# Periodic Solutions for a Semi-Ratio-Dependent Predator-Prey System with Delays on Time Scales

Authors: Xiaoquan Ding; Gaifang Zhao
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

This paper is devoted to the existence of periodic solutions for a semi-ratio-dependent predator-prey system with time delays on time scales. With the help of a continuation theorem based on coincidence degree theory, we establish necessary and sufficient conditions for the existence of periodic solutions. Our results show that for the most monotonic prey growth such as the logistic, the Gilpin, and the Smith growth, and the most celebrated functional responses such as the Holling type, the sigmoidal type, the Ivlev type, the Monod-Haldane type, and the Beddington-DeAngelis type, the system always has at least one periodic solution. Some known results are shown to be special cases of the present paper.


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## Body

## 1. Introduction

In the past decades, many authors have investigated the existence of periodic solutions for population models governed by the differential and difference equations [1–7]. In particular, the existence of periodic solutions for semi-ratio-dependent predator-prey systems has been studied extensively in the literature and seen great progress [8–16].Recently, in order to unify differential and difference equations, people have done a lot of research about dynamic equations on time scales. In fact, continuous and discrete systems are very important in implementing and applications. But it is troublesome to study the existence of periodic solutions for continuous and discrete systems, respectively. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations. For the theory of dynamic equations on time scales, we refer the reader to [17, 18]. For the research on periodic solutions of dynamic equations on time scales describing population dynamics, one may consult [19–26], and so forth.In this paper, we consider the following periodic semi-ratio-dependent predator-prey system with time delays on a time scale𝕋: (1.1)u1Δ(t)=g(t,eu1(t-τ1(t)))-h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t),u2Δ(t)=c(t)-d(t)eu2(t)-u1(t-τ3(t)).Here𝕋 is a periodic time scale which has the subspace topology inherited from the standard topology on ℝ. The symbol Δ stands for the delta derivative which gives the ordinary derivative if 𝕋=ℝ and the forward difference operator if 𝕋=ℤ.In system (1.1), set x(t)=exp[u1(t)], y(t)=exp[u2(t)]. If 𝕋=ℝ, then system (1.1) reduces to the standard semi-ratio-dependent predator-prey system governed by the ordinary differential equations:
(1.2)x′(t)=x(t)g(t,x(t-τ1(t)))-h(t,x(t),y(t))y(t-τ2(t)),y′(t)=y(t)[c(t)-d(t)y(t)x(t-τ3(t))],
where x(t) and y(t) stand for the population of the prey and the predator, respectively. The function g(t,x) is the growth rate of the prey in the absence of the predator. The predator consumes the prey according to the functional response h(t,x,y) and grows logistically with growth rate c(t) and carrying capacity x(t)/d(t) proportional to the population size of the prey. The function d(t) is a measure of the food quality that the prey provides for conversion into the predator birth. If 𝕋=ℤ, then system (1.1) is reformulated as
(1.3)x(k+1)=x(k)exp⁡[g(k,x(k-τ1(k)))-h(k,x(k),y(k))y(k-τ2(k))x(k)],y(k+1)=y(k)exp⁡[c(k)-d(k)y(k)x(k-τ3(k))],
which is the discrete time semi-ratio-dependent predator-prey system and is a discrete analogue of (1.2).We note that Ding and Jiang [8, 9], Ding et al. [10], Liu [11], Liu and Huang [12], and Wang et al. [13] studied some special cases of system (1.2). Fan and Wang [14], Fazly and Hesaaraki [15], and Liu [16] discussed some special cases of system (1.3). Bohner et al. [19], Fazly and Hesaaraki [21], and Zhuang [26] investigated some special cases of system (1.1). So far as we know, there is no published paper concerned system (1.1).The main purpose of this paper is, by using the coincidence degree theory developed by Gaines and Mawhin [27], to derive necessary and sufficient conditions for the existence of periodic solutions of system (1.1). Furthermore, we will see that our result for the above system can be easily extended to the one with distributed or state-dependent delays. Our result generalizes some theorems in [8, 9, 11, 12, 15, 16, 21], improves and generalizes some theorems in [10, 13, 14, 19, 26].


