t),x2′(t)=x2(t)(-d(t)+c2(t)x1(t)x2(t)+α(t)x1(t)+β(t)x22(t))-h2(t).
If 𝕋=ℤ, then (1.3) is reformulated as
(1.5)x1(k+1)=x1(k)(a(k)-b(k)x1(k)-c1(k)x2(k)x2(k)+α(k)x1(k)+β(k)x22(k))-h1(k),x2(k+1)=x2(k)(-d(k)+c2(k)x1(k)x2(k)+α(k)x1(k)+β(k)x22(k))-h2(k),
which is the discrete time ratio-dependent predator-prey system with Beddington-DeAngelis functional response and harvesting terms and is also a discrete analogue of (1.3).


## 2. Preliminaries

A time scale𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. Throughout this paper, we assume that the time scale 𝕋 is unbounded above and below, such as ℝ, ℤ, and ⋃k∈ℤ[2k,2k+1]. The following definitions and lemmas can be found in [7].Definition 2.1 (see [8]).
One says that a time scale𝕋 is periodic if there exists p>0 such that if t∈𝕋, then t±p∈𝕋. For 𝕋≠ℝ, the smallest positive p is called the period of the time scale.Definition 2.2 (see [8]).
Let𝕋≠ℝ be a periodic time scale with period p. One says that the function f:𝕋→ℝ is periodic with period ω if there exists a natural number n such that ω=np, f(t+ω)=f(t) for all t∈𝕋, and ω is the smallest number such that f(t+ω)=f(t). If 𝕋=ℝ, one says that f is periodic with period ω>0 if ω is the smallest positive number such that f(t+ω)=f(t) for all t∈𝕋.Notation 2.3.
To facilitate the discussion below, we now introduce some notation to be used throughout this paper. Let𝕋 be ω-periodic; that is, t∈𝕋 implies t+ω∈𝕋,
(2.1)κ=min{[0,+∞)∩𝕋},Iω=[κ,κ+ω]∩𝕋,fM=supt∈𝕋f(t),fL=inft∈𝕋f(t),
where f∈Crd(𝕋,ℝ) is an ω-periodic function.Notation 2.4.
LetX, Z be two Banach spaces, let L:DomL⊂X→Z be a linear mapping, and let N:X→Z be a continuous mapping. If L is a Fredholm mapping of index zero and there exist continuous projectors P:X→X and Q:Z→Z such that ImP=KerL, KerQ=ImL=Im(I-Q), then the restriction L|DomL∩KerP:(I-P)X→ImL is invertible. Denote the inverse of that map by KP. If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω¯ if QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J:ImQ→KerL.Lemma 2.5 (Continuation theorem [9]).
LetX, Z be two Banach spaces, and let L be a Fredholm mapping of index zero. Assume that N:Ω¯→Z is L-compact on Ω¯ with Ω being open bounded in X. Furthermore assume the following:(a)
for eachλ∈(0,1), x∈∂Ω∩DomL, Lx≠λNx;(b)
for eachx∈∂Ω∩KerL, QNx≠0;(c)
deg{JQN,Ω∩KerL,0}≠0.Then the operator equation Lx=Nx has at least one solution in DomL∩Ω¯.


## 3. Existence of Periodic Solutions

Our main result on the existence of two positive periodic solutions for system (1.3) is stated in the following theorem.Theorem 3.1.
Assume that the following holds:(i)
aL-c1M>2bMh1M,(ii)
c2M>dLαL,(iii)
(dLαL-c2M)h1L+h2LaM>2aMdLh1LαLh2L,then system (1.3) has at least two positive ω-periodic solutions.Proof.
