Permanence and Extinction of a Generalized Gause-Type Predator-Prey System with Periodic Coefficients

Abstract

We study the permanence and extinction of a generalized Gause-type predator-prey system with periodic coefficients. We provide a sufficient and necessary condition to guarantee the predator and prey species to be permanent and a sufficient condition for the existence of a periodic solution. In addition we prove that when the predator population tends to extinction, the prey population keeps oscillating above a positive population level.

1. Introduction

Permanence of a dynamical system has always been a hot issue in the past few decades. The concept of permanence has been introduced and investigated by several authors, each using his own terminology: “cooperativity” in the earlier papers of Schuster et al. [1], and Hofbauer [2], “permanent coexistence” by Hutson and Vickers [3], “uniform persistence” in Butler et al. [4], and “ecological stability” by Svirezhev and Logofet [5–7] (for more detailed statements of the concept see [8]).

Many important results have been found in recent years [1–37]. Some authors (see [21, 28, 31, 33]) have considered the following two species periodic Lotka-Volterra predator-prey system 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑏₁(𝑡)−𝑎₁₁(𝑡)𝑥₁(𝑡)−𝑎₁₂(𝑡)𝑥₁(𝑡)𝑥₂(𝑡),𝑥̇₂(𝑡)=𝑥₂(𝑡)−𝑏₂(𝑡)−𝑎₂₂(𝑡)𝑥₂(𝑡)+𝑎₂₁(𝑡)𝑥₁(𝑡)𝑥₂(𝑡), (1.1) where 𝑏_𝑖(𝑡) and 𝑎_𝑖𝑗(𝑡)(𝑖,𝑗=1,2) are periodic functions on 𝑅 with common period 𝜔>0 and 𝑎_𝑖𝑗(𝑡)≥0 for all 𝑡∈𝑅. They have established sufficient and necessary conditions for the existence of positive 𝜔-periodic solutions of the system by using different methods, respectively. Teng [31] has given sufficient and necessary conditions for the uniform persistence of the system.

Cui [16] has considered the permanence of the following Lotka-Volterra predator-prey model with periodic coefficients: ̇𝑎(𝑡)(𝑡)𝑐(𝑡)𝑦𝑥=𝑥−𝑏𝑥−𝑝(𝑡),̇(𝑡)+𝑒(𝑡)𝑥+𝑥𝑦=𝑦−𝑑𝑝(𝑡)(𝑡)𝑦.+𝑥−𝑓 (1.2) He provided a sufficient and necessary condition to guarantee the predator and prey species to be permanent. In Theorems 2.2 and 2.3 he set a precondition that 𝑓(𝑡)≢0. This restricts the application of the theorems more or less, since many researchers often neglect the logistic term in the predator equation when the population level of the predator is relatively low and the competition between predators can be ignored, and it proved to be an unnecessary precondition in our paper. However, the research methods in his work inspired me, and many proofs, especially in the first half of this paper, are analogous to [16].

In this paper we consider the permanence of the following generalized Gause-type predator-prey system,̇𝑓()()𝑦,̇𝛾(𝑡)𝑔()(𝑡)(𝑡)𝑦,𝑥=𝑥𝑡,𝑥−𝑔𝑡,𝑥𝑦=𝑦𝑡,𝑥𝑥−𝜇−ℎ (1.3) where 𝑓(𝑡+𝜔,𝑥)=𝑓(𝑡,𝑥), 𝑔(𝑡+𝜔,𝑥)=𝑔(𝑡,𝑥) for all 𝑡 and 𝛾(⋅), 𝜇(⋅), ℎ(⋅) are all periodic continuous functions with common period 𝜔>0; 𝛾(⋅), 𝜇(⋅) are positive, and ℎ(⋅) is nonnegative. We emphasize that our model includes the case when ℎ(𝑡)≡0.