## 2. Preliminaries

In this section, we briefly give some elements of the time scale calculus, recall the continuation theorem from coincidence degree theory, and state an auxiliary result that will be used in this paper.First, let us present some foundational definitions and results from the calculus on time scales so that the paper is self-contained. For more details, we refer the reader to [17, 18].A time scale𝕋 is an arbitrary nonempty closed subset 𝕋 of the real numbers ℝ, which inherits the standard topology of ℝ. Thus, the real numbers ℝ, the integers ℤ, and the natural numbers ℕ are examples of time scales, while the rational numbers ℚ and the open interval (1,2) are no time scales.Letω>0. Throughout this paper, the time scale 𝕋 is assumed to be ω-periodic; that is, t∈𝕋 implies t+ω∈𝕋. In particular, the time scale 𝕋 under consideration is unbounded above and below.Fort∈𝕋, the forward and backward jump operators σ,ρ:𝕋→𝕋 are defined by
(2.1)σ(t)=inf⁡{s∈T:s>t},ρ(t)=sup⁡{s∈T:s<t}
respectively.Ifσ(t)=t, t is called right-dense (otherwise: right-scattered), and if ρ(t)=t, then t is called left-dense (otherwise left-scattered).A functionf:𝕋→ℝ is said to be rd-continuous if 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 is denoted by Crd(𝕋).Forf:𝕋→ℝ and t∈𝕋 we define fΔ(t), the delta-derivative of f at t, to be the number (provided it exists) with the property that, given any ε>0, there is a neighborhood U of t (i.e., U=(t-δ,t+δ)∩𝕋 for some δ>0) in 𝕋 such that
(2.2)|[f(σ(t))-f(s)]-fΔ(t)[σ(t)-s]|≤ε|σ(t)-s|,∀s∈U.f is said to be delta-differentiable if its delta-derivative exists for all t∈𝕋. The set of functions f:𝕋→ℝ that are delta-differentiable and whose delta-derivative is rd-continuous functions is denoted by Crd1(𝕋).A functionF:𝕋→ℝ is called a delta-antiderivative of f:𝕋→ℝ provided FΔ(t)=f(t), for all t∈𝕋. Then, we define the delta integral by
(2.3)∫abf(t)Δt=F(b)-F(a),∀a,b∈T.Lemma 2.1.
Every delta differentiable function is continuous.Lemma 2.2.
Every rd-continuous function has a delta-antiderivative.Lemma 2.3.
Ifa, b, c∈𝕋, α, β∈ℝ and f, g∈Crd(𝕋), then (a)
∫ab‍[αf(t)+βg(t)]Δt=α∫ab‍f(t)Δt+β∫ab‍βg(t)Δt,(b)
∫ab‍f(t)Δt=∫ac‍f(t)Δt+∫cb‍f(t)Δt,(c)
iff(t)≥0 for all a≤t<b, then ∫ab‍f(t)Δt≥0,(d)
if|f(t)|≤g(t) on [a,b):={t∈𝕋:a≤t<b}, then |∫ab‍f(t)Δt|≤∫ab‍g(t)Δt.Next, let us recall the continuation theorem in coincidence degree theory. To do so, we need to introduce the following notation.LetX, Y be real Banach spaces, let L:DomL⊂X→Y be a linear mapping, and let N:X→Y be a continuous mapping.The mappingL is said to be a Fredholm mapping of index zero, if dimKerL=codimImL<+∞ and ImL is closed in Y.IfL is a Fredholm mapping of index zero, then there exist continuous projectors P:X→X and Q:Y→Y, such that ImP=KerL, KerQ=ImL=Im(I-Q). It follows that the restriction LP of L to DomL∩KerP:(I-P)X→ImL is invertible. Denote the inverse of LP by KP.The mappingN is said to be L-compact on Ω¯, if Ω is an open bounded subset of X, QN(Ω¯) is bounded, and KP(I-Q)N:Ω¯→X is compact.SinceImQ is isomorphic to KerL, there exists an isomorphism J:ImQ→KerL.Here we state the Gaines-Mawhin theorem, which is a main tool in the proof of our main result.Lemma 2.42.4 (continuation theorem [27, page 40]).
