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

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.

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 scale 𝕋: 𝑢^Δ₁(𝑡)=𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}−ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)},𝑢^Δ₂(𝑡)=𝑐(𝑡)−𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}. (1.1)

Here 𝕋 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: 𝑥^′(𝑡)=𝑥(𝑡)𝑔𝑡,𝑥𝑡−𝜏₁(𝑡)−ℎ𝑡,𝑥(𝑡),𝑦(𝑡)𝑦𝑡−𝜏₂(𝑡),𝑦^′(𝑡)(𝑡𝑐(𝑡)(𝑡)𝑦(𝑡)=𝑦)−𝑑𝑥𝑡−𝜏₃(𝑡), (1.2) 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 𝑥(𝑘+1)=𝑥(𝑘)exp𝑔𝑘,𝑥𝑘−𝜏₁(𝑘)(𝑘)(𝑘)𝑦−ℎ𝑘,𝑥,𝑦𝑘−𝜏₂(𝑘)𝑥(𝑘),𝑦()(𝑘)𝑐(𝑘)(𝑘)𝑦(𝑘)𝑘+1=𝑦exp−𝑑𝑥𝑘−𝜏₃(𝑘), (1.3) 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 scale 𝕋 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.

Let 𝜔>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.

For 𝑡∈𝕋, the forward and backward jump operators 𝜎,𝜌∶𝕋→𝕋 are defined by 𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡} (2.1) respectively.

If 𝜎(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 function 𝑓∶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 𝐶_rd(𝕋).

For 𝑓∶𝕋→ℝ and 𝑡∈𝕋 we define 𝑓^Δ(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 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]≤𝜀|𝜎(𝑡)−𝑠|,∀𝑠∈𝑈. (2.2)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 𝐶¹_rd(𝕋).

A function 𝐹∶𝕋→ℝ is called a delta-antiderivative of 𝑓∶𝕋→ℝ provided 𝐹^Δ(𝑡)=𝑓(𝑡), for all 𝑡∈𝕋. Then, we define the delta integral by ^𝑏_𝑎𝑓(𝑡)Δ𝑡=𝐹(𝑏)−𝐹(𝑎),∀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. If 𝑎, 𝑏, 𝑐∈𝕋, 𝛼, 𝛽∈ℝ and 𝑓, 𝑔∈𝐶_rd(𝕋), then (a)∫^𝑏_𝑎[𝛼𝑓(𝑡)+𝛽𝑔(𝑡)]Δ𝑡=𝛼∫^𝑏_𝑎𝑓(𝑡)Δ𝑡+𝛽∫^𝑏_𝑎𝛽𝑔(𝑡)Δ𝑡, (b)∫^𝑏_𝑎𝑓(𝑡)Δ𝑡=∫^𝑐_𝑎𝑓(𝑡)Δ𝑡+∫^𝑏_𝑐𝑓(𝑡)Δ𝑡, (c) if 𝑓(𝑡)≥0 for all 𝑎≤𝑡<𝑏, then ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡≥0, (d) if |𝑓(𝑡)|≤𝑔(𝑡) on [𝑎,𝑏)∶={𝑡∈𝕋∶𝑎≤𝑡<𝑏}, then |∫^𝑏_𝑎𝑓(𝑡)Δ𝑡|≤∫^𝑏_𝑎𝑔(𝑡)Δ𝑡.

Next, let us recall the continuation theorem in coincidence degree theory. To do so, we need to introduce the following notation.

Let 𝑋, 𝑌 be real Banach spaces, let 𝐿∶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 dimKer𝐿=codimIm𝐿<+∞ and Im𝐿 is closed in 𝑌.

If 𝐿 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 𝐿_𝑃 of 𝐿 to Dom𝐿∩Ker𝑃∶(𝐼−𝑃)𝑋→Im𝐿 is invertible. Denote the inverse of 𝐿_𝑃 by 𝐾_𝑃.

