Periodic Solutions for Gause-Type Ratio-Dependent Predator-Prey Systems with Delays on Time Scales

Abstract

This paper is devoted to periodic Gause-type ratio-dependent predator-prey systems with monotonic or nonmonotonic numerical responses on time scales. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solutions. In particular, our results improve and generalize some known ones.

1. Introduction

In this paper, we consider the following periodic Gause-type ratio-dependent predator-prey system with time delays on a time scale 𝕋: 𝑢^Δ₁(𝑡)=𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}−𝑝𝑡,e^{𝑢1(𝑡)−𝑢2(𝑡)},𝑢^Δ₂(𝑡)=−𝑑(𝑡)+ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))}. (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.

The theory of dynamic equations on time scales, which was introduced in order to unify differential and difference equations, has recently received a lot of attention, and many results on the issue have been documented in monographs [1, 2] and papers [3–12].

In system (1), set 𝑥(𝑡)=exp[𝑢₁(𝑡)], 𝑦(𝑡)=exp[𝑢₂(𝑡)]. If 𝕋=ℝ, then system (1) reduces to the standard Gause-type ratio-dependent predator-prey system governed by the ordinary differential equations 𝑥^′(𝑡)(𝑡)𝑓(((𝑡(𝑡)𝑔𝑥(𝑡)=𝑥𝑡,𝑥𝑡−𝜎)))−𝑦𝑡,𝑦(𝑡),𝑦^′(𝑡)(𝑡(𝑡)𝑥((𝑡=𝑦)−𝑑+ℎ𝑡,𝑡−𝜏))𝑦((𝑡,𝑡−𝜏)) (2) where 𝑥(𝑡) and 𝑦(𝑡) stand for the population of the prey and the predator, respectively. The function 𝑓(𝑡,𝑣) is the growth rate of the prey in the absence of the predator. The function 𝑑(𝑡) is the death rate of the predator. The function 𝑔(𝑡,𝑣)=∶𝑣𝑝(𝑡,𝑣), called functional response of predator to prey, describes the change in the rate of exploitation of prey by a predator as a result of a change in the prey density. The function ℎ(𝑡,𝑣), called numerical response of predator to prey, describes the change in reproduction rate with changing prey density. If 𝕋=ℤ, then system (1) is reformulated as 𝑥()(𝑘)𝑓(((𝑘𝑥(𝑘)𝑘+1=𝑥exp𝑘,𝑥𝑘−𝜎)))−𝑝𝑘,𝑦(𝑘),𝑦()(𝑘)(𝑘)𝑥((𝑘𝑘+1=𝑦exp−𝑑+ℎ𝑘,𝑘−𝜏))𝑦((𝑘,𝑘−𝜏)) (3) which is the discrete time ratio-dependent predator-prey system and is a discrete analogue of (2).

We note that Ding [13], Ding and Jiang [14], Fan and Li [15], Fan et al. [16], Fan and Wang [17], Fan et al. [18], Hsu et al. [19], Kuang [20], Wang and Li [21], and Xia and Han [22] considered some special cases of system (2). Fan and Wang [23], and Xia et al. [24] discussed some special cases of system (3). Bohner et al. [4], and Shao [10] investigated some special cases of system (1). So far as we know, there is no published paper concerned system (1).

The main purpose of this paper is, by using the coincidence degree theory developed by Gaines and Mawhin [25], to derive sufficient conditions for the existence of periodic solutions of system (1). Furthermore, we will see that our results for the above system can be easily extended to the one with distributed or state-dependent delays. Our results improve or generalize some theorems in [10, 13–18, 21, 23].

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 [1, 2].

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→ℝ are defined by 𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡}, (4) 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(𝑡𝜎(𝑡)|𝜎(𝑡)|)[−𝑠]≤𝜀−𝑠,∀𝑠∈𝑈. (5)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 are 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. (6)

If 𝑓 is continuous, then 𝑓 is rd-continuous, and if 𝑓 is delta-differentiable at 𝑡, then 𝑓 is continuous at 𝑡. Moreover, every rd-continuous function on 𝕋 has a delta-antiderivative.