LetΩ⊂X be an open bounded set, let L be a Fredholm mapping of index zero and let N be L-compact on Ω¯. Assume    (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 Lx=Nx has at least one solution in Ω¯∩DomL.For convenience and simplicity in the following discussion, we always use the following notation:(2.4)κ=min⁡{[0,+∞)∩T},Iω=[κ,κ+ω]∩T,a^=1ω∫κκ+ωa(t)Δt,A^=1ω∫κκ+ω|a(t)|Δt,b~(x)=1ω∫κκ+ωb(t,x)Δt,B~(x)=1ω∫κκ+ω|b(t,x)|Δt,φ¯(x,y)=1ω∫κκ+ωφ(t,x,y)Δt,
where a∈Crd(𝕋) is an ω-periodic function, b:𝕋×ℝ→ℝ and φ:𝕋×ℝ2→ℝ are rd-continuous and ω-periodic in their first variable.In order to achieve the priori estimation in the case of dynamic equations on a time scale𝕋, we now give the following inequality which is proved in [19, Lemma 2.4].Lemma 2.5.
Lett1, t2∈Iω and t∈𝕋. If φ∈Crd1(𝕋) is an ω-periodic real function, then
(2.5)φ(t)≤φ(t1)+∫κκ+ω|φΔ(t)|Δt,φ(t)≥φ(t2)-∫κκ+ω|φΔ(t)|Δt.


## 3. Existence of Periodic Solutions

In this section, we study the existence of periodic solutions of system (1.1). For the sake of generality, we make the following fundamental assumptions for system (1.1).(H1)
τi:𝕋→ℝ+ is rd-continuous and ω-periodic such that t-τi(t)∈𝕋 for i=1, 2, 3, and t∈𝕋.(H2)
c:𝕋→ℝ and d:𝕋→(0,+∞) are rd-continuous and ω-periodic.(H3)
g:𝕋×ℝ→ℝ is rd-continuous and ω-periodic in the first variable and is continuously differentiable in the second variable and (∂g/∂x)(t,x)<0, limx→+∞g(t,x)<0 for all t∈𝕋, x>0.(H4)
h:𝕋×ℝ2→ℝ+ is rd-continuous and ω-periodic in the first variable and is continuously differentiable in the last two variables. In addition, there exist a positive integer m and ω-periodic rd-continuous functions ai:𝕋→ℝ+, i=0,1,…,m-1, such that
(3.1)h(t,x,y)≤a0(t)xm+a1(t)xm-1+⋯+am-1(t)x,∀t∈T,x>0,y>0.Readers familiar with predator-prey models may notice that the above assumptions are reasonable for population models. Under the above assumptions, system (1.1) covers many models that have appeared in the literature. For instance, g(t,x) can be taken as the logistic growth a-bx, the Gilpin growth a-bxθ, and the Smith growth (a-bx)/(D+x). h(t,x,y) can be taken as functional responses of the Lotka-Volterra type mx, the Holling type mxn/(A+xn)(n≥1), the Ivlev type m(1-e-Ax), the sigmoidal type mx2/[(A+x)(B+x)], the Monod-Haldane type mx/(A+Bx+x2), and the Beddington-DeAngelis type mx/(A+Bx+Cy), and so forth.By (H3), we have
(3.2)g~′(x)=1ω∫κκ+ω∂g∂x(t,x)Δt<0,lim⁡x→+∞g~(x)=1ω∫κκ+ωlim⁡x→+∞g(t,x)Δt<0.
Thus g~(x) is strictly decreasing on [0,+∞).We are now in a position to state and prove our main result.Theorem 3.1.
Under the assumptions (H1)–(H4), system (1.1) has at least one ω-periodic solution if and only if (H5)
g~(0)>0,(H6)
c^>0hold.Proof.
