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

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Discrete Dynamics in Nature and Society Volume 2012, Article ID 928704,15pages doi:10.1155/2012/928704

Research Article

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

Xiaoquan Ding and Gaifang Zhao

School of Mathematics and Statistics, Henan University of Science and Technology, Henan, Luoyang 471003, China

Correspondence should be addressed to Xiaoquan Ding,[email protected] Received 30 March 2012; Accepted 11 May 2012

Academic Editor: Ugurhan Mugan

Copyrightq2012 X. Ding and G. Zhao. 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.

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.

1. Introduction

In the past decades, many authors have investigated the existence of periodic solutions for population models governed by the diﬀerential and diﬀerence equations1–7. In particular, the existence of periodic solutions for semi-ratio-dependent predator-prey systems has been studied extensively in the literature and seen great progress8–16.

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 consult19–26, and so forth.

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In this paper, we consider the following periodic semi-ratio-dependent predator-prey system with time delays on a time scaleT:

u^Δ₁t g

t,e^u¹^t−τ¹^t

−h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^t, u^Δ₂t ct−dte^u²^t−u¹^t−τ³^t.

1.1

HereTis 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 ifT Rand the forward diﬀerence operator ifT Z.

In system 1.1, set xt expu1t, yt expu2t. If T R, then system 1.1reduces to the standard semi-ratio-dependent predator-prey system governed by the ordinary diﬀerential equations:

xt xtgt, xt−τ1t−h

t, xt, yt

yt−τ2t, yt yt

ct−dt yt

xt−τ3t

,

1.2

wherextandytstand for the population of the prey and the predator, respectively. The function gt, xis the growth rate of the prey in the absence of the predator. The predator consumes the prey according to the functional responseht, x, yand grows logistically with growth ratectand carrying capacityxt/dtproportional to the population size of the prey. The functiondtis a measure of the food quality that the prey provides for conversion into the predator birth. IfT Z, then system1.1is reformulated as

xk1 xkexp

gk, xk−τ₁k−h

k, xk, ykyk−τ₂k xk

, yk1 ykexp

ck−dk yk

xk−τ₃k

,

1.3

which is the discrete time semi-ratio-dependent predator-prey system and is a discrete analogue of1.2.

We note that Ding and Jiang8,9, Ding et al.10, Liu11, Liu and Huang12, and Wang et al.13studied some special cases of system 1.2. Fan and Wang14, Fazly and Hesaaraki15, and Liu16discussed some special cases of system1.3. Bohner et al.19, Fazly and Hesaaraki21, and Zhuang26investigated some special cases of system1.1.

So far as we know, there is no published paper concerned system1.1.

The main purpose of this paper is, by using the coincidence degree theory developed by Gaines and Mawhin27, to derive necessary and suﬃcient conditions for the existence of periodic solutions of system1.1. 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 in8,9,11,12,15,16,21, improves and generalizes some theorems in10,13,14,19,26.

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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 to17,18.

A time scaleTis an arbitrary nonempty closed subsetTof the real numbersR, which inherits the standard topology ofR. Thus, the real numbersR, the integersZ, and the natural numbersNare examples of time scales, while the rational numbersQand the open interval 1,2are no time scales.

Letω > 0. Throughout this paper, the time scaleTis assumed to beω-periodic; that is,t∈Timpliestω ∈T. In particular, the time scaleTunder consideration is unbounded above and below.

Fort∈T, the forward and backward jump operatorsσ, ρ:T → Tare defined by σt inf{s∈T:s > t}, ρt sup{s∈T:s < t} 2.1 respectively.

Ifσt t,tis called right-denseotherwise: right-scattered, and ifρt t, thentis called left-denseotherwise left-scattered.

A functionf : T → Ris 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 byCrdT.

For f : T → Rand t ∈ Twe definef^Δt, the delta-derivative of f at t, to be the numberprovided it existswith the property that, given anyε >0, there is a neighborhood Uofti.e.,U t−δ, tδ∩Tfor someδ >0inTsuch that

fσt−fs

−f^Δtσt−s≤ε|σt−s|, ∀s∈U. 2.2 fis said to be delta-diﬀerentiable if its delta-derivative exists for allt∈T. The set of functions f:T → Rthat are delta-diﬀerentiable and whose delta-derivative is rd-continuous functions is denoted byC_rd¹ T.