In the absence of predators, system (1.3) becomes𝑥̇=𝑥𝑓(𝑡,𝑥), (1.4) where 𝑓 is a real-valued function defined on 𝐑²₊₀=(𝑡,𝑥)∈𝐑²∶𝑡≥0,𝑥≥0. (1.5)

Vance and Coddington [35] have studied system (1.4) and proved the existence of a unique periodic solution under some assumptions. Apart from the assumption we mentioned above that 𝑓(⋅,𝑥) is 𝜔-periodic, the other assumptions with a mild modification are as follows. (A1)Function 𝑓 is continuous and differentiable with respect to 𝑥 on 𝐑²₊₀, and 𝜕𝑓/𝜕𝑥 is continuous on 𝐑²₊₀. (A2)There are continuous functions 𝑝 and 𝜆 with 𝑝(𝑥)>0 for 𝑥>0 and 𝜆(𝑡)≥0 for 𝑡≥0, such that 𝜕𝑓(𝑥)𝜆(𝑡)𝜕𝑥≤−𝑝for𝑡≥0,𝑥≥0,_∞0𝜆(𝑠)𝑑𝑠=∞. (1.6)(A3)There exist constants 𝛽≥0 and 𝐾>0, such that (i)𝑓(𝑡,𝐾)≤𝛽 for 𝑡≥0,(ii)∫_𝑡^𝑡+𝜔𝑓(𝑠,𝐾)𝑑𝑠≤0 for 𝑡≥0. (A4)There exist constants 𝛼≥0 and0<𝛿≤𝐾, such that(i)𝑓(𝑡,𝛿)≥−𝛼for 𝑡≥0,(ii)∫_𝑡^𝑡+𝜔𝑓(𝑠,𝛿)𝑑𝑠>0 for 𝑡≥0.

In addition, we assume that the 𝜔-periodic function 𝑔(⋅,𝑥) satisfies the following. (A5)Function 𝑔(𝑡,𝑥) is continuous with respect to 𝑡, and 𝐺(𝑡,𝑥)∶=𝑥𝑔(𝑡,𝑥) is strictly monotonely increasing with respect to 𝑥, (A6)𝑔(𝑡,𝑥) is nonnegative and is positive when 𝑥>0, (A7)𝑔(𝑡,𝑥) is bounded on 𝐑²₊₀, that is there is a constant 𝑀_𝑔 such that 𝑔(𝑡,𝑥)≤𝑀_𝑔 for (𝑡,𝑥)∈𝐑²₊₀.

Traditionally 𝐺(𝑡,𝑥) represents a grazing rate. Usually when the amount of prey increases, the grazing rate increases and eventually tends to a maximal value as the prey population tends to infinity. But here we do not emphasize that 𝐺(𝑡,𝑥) is bounded because its unnecessary theoretically. Assumption (A5) means that there is a higher capture rate when there is a larger amount of prey. 𝑔(𝑡,𝑥) is directly proportional to the mean possibility density for each individual prey being captured, and (A6), (A7) suggest that there is a possibility, but its not definitely for each individual prey being captured. In ecology 𝐺(𝑡,𝑥) is called the functional response or grazing function, for example, the Holling-type grazing function:𝑅_𝑚𝑥^𝑛𝛼+𝑥^𝑛,𝑛≥1, (1.7) where 𝑅_𝑚,𝛼>0 or the generalized form 𝑅_𝑚(𝑡)𝑥^𝑛𝛼(𝑡)+𝑥^𝑛,𝑛≥1, (1.8) where 𝑅_𝑚(𝑡),𝛼(𝑡)>0; the Ivlev grazing function 𝑅_𝑚1−𝑒^−𝜆𝑥, (1.9) where 𝑅_𝑚,𝜆>0. Obviously all of these functions satisfy (A5)–(A7).

In this paper we will establish sufficient and necessary conditions for the permanence of system (1.3). In the next section we state our main results. These results are proved in Section 3. Two applications are given in Section 4.