The mapping 𝑁 is said to be 𝐿-compact on Ω, if Ω is an open bounded subset of 𝑋, 𝑄𝑁(Ω) is bounded, and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact.

Since Im𝑄 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 2.4 2.4 (continuation theorem [27, page 40]). Let Ω⊂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 each 𝜆∈(0,1),𝑥∈𝜕Ω∩Dom𝐿,𝐿𝑥≠𝜆𝑁𝑥; (b)for each 𝑥∈𝜕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: 𝜅=min{[0,+∞)∩𝕋},𝐼_𝜔=[]1𝜅,𝜅+𝜔∩𝕋,𝑎=𝜔_𝜅^𝜅+𝜔𝑎(𝑡)1Δ𝑡,𝐴=𝜔_𝜅^𝜅+𝜔|𝑎(𝑡𝑏̃(𝑥)=1)|Δ𝑡,𝜔_𝜅^𝜅+𝜔𝑏()𝐵(𝑥)=1𝑡,𝑥Δ𝑡,𝜔_𝜅^𝜅+𝜔|𝑏(𝑡,𝑥)|Δ𝑡,𝜑=1𝑥,𝑦𝜔_𝜅^𝜅+𝜔𝜑𝑡,𝑥,𝑦Δ𝑡, (2.4) where 𝑎∈𝐶_rd(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 scale 𝕋, we now give the following inequality which is proved in [19, Lemma 2.4].

Lemma 2.5. Let 𝑡₁, 𝑡₂∈𝐼_𝜔 and 𝑡∈𝕋. If 𝜑∈𝐶¹_rd(𝕋) is an 𝜔-periodic real function, then 𝜑(𝑡)≤𝜑𝑡₁+_𝜅^𝜅+𝜔𝜑^Δ(𝑡)Δ𝑡,𝜑(𝑡)≥𝜑𝑡₂−_𝜅^𝜅+𝜔𝜑^Δ(𝑡)Δ𝑡. (2.5)