“Only if” part: Suppose that(u1(t),u2(t))T is an ω-periodic solution of system (1.1). Then by integrating (1.1) on both side from κ to κ+ω, we have
(3.3)∫κκ+ω[g(t,eu1(t-τ1(t)))-h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)]Δt=∫κκ+ωu1Δ(t)Δt=0,(3.4)∫κκ+ω[c(t)-d(t)eu2(t)-u1(t-τ3(t))]Δt=∫κκ+ωu2Δ(t)Δt=0.
By (H4) and the monotonicity of function g~(x), we obtain from (3.3) that
(3.5)g~(0)>1ω∫κκ+ωg(t,eu1(t-τ1(t)))Δt=1ω∫κκ+ωh(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)Δt≥0,
which is (H5).
By (H2) and (3.4), we have
(3.6)c^=1ω∫κκ+ω‍d(t)eu2(t)-u1(t-τ3(t))Δt>0,
which gives (H6).
“If” part: Take(3.7)X=Y={u=(u1(t),u2(t))T∣ui∈Crd⁡(T),ui(t+ω)=ui(t),i=1,2},‖u‖=‖(u1(t),u2(t))T‖=max⁡t∈Iω|u1(t)|+max⁡t∈Iω|u2(t)|
Then X and Y are Banach spaces with the norm ∥·∥. Set
(3.8)L:Dom⁡L⊂X⟶Y,L(u1(t)u2(t))=(u1Δ(t)u2Δ(t)),
where DomL={u=(u1(t),u2(t))T∈X∣ui∈Crd1(𝕋),i=1,2} and
(3.9)N:X⟶Y,N(u1(t)u2(t))=(g(t,eu1(t-τ1(t)))-h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)c(t)-d(t)eu2(t)-u1(t-τ3(t))).
With these notations system (1.1) can be written in the form
(3.10)Lu=Nu,u∈X.
Obviously, KerL=ℝ2, ImL={(u1(t),u2(t))T∈Y:∫κκ+ω‍ui(t)Δt=0,i=1,2} is closed in Y, and dim KerL=codim ImL=2. Therefore L is a Fredholm mapping of index zero. Now define two projectors P:X→X and Q:Y→Y as
(3.11)P(u1(t)u2(t))=Q(u1(t)u2(t))=(u^1u^2),(u1(t)u2(t))∈X=Y.
Then P and Q are continuous projectors such that
(3.12)Im⁡P=Ker⁡L,Ker⁡Q=Im⁡L=Im⁡(I-Q).
Furthermore, through an easy computation we find that the generalized inverse KP of LP has the form
(3.13)KP:Im⁡L⟶Dom⁡L∩Ker⁡P,KP(u)=∫κtu(s)Δs-1ω∫κκ+ω∫κtu(s)ΔsΔt.
Then QN:X→Y and KP(I-Q)N:X→X read as
(3.14)QNu=1ω(∫κκ+ω[g(t,eu1(t-τ1(t)))-h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)]Δt∫κκ+ω[c(t)-d(t)eu2(t)-u1(t-τ3(t))]Δt),(3.15)KP(I-Q)Nu=(∫κt[g(s,eu1(s-τ1(s)))-h(s,eu1(s),eu2(s))eu2(s-τ2(s))-u1(s)]Δs∫κt[c(s)-d(s)eu2(s)-u1(s-τ3(s))]Δs)-1ω(t-κ-1ω∫κκ+ω(s-κ)Δs)×(∫κκ+ω[g(s,eu1(s-τ1(s)))-h(s,eu1(s),eu2(s))eu2(s-τ2(s))-u1(s)]Δs∫κκ+ω[c(s)-d(s)eu2(s)-u1(s-τ3(s))]Δs)-1ω(∫κκ+ω∫κt[g(s,eu1(s-τ1(s)))-h(s,eu1(s),eu2(s))eu2(s-τ2(s))-u1(s)]ΔsΔt∫κκ+ω∫κt[c(s)-d(s)eu2(s)-u1(s-τ3(s))]ΔsΔt).