2. Main Results

Throughout this paper, we will assume that all the functions 𝑓(⋅,𝑥), 𝑔(⋅,𝑥), 𝛾(⋅), 𝜇(⋅), and ℎ(⋅) are continuous and periodic with common period 𝜔>0. For any continuous 𝜔-periodic function 𝑢(𝑡) defined on 𝑅, we denote1𝑢(𝑡)=𝜔^𝜔₀𝑢(𝑡)𝑑𝑡,𝑢^𝑀=max_{[]𝑡∈0,𝜔}𝑢(𝑡),𝑢^𝐿=min_{[]𝑡∈0,𝜔}𝑢(𝑡). (2.1) In order to describe our main results, we first introduce a lemma.

Lemma 2.1 (see [35]). Suppose that 𝑓 satisfies (A1)–(A4). Then system (1.4) possesses a unique 𝜔-periodic positive solution 𝑥^∗(𝑡) which is globally asymptotically stable with respect to the positive 𝑥-axis.

Theorem 2.2. Suppose that 𝑓 satisfies (A1)–(A4), 𝑔 satisfies (A5)–(A7). Then system (1.3) is permanent provided that 𝛾(𝑡)𝑔(𝑡,𝑥^∗(𝑡))𝑥^∗(𝑡)−𝜇(𝑡)>0, (2.2) where 𝑥^∗(𝑡) is the unique periodic solution of (1.4) given by Lemma 2.1.

Theorem 2.3. Suppose that 𝑓 satisfies (A1)–(A4), 𝑔 satisfies (A5)–(A7), and 𝛾(𝑡)𝑔(𝑡,𝑥^∗(𝑡))𝑥^∗(𝑡)−𝜇(𝑡)≤0, (2.3) then(i)lim_𝑡→∞𝑦(𝑡)=0,(ii)lim_𝑡→∞inf𝑥(𝑡)>0, for any solution (𝑥(𝑡),𝑦(𝑡)) of system (1.3) with positive initial conditions, where 𝑥^∗(𝑡) is the unique periodic solution of (1.4) given by Lemma 2.1.

By Theorems 2.2 and 2.3, we have the following corollary.

Corollary 2.4. Suppose that 𝑓 satisfies (A1)–(A4), 𝑔 satisfies (A5)–(A7). Then system (1.3) is permanent if and only if (2.2) holds.

Lemma 2.5 (see Theorem 15.5 in [30, 37]). Consider a periodic system 𝑥̇=𝐹(𝑡,𝑥),𝐹(𝑡+𝜔,𝑥)=𝐹(𝑡,𝑥),𝜔>0, (2.4) where 𝐹(𝑡,𝑥)∈𝐶(𝑅×𝑅²,𝑅²) satisfies a local Lipschitz condition. If all solutions of the above system exist in the future and one of them is bounded, then there exists a periodic solution of period 𝜔.

By Theorem 2.2 and Lemma 2.5, we have the following corollary (see proof in Section 3).

Corollary 2.6. Suppose 𝑓 satisfies (A1)–(A4), 𝑔 satisfies (A5)–(A7), and condition (2.2) holds. If in addition 𝑔(𝑡,𝑥) satisfies a local Lipschitz condition with respect to 𝑥, then system (1.3) has a positive 𝜔-periodic solution.

3. Proof of Our Main Results

Lemma 3.1 (see [31]). If 𝑎(𝑡), 𝑏(𝑡) are 𝜔-periodic functions, 𝑏(𝑡)≥0, for all 𝑡∈𝑅and 𝑏(𝑡)>0, then 𝑢̇=𝑢(𝑎(𝑡)−𝑏(𝑡)𝑢) (3.1) has a unique nonnegative 𝜔-periodic solution 𝑢^∗ which is globally asymptotically stable with respect to the positive 𝑢-axis. Moreover, if 𝑎(𝑡)>0, then 𝑢^∗(𝑡)>0, for all 𝑡∈𝑅 and if 𝑎(𝑡)≤0, then 𝑢^∗(𝑡)=0.