Let(3.1)X=Z={z=(z1,z2)T∈Crd(𝕋,ℝ2)∣zi(t+ω)=zi(t),i=1,2,∀t∈𝕋},
and define
(3.2)∥z∥=∥(z1,z2)T∥=∑i=12maxt∈Iω|zi(t)|,z∈X(orZ),
where |·| is the Euclidean norm. Then X and Z are both Banach spaces with the above norm ∥·∥. Let
(3.3)Nz(t)=(f1(t)f2(t))=(a(t)-b(t)ez1(t)-c1(t)ez2(t)ez2(t)+α(t)ez1(t)+β(t)e2z2(t)-h1(t)e-z1(t)-d(t)+c2(t)ez1(t)ez2(t)+α(t)ez1(t)+β(t)e2z2(t)-h2(t)e-z2(t)),Lz(t)=zΔ(t),Pz(t)=Qz(t)=1ω∫Iωz(t)Δt,z∈X.
Then
(3.4)KerL={z∈X∣z=(z1(t),z2(t))T∈ℝ2fort∈𝕋},ImL={(z1,z2)T∈Z∣∫Iωzi(t)Δt=0,i=1,2,fort∈𝕋},
and dimKerL=codimImL=2. Since ImL is closed in Z, then L is a Fredholm mapping of index zero. It is easy to show that P, Q are continuous projectors such that ImP=KerL and KerQ=ImL=Im(I-Q). Furthermore, the generalized inverse (to L) KP:ImL→KerP∩DomL exists and is given by
(3.5)KPz=KP(z1z2)=(∫κtz1(s)Δs-1ω∫κκ+ω∫κtz1(s)ΔsΔt∫κtz2(s)Δs-1ω∫κκ+ω∫κtz2(s)ΔsΔt).
Thus
(3.6)QNz=QN(z1z2)=(1ω∫κκ+ωf1(t)Δt1ω∫κκ+ωf2(t)Δt),KP(I-Q)Nz=(∫κtf1(s)Δs-1ω∫κκ+ω∫κtf1(s)ΔsΔt+(12-tω)∫κκ+ω(t-κ)f1(s)Δs∫κtf2(s)Δs-1ω∫κκ+ω∫κtf2(s)ΔsΔt+(12-tω)∫κκ+ω(t-κ)f2(s)Δt).
Obviously,QN:X→Z, KP(I-Q)N:X→X are continuous. Since X is a Banach space, using the Arzela-Ascoli theorem, it is easy to show that KP(I-Q)N(Ω)¯ is compact for any open bounded set Ω⊂X. Moreover, QN(Ω¯) is bounded, thus, N is L-compact on Ω¯ for any open bounded set Ω⊂X. Corresponding to the operator equation Lz=λNz, λ∈(0,1), we have
(3.7)ziΔ(t)=λfi(t),i=1,2.
Suppose that(z1,z2)T∈X is a solution of (3.7) for certain λ∈(0,1). Then there exist ξi,ηi∈Iω,i=1,2, such that
(3.8)zi(ξi)=mint∈Iω{zi(t)},zi(ηi)=maxt∈Iω{zi(t)},i=1,2.
It is clear that ziΔ(ξi)=0,ziΔ(ηi)=0,i=1,2. From this and system (3.7),  we have
(3.9a)f1(ξ1)=0,(3.9b)f2(ξ2)=0,(3.10a)f1(η1)=0,(3.10b)f2(η2)=0.
It follows from (3.9a) that
(3.11)z1(ξ1)<lnaMbL.
From (3.9b), we obtain
(3.12)dL≤d(ξ2)<c2(ξ2)ez1(ξ2)ez2(ξ2)+α(ξ2)ez1(ξ2)+β(ξ2)e2z2(ξ2)<c2MaMbL(ez2(ξ2)+βLe2z2(ξ2)),
which leads to
(3.13)z2(ξ2)<ln-bLdL+(bLdL)2+4bLdLβLc2MaM2bLdLβL=:lnρ.
Equation (3.10a) yields
(3.14)z1(η1)>lnh1LaM.