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, 𝐿∶Dom𝐿⊂𝑋→𝑌 be a linear mapping, and 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 1 (Continuation Theorem [25, page 40]). Let Ω⊂𝑋 be an open bounded set, 𝐿 a Fredholm mapping of index zero, and 𝑁  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, (7) where 𝑎∈𝐶_rd(𝕋) is an 𝜔-periodic function, 𝑏∶𝕋×ℝ→ℝ is rd-continuous and 𝜔-periodic in its first variable.

In order to achieve the prior estimation in the case of dynamic equations on a time scale 𝕋, we also require the following inequality which is proved in [11, Theorem 2.1].

Lemma 2. Let 𝑡₁, 𝑡₂∈𝐼_𝜔, and 𝑡∈𝕋. If 𝜑∈𝐶¹_rd(𝕋) is an 𝜔-periodic real function, then 𝜑(𝑡)≤𝜑𝑡₁+12_𝜅^𝜅+𝜔𝜑^Δ(𝑡𝜑(𝑡)𝑡)Δ𝑡,≥𝜑₂−12_𝜅^𝜅+𝜔𝜑^Δ(𝑡)Δ𝑡. (8)

3. Existence of Periodic Solutions

In this section, we study the existence of periodic solutions of system (1). For the sake of generality, we make the following fundamental assumptions for system (1): (H₁)𝜎,𝜏∶𝕋→ℝ⁺ are rd-continuous and 𝜔-periodic such that 𝑡−𝜎(𝑡),𝑡−𝜏(𝑡)∈𝕋 for all 𝑡∈𝕋.  (H₂)𝑑∶𝕋→ℝ is rd-continuous and 𝜔-periodic such that 𝑑>0.  (H₃)𝑓∶𝕋×ℝ→ℝ is rd-continuous and 𝜔-periodic in the first variable and is continuously differentiable in the second variable such that 𝑓(0)>0, lim_𝑥→+∞𝑓(𝑥)<0, and (𝜕𝑓/𝜕𝑥)(𝑡,𝑥)<0 for all 𝑡∈𝕋, 𝑥>0.  (H₄)𝑝∶𝕋×ℝ→ℝ⁺ is rd-continuous and 𝜔-periodic in the first variable and is continuously differentiable in the second variable.  (H₅)ℎ∶𝕋×ℝ→ℝ⁺ is rd-continuous and 𝜔-periodic in the first variable and is continuously differentiable in the second variable such that ℎ(𝑡,0)=0 for all 𝑡∈𝕋.  (H₆)𝑓(0)>𝑐̂ and sup_𝑥≥0ℎ(𝑥)>𝑑, where 𝑐(𝑡)=∶sup_𝑥>0𝑝(𝑡,𝑥) for all 𝑡∈𝕋.

From (H₃), we have 𝑓^′(𝑥)=1𝜔_𝜅^𝜅+𝜔𝜕𝑓()𝜕𝑥𝑡,𝑥Δ𝑡<0. (9) Thus 𝑓(𝑥) is strictly decreasing on [0,+∞). From this, (H₃), and (H₆), one can easily see that each of the equations 𝑓(𝑥)=1𝜔_𝜅^𝜅+𝜔sup_𝑥>0𝑝(𝑡,𝑥)Δ𝑡,𝑓(𝑥)=0, (10) has a unique positive solution. Let 𝑉₁, 𝑉₂ be their solutions, respectively, then 𝑉₁<𝑉₂.

3.1. Monotone Case

In this section, we study the existence of 𝜔-periodic solution of system (1) when the numerical response function ℎ(𝑡,𝑥) satisfies the following monotone condition:  (H₇)(𝜕ℎ/𝜕𝑥)(𝑡,𝑥)>0 for 𝑡∈𝕋, 𝑥>0.