Lemma 3.2. Suppose that 𝑓 satisfies (A1)–(A4), then there exists an 𝜀₀>0, such that for any 0<𝜀≤𝜀₀, system 𝑥̇=𝑥𝑓(𝑡,𝑥)−𝜀 (3.2) possesses an 𝜔-periodic positive solution 𝑥^∗_𝜀(𝑡) which is globally asymptotically stable with respect to the positive 𝑥-axis.

Proof. It suffices to show that if 𝑓 satisfies (A1)–(A4), then there exists an 𝜀₀>0 such that for any 0<𝜀≤𝜀₀, 𝑓=𝑓−𝜀 satisfies (A1)–(A4). Obviously 𝑓 satisfies (A1)–(A3). Taking 𝜀₀=(1/2𝜔)∫^𝜔₀𝑓(𝑠,𝛿)𝑑𝑠, its easy to see 𝑓 satisfies (A4).

From a comparison theorem (see Theorem 1.1 [37]) and Lemma 2.1, we have the following lemma.

Lemma 3.3. Suppose that 𝑓 satisfies (A1)–(A4), 𝑥(𝑡) is a solution of system (1.4), and 𝑥^∗(𝑡) is the periodic solution of system (1.4) given by Lemma 2.1. Let 𝑣(𝑡) be a right maximal (right minimal) solution of a scalar differential equation 𝑣̇=𝑤(𝑡,𝑣) on [𝑡₀,+∞)⊂[0,+∞), where 𝑤(𝑡,𝑣) is continuous and satisfies 𝑤(𝑡,𝑥)≤(≥)𝑥𝑓(𝑡,𝑥) for all (𝑡,𝑥)∈𝑅²₊₀. Then the following conclusions hold.(i)If 𝑣(𝑡₀)≤(≥)𝑥(𝑡₀), then 𝑣(𝑡)≤(≥)𝑥(𝑡) for 𝑡≥𝑡₀.(ii)For all 𝜀>0 there exist a 𝜏≥𝑡₀ such that 𝑣(𝑡)<𝑥^∗(𝑡)+𝜀(𝑣(𝑡)>𝑥^∗(𝑡)−𝜀) for 𝑡≥𝜏.

Proposition 3.4. Under assumptions (A1)–(A7), there exist 𝑀_𝑥 and 𝑀_𝑦, such that lim_𝑡→∞sup𝑥(𝑡)≤𝑀_𝑥,lim_𝑡→∞sup𝑦(𝑡)≤𝑀_𝑦 (3.3) for all solutions (𝑥(𝑡),𝑦(𝑡)) of system (1.3) with positive initial values.