Clearly, QN and KP(I-Q)N are continuous. By using the Arzela-Ascoli theorem, it is not difficult to prove that KP(I-Q)N(Ω¯)¯ is compact for any open bounded set Ω⊂X. Moreover, QN(Ω¯) is bounded. Therefore N is L-compact on Ω¯ with any open bounded set Ω⊂X.
In order to apply Lemma2.4, we need to find appropriate open, bounded subsets in X. Corresponding to the operator equation Lu=λNu,λ∈(0,1), we have
(3.16)u1Δ(t)=λ[g(t,eu1(t-τ1(t)))-h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)],u2Δ(t)=λ[c(t)-d(t)eu2(t)-u1(t-τ3(t))].
Suppose that (u1(t),u2(t))T∈X is a solution of (3.16) for a certain λ∈(0,1). Integrating (3.16) on both side from κ to κ+ω leads to
(3.17)∫κκ+ωλ[g(t,eu1(t-τ1(t)))-h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)]Δt=∫κκ+ωu1Δ(t)Δt=0,∫κκ+ωλ[c(t)-d(t)eu2(t)-u1(t-τ3(t))]Δt=∫κκ+ωu2Δ(t)Δt=0.
That is
(3.18)∫κκ+ωg(t,eu1(t-τ1(t)))Δt=∫κκ+ωh(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)Δt(3.19)∫κκ+ωd(t)eu2(t)-u1(t-τ3(t))Δt=∫κκ+ωc(t)Δt=c^ω.
From (3.18), we have
(3.20)g~(0)ω=∫κκ+ω{[g(t,0)-g(t,eu1(t-τ1(t)))]+h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)}Δt.
It follows from (3.16), (3.18), (3.19), (3.20), and (H2)–(H4) that
(3.21)∫κκ+ω|u1Δ(t)|Δt≤λ∫κκ+ω|g(t,eu1(t-τ1(t)))-h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)|Δt<∫κκ+ω{[g(t,0)-g(t,eu1(t-τ1(t)))]+h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t)}Δt+∫κκ+ω|g(t,0)|Δt=[G~(0)+g~(0)]ω,(3.22)∫κκ+ω|u2Δ(t)|Δt≤λ∫κκ+ω|c(t)-d(t)eu2(t)-u1(t-τ3(t))|Δt<∫κκ+ω|c(t)|Δt+∫κκ+ωd(t)eu2(t)-u1(t-τ3(t))Δt=[C^+c^]ω.
Since(u1(t),u2(t))T∈X, there exist ξi,ηi∈Iω such that
(3.23)ui(ξi)=min⁡t∈Iωui(t),ui(ηi)=max⁡t∈Iωui(t),i=1,2.
Then from (3.19) and (H2), we have
(3.24)c^≤1ω∫κκ+ωd(t)eu2(η2)-u1(ξ1)Δt=d^eu2(η2)-u1(ξ1),c^≥1ω∫κκ+ωd(t)eu2(ξ2)-u1(η1)Δt=d^eu2(ξ2)-u1(η1).
These, together with (H6), yield
(3.25)u2(η2)≥ln⁡[c^d^]+u1(ξ1),(3.26)u2(ξ2)≤ln⁡[c^d^]+u1(η1).
From (3.18), (H4), and the monotonicity of function g~(x), we have
(3.27)g~(eu1(ξ1))≥1ω∫κκ+ωg(t,eu1(t-τ1(t)))Δt≥0.
In view of (H3), (H5), and the continuity of function g~(x), it is easy to see that there exists a positive constant α1 such that
(3.28)g~(α1)=0.
Then, from (3.27), (3.28), and the monotonicity of function g~(x), we have
(3.29)u1(ξ1)≤ln⁡α1.
By Lemma 2.5, we obtain from (3.21) and (3.29) that for all t∈Iω(3.30)u1(t)≤u1(ξ1)+∫κκ+ω|u1Δ(t)|Δt≤ln⁡α1+[G~(0)+g~(0)]ω:=β1.