From (H₅) and (H₇), we have ℎ^′(𝑥)=1𝜔_𝜅^𝜅+𝜔𝜕ℎ()𝜕𝑥𝑡,𝑥Δ𝑡>0. (11) Then ℎ(𝑥) is strictly increasing on [0,+∞). From this, (H₅), and (H₆), one can easily see that the equation ℎ(𝑥)=𝑑 has a unique positive solution. Let 𝑣₀ be its solution.

Theorem 3. Suppose that (H₁)–(H₇) hold. Then system (1) has at least one 𝜔-periodic solution.

Proof. Take 𝑋=𝑌=𝑢=𝑢₁(𝑡),𝑢₂(𝑡)^T∣𝑢_𝑖(𝕋),𝑢∈𝐶_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), (12) 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), (13) where Dom𝐿={𝑢=(𝑢₁(𝑡),𝑢₂(𝑡))^T∈𝑋∣𝑢_𝑖∈𝐶¹_rd(𝕋),𝑖=1,2} and 𝑁𝑢𝑁∶𝑋→𝑌,₁(𝑡)𝑢₂(𝑡)=𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}−𝑝𝑡,e^{𝑢1(𝑡)−𝑢2(𝑡)}−𝑑(𝑡)+ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))}. (14) With these notations system (1) can be written in the form 𝐿𝑢=𝑁𝑢,𝑢∈𝑋. (15) Obviously, Ker𝐿=ℝ², Im𝐿={(𝑢₁(𝑡),𝑢₂(𝑡))^T∈𝑌∶𝑢_𝑖=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, (16) then 𝑃 and 𝑄 are continuous projectors such that Im𝑃=Ker𝐿,Ker𝑄=Im𝐿=Im(𝐼−𝑄). (17) 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. (18) Then 𝑄𝑁∶𝑋→𝑌 and 𝐾_𝑃(𝐼−𝑄)𝑁∶𝑋→𝑋 read as =1𝑄𝑁𝑢𝜔∫_𝜅^𝜅+𝜔𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}−𝑝𝑡,e^{𝑢1(𝑡)−𝑢2(𝑡)}∫Δ𝑡_{𝜅𝜅+𝜔}−𝑑(𝑡)+ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))},𝐾Δ𝑡_𝑃()=𝐼−𝑄𝑁𝑢_𝑡𝜅(𝑠)1𝑁𝑢Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅(𝑠)−1𝑁𝑢Δ𝑠Δ𝑡𝜔1𝑡−𝜅−𝜔_𝜅^𝜅+𝜔(𝑠−𝜅)Δ𝑠_𝜅^𝜅+𝜔𝑁𝑢(𝑠)Δ𝑠. (19) 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 1, we need to find an appropriate open, bounded subset in 𝑋. Corresponding to the operator equation 𝐿𝑢=𝜆𝑁𝑢,𝜆∈(0,1), we have 𝑢^Δ₁(𝑡)=𝜆𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}−𝑝𝑡,e^{𝑢1(𝑡)−𝑢2(𝑡)},𝑢^Δ₂(𝑡)=𝜆−𝑑(𝑡)+ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))}. (20) Suppose that (𝑢₁(𝑡),𝑢₂(𝑡))^T∈𝑋 is a solution of (20) for a certain 