Proof. Obviously, 𝑅²₊={(𝑥,𝑦)∶𝑥≥0,𝑦≥0}is a positively invariant set of system (1.3). Given any solution (𝑥(𝑡),𝑦(𝑡)) of (1.3) with positive initial values, from system (1.3) we have 𝑥̇≤𝑥𝑓(𝑡,𝑥). (3.4) The following equation 𝑣̇=𝑣𝑓(𝑡,𝑣), (3.5) has a globally asymptotically stable positive 𝜔-periodic solution 𝑣^∗(𝑡) by Lemma 2.1. By Lemma 3.3, there exists 𝑇₁>0, such that 𝑥(𝑡)<𝑣^∗(𝑡)+1,for𝑡>𝑇₁. (3.6) Let 𝑀_𝑥=max_{0≤𝑡≤𝜔}{𝑣^∗(𝑡)+1}. We have lim_𝑡→∞sup𝑥(𝑡)≤𝑀_𝑥. (3.7) By (1.3), 𝛾^𝑀𝑥̇+𝑦̇=𝛾^𝑀𝑥𝑓(𝑡,𝑥)−𝛾^𝑀−𝛾(𝑡)𝑔(𝑡,𝑥)𝑥𝑦−𝜇(𝑡)𝑦−ℎ(𝑡)𝑦²≤𝛾^𝑀𝑥𝑓(𝑡,𝑥)−𝜇(𝑡)𝑦≤𝛾^𝑀𝑥𝑓(𝑡,𝑥)−𝜇^𝐿𝑦=𝛾^𝑀𝑥𝑓(𝑡,𝑥)+𝜇^𝐿−𝜇^𝐿𝛾^𝑀𝑥+𝑦. (3.8) Denote 𝑤(𝑡)=𝛾^𝑀𝑥(𝑡)+𝑦(𝑡). Then we have 𝑤̇+𝜇^𝐿𝑤≤𝛾^𝑀𝑥𝑓(𝑡,𝑥)+𝜇^𝐿. (3.9) Since 𝑥(𝑡)≤max_{0≤𝑡≤𝜔}{𝑣^∗(𝑡)+1}, for𝑡≥𝑇₁, there is a constant 𝐵>0, such that 𝛾^𝑀𝑥𝑓(𝑡,𝑥)+𝜇^𝐿≤𝐵for𝑡≥𝑇₁, (3.10) by the continuity of 𝑓. It follows that 𝑤̇+𝜇^𝐿𝑤≤𝐵,for𝑡≥𝑇₁. (3.11) Notice that 𝜇^𝐿>0. It's easy to show that there is an 𝐴>0, such that lim_𝑡→∞sup𝑤(𝑡)≤𝐴. (3.12) By the notation 𝑤=𝛾^𝑀𝑥+𝑦, taking 𝑀_𝑥=𝐴/𝛾^𝑀 and 𝑀_𝑦=𝐴, Proposition 3.4 is proved.

Proposition 3.5. Under assumption (A1)–(A7), there exists a positive constant 𝜂_𝑥, such that lim_𝑡→∞sup𝑥(𝑡)≥𝜂_𝑥, (3.13) for all solutions (𝑥(𝑡),𝑦(𝑡)) of (1.3) with positive initial values.