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)𝜏_𝑖∶𝕋→ℝ⁺ is rd-continuous and 𝜔-periodic such that 𝑡−𝜏_𝑖(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, lim_𝑥→+∞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 𝑎_𝑖∶𝕋→ℝ⁺, 𝑖=0,1,…,𝑚−1, such that ℎ𝑡,𝑥,𝑦≤𝑎₀(𝑡)𝑥^𝑚+𝑎₁(𝑡)𝑥^𝑚−1+⋯+𝑎_𝑚−1(𝑡)𝑥,∀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, 𝑔(𝑡,𝑥) can be taken as the logistic growth 𝑎−𝑏𝑥, the Gilpin growth 𝑎−𝑏𝑥^𝜃, 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 𝑚𝑥^𝑛/(𝐴+𝑥^𝑛)(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 𝑔^′(𝑥)=1𝜔_𝜅^𝜅+𝜔𝜕𝑔()𝜕𝑥𝑡,𝑥Δ𝑡<0,lim_𝑥→+∞(𝑥)=1𝑔𝜔_𝜅^𝜅+𝜔lim_𝑥→+∞𝑔(𝑡,𝑥)Δ𝑡<0. (3.2) 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)𝑐̂>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 _𝜅^𝜅+𝜔𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}−ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)}Δ𝑡=_𝜅^𝜅+𝜔𝑢^Δ₁(𝑡)Δ𝑡=0,_{𝜅𝜅+𝜔}𝑐(𝑡)−𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}Δ𝑡=_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡=0. (3.3)(3.4) By (H₄) and the monotonicity of function 𝑔(𝑥), we obtain from (3.3) that (0)>1𝑔𝜔_𝜅^𝜅+𝜔𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}1Δ𝑡=𝜔_𝜅^𝜅+𝜔ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)}Δ𝑡≥0, (3.5) which is (H₅).
By (H₂) and (3.4), we have 1𝑐̂=𝜔_𝜅^𝜅+𝜔𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}Δ𝑡>0, (3.6) which gives (H₆).
“If” part: Take 𝑋=𝑌=𝑢=𝑢₁(𝑡),𝑢₂(𝑡)^𝑇∣𝑢_𝑖∈𝐶_rd(𝕋),𝑢_𝑖(𝑡+𝜔)=𝑢_𝑖(𝑡),‖𝑢‖=𝑢,𝑖=1,2₁(𝑡),𝑢₂(𝑡)^𝑇=max_{𝑡∈𝐼𝜔}𝑢₁(𝑡)+max_{𝑡∈𝐼𝜔}𝑢₂(𝑡) (3.7) Then 𝑋 and 𝑌 are Banach spaces with the norm ‖⋅‖. Set 𝑢𝐿∶Dom𝐿⊂𝑋⟶𝑌,𝐿¹(𝑡)𝑢₂(𝑡)=𝑢^Δ₁(𝑡)𝑢^Δ₂(𝑡), (3.8) where Dom𝐿={𝑢=(𝑢₁(𝑡),𝑢₂(𝑡))^𝑇∈𝑋∣𝑢_𝑖∈𝐶¹_rd(𝕋),𝑖=1,2} and 𝑢𝑁∶𝑋⟶𝑌,𝑁¹(𝑡)𝑢₂(𝑡)=𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}−ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)}𝑐(𝑡)−𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}. (3.9) With these notations system (1.1) can be written in the form 𝐿𝑢=𝑁𝑢,𝑢∈𝑋. (3.10) Obviously, Ker𝐿=ℝ², Im𝐿={(𝑢₁(𝑡),𝑢₂(𝑡))^T∈𝑌∶∫_𝜅^𝜅+𝜔𝑢_𝑖(𝑡)Δ𝑡=0,𝑖=1,2} is closed in 