Equation (3.10b) deduces
(3.15)c2MαL>c2(η2)ez1(η2)ez2(η2)+α(η2)ez1(η2)+β(η2)e2z2(η2)>h2Lez1(η2),
which implies
(3.16)z2(η2)>lnαLh2Lc2M.
From (3.9a), we also have
(3.17)b(ξ1)e2z1(ξ1)+c1(ξ1)ez1(ξ1)+z2(ξ1)ez2(ξ1)+α(ξ1)ez1(ξ1)+β(ξ1)e2z2(ξ1)+h1(ξ1)-a(ξ1)ez1(ξ1)=0,
which implies that
(3.18)bMe2z1(ξ1)+(c1M-aL)ez1(ξ1)+h1M>0.
Set
(3.19)u±=(aL-c1M)±(c1M-aL)2-4bMh1M2bM,
then
(3.20)z1(ξ1)>lnu+orz1(ξ1)<lnu-.
From (3.10a), a parallel argument to (3.20) gives
(3.21)z1(η1)>lnu+orz1(η1)<lnu-.
From (3.9b), we also obtain
(3.22)c2(ξ2)ez1(ξ2)ez2(ξ2)+α(ξ2)ez1(ξ2)-d(ξ2)-h2(ξ2)e-z2(ξ2)≥0,
which implies
(3.23)aMdLe2z2(ξ2)+[(dLαL-c2M)h1L+h2LaM]ez2(ξ2)+h1Lh2LαL≤0.
Set
(3.24)v±=(c2M-dLαL)h1L-h2LaM±((c2M-dLαL)h1L-h2LaM)2-4aMdLh1Lh2LαL2aMdL,
then
(3.25)lnv-<z2(ξ2)<lnv+orlnv+<z2(η2)<lnv-.
From (3.11), (3.14), (3.20), and (3.21), we obtain for all t∈Iω,
(3.26)lnh1LaM<z1(t)<lnu-
or
(3.27)lnu+<z1(t)<lnaMbL.
From (3.13), (3.16), and (3.25), we obtain that for all t∈Iω,
(3.28)A:=max{lnv-,lnαLh2Lc2M}<z2(t)<min{lnv+,lnρ}=:B.
Obviously, ln(h1L/aM), ln(aM/bL), lnu±, A, B are independent of λ. Now take
(3.29)Ω1={z=(z1,z2)T∈X∣z1(t)∈(lnh1LaM,lnu-),z2(t)∈(A,B)},Ω2={z=(z1,z2)T∈X∣z1(t)∈(lnu+,lnaMbL),z2(t)∈(A,B)},
then Ω1 and Ω2 are bounded open subsets of X, and Ω1∩Ω2=Ø. Thus Ω1 and Ω2 satisfy the requirement (a) in Lemma 2.5.
Now we show that (b) of Lemma2.5 holds; that is, we need to prove when z=(z1,z2)T∈∂Ωi∩KerL=∂Ωi∩ℝ2, QNz≠(0,0)T, i=1,2. If it is not true, then when z∈∂Ωi∩KerL=∂Ωi∩ℝ2, i=1,2, constant vector z with z∈∂Ωi, i=1,2, satisfies QNz=(0,0)T. Thus there exist two points ti∈Iω(i=1,2) such that
(3.30)a(t1)-b(t1)ez1-c1(t1)ez2ez2+α(t1)ez1+β(t1)e2z2-h1(t1)e-z1=0,-d(t2)+c2(t2)ez1ez2+α(t2)ez1+β(t2)e2z2-h2(t2)e-z2=0.
From this and following the arguments of (3.26)–(3.28), we have
(3.31)lnh1LaM<z1<lnu-,A<z2<Borlnu+<z1<lnaMbL,A<z2<B.
Thus z∈Ω1∩ℝ2, or z∈Ω2∩ℝ2. This contradicts the fact that z∈∂Ωi∩ℝ2, i=1,2. Hence condition (b) in Lemma 2.5 holds.