𝜆∈(0,1). Integrating (20) on both sides from 𝜅 to 𝜅+𝜔 leads to _𝜅^𝜅+𝜔𝜆𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}−𝑝𝑡,e^{𝑢1(𝑡)−𝑢2(𝑡)}Δ𝑡=0,_{𝜅𝜅+𝜔}𝜆−𝑑(𝑡)+ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))}Δ𝑡=0. (21) That is _𝜅^𝜅+𝜔𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}Δ𝑡=_𝜅^𝜅+𝜔𝑝𝑡,e^{𝑢1(𝑡)−𝑢2(𝑡)}Δ𝑡,_{𝜅𝜅+𝜔}ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))}Δ𝑡=_𝜅^𝜅+𝜔𝑑(𝑡)Δ𝑡=𝑑𝜔. (22)(23) From (22), we have 𝑓(0)𝜔=_𝜅^𝜅+𝜔𝑓(𝑡,0)−𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}+𝑝𝑡,e^{𝑢1(𝑡)−𝑢2(𝑡)}Δ𝑡. (24) It follows from (20), (22), (23), (24), and (H₃)–(H₅) that _𝜅^𝜅+𝜔𝑢^Δ₁(𝑡)Δ𝑡≤𝜆_𝜅^𝜅+𝜔𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}−𝑚𝑡,e^{𝑢1(𝑡)−𝑢2(𝑡)}Δ𝑡<_𝜅^𝜅+𝜔𝑓(𝑡,0)−𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}+𝑝𝑡,e^{𝑢1(𝑡)−𝑢2(𝑡)}+Δ𝑡_{𝜅𝜅+𝜔}𝑓(𝑡,0)Δ𝑡=𝐹(0)+𝑓(0)𝜔,_{𝜅𝜅+𝜔}𝑢^Δ₂(𝑡)Δ𝑡≤𝜆_𝜅^𝜅+𝜔−𝑑(𝑡)+ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))}<Δ𝑡_{𝜅𝜅+𝜔}|𝑑(𝑡)|Δ𝑡+_𝜅^𝜅+𝜔ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))}=𝑑Δ𝑡𝐷+𝜔. (25)(26) Since (𝑢₁(𝑡),𝑢₂(𝑡))^T∈𝑋, there exist 𝜉_𝑖,𝜂_𝑖∈𝐼_𝜔 such that 𝑢_𝑖𝜉_𝑖=max_{𝑡∈𝐼𝜔}𝑢_𝑖(𝑡),𝑢_𝑖𝜂_𝑖=min_{𝑡∈𝐼𝜔}𝑢_i(𝑡),𝑖=1,2. (27) Then from (22), (27), (H₃), (H₄), and the monotonicity of function 𝑓, we have 𝑓e^𝑢1(𝜉1)=1𝜔_𝜅^𝜅+𝜔𝑓𝑡,e^𝑢1(𝜉1)≤1Δ𝑡𝜔_𝜅^𝜅+𝜔𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}≤1Δ𝑡𝜔_𝜅^𝜅+𝜔sup_𝑥>0𝑝(𝑡,𝑥)Δ𝑡, (28) which, together with the monotonicity of function 𝑓, leads to 𝑢₁𝜉₁≥ln𝑉₁. (29) By Lemma 2, we obtain from (25) and (29) that for all 𝑡∈𝐼_𝜔𝑢₁(𝑡)≥𝑢₁𝜉₁−12_𝜅^𝜅+𝜔𝑢^Δ₁(𝑡)Δ𝑡>ln𝑉₁−𝐹(0)+𝑓(0)2𝜔∶=𝐵₁. (30) From (22), (27), (H₃), and the monotonicity of function 𝑓, we have 𝑓e^𝑢1(𝜂1)=1𝜔_𝜅^𝜅+𝜔𝑓𝑡,e^𝑢1(𝜂1)≥1Δ𝑡𝜔_𝜅^𝜅+𝜔𝑓𝑡,e^{𝑢1(𝑡−𝜎(𝑡))}Δ𝑡>0, (31) which, together with the monotonicity of function 𝑓, leads to 𝑢₁𝜂₁≤ln𝑉₂. (32) By Lemma 2, we get from (25) and (32) that