A functionF :T → Ris called a delta-antiderivative off :T → RprovidedF^Δt ft, for allt∈T. Then, we define the delta integral by

_b

a

ftΔt Fb−Fa, ∀a, b∈T. 2.3

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∈Randf,g∈C_rdT, then a _b

aαft βgt Δt α_b

aft Δtβ_b

aβgt Δt, b_b

aft Δt _c

aft Δt_b

cft Δt,

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cifft≥0 for alla≤t < b, then_b

aft Δt≥0, dif|ft| ≤gtona, b: {t∈T:a≤t < b}, then|_b

aft Δt| ≤_b

agt 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, letL : DomL⊂ X → Y be a linear mapping, and let N:X → Y be a continuous mapping.

The mapping L is said to be a Fredholm mapping of index zero, if dim KerL codim ImL <∞and ImLis closed inY.

IfLis a Fredholm mapping of index zero, then there exist continuous projectorsP : X → XandQ:Y → Y, such that ImP KerL, KerQ ImL ImI−Q. It follows that the restrictionL_P ofLto DomL∩KerP:I−PX → ImLis invertible. Denote the inverse ofL_P byKP.

The mappingN is said to beL-compact onΩ, ifΩis an open bounded subset ofX, QNΩis bounded, andKPI−QN:Ω → Xis compact.

Since ImQis isomorphic to KerL, there exists an isomorphismJ: ImQ → KerL.

Here we state the Gaines-Mawhin theorem, which is a main tool in the proof of our main result.

Lemma 2.4continuation theorem27, page 40. LetΩ⊂X be an open bounded set, letLbe a Fredholm mapping of index zero and letNbeL-compact onΩ. Assume

afor eachλ∈0,1, x∈∂Ω∩DomL, Lx /λNx;

bfor eachx∈∂Ω∩KerL, QNx /0;

cdegJQN,Ω∩KerL,0/0.

ThenLx Nxhas at least one solution inΩ∩DomL.

For convenience and simplicity in the following discussion, we always use the following notation:

κ min{0,∞∩T}, I_ω κ, κω∩T, a 1 ω

_κω

κ

atΔt, A 1

ω _κω

κ

|at|Δt, bx 1

ω _κω

κ

bt, xΔt, Bx 1 ω

_κω

κ

|bt, x|Δt, ϕ

x, y 1 ω

_κω

κ

ϕ t, x, y

Δt, 2.4 wherea ∈ CrdTis anω-periodic function,b : T×R → Randϕ : T×R² → Rare rd- continuous andω-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 in19, Lemma 2.4.

Lemma 2.5. Lett₁,t₂∈I_ωandt∈T. Ifϕ∈C¹_rdTis anω-periodic real function, then

ϕt≤ϕt1

_κω

κ

ϕ^ΔtΔt, ϕt≥ϕt2− _κω

κ

ϕ^ΔtΔt. 2.5

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3. Existence of Periodic Solutions

In this section, we study the existence of periodic solutions of system1.1. For the sake of generality, we make the following fundamental assumptions for system1.1.

H1τ_i :T → R is rd-continuous andω-periodic such thatt−τ_it ∈Tfori 1, 2, 3, andt∈T.

H2c:T → Randd:T → 0,∞are rd-continuous andω-periodic.

H3g : T × R → R is rd-continuous and ω-periodic in the first variable and is continuously diﬀerentiable in the second variable and ∂g/∂xt, x < 0, limx→∞gt, x<0 for allt∈T,x >0.

H4h : 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 a_i : T → R, i 0,1, . . . , m−1, such that

h t, x, y

≤a₀tx^ma₁tx^m−1· · ·a_m−1tx, ∀t∈T, x >0, y >0. 3.1

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,gt, xcan be taken as the logistic growtha−bx, the Gilpin growtha−bx^θ, and the Smith growtha−bx/Dx. ht, x, ycan be taken as functional responses of the Lotka-Volterra typemx, the Holling type mxⁿ/Axⁿn≥1 , the Ivlev typem1−e^−Ax, the sigmoidal typemx²/AxBx, the Monod-Haldane typemx/ABxx², and the Beddington-DeAngelis typemx/A BxCy, and so forth.

ByH3, we have

gx 1 ω

_κω

κ

∂g

∂xt, x Δt <0, lim

x→∞gx 1 ω

_κω

κ

xlim→∞gt, x Δt <0. 3.2

Thusgx is strictly decreasing on0,∞.

We are now in a position to state and prove our main result.

Theorem 3.1. Under the assumptions (H1)–(H₄), system1.1has at least oneω-periodic solution if and only if

H5 g0 >0, H6 c > 0 hold.