Proof. Suppose that (3.13) is not true. Then there is a sequence {𝑧_𝑚}⊂𝑅²₊, such that lim_𝑡→∞sup𝑥𝑡,𝑧_𝑚<1𝑚,𝑚=1,2,…, (3.14) where (𝑥(𝑡,𝑧_𝑚),𝑦(𝑡,𝑧_𝑚)) is the solution of (1.3) with (𝑥(0,𝑧_𝑚),𝑦(0,𝑧_𝑚))=𝑧_𝑚. By assumption (A5), we can choose sufficiently small positive numbers 𝜀_𝑥<1 and 𝜀_𝑦<1, such that 𝛾(𝑡)𝑔𝑡,𝜀_𝑥𝜀_𝑥−𝜇(𝑡)<0,𝜙_𝜀(𝑡)>0, (3.15)(3.16) where 𝜙_𝜀(𝑡)=𝑓𝑡,𝜀_𝑥−𝑀_𝑔𝜀_𝑦exp(𝛼𝜔),𝛼=max_{0≤𝑡≤𝜔}𝜇(𝑡)+𝛾(𝑡)𝑔𝑡,𝜀_𝑥𝜀_𝑥+ℎ(𝑡)𝜀_𝑦. (3.17)
By (3.14), for the given 𝜀_𝑥>0, there exists a positive integer 𝑁₀, such thatlim_𝑡→∞sup𝑥𝑡,𝑧_𝑚<1𝑚<𝜀_𝑥,𝑚>𝑁₀. (3.18) For the rest of this proof, we assume that 𝑚>𝑁₀. Equation (3.18) implies there exists 𝜏₁^(𝑚)>0, such that 𝑥𝑡,𝑧_𝑚<𝜀_𝑥,𝑡≥𝜏₁^(𝑚), (3.19) and further 𝑦̇𝑡,𝑧_𝑚≤𝑦𝑡,𝑧_𝑚𝛾(𝑡)𝑔𝑡,𝜀_𝑥𝜀_𝑥−𝜇(𝑡)−ℎ(𝑡)𝑦𝑡,𝑧_𝑚,for𝑡≥𝜏₁^(𝑚). (3.20)
By (3.15), and Lemma 3.1, any solution 𝑣(𝑡) of the following equation, 𝑣̇=𝑣𝛾(𝑡)𝑔𝑡,𝜀_𝑥𝜀_𝑥−𝜇(𝑡)−ℎ(𝑡)𝑣 (3.21) with positive initial conditions satisfies lim_𝑡→∞𝑣(𝑡)=0. (3.22) Hence, lim_𝑡→∞𝑦𝑡,𝑧_𝑚=0 (3.23) by Lemma 3.3. So there is a 𝜏₂^(𝑚)>𝜏₁^(𝑚), such that 𝑦𝑡,𝑧_𝑚<𝜀_𝑦,for𝑡≥𝜏₂^(𝑚). (3.24) It follows that 𝑥̇𝑡,𝑧_𝑚≥𝑥𝑡,𝑧_𝑚𝑓𝑡,𝑥𝑡,𝑧_𝑚−𝑔𝑡,𝜀_𝑥𝜀_𝑦,𝑡≥𝜏₂^(𝑚). (3.25) By (3.16), the equation 𝑥̇=𝑥𝑓𝑡,𝑥𝑡,𝑧_𝑚−𝑔𝑡,𝜀_𝑥𝜀_𝑦 (3.26) has an 𝜔-periodic positive solution 𝑥^∗(𝑡) which is globally asymptotically stable. Hence, 𝑥𝑡,𝑧_𝑚>𝑥^∗(𝑡)2, (3.27) for sufficiently large 𝑡>0 and 𝑚>𝑁₀, which is a contradiction with (3.14). This completes the proof of Proposition 3.5.