for all 𝑡∈𝐼_𝜔𝑢₁(𝑡)≤𝑢₁𝜂₁+12_𝜅^𝜅+𝜔𝑢^Δ₁(𝑡)Δ𝑡<ln𝑉₂+𝐹(0)+𝑓(0)2𝜔∶=𝐵₂. (33) From (23), (27), (H₅), and the monotonicity of function ℎ, we have ℎe^{𝑢1(𝜉1)−𝑢2(𝜂2)}≥1𝜔_𝜅^𝜅+𝜔ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))}ℎeΔ𝑡=𝑑,_{𝑢1(𝜂1)−𝑢2(𝜉2)}≤1𝜔_𝜅^𝜅+𝜔ℎ𝑡,e^{𝑢1(𝑡−𝜏(𝑡))−𝑢2(𝑡−𝜏(𝑡))}Δ𝑡=𝑑, (34) which, together with the monotonicity of function ℎ, lead to e^{𝑢1(𝜉1)−𝑢2(𝜂2)}≥𝑣₀,e^{𝑢1(𝜂1)−𝑢2(𝜉2)}≤𝑣₀. (35) That is 𝑢₂𝜂₂≤𝑢₁𝜉₁−ln𝑣₀,𝑢₂𝜉₂≥𝑢₁𝜂₁−ln𝑣₀. (36) By Lemma 2, we get from (26), (30), (33), and (36) that for all 𝑡∈𝐼_𝜔𝑢₂(𝑡)≤𝑢₂𝜂₂+12_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡<𝐵₁−ln𝑣₀+𝐷+𝑑2𝜔∶=𝐵₃,𝑢₂(𝑡)≥𝑢₂𝜉₂−12_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡>𝐵₂−ln𝑣₀−𝐷+𝑑2𝜔∶=𝐵₄. (37) In view of (30), (33), (37), we obtain max_{𝑡∈𝐼𝜔}𝑢₁(𝑡)≤max𝐵₁,𝐵₂∶=𝐵₅,max_{𝑡∈𝐼𝜔}𝑢₂(𝑡)≤max𝐵₃,𝐵₄∶=𝐵₆. (38)(39) Clearly, 𝐵₅ and 𝐵₆ are independent of 𝜆.
From (H₃), (H₆), and the monotonicity of functions 𝑓 and ℎ, it is easy to show that the algebraic system 𝑓e^𝑢1−𝑝e^{𝑢1−𝑢2}−ℎe=0,𝑑+^{𝑢1−𝑢2}=0, (40) has a unique solution (𝑢^∗₁,𝑢^∗₂)^T∈ℝ².
We now take Ω={(𝑢₁(𝑡),𝑢₂(𝑡))^T∈𝑋∶‖(𝑢₁(𝑡),𝑢₂(𝑡))^T‖<𝐵₅+𝐵₆+𝐵₀}, where 𝐵₀ is taken sufficiently large such that ‖(𝑢^∗₁,𝑢^∗₂)^T‖=|𝑢^∗₁|+|𝑢^∗₂|<𝐵₀. With the help of (38) and (39), it is easy to see that Ω satisfies condition (a) in Lemma 1. When (𝑢₁(𝑡),𝑢₂(𝑡))^T∈𝜕Ω∩Ker𝐿=𝜕Ω∩ℝ², (𝑢₁(𝑡),𝑢₂(𝑡))^T is a constant vector in ℝ² with |𝑢₁|+|𝑢₂|=𝐵₅+𝐵₆+𝐵₀. Thus, we have 𝑢𝑄𝑁¹𝑢₂=𝑓e^𝑢1−𝑝e^{𝑢1−𝑢2}−𝑑+ℎe^{𝑢1−𝑢2}≠00. (41) This proves that condition (b) in Lemma 1 is satisfied.
Taking 𝐽=𝐼∶Im𝑄→Ker𝐿, (𝑢₁,𝑢₂)^T→(𝑢₁,𝑢₂)^T, a direct calculation shows that ()−𝑓deg𝐽𝑄𝑁,Ω∩Ker𝐿,0=sign^′e^𝑢∗1ℎ^′e^{𝑢∗1−𝑢∗2}e^{2𝑢∗1−𝑢∗2}=1≠0. (42) By now we have proved that Ω satisfies all the requirements in Lemma 1. Hence, (1) has at least one 𝜔-periodic solution. This completes the proof.