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Proof. “Only if” part: Suppose thatu1t, u2t^T is anω-periodic solution of system1.1.

Then by integrating1.1on both side fromκtoκω, we have _κω

κ

g

t,e^u¹^t−τ¹^t

−h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^t Δt

_κω

κ

u^Δ₁t Δt 0, 3.3 _κω

κ

ct−dte^u²^t−u¹^t−τ³^t Δt

_κω

κ

u^Δ₂t Δt 0. 3.4

ByH4and the monotonicity of functiongx, we obtain from 3.3that

g0> 1

ω _κω

κ

g

t,e^u¹^t−τ¹^t Δt 1

ω _κω

κ

h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^t Δt≥0, 3.5

which isH5.

ByH2and3.4, we have

c 1

ω _κω

κ

dte^u²^t−u¹^t−τ³^t Δt >0, 3.6

which givesH6.

“If” part: Take

X Y

u u1t, u2t^T |ui∈CrdT, uitω uit, i 1,2 , u u1t, u2t^T max

t∈Iω

|u1t|max

t∈Iω

|u2t| 3.7

ThenXandY are Banach spaces with the norm · . Set

L: DomL⊂X−→Y, L u1t

u2t

u^Δ₁t u^Δ₂t

, 3.8

where DomL {u u1t, u2t^T ∈X|u_i ∈C¹_rdT, i 1,2}and

N:X−→Y, N u1t

u₂t

g

t,e^u¹^t−τ¹^t

−h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^t ct−dte^u²^t−u¹^t−τ³^t

. 3.9

With these notations system1.1can be written in the form

Lu Nu, u∈X. 3.10

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Obviously, KerL R², ImL {u1t, u2t^T ∈ Y : _κω

κ uit Δt 0, i 1,2}is closed in Y, and dim KerL codim ImL 2. ThereforeLis a Fredholm mapping of index zero. Now define two projectorsP :X → XandQ:Y → Yas

P u₁t

u2t

Q u₁t

u2t

u₁

u2

,

u₁t u2t

∈X Y. 3.11

ThenP andQare continuous projectors such that

ImP KerL, KerQ ImL ImI−Q. 3.12

Furthermore, through an easy computation we find that the generalized inverseK_PofL_Phas the form

KP : ImL−→DomL∩KerP, K_Pu

_t

κ

usΔs− 1 ω

_κω

κ

_t

κ

usΔsΔt. 3.13

ThenQN :X → Y andK_PI−QN:X → Xread as

QNu 1

ω _κω

κ g

t,e^u¹^t−τ¹^t

−h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^t _κω Δt

κ ct−dte^u²^t−u¹^t−τ³^t Δt

, 3.14

K_PI−QNu _t

κ g

s,e^u¹^s−τ¹^s

−h

s,e^u¹^s,e^u²^s

e^u²^s−τ²^s−u¹^s _t Δs

κ cs−dse^u²^s−u¹^s−τ³^s Δs

− 1 ω

t−κ− 1 ω

_κω

κ

s−κ Δs

× _κω

κ g

s,e^u¹^s−τ¹^s

−h

s,e^u¹^s,e^u²^s

e^u²^s−τ²^s−u¹^s _κω Δs

κ cs−dse^u²^s−u¹^s−τ³^s Δs

− 1 ω

_κω

κ

_t

κ g

s,e^u¹^s−τ¹^s

−h

s,e^u¹^s,e^u²^s

e^u²^s−τ²^s−u¹^s _κω ΔsΔt

κ

_t

κ cs−dse^u²^s−u¹^s−τ³^s ΔsΔt

. 3.15 Clearly,QN andK_PI−QN are continuous. By using the Arzela-Ascoli theorem, it is not diﬃcult to prove thatK_PI−QNΩis compact for any open bounded setΩ⊂X. Moreover, QNΩis bounded. ThereforeNisL-compact onΩwith any open bounded setΩ⊂X.

In order to apply Lemma2.4, we need to find appropriate open, bounded subsets in X. Corresponding to the operator equationLu λNu , λ∈0,1, we have

u^Δ₁t λ g

t,e^u¹^t−τ¹^t

−h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^t , u^Δ₂t λ

ct−dte^u²^t−u¹^t−τ³^t .