Finally, we show that (c) in Lemma2.5 is valid. Noticing that the system of algebraic equations
(3.32)a(t1)-b(t1)ex-h1(t1)ex=0,-d(t2)+c2(t2)exey+α(t2)ex+β(t2)e2y-h2(t2)ey=0
has two distinct solutions (x1*,y1*) and (x2*,y2*) since the conditions hold in Theorem 3.1, and we have
(3.33)(x1*,y1*)=(lnx-,lny-),(x2*,y2*)=(lnx+,lny+),
where
(3.34)x±=a(t1)±(a(t1))2-4b(t1)h1(t1)2b(t1),y±=12a(t2)d(t2){((c2(t2)-d(t2)α(t2))h1(t2)-h2(t2)a(t2))2-4a(t2)d(t2)h1(t2)h2(t2)α(t2)(c2(t2)-d(t2)α(t2))h1(t2)-h2(t2)a(t2)±((c2(t2)-d(t2)α(t2))h1(t2)-h2(t2)a(t2))2-4a(t2)d(t2)h1(t2)h2(t2)α(t2)}.
It is easy to verify that
(3.35)lnh1LaM<lnx-<lnu-<lnu+<lnx+<lnaMbL,lnαLh1Lc2M<lny-<lnv-<lnv+<lny+<lnρ.
Therefore, (x1*,y1*)∈Ω1, (x2*,y2*)∈Ω2.
Define the homotopyHμ(z1,z2):DomL×[0,1]→ℝ2 by
(3.36)Hμ(z1,z2)=μQN(z1,z2)+(1-μ)G(z1,z2),forμ∈[0,1],
where
(3.37)G(z1z2)=(a(t1)-b(t1)u-h1(t1)u-d(t2)+c2(t2)uv+α(t2)u+β(t2)v2-h2(t2)v),
and μ∈[0,1] is a parameter, u=ez1, v=ez2, and (u,v)T is a constant vector in ℝ2. It is easy to show that 0∉Hμ(∂Ωi∩KerL)(i=1,2). In fact, if there are a certain z*=(z1*,z2*) and a certain μ*∈[0,1] such that Hμ*(z1*,z2*)=0, where zi*∈∂Ωi∩KerL(i=1,2), then there exist ti*∈Iω(i=1,2) such that
(3.38)a(t1*)-b(t1*)ez1*-μ*c1(t1*)ez2*ez2*+α(t1*)ez1*+β(t1*)e2z2*-h1(t1*)e-z1*=0,-d(t2*)+c2(t2*)ez1*ez2*+α(t2*)ez1*+β(t2*)e2z2*-h2(t2*)e-z2*=0.
By carrying out similar arguments as above, we also obtain conclusions as same as (3.26)–(3.28), that is,
(3.39)lnh1LaM<z1*<lnu-,A<z2*<Borlnu+<z1*<lnaMbL,A<z2*<B.
This contradicts the fact that zi*∈∂Ωi∩KerL(i=1,2).