Noticing that both systems (2) and (3) are special cases of system (1), by Theorem 3, we can obtain the following results.

Theorem 4. Suppose that (H₁)–(H₇) hold. Then system (2) has at least one positive 𝜔-periodic solution.

Theorem 5. Suppose that (H₁)–(H₇) hold. Then system (3) has at least one positive 𝜔-periodic solution.

3.2. Nonmonotone Case

In this section, we study the existence of 𝜔-periodic solution of system (1) when the numerical response function ℎ(𝑡,𝑥) satisfies the following nonmonotone condition:  (H₈)lim_𝑣→+∞ℎ(𝑡,𝑥)=0 for 𝑡∈ℝ, and there exists a positive constant 𝑣 such that 𝑥−𝑣𝜕ℎ()𝜕𝑥𝑡,𝑥<0,for𝑡∈𝕋,𝑥>0,𝑥≠𝑣. (43)By (H₅) and (H₈), we have 𝑥−𝑣ℎ^′(𝑥)=1𝜔_𝜅^𝜅+𝜔𝑥−𝑣𝜕ℎ()𝜕𝑥𝑡,𝑥Δ𝑡<0,lim_𝑣→+∞ℎ(𝑥)=1𝜔_𝜅^𝜅+𝜔lim_𝑥→+∞ℎ(𝑡,𝑥)Δ𝑡=0, (44) then ℎ(𝑥) is strictly increasing on [0,𝑣] and strictly decreasing on [𝑣,+∞). By these, (H₅) and (H₆), one can easily see that the equation ℎ(𝑥)=𝑑, (45) has two distinct positive solutions, namely, 𝑣₁, 𝑣₂. Without loss of generality, we suppose that 𝑣₁<𝑣₂, then 𝑣₁<𝑣<𝑣₂.

Theorem 6. In addition to (H₁)–(H₆) and (H₈), suppose further that the following condition holds:(H₉)(𝑉₂/𝑉₁)e^[𝐹₍₀₎₊𝑓₍₀₎₊𝐷₊𝑑_]𝜔<𝑣₂/𝑣₁. Then system (1) has at least two 𝜔-periodic solutions.