3.16

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Suppose thatu1t, u2t^T ∈Xis a solution of3.16for a certainλ∈0,1. Integrating3.16 on both side fromκtoκωleads to

_κω

κ

λ g

t,e^u¹^t−τ¹^t

−h

t,e^u¹^t,e^u²^t

_κω

κ

u^Δ₁t Δt 0, _κω

κ

λ

ct−dte^u²^t−u¹^t−τ³^t Δt

_κω

κ

u^Δ₂t Δt 0.

3.17

That is

_κω

κ

g

t,e^u¹^t−τ¹^t Δt

_κω

κ

h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^t Δt 3.18 _κω

κ

dte^u²^t−u¹^t−τ³^tΔt

_κω

κ

ctΔt cω. 3.19

From3.18, we have

g0ω

_κω

κ

gt,0−g

t,e^u¹^t−τ¹^t h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^t

Δt . 3.20

It follows from3.16,3.18,3.19,3.20, andH2–H4that _κω

κ

u^Δ₁tΔt≤λ _κω

κ

g

t,e^u¹^t−τ¹^t

−h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^tΔt

<

_κω

κ

gt,0−g

t,e^u¹^t−τ¹^t h

t,e^u¹^t,e^u²^t

_κω

κ

gt,0Δt G0 g0

ω,

3.21 _κω

κ

u^Δ₂tΔt≤λ _κω

κ

ct−dte^u²^t−u¹^t−τ³^tΔt

<

_κω

κ

|ct|Δt _κω

κ

dte^u²^t−u¹^t−τ³^t Δt Cc

ω.

3.22

Sinceu1t, u2t^T∈X, there existξi, ηi∈Iωsuch that u_iξi min

t∈Iω

u_it, u_i η_i

maxt∈Iω

u_it, i 1,2. 3.23

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Then from3.19andH2, we have

c≤ 1

ω _κω

κ

dte^u²^η²^−u¹^ξ¹ Δt de^u²^η²^−u¹^ξ¹,

c≥ 1 ω

_κω

κ

dte^u²^ξ²^−u¹^η¹ Δt de^u²^ξ²^−u¹^η¹.

3.24

These, together withH6, yield

u2

η2

≥ln c

d

u1ξ1, 3.25

u₂ξ2≤ln c

d

u₁ η₁

. 3.26

From3.18,H4, and the monotonicity of functiongx, we have

g e^u¹^ξ¹

≥ 1 ω

_κω

κ

g

t,e^u¹^t−τ¹^t

Δt≥0. 3.27

In view ofH3,H5, and the continuity of functiongx, it is easy to see that there exists a positive constantα1such that

gα1 0. 3.28

Then, from3.27,3.28, and the monotonicity of functiongx, we have

u1ξ1≤ lnα1. 3.29

By Lemma2.5, we obtain from3.21and3.29that for allt∈I_ω

u₁t≤u₁ξ1

_κω

κ

u^Δ₁tΔt≤lnα₁

G0 g0

ω: β₁. 3.30

By Lemma2.5, we also obtain from3.22,3.26, and3.30that for allt∈I_ω

u₂t≤u₂ξ2

_κω

κ

u^Δ₂tΔt≤β₁ln c

d

Cc

ω: β₂. 3.31

It follows fromH4and3.30that

0≤ 1 ω

_κω

κ

h

t,e^u¹^t,e^u²^t

e^−u¹^t Δt≤a₀e^m−1β¹a₁e^m−2β¹· · ·a_m−1: c₀. 3.32

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In order to obtain β3 and β4 such that u1t ≥ β3 andu2t ≥ β4 for allt ∈ Iω, we consider the following two cases.

Case 1. Ifu₂η2 ≥ u₁η1, then from3.18,3.23,3.32,H4, and monotonicity of functiongx, we have

g

e^u²^η²

≤g e^u¹^η¹

≤ 1 ω

_κω

κ

g

t,e^u¹^t−τ¹^t Δt

≤ e^u²^η² ω

_κω

κ

h

t,e^u¹^t,e^u²^t

e^−u¹^tΔt≤c₀e^u²^η².