Note thatJ=I (identical mapping), since ImQ=KerL, according to the invariance property of homotopy, direct calculation produces
(3.40)deg{JQN(z1,z2)T,Ωi∩KerL,(0,0)T}=deg{G(z1,z2)T,Ωi∩KerL,(0,0)T}=sign|-b(t1)+h1(t1)ui*20c2(t2)(vi*+β(t2)vi*2)(vi*+α(t2)ui*+β(t2)vi*2)2h2(t2)vi*2-c2(t2)ui*(1+2β(t2)vi*)(vi*+α(t2)ui*+β(t2)vi*2)2|=sign|-b(t1)ui*+h1(t1)ui*0c2(t2)vi*(vi*+β(t2)vi*2)(vi*+α(t2)ui*+β(t2)vi*2)2h2(t2)vi*-c2(t2)ui*vi*(1+2β(t2)vi*)(vi*+α(t2)ui*+β(t2)vi*2)2|=sign[(-b(t1)ui*+h1(t1)ui*)(h2(t2)vi*-c2(t2)ui*vi*(1+2β(t2)vi*)(vi*+α(t2)ui*+β(t2)vi*2)2)]=sign[(a(t1)-2b(t1)ui*)(h2(t2)vi*-(vi*d(t2)+h2(t2))(1+2β(t2)vi*)c2(t2)ui*(h2(t2)vi*+d(t2)))],
where deg{·,·,·} is the Brouwer degree, and ui*, vi* are positive solution for (3.32) in Ωi, respectively. And
(3.41)h2(t2)vi*-(vi*d(t2)+h2(t2))(1+2β(t2)vi*)c2(t2)ui*(h2(t2)vi*+d(t2))≠0,
otherwise, we have c2(t2)ui*h2(t2)=(1+2β(t2)vi*)(vi*d(t2)+h2(t2))2. After test and verification, it is not possible. Thus
(3.42)deg{JQN(z1,z2)T,Ω1∩KerL,(0,0)T}=deg{G(z1,z2)T,Ω1∩KerL,(0,0)T}=sign[(a(t1)-2b(t1)x-)(h2(t2)y--(y-d(t2)+h2(t2))(1+2β(t2)y-)c2(t2)x-(h2(t2)y-+d(t2)))]=-1,deg{JQN(z1,z2)T,Ω2∩KerL,(0,0)T}=deg{G(z1,z2)T,Ω2∩KerL,(0,0)T}=sign[(a(t1)-2b(t1)x+)(h2(t2)y+-(y+d(t2)+h2(t2))(1+2β(t2)y+)c2(t2)x+(h2(t2)y++d(t2)))]=1.
By now we have proved thatΩi(i=1,2) verifies all requirements of Lemma 2.5. Therefore, system (1.3) has at least two ω-periodic solutions in DomL∩Ωi¯(i=1,2), respectively. The proof is complete.Corollary 3.2.
If the conditions in Theorem3.1 hold, then both the corresponding continuous model (1.4) and the discrete model (1.5) have at least two ω-periodic solutions.Remark 3.3.
Ifβ(t)≡0 in system (1.3), then the system is a ratio-dependent predator-prey model with Holling-type functional response and harvesting terms on time scales:
(3.43)z1Δ(t)=a(t)-b(t)ez1(t)-c1(t)ez2(t)ez2(t)+α(t)ez1(t)-h1(t)e-z1(t),z2Δ(t)=-d(t)+c2(t)ez1(t)ez2(t)+α(t)ez1(t)-h2(t)e-z2(t).
Therefore, we have the following results.Corollary 3.4.
If the conditions in Theorem3.1 hold, then system (3.43) has at least two ω-periodic solutions. Specifically, both the corresponding continuous model and the discrete model of the system (3.43) have at least two ω-periodic solutions.Remark 3.5.
Ifh1(t)≡0, h2(t)≡0 in system (1.3), then the system is a general ratio-dependent predator-prey model with Beddington-DeAngelis functional response on time scales:
(3.44)z1Δ(t)=a(t)-b(t)ez1(t)-c1(t)ez2(t)ez2(t)+α(t)ez1(t)+β(t)e2z2(t),z2Δ(t)=-d(t)+c2(t)ez1(t)ez2(t)+α(t)ez1(t)+β(t)e2z2(t).
Using Lemma 2.5, we can obtain another important conclusion as follows.Theorem 3.6.
Assume that the following holds:(i)
aL>c1M,(ii)
c2M>dLαL,(iii)
aL>c1M/βL,then system (3.44) has at least a positive ω-periodic solutions.Corollary 3.7.
If the conditions in Theorem3.6 hold, then both the corresponding continuous model and the discrete model of the system (3.44) have at least a positive ω-periodic solutions.



---

*Source: 243974-2009-10-08.xml*