By Lemma 2.5, we also obtain from (3.22), (3.26), and (3.30) that for all t∈Iω(3.31)u2(t)≤u2(ξ2)+∫κκ+ω|u2Δ(t)|Δt≤β1+ln⁡[c^d^]+[C^+c^]ω:=β2.
It follows from (H4)and (3.30) that
(3.32)0≤1ω∫κκ+ωh(t,eu1(t),eu2(t))e-u1(t)Δt≤a^0e(m-1)β1+a^1e(m-2)β1+⋯+a^m-1:=c0.
In order to obtainβ3 and β4 such that u1(t)≥β3 and u2(t)≥β4 for all t∈Iω, we consider the following two cases.
Case  1. If u2(η2)≥u1(η1), then from (3.18), (3.23), (3.32), (H4), and monotonicity of function g~(x), we have
(3.33)g~(eu2(η2))≤g~(eu1(η1))≤1ω∫κκ+ωg(t,eu1(t-τ1(t)))Δt≤eu2(η2)ω∫κκ+ωh(t,eu1(t),eu2(t))e-u1(t)Δt≤c0eu2(η2).
From (H3), (H5), and the continuity of function g~(x), one can easily see that there exists a positive constant α2 such that
(3.34)g~(α2)-c0α2=0.
Then, from (3.33), (3.34), and the monotonicity of function g~(x)-c0x, we have
(3.35)u2(η2)≥ln⁡α2.
By Lemma 2.5, we obtain from (3.22) and (3.35) that for all t∈Iω(3.36)u2(t)≥u2(η2)-∫κκ+ω|u2Δ(t)|Δt≥ln⁡α2-[C^+c^]ω:=β41.
By Lemma 2.5, we also obtain from (3.21), (3.26) that for all t∈Iω(3.37)u1(t)≥u1(η1)-∫κκ+ω|u1Δ(t)|Δt≥β41-ln⁡[c^d^]-[G~(0)+g~(0)]ω:=β31.
Case  2. If u2(η2)<u1(η1), then from (3.18), (3.23), (3.32), (H4), and monotonicity of function g~(x), we have
(3.38)g~(eu1(η1))≤1ω∫κκ+ωg(t,eu1(t-τ1(t)))Δt≤eu2(η2)ω∫κκ+ωh(t,eu1(t),eu2(t))e-u1(t)Δt≤c0eu2(η2)≤c0eu1(η1).
Then, from (3.34) and the monotonicity of function g~(x)-c0x, we have
(3.39)u1(η1)≥ln⁡α2.
By Lemma 2.5, we obtain from (3.21), (3.39) that for all t∈Iω(3.40)u1(t)≥u1(η1)-∫κκ+ω|u1Δ(t)|Δt≥ln⁡α2-[G~(0)+g~(0)]ω:=β32.
By Lemma 2.5, we also obtain from (3.22), (3.25), and (3.40) that for all t∈Iω(3.41)u2(t)≥u2(η2)-∫κκ+ω|u2Δ(t)|Δt≥β32+ln⁡[c^d^]-[C^+c^]ω:=β42.
Now, we take β3=:min{β31,β32} and β4=:min{β41,β42}. Then it follows from (3.36), (3.37), (3.40), and (3.41) that for all t∈Iω, u1(t)≥β3 and u2(t)≥β4. Hence from these, (3.30), and (3.31), we have
(3.42)sup⁡t∈Iω|u1(t)|≤max⁡{|β1|,|β3|}:=β5,sup⁡t∈Iω|u1(t)|≤max⁡{|β2|,|β4|}:=β6.
Clearly, β5 and β6 are independent of λ.
On the other hand, forμ∈[0,1], we consider the following algebraic system:
(3.43)g~(eu1)-μh¯(eu1,eu2)eu2-u1=0,c^-d^eu2-u1=0,
where (u1,u2)T∈ℝ2. From the second equation of (3.43) and (H6), we have
(3.44)u2=u1+ln⁡[c^d^].