𝑌, and dimKer𝐿=codimIm𝐿=2. Therefore 𝐿 is a Fredholm mapping of index zero. Now define two projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑌→𝑌 as 𝑃𝑢₁(𝑡)𝑢₂(𝑡)𝑢=𝑄¹(𝑡)𝑢₂(𝑡)=𝑢₁𝑢₂,𝑢₁(𝑡)𝑢₂(𝑡)∈𝑋=𝑌. (3.11) Then 𝑃 and 𝑄 are continuous projectors such that Im𝑃=Ker𝐿,Ker𝑄=Im𝐿=Im(𝐼−𝑄). (3.12) Furthermore, through an easy computation we find that the generalized inverse 𝐾_𝑃 of 𝐿_𝑃 has the form 𝐾_𝑃𝐾∶Im𝐿⟶Dom𝐿∩Ker𝑃,_𝑃(𝑢)=^𝑡_𝜅𝑢(𝑠)1Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝑢(𝑠)Δ𝑠Δ𝑡. (3.13) Then 𝑄𝑁∶𝑋→𝑌 and 𝐾_𝑃(𝐼−𝑄)𝑁∶𝑋→𝑋 read as 1𝑄𝑁𝑢=𝜔∫_𝜅^𝜅+𝜔𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}−ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)}∫Δ𝑡_{𝜅𝜅+𝜔}𝑐(𝑡)−𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}𝐾Δ𝑡_𝑃,()∫𝐼−𝑄𝑁𝑢=^𝑡𝜅𝑔𝑠,e^{𝑢1(𝑠−𝜏1(𝑠))}−ℎ𝑠,e^𝑢1(𝑠),e^𝑢2(𝑠)e^{𝑢2(𝑠−𝜏2(𝑠))−𝑢1(𝑠)}∫Δ𝑠_𝑡𝜅𝑐(𝑠)−𝑑(𝑠)e^{𝑢2(𝑠)−𝑢1(𝑠−𝜏3(𝑠))}−1Δ𝑠𝜔1𝑡−𝜅−𝜔_𝜅^𝜅+𝜔()×∫𝑠−𝜅Δ𝑠_{𝜅𝜅+𝜔}𝑔𝑠,e^{𝑢1(𝑠−𝜏1(𝑠))}−ℎ𝑠,e^𝑢1(𝑠),e^𝑢2(𝑠)e^{𝑢2(𝑠−𝜏2(𝑠))−𝑢1(𝑠)}∫Δ𝑠_{𝜅𝜅+𝜔}𝑐(𝑠)−𝑑(𝑠)e^{𝑢2(𝑠)−𝑢1(𝑠−𝜏3(𝑠))}−1Δ𝑠𝜔∫_𝜅^𝜅+𝜔∫^𝑡_𝜅𝑔𝑠,e^{𝑢1(𝑠−𝜏1(𝑠))}−ℎ𝑠,e^𝑢1(𝑠),e^𝑢2(𝑠)e^{𝑢2(𝑠−𝜏2(𝑠))−𝑢1(𝑠)}∫Δ𝑠Δ𝑡_{𝜅𝜅+𝜔}∫^𝑡_𝜅𝑐(𝑠)−𝑑(𝑠)e^{𝑢2(𝑠)−𝑢1(𝑠−𝜏3(𝑠))}.Δ𝑠Δ𝑡 (3.14)(3.15) Clearly, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous. By using the Arzela-Ascoli theorem, it is not difficult to prove that 𝐾_𝑃(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 Lemma 2.4, we need to find appropriate open, bounded subsets in 𝑋. Corresponding to the operator equation 𝐿𝑢=𝜆𝑁𝑢,𝜆∈(0,1), we have 𝑢^Δ₁(𝑡)=𝜆𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}−ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)},𝑢^Δ₂(𝑡)=𝜆𝑐(𝑡)−𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}. (3.16) 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 _𝜅^𝜅+𝜔𝜆𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}−ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)}Δ𝑡=_𝜅^𝜅+𝜔𝑢^Δ₁(𝑡)Δ𝑡=0,_{𝜅𝜅+𝜔}𝜆𝑐(𝑡)−𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}Δ𝑡=_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡=0. (3.17) That is _𝜅^𝜅+𝜔𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}Δ𝑡=_𝜅^𝜅+𝜔ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)}Δ𝑡_{𝜅𝜅+𝜔}𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}Δ𝑡=_𝜅^𝜅+𝜔𝑐(𝑡)Δ𝑡=𝑐𝜔̂. (3.18)(3.19) From (3.18), we have 𝑔(0)𝜔=_𝜅^𝜅+𝜔𝑔(𝑡,0)−𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}+ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)}Δ𝑡. (3.20) It follows from (3.16), (3.18), (3.19), (3.20), and (H₂)–(H₄) that _𝜅^𝜅+𝜔𝑢^Δ₁(𝑡)Δ𝑡≤𝜆_𝜅^𝜅+𝜔𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}−ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)}<Δ𝑡_{𝜅𝜅+𝜔}𝑔(𝑡,0)−𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}+ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)}+Δ𝑡_{𝜅𝜅+𝜔}𝑔(=𝐺(0)(0𝑡,0)Δ𝑡+𝑔)𝜔,_{𝜅𝜅+𝜔}𝑢^Δ₂(𝑡)Δ𝑡≤𝜆_𝜅^𝜅+𝜔𝑐(𝑡)−𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}<Δ𝑡_{𝜅𝜅+𝜔}|𝑐(𝑡)|Δ𝑡+_𝜅^𝜅+𝜔𝑑(𝑡)e^{𝑢2(𝑡)−𝑢1(𝑡−𝜏3(𝑡))}=Δ𝑡𝐶+𝑐̂𝜔. (3.21)(3.22)
Since (𝑢₁(𝑡),𝑢₂(𝑡))^T∈𝑋, there exist 𝜉_𝑖,𝜂_𝑖∈𝐼_𝜔 such that 𝑢_𝑖𝜉_𝑖=min_{𝑡∈𝐼𝜔}𝑢_𝑖(𝑡),𝑢_𝑖𝜂_𝑖=max_{𝑡∈𝐼𝜔}𝑢_𝑖(𝑡),𝑖=1,2. (3.23) Then from (3.19) and (H₂), we have 1𝑐̂≤𝜔_𝜅^𝜅+𝜔𝑑(𝑡)e^{𝑢2(𝜂2)−𝑢1(𝜉1)}Δ𝑡=𝑑e^{𝑢2(𝜂2)−𝑢1(𝜉1)},1𝑐̂≥𝜔_𝜅^𝜅+𝜔𝑑(𝑡)e^{𝑢2(𝜉2)−𝑢1(𝜂1)}Δ𝑡=𝑑e^{𝑢2(𝜉2)−𝑢1(𝜂1)}. (3.24) These, together with (H₆), yield 𝑢₂𝜂₂≥ln𝑐̂𝑑+𝑢₁𝜉₁,𝑢₂𝜉₂≤ln𝑐̂𝑑+𝑢₁𝜂₁. (3.25)(3.26) From (3.18), (H₄), and the monotonicity of function 𝑔(𝑥), we have 𝑔e^𝑢1(𝜉1)≥1𝜔_𝜅^𝜅+𝜔𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}Δ𝑡≥0. (3.27) 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 𝑔𝛼₁=0. (3.28) Then, from (3.27), (3.28), and the monotonicity of function 𝑔(𝑥), we have 𝑢₁𝜉₁≤ln𝛼₁. (3.29) By Lemma 2.5, we obtain from (3.21) and (3.29) that for all 𝑡∈𝐼_𝜔𝑢₁(𝑡)≤𝑢₁𝜉₁+_𝜅^𝜅+𝜔𝑢^Δ₁(𝑡)Δ𝑡≤ln𝛼₁+𝐺(0)+𝑔(0)𝜔∶=𝛽₁. (3.30) By Lemma 2.5, we also obtain from (3.22), (3.26), and (3.30) that for all 𝑡∈𝐼_𝜔𝑢₂(𝑡)≤𝑢₂𝜉₂+_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡≤𝛽₁+ln𝑐̂𝑑+𝐶+𝑐̂𝜔∶=𝛽₂. (3.31) It follows from (H₄)and (3.30) that 10≤𝜔_𝜅^𝜅+𝜔ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{−𝑢1(𝑡)}Δ𝑡≤𝑎₀e^{(𝑚−1)𝛽1}+𝑎₁e^{(𝑚−2)𝛽1}+⋯+𝑎_𝑚−1∶=𝑐₀. (3.32)