3.33

FromH3,H5, and the continuity of functiongx, one can easily see that there exists a positive constantα₂such that

gα2−c0α2 0. 3.34

Then, from3.33,3.34, and the monotonicity of functiongx −c₀x, we have u2

η2

≥ lnα2. 3.35

By Lemma2.5, we obtain from3.22and3.35that for allt∈I_ω u2t≥u2

η2

− _κω

κ

u^Δ₂tΔt≥lnα2− Cc

ω: β¹₄. 3.36

By Lemma2.5, we also obtain from3.21,3.26that for allt∈Iω

u1t≥u1

η1

− _κω

κ

u^Δ₁tΔt≥β¹₄−ln c

d

−

G0 g0

ω: β¹₃. 3.37

Case 2. Ifu₂η2 < u₁η1, then from3.18,3.23,3.32,H4, and monotonicity of functiongx, we have

g

e^u¹^η¹

≤ 1 ω

_κω

κ

g

t,e^u¹^t−τ¹^t

Δt≤ e^u²^η² ω

_κω

κ

h

t,e^u¹^t,e^u²^t

e^−u¹^t Δt

≤c₀e^u²^η²≤c₀e^u¹^η¹.

3.38

Then, from3.34and the monotonicity of functiongx−c0x, we have u₁

η₁

≥lnα₂. 3.39

By Lemma2.5, we obtain from3.21,3.39that for allt∈Iω

u₁t≥u₁ η₁

− _κω

κ

u^Δ₁tΔt≥lnα₂−

G0 g0

ω: β₃². 3.40

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By Lemma2.5, we also obtain from3.22,3.25, and3.40that for allt∈Iω

u₂t≥u₂ η₂

− _κω

κ

u^Δ₂tΔt≥β²₃ln c

d

− Cc

ω: β²₄. 3.41

Now, we take β3 : min{β¹₃, β₃²}and β4 : min{β¹₄, β₄²}. Then it follows from3.36,3.37, 3.40, and3.41that for allt∈I_ω,u₁t≥ β₃andu₂t ≥β₄. Hence from these,3.30, and 3.31, we have

sup

t∈Iω

|u1t| ≤maxβ₁,β₃: β₅, sup

t∈Iω

|u1t| ≤maxβ₂,β₄: β₆. 3.42 Clearly,β5andβ6are independent ofλ.

On the other hand, forμ∈0,1, we consider the following algebraic system:

geû¹−μheû¹,eû²eû²^−u¹ 0,

c−de ^u²^−u¹ 0,

3.43

whereu1, u2^T∈R². From the second equation of3.43andH6, we have u2 u1ln

c d

. 3.44

From the first equation of3.43andH4, we also have

geû¹ μheû¹,eû²eû²^−u¹ ≥0. 3.45 Then, from3.44and the monotonicity of functiongx, we obtain

u₁≤ lnα₁. 3.46

Substituting3.44into the first equation of3.43, we can get fromH4,3.30,3.32, and 3.46that

ge^u¹ μc

dhe^u¹,e^u²≤ c

dhe^u¹,e^u²≤ c d

a0e^m−1u¹a1e^m−2u¹· · ·a_m−1 e^u¹

≤ c d

a₀e^m−1β¹a₁e^m−2β¹· · ·a_m−1

e^u¹ c₀c de^u¹.

3.47

In view ofH3,H5, and the continuity of functiongx, it is easy to see that there exists a positive constantα3such that

gα3−c₀c

dα₃ 0. 3.48

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Then, from3.47,3.48, and the monotonicity of functiongx, we obtain

u1≥ lnα3. 3.49

It follows from3.44,3.46, and3.49that

|u1| ≤ max{|lnα1|,|lnα3|}: β7, |u2| ≤β7lnc−lnd: β8. 3.50 Clearly,β₇andβ₈are also independent ofμ.

We takeΩ {u u1t, u2t^T ∈ X | u < β}, hereβ β5β6β7β8. Now we check the conditions of Lemma2.4.

aBy3.42, one can conclude that for eachλ∈0,1,u∈∂Ω ∩DomL,Lu /λNu.

bWhen u1t, u2t^T ∈ ∂Ω ∩ KerL, u1t, u2t^T is a constant vector inR², we denote it byu1, u2^Tand|u1||u2| β. If

QNu

ge û¹−heû¹,eû²eû²^−u¹

c−de ^u²^−u¹

0, 3.51

thenu1, u₂^Tis a constant solution of system3.43withμ 1. By3.50, we have|u1||u2| ≤ β7β8< β. This contradiction implies that for eachu∈∂Ω∩KerL, QNu /0.