From the first equation of (3.43) and (H4), we also have
(3.45)g~(eu1)=μh¯(eu1,eu2)eu2-u1≥0.
Then, from (3.44) and the monotonicity of function g~(x), we obtain
(3.46)u1≤ln⁡α1.
Substituting (3.44) into the first equation of (3.43), we can get from (H4), (3.30), (3.32), and (3.46) that
(3.47)g~(eu1)=μc^d^h¯(eu1,eu2)≤c^d^h¯(eu1,eu2)≤c^d^[a^0e(m-1)u1+a^1e(m-2)u1+⋯+a^m-1]eu1≤c^d^[a^0e(m-1)β1+a^1e(m-2)β1+⋯+a^m-1]eu1=c0c^d^eu1.
In view of (H3), (H5), and the continuity of function g~(x), it is easy to see that there exists a positive constant α3 such that
(3.48)g~(α3)-c0c^d^α3=0.
Then, from (3.47), (3.48), and the monotonicity of function g~(x), we obtain
(3.49)u1≥ln⁡α3.
It follows from (3.44), (3.46), and (3.49) that
(3.50)|u1|≤max⁡{|ln⁡α1|,|ln⁡α3|}:=β7,|u2|≤β7+|ln⁡c^-ln⁡d^|:=β8.
Clearly, β7 and β8 are also independent of μ.
We takeΩ={u=(u1(t),u2(t))T∈X∣∥u∥<β}, here β=β5+β6+β7+β8. Now we check the conditions of Lemma 2.4.(a)
By (3.42), one can conclude that for each λ∈(0,1), u∈∂Ω∩DomL, Lu≠λNu.(b)
When(u1(t),u2(t))T∈∂Ω∩KerL, (u1(t),u2(t))T is a constant vector in ℝ2, we denote it by (u1,u2)T and |u1|+|u2|=β. If
(3.51)QNu=(g~(eu1)-h¯(eu1,eu2)eu2-u1c^-d^eu2-u1)=0, then (u1,u2)T is a constant solution of system (3.43) with μ=1. By (3.50), we have |u1|+|u2|≤β7+β8<β. This contradiction implies that for each u∈∂Ω∩KerL,QNu≠0.(c)
In order to verify the condition (c) in Lemma2.4, we define ϕ:ℝ2×[0,1]→ℝ2 by
(3.52)ϕ(u1,u2,μ)=(g~(eu1)c^-d^eu2-u1)+μ(-h-(eu1,eu2)eu2-u10),where μ∈[0,1] is a parameter. When (u1,u2)T∈∂Ω∩KerL, (u1,u2)T is a constant vector in ℝ2 with |u1|+|u2|=β. By (3.50) we know ϕ(u1,u2,μ)≠(0,0)T on ∂Ω∩KerL. Thus, ϕ is a homotopy mapping. Moreover, it is not difficult to see that the following algebraic system:
(3.53)g~(eu1)=0,c^-d^eu2-u1=0,
has a unique solution (u1†,u2†)T∈∂Ω∩KerL. So, due to homotopy invariance theorem of topology degree and taking J=I:ImQ→KerL,(u1,u2)T→(u1,u2)T, we obtain
(3.54)deg⁡(JQN,Ω∩Ker⁡L,(0,0)T)=deg⁡(ϕ(u1,u2,1),Ω∩Ker⁡L,(0,0)T)=deg⁡(ϕ(u1,u2,0),Ω∩Ker⁡L,(0,0)T)=sign⁡{-g~(eu1†)eu2†}≠0.
By now we have proved that Ω satisfies all the requirements in Lemma 2.4. Hence, system (1.1) has at least one ω-periodic solution. This completes the proof.Noticing that both systems (1.2) and (1.3) are special cases of system (1.1), by Theorem 3.1, we can obtain the following results.Theorem 3.2.
Under the assumptions (H1)–(H4), system (1.2) has at least one positive ω-periodic solution if and only if (H5) and (H6) hold.Theorem 3.3.