Proof. The proof is similar to that of Theorem 3. To complete the proof, we need to find two disjoint open bounded subsets in 𝑋. Under condition (H₉), (34) is no longer valid. So, we need to make a corresponding change.
Designating 𝑧(𝑡)=𝑢₁(𝑡)−𝑢₂(𝑡), there also exist 𝜉₀, 𝜂₀∈𝐼_𝜔 such that 𝑧𝜉₀=max_{𝑡∈𝐼𝜔}𝑧(𝑡),𝑧𝜂₀=min_{𝑡∈𝐼𝜔}𝑧(𝑡). (46) From (23), (27), and the monotonicity of functions ℎ and ℎ, we will show that 𝑧(𝜉₀) and 𝑧(𝜂₀) can not simultaneously lie in (−∞,ln𝑣₁),(ln𝑣₁,ln𝑣₂) or (ln𝑣₂,+∞). In fact, if 𝑧(𝜂₀)≤𝑧(𝜉₀)<ln𝑣₁, then 1𝑑=𝜔_𝜅^𝜅+𝜔ℎ𝑡,e^{𝑧(𝑡−𝜏(𝑡))}≤ℎeΔ𝑡_𝑧(𝜉0)<ℎ𝑣₁=𝑑. (47) This is a contradiction. If ln𝑣₂<𝑧(𝜂₀)≤𝑧(𝜉₀), then 1𝑑=𝜔_𝜅^𝜅+𝜔ℎ𝑡,e^{𝑧(𝑡−𝜏(𝑡))}≤ℎeΔ𝑡_𝑧(𝜂0)<ℎ𝑣₂=𝑑. (48) This is also a contradiction. If ln𝑣₁<𝑧(𝜂₀)≤𝑧(𝜉₀)<ln𝑣₂, then 1𝑑=𝜔_𝜅^𝜅+𝜔ℎ𝑡,e^{𝑧(𝑡−𝜏(𝑡))}ℎeΔ𝑡≥min^𝑧(𝜉0),ℎe^𝑧(𝜂0)>ℎ𝑣₁=ℎ𝑣₂=𝑑. (49) This is also a contradiction. Consequently, the distribution of 𝑧(𝜉₀) and 𝑧(𝜂₀) only have the following two cases.
Case  1. 𝑧(𝜂₀)≤ln𝑣₂≤𝑧(𝜉₀). Noticing that 𝑧𝜂₀≥𝑢₁𝜂₁−𝑢₂𝜉₂,𝑧𝜉₀≤𝑢₁𝜉₁−𝑢₂𝜂₂, (50) from (27), (30) and (33), we obtain 𝑢₂𝜉₂≥𝑢₁𝜂₁−𝑧𝜂₀≥𝑢₁𝜂₁−ln𝑣₂>𝐵₁−ln𝑣₂,𝑢₂𝜂₂≤𝑢₁𝜉₁−𝑧𝜉₀≤𝑢₁𝜉₁−ln𝑣₂<𝐵₂−ln𝑣₂. (51) By Lemma 2, we find from (26) and (51) that for all 𝑡∈𝐼_𝜔𝑢₂(𝑡)≥𝑢₂𝜉₂−12_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡>𝐵₁−ln𝑣₂−𝐷+𝑑2𝜔∶=𝐵₇,𝑢₂(𝑡)≤𝑢₂𝜂₂+12_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡<𝐵₂−ln𝑣₂+𝐷+𝑑2𝜔∶=𝐵₈. (52)
Case  2. 𝑧(𝜂₀)≤ln𝑣₁≤𝑧(𝜉₀). From (27), (30) and (33), we also obtain 𝑢₂𝜉₂≥𝑢₁𝜂₁−𝑧𝜂₀≥𝑢₁𝜂₁−ln𝑣₁>𝐵₁−ln𝑣₁,𝑢₂𝜂₂≤𝑢₁𝜉₁−𝑧𝜉₀≤𝑢₁𝜉₁−ln𝑣₁<𝐵₂−ln𝑣₁. (53) By Lemma 2, we find from (26) and (53) that for all 𝑡∈𝐼_𝜔𝑢₂(𝑡)≥𝑢₂𝜉₂−12_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡>𝐵₁−ln𝑣₁−𝐷+𝑑2𝜔∶=𝐵₉,𝑢₂(𝑡)≤𝑢₂𝜂₂+12_𝜅^𝜅+𝜔𝑢^Δ₂(𝑡)Δ𝑡<𝐵₂−ln𝑣₁+𝐷+𝑑2𝜔∶=𝐵₁₀. (54) By (H₉), we know that 𝐵₇<𝐵₈<𝐵₉<𝐵₁₀. (55) Clearly, 𝐵₇, 𝐵₈, 𝐵₉, and 𝐵₁₀ are independent of 𝜆.
By the monotonicity of functions 𝑓 and ℎ, (H₃), (H₅), and (H₆), it is easy to show that algebraic system (40) has two distinct solutions: 𝑢^(𝑖)=ln𝑉₀^(𝑖),ln𝑉₀^(𝑖)−ln𝑣_𝑖^T,𝑖=1,2, (56) where 𝑉₀^(𝑖) is the unique solution of equation 𝑓(𝑣)−𝑝(𝑣_𝑖)=0. Obviously, 𝑉₁≤𝑉₀^(𝑖)<𝑉₂. (57)
We now take Ω₁=𝑢₁(𝑡),𝑢₂(𝑡)^T∈𝑋∶𝑢₁(𝑡)<𝐵₅+𝛽₀,𝑢₂(𝑡)∈𝐵₇,𝐵₈,Ω₂=𝑢₁(𝑡),𝑢₂(𝑡)^T∈𝑋∶𝑢₁(𝑡)<𝐵₅+𝛽₀,𝑢₂(𝑡)∈𝐵₉,𝐵₁₀, (58) where 𝛽₀ is taken sufficiently large such that 𝛽₀>maxln𝑉₀⁽¹⁾,ln𝑉₀⁽²⁾, (59) Then both Ω₁ and Ω₂ are bounded open subsets of 𝑋. It follows from (55), (57), and (59) that 𝑢⁽¹⁾∈Ω₁, 𝑢⁽²⁾∈Ω₂, and Ω₁∩Ω₂=Ø. With the help of (38), (52), (54), and (59), it is easy to see that Ω₁ and Ω₂ satisfy condition (a) in Lemma 1. When (𝑢₁(𝑡),𝑢₂(𝑡))^T∈𝜕Ω_𝑖∩Ker𝐿=𝜕Ω_𝑖∩ℝ²(𝑖=1,2), (𝑢₁(𝑡),𝑢₂(𝑡))^T is a constant vector in ℝ². Thus, we have 𝑢𝑄𝑁¹𝑢₂=𝑓e^𝑢1−𝑝e^{𝑢1−𝑢2}−𝑑+ℎe^{𝑢1−𝑢2}≠00. (60) This proves that condition (b) in Lemma 1 is satisfied. A direct calculation shows that deg𝐽𝑄𝑁,Ω_𝑖∩Ker𝐿,0=sign−𝑣_𝑖𝑉₀^(𝑖)𝑓^′𝑉₀^(𝑖)ℎ^′𝑣_𝑖=(−1)^𝑖−1≠0,𝑖=1,2. (61)
By now we have proved that Ω₁ and Ω₂ satisfy all the requirements in Lemma 1. Hence, system (1) has at least two 𝜔-periodic solutions 𝑢^∗(𝑡) and 𝑢^†(𝑡) in Dom𝐿∩Ω₁ and Dom𝐿∩Ω₂, respectively. This completes the proof.