In order to obtain 𝛽₃ and 𝛽₄ such that 𝑢₁(𝑡)≥𝛽₃ and 𝑢₂(𝑡)≥𝛽₄ for all 𝑡∈𝐼_𝜔, 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 𝑔e^𝑢2(𝜂2)≤𝑔e^𝑢1(𝜂1)≤1𝜔_𝜅^𝜅+𝜔𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}≤eΔ𝑡_𝑢2(𝜂2)𝜔_𝜅^𝜅+𝜔ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{−𝑢1(𝑡)}Δ𝑡≤𝑐₀e^𝑢2(𝜂2). (3.33) From (H₃), (H₅), and the continuity of function 𝑔(𝑥), one can easily see that there exists a positive constant 𝛼₂ such that 𝑔𝛼₂−𝑐₀𝛼₂=0. (3.34) Then, from (3.33), (3.34), and the monotonicity of function 𝑔(𝑥)−𝑐₀𝑥, we have 𝑢₂𝜂₂≥ln𝛼₂. (3.35) By Lemma 2.5, we obtain from (3.22) and (3.35) that for all 𝑡∈𝐼_𝜔𝑢₂(𝑡)≥𝑢₂𝜂₂−_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡≥ln𝛼₂−𝐶+𝑐̂𝜔∶=𝛽¹₄. (3.36) By Lemma 2.5, we also obtain from (3.21), (3.26) that for all 𝑡∈𝐼_𝜔𝑢₁(𝑡)≥𝑢₁𝜂₁−_𝜅^𝜅+𝜔𝑢^Δ₁(𝑡)Δ𝑡≥𝛽¹₄−ln𝑐̂𝑑−𝐺(0)+𝑔(0)𝜔∶=𝛽¹₃. (3.37)
Case  2. If 𝑢₂(𝜂₂)<𝑢₁(𝜂₁), then from (3.18), (3.23), (3.32), (H₄), and monotonicity of function 𝑔(𝑥), we have 𝑔e^𝑢1(𝜂1)≤1𝜔_𝜅^𝜅+𝜔𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}eΔ𝑡≤^𝑢2(𝜂2)𝜔_𝜅^𝜅+𝜔ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{−𝑢1(𝑡)}Δ𝑡≤𝑐₀e^𝑢2(𝜂2)≤𝑐₀e^𝑢1(𝜂1). (3.38) Then, from (3.34) and the monotonicity of function 𝑔(𝑥)−𝑐₀𝑥, we have 𝑢₁𝜂₁≥ln𝛼₂. (3.39) By Lemma 2.5, we obtain from (3.21), (3.39) that for all 𝑡∈𝐼_𝜔𝑢₁(𝑡)≥𝑢₁𝜂₁−_𝜅^𝜅+𝜔𝑢^Δ₁(𝑡)Δ𝑡≥ln𝛼₂−𝐺(0)+𝑔(0)𝜔∶=𝛽²₃. (3.40) By Lemma 2.5, we also obtain from (3.22), (3.25), and (3.40) that for all 𝑡∈𝐼_𝜔𝑢₂(𝑡)≥𝑢₂𝜂₂−_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡≥𝛽²₃+ln𝑐̂𝑑−𝐶+𝑐̂𝜔∶=𝛽²₄. (3.41) Now, we take 𝛽₃=∶min{𝛽¹₃,𝛽²₃} and 𝛽₄=∶min{𝛽¹₄,𝛽²₄}. Then it follows from (3.36), (3.37), (3.40), and (3.41) that for all 𝑡∈𝐼_𝜔, 𝑢₁(𝑡)≥𝛽₃ and 𝑢₂(𝑡)≥𝛽₄. Hence from these, (3.30), and (3.31), we have sup_{𝑡∈𝐼𝜔}𝑢₁(𝑡)≤max𝛽₁,𝛽₃∶=𝛽₅,sup_{𝑡∈𝐼𝜔}𝑢₁(𝑡)≤max𝛽₂,𝛽₄∶=𝛽₆. (3.42) Clearly, 𝛽₅ and 𝛽₆ are independent of 𝜆.