Under the assumptions (H1)–(H4), system (1.3) has at least one positive ω-periodic solution if and only if (H5) and (H6) hold.The proof of Theorem3.1 shows that it remains valid for the following periodic semi-ratio-dependent predator-prey system on a time scale:
(3.55)u1Δ(t)=g(t,eu1(t-τ1(t)))-h(t,eu1(t),eu2(t))eu2(t-τ2(t))-u1(t),u2Δ(t)=c(t)-d(t)eu2(t-τ3(t))-u1(t-τ3(t)).(3.56)x′(t)=x(t)g(t,x(t-τ1(t)))-h(t,x(t),y(t))y(t-τ2(t)),y′(t)=y(t)[c(t)-d(t)y(t-τ3(t))x(t-τ3(t))],(3.57)
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,Remark 3.4.
One can easily verify that if their parameters are positiveω-periodic functions, all the prey growth types and the functional responses mentioned previously satisfy the assumptions of Theorem 3.1. Therefore, by Theorem 3.1, the system (1.1) with the logistic, the Gilpin, or the Smith prey growth and with the Lotka-Volterra, the Holling, the sigmoidal, the Ivlev, the Monod-Haldane, or the Beddington-DeAngelis functional responses always has at least one ω-periodic solution.Remark 3.5.
Similarly, by Theorems3.2 and 3.3, the systems (1.2) and (1.3) with the logistic, the Gilpin, or the Smith prey growth and with the Lotka-Volterra, the Holling, the sigmoidal, the Ivlev, the Monod-Haldane, or the Beddington-DeAngelis functional responses, always have at least one positive ω-periodic solution, respectively.Remark 3.6.
Bohner et al. [19], Fazly and Hesaaraki [21] studied the special cases of system (1.1) for τ1(t)=τ2(t)=τ3(t)=0, g(t,x)=a(t)-b(t)x, and h(t,x,y)=p(t,x). Zhuang [26] studied the special case of system (1.1) for τ2(t)=0, g(t,x)=a(t)-b(t)x, and h(t,x,y)=k(t)x/(m2+x2). Therefore, our Theorem 3.1 generalizes and improves Theorem 3.4 in [19] and Theorem 3.1 in [26] and generalizes Theorem 1 in [21].Remark 3.7.
Wang et al. [13] studied the special case of system (1.2) for τ1(t)=τ2(t)=τ3(t)=0, g(t,x)=a(t)-b(t)x, and h(t,x,y)=p(t,x). Ding et al. [10] studied the special case of system (1.2) for τ2(t)=τ3(t)=0, g(t,x)=a(t)-b(t)x, and h(t,x,y)=k(t)x/(m2+x2). Ding and Jiang [8] studied the special case of system (1.2) for h(t,x,y)=p(t,x). Liu [11] studied the special case of system (1.2) for τ1(t)=τ2(t)=τ3(t)=0 and g(t,x)=a(t)-b(t)x. Liu and Huang [12] studied the special case of system (3.56) for τ1(t)=τ2(t)=0 and g(t,x)=a(t)-b(t)x. Ding and Jiang [9] studied the special case of system (3.56) for τ1(t)=0. Therefore, our Theorem 3.2 generalizes and improves Theorem 3.3 in [13] and Theorem 2.1 in [10] and generalizes Theorem 2.2 in [8], Theorem 2.2 in [9], Theorem 2.1 in [11], and Corollary 3.1 in [12].Remark 3.8.
Fan and Wang [14], Fazly and Hesaaraki [15] studied the special cases of system (1.3) for τ1(k)=τ2(k)=τ3(k)=0, g(k,x)=a(k)-b(k)x, and h(k,x,y)=p(k,x). Liu [16] studied the special case of system (1.3) for τ1(k)=τ2(k)=τ3(k)=0 and g(k,x)=a(k)-b(k)x. Therefore, our Theorem 3.3 generalizes and improves Theorem 2.1 in [14] and generalizes Theorem 1 in [15] and Theorem 2.1 in [16].



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*Source: 928704-2012-07-05.xml*