Noticing that both systems (2) and (3) are special cases of system (1), by Theorem 6, we can obtain the following results.

Theorem 7. Suppose that (𝐻₁)–(𝐻₆) and (𝐻₈)-(𝐻₉) hold. Then system (2) has at least two positive 𝜔-periodic solutions.

Theorem 8. Suppose that (𝐻₁)–(𝐻₆) and (𝐻₈)-(𝐻₉) hold. Then system (3) has at least two positive 𝜔-periodic solutions.

Remark 9. The proof of Theorems 3 and 6 shows that all the above results also remain valid if some delay terms are replaced by terms with distributed or state-dependent delays.

Remark 10. In their Theorem 2.2, Ding and Jiang [14] proved that (2) has at least two positive 𝜔-periodic solutions if (H₁)–(H₆), (H₈), and(H^′₉)(𝑉₂/𝑉₁)e^2[𝐹₍₀₎₊𝑓₍₀₎₊𝐷₊𝑑_]𝜔<𝑣₂/𝑣₁, hold. Obviously, the condition (H^′₉) implies (H₉). Hence, our Theorem 7 improves Theorem 2.2 of [14].

Remark 11. Shao [10] studied a special case of system (1) for 𝑓(𝑡,𝑥)=𝑎(𝑡)−𝑏(𝑡)𝑥. Therefore, our Theorem 3 generalizes Theorem 1 of [10].

Remark 12. Fan and Wang [17], Wang and Li [21], and Fan et al. [16, 18] studied some special cases of system (2) for 𝑓(𝑡,𝑥)=𝑎(𝑡)−𝑏(𝑡)𝑥. Therefore, our Theorem 4 generalizes Theorem 2.1 of [17], Theorem 3.1 of [16], Theorem 3.5 of [18], and Theorem 2.1 of [21].

Remark 13. Fan and Wang [23] and Xia et al. [24] studied some special cases of system (3) for 𝑓(𝑡,𝑥)=𝑎(𝑡)−𝑏(𝑡)𝑥. Therefore, our Theorem 5 generalizes Theorem 2.1 of [23], and Theorem 3.1 of [24].

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 11271110, 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.