cIn order to verify the conditioncin Lemma2.4, we defineφ:R²×0,1 → R²by

φ

u₁, u₂, μ ge^u¹

c−de ^u²^−u¹

μ

−heû¹,eû²eû²^−u¹ 0

, 3.52

whereμ ∈ 0,1is a parameter. Whenu1, u2^T ∈ ∂Ω∩KerL,u1, u2^T is a constant vector inR²with|u1||u2| β. By3.50we knowφu1, u₂, μ/ 0,0^T on∂Ω∩KerL. Thus,φis a homotopy mapping. Moreover, it is not diﬃcult to see that the following algebraic system:

ge^u¹ 0, c−de ^u²^−u¹ 0, 3.53

has a unique solutionu^†₁, u^†₂^T ∈ ∂Ω∩KerL. So, due to homotopy invariance theorem of topology degree and takingJ I: ImQ → KerL,u1, u2^T → u1, u2^T, we obtain

deg

JQN,Ω∩KerL,0,0^T

deg

φu1, u₂,1,Ω∩KerL,0,0^T deg

φu1, u₂,0,Ω∩KerL,0,0^T

sign

−g e^u^†¹

e^u^†² /0.

3.54 By now we have proved thatΩsatisfies all the requirements in Lemma2.4. Hence, system 1.1has at least oneω-periodic solution. This completes the proof.

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Noticing that both systems 1.2 and 1.3 are special cases of system 1.1, by Theorem3.1, we can obtain the following results.

Theorem 3.2. Under the assumptions (H1)–(H₄), system1.2has at least one positive ω-periodic solution if and only if (H5) and (H6) hold.

Theorem 3.3. Under the assumptions (H1)–(H₄), system1.3has at least one positive ω-periodic solution if and only if (H₅) and (H₆) hold.

The proof of Theorem3.1shows that it remains valid for the following periodic semi- ratio-dependent predator-prey system on a time scale:

u^Δ₁t g

t,e^u¹^t−τ¹^t

−h

t,e^u¹^t,e^u²^t

e^u²^t−τ²^t−u¹^t, u^Δ₂t ct−dte^u²^t−τ³^t−u¹^t−τ³^t.

3.55

xt xtgt, xt−τ1t−h

t, xt, yt

yt−τ2t, yt yt

ct−dtyt−τ3t xt−τ3t

, 3.56

xk1 xkexp

gk, xk−τ₁k−h

k, xk, ykyk−τ2k xk

, yk1 ykexp

ck−dkyk−τ₃k xk−τ₃k

,

3.57

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 Theorem3.1. Therefore, by Theorem3.1, the system1.1with 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 oneω-periodic solution.

Remark 3.5. Similarly, by Theorems3.2and3.3, the systems1.2and1.3with 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 positiveω-periodic solution, respectively.

Remark 3.6. Bohner et al.19, Fazly and Hesaaraki21studied the special cases of system 1.1forτ₁t τ₂t τ₃t 0,gt, x at−btx, andht, x, y pt, x. Zhuang26 studied the special case of system1.1forτ₂t 0, gt, x at−btx, andht, x, y ktx/m²x². Therefore, our Theorem 3.1generalizes and improves Theorem 3.4 in19 and Theorem 3.1 in26and generalizes Theorem 1 in21.

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Remark 3.7. Wang et al.13studied the special case of system1.2forτ1t τ2t τ3t 0,gt, x at−btx, andht, x, y pt, x. Ding et al.10studied the special case of system1.2forτ₂t τ₃t 0,gt, x at−btx, andht, x, y ktx/m²x². Ding and Jiang8studied the special case of system1.2forht, x, y pt, x. Liu11studied the special case of system1.2forτ1t τ2t τ3t 0 andgt, x at−btx. Liu and Huang12studied the special case of system3.56forτ₁t τ₂t 0 andgt, x at−btx. Ding and Jiang9studied the special case of system3.56forτ1t 0. Therefore, our Theorem3.2generalizes and improves Theorem 3.3 in13and Theorem 2.1 in10and generalizes Theorem 2.2 in8, Theorem 2.2 in9, Theorem 2.1 in11, and Corollary 3.1 in 12.

Remark 3.8. Fan and Wang14, Fazly and Hesaaraki15studied the special cases of system 1.3forτ₁k τ₂k τ₃k 0,gk, x ak−bkx, andhk, x, y pk, x. Liu16 studied the special case of system1.3forτ1k τ2k τ3k 0 andgk, x ak−bkx.

Therefore, our Theorem3.3generalizes and improves Theorem 2.1 in14and generalizes Theorem 1 in15and Theorem 2.1 in16.

Acknowledgments

This work is supported by the Key Programs for Science and Technology of the Education Department of Henan Province under Grant 12A110007, and the Scientific Research Start-up Funds of Henan University of Science and Technology.

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