On the other hand, for 𝜇∈[0,1], we consider the following algebraic system: 𝑔e^𝑢1−𝜇ℎe^𝑢1,e^𝑢2e^{𝑢2−𝑢1}=0,𝑐̂−𝑑e^{𝑢2−𝑢1}=0, (3.43) where (𝑢₁,𝑢₂)^𝑇∈ℝ². From the second equation of (3.43) and (H₆), we have 𝑢₂=𝑢₁+ln𝑐̂𝑑. (3.44) From the first equation of (3.43) and (H₄), we also have 𝑔e^𝑢1=𝜇ℎe^𝑢1,e^𝑢2e^{𝑢2−𝑢1}≥0. (3.45) Then, from (3.44) and the monotonicity of function 𝑔(𝑥), we obtain 𝑢₁≤ln𝛼₁. (3.46) Substituting (3.44) into the first equation of (3.43), we can get from (H₄), (3.30), (3.32), and (3.46) that 𝑔e^𝑢1=𝜇𝑐̂𝑑ℎe^𝑢1,e^𝑢2≤𝑐̂𝑑ℎe^𝑢1,e^𝑢2≤𝑐̂𝑑𝑎₀e^{(𝑚−1)𝑢1}+𝑎₁e^{(𝑚−2)𝑢1}+⋯+𝑎_𝑚−1e^𝑢1≤𝑐̂𝑑𝑎₀e^{(𝑚−1)𝛽1}+𝑎₁e^{(𝑚−2)𝛽1}+⋯+𝑎_𝑚−1e^𝑢1=𝑐₀𝑐̂𝑑e^𝑢1. (3.47) 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 𝑔𝛼₃−𝑐₀𝑐̂𝑑𝛼₃=0. (3.48) Then, from (3.47), (3.48), and the monotonicity of function 𝑔(𝑥), we obtain 𝑢₁≥ln𝛼₃. (3.49) It follows from (3.44), (3.46), and (3.49) that 𝑢₁≤maxln𝛼₁,ln𝛼₃∶=𝛽₇,𝑢₂≤𝛽₇+ln𝑐̂−ln𝑑∶=𝛽₈. (3.50) Clearly, 𝛽₇ and 𝛽₈ are also independent of 𝜇.
We take Ω={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 e𝑄𝑁𝑢=𝑔^𝑢1−ℎe^𝑢1,e^𝑢2e^{𝑢2−𝑢1}𝑐̂−𝑑e^{𝑢2−𝑢1}=0, (3.51) 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 Lemma 2.4, we define 𝜙∶ℝ²×[0,1]→ℝ² by 𝜙𝑢₁,𝑢₂=e,𝜇𝑔^𝑢1𝑐̂−𝑑e^{𝑢2−𝑢1}−+𝜇ℎe^𝑢1,e^𝑢2e^{𝑢2−𝑢1}0, (3.52)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: 𝑔e^𝑢1=0,𝑐̂−𝑑e^{𝑢2−𝑢1}=0, (3.53) has a unique solution (𝑢^†₁,𝑢^†₂)^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 deg𝐽𝑄𝑁,Ω∩Ker𝐿,(0,0)^T=deg𝜙𝑢₁,𝑢₂,1,Ω∩Ker𝐿,(0,0)^T=deg𝜙𝑢₁,𝑢₂,0,Ω∩Ker𝐿,(0,0)^T=sign−𝑔e^𝑢†1e^𝑢†2≠0. (3.54) 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 Theorem 3.1 shows that it remains valid for the following periodic semi-ratio-dependent predator-prey system on a time scale: 𝑢^Δ₁(𝑡)=𝑔𝑡,e^{𝑢1(𝑡−𝜏1(𝑡))}−ℎ𝑡,e^𝑢1(𝑡),e^𝑢2(𝑡)e^{𝑢2(𝑡−𝜏2(𝑡))−𝑢1(𝑡)},𝑢^Δ₂(𝑡)=𝑐(𝑡)−𝑑(𝑡)e^{𝑢2(𝑡−𝜏3(𝑡))−𝑢1(𝑡−𝜏3(𝑡))}. (3.55)𝑥^′(𝑡)=𝑥(𝑡)𝑔𝑡,𝑥𝑡−𝜏₁(𝑡)−ℎ𝑡,𝑥(𝑡),𝑦(𝑡)𝑦𝑡−𝜏₂(𝑡),𝑦^′(𝑡)(𝑡𝑐(𝑡)(𝑡)𝑦=𝑦)−𝑑𝑡−𝜏₃(𝑡)𝑥𝑡−𝜏₃(𝑡), (3.56)

𝑥(𝑘+1)=𝑥(𝑘)exp𝑔𝑘,𝑥𝑘−𝜏₁(𝑘)(𝑘)(𝑘)𝑦−ℎ𝑘,𝑥,𝑦𝑘−𝜏₂(𝑘)𝑥(𝑘),𝑦()(𝑘)𝑐(𝑘)(𝑘)𝑦𝑘+1=𝑦exp−𝑑𝑘−𝜏₃(𝑘)𝑥𝑘−𝜏₃(𝑘), (3.57)

Remark 3.4. One can easily verify that if their parameters are positive 𝜔-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 Theorems 3.2 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].

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.