Proposition 3.6. Under assumption (A1)–(A7), there exists a positive constant 𝛾_𝑥, such that lim_𝑡→∞inf𝑥(𝑡)≥𝛾_𝑥 (3.28) for all solutions (𝑥(𝑡),𝑦(𝑡)) of system (1.3) with positive initial values.

Proof. Suppose that (3.28) is not true, then there exists a sequence {𝑧_𝑚}⊂𝑅²₊ such that lim_𝑡→∞inf𝑥𝑡,𝑧_𝑚<𝜂_𝑥2𝑚²,𝑚=1,2,…. (3.29) On the other hand, by Proposition 3.5, we have lim_𝑡→∞sup𝑥𝑡,𝑧_𝑚>𝜂_𝑥,𝑚=1,2,…. (3.30) Hence, there are time sequences {𝑠_𝑞^(𝑚)} and {𝑡_𝑞^(𝑚)} satisfying 0<𝑠₁^(𝑚)<𝑡₁^(𝑚)<𝑠₂^(𝑚)<𝑡₂^(𝑚)<⋯<𝑠_𝑞^(𝑚)<𝑡_𝑞^(𝑚)𝑠<⋯,_𝑞(𝑚)⟶∞,𝑡_𝑞^(𝑚)𝑥𝑡⟶∞,as𝑞⟶∞,_𝑞(𝑚),𝑧_𝑚=𝜂_𝑥𝑚²<𝑥𝑡,𝑧_𝑚<𝜂_𝑥𝑚=𝑥𝑠_𝑞^(𝑚),𝑧_𝑚,𝑡∈𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚). (3.31) By Proposition 3.4, for a given positive integer 𝑚, there is a 𝑇^(𝑚)>0, such that 𝑥𝑡,𝑧_𝑚≤𝑀_𝑥,𝑦𝑡,𝑧_𝑚≤𝑀_𝑦,for𝑡≥𝑇₁^(𝑚). (3.32) Because of 𝑠_𝑞^(𝑀)→∞ as 𝑞→∞, there is a positive integer 𝐾^(𝑚) such that 𝑠_𝑞^(𝑚)>𝑇₁^(𝑚) since 𝑞≥𝐾(𝑚), and hence 𝑥̇≥𝑥𝑡,𝑧_𝑚𝑓𝑡,𝑀_𝑥−𝑀_𝑔𝑀_𝑦∶=𝜁(𝑡)𝑥𝑡,𝑧_𝑚 (3.33) for 𝑞≥𝐾^(𝑚). Integrating (3.33) from 𝑠_𝑞^(𝑚) to 𝑡_𝑞^(𝑚) yields 𝑥𝑡_𝑞^(𝑚),𝑧_𝑚≥𝑥𝑠_𝑞^(𝑚),𝑧_𝑚exp^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)}𝜁(𝑡)𝑑𝑡 (3.34) or −^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)}𝜁(𝑡)𝑑𝑡≥ln𝑚,for𝑞≥𝐾^(𝑚). (3.35) If 𝜁(𝑡)≥0, it leads to a contradiction. Otherwise, 𝜁(𝑡)<0, we have 𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)⟶∞,as𝑚⟶∞,𝑞≥𝐾^(𝑚), (3.36) since 𝜁(𝑡) is bounded. By (3.15) and (3.16), there are constants 𝑃>0 and 𝑁₀>0, such that 𝜂_𝑥𝑚<𝜀_𝑥,𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)𝑀>2𝑃,_𝑦exp^𝑃₀𝛾(𝑡)𝑔𝑡,𝜀_𝑥𝜀_𝑥−𝜇(𝑡)−ℎ(𝑡)𝜀_𝑦𝑑𝑡<𝜀_𝑦,^𝑃₀𝜙_𝜀(𝑡)𝑑𝑡>0 (3.37)(3.38) for 𝑚≥𝑁₀, 𝑞≥𝐾^(𝑚), and 𝑎≥𝑃. Equation (3.37) implies 𝑥𝑡,𝑧_𝑚<𝜀_𝑥,𝑡∈𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚) (3.39) for 𝑚≥𝑁₀, 𝑞≥𝐾^(𝑚). For the positive 𝜀_𝑦 satisfying (3.16) and (3.38), we have the following two cases:(i)𝑦(𝑡,𝑧_𝑚)≥𝜀_𝑦 for all 𝑡∈[𝑠_𝑞^(𝑚),𝑠_𝑞^(𝑚)+𝑃];(ii)there exists 𝜏_𝑞^(𝑚)₁∈[𝑠_𝑞^(𝑚),𝑠_𝑞^(𝑚)+𝑃], such that 𝑦(𝜏_𝑞^(𝑚)₁,𝑧_𝑚)<𝜀_𝑦.If (i) holds, by (3.38) and (3.39) we have 𝜀_𝑦≤𝑦𝑠_𝑞^(𝑚)+𝑃,𝑧_𝑚≤𝑦𝑠_𝑞^(𝑚),𝑧_𝑚exp^{𝑠𝑞(𝑚)}_𝑠^+𝑃_𝑞(𝑚)𝛾(𝑡)𝑔𝑡,𝜀_𝑥𝜀_𝑥−𝜇(𝑡)−ℎ(𝑡)𝜀_𝑦𝑑𝑡≤𝑀_𝑦exp^𝑃₀𝛾(𝑡)𝑔𝑡,𝜀_𝑥𝜀_𝑥−𝜇(𝑡)−ℎ(𝑡)𝜀_𝑦𝑑𝑡<𝜀_𝑦. (3.40) This is a contradiction.
If (ii) holds, we now claim that𝑦𝑡,𝑧_𝑚≤𝜀_𝑦exp(𝛼𝜔),𝑡∈𝜏_𝑞^(𝑚)₁,𝑡_𝑞^(𝑚). (3.41) Otherwise, there exists 𝜏_𝑞^(𝑚)₂∈(𝜏_𝑞^(𝑚)₁,𝑡_𝑞^(𝑚)] such that 𝑦𝜏_𝑞^(𝑚)₂,𝑧_𝑚>𝜀_𝑦exp(𝛼𝜔). (3.42)
By the continuity of 𝑦(𝑡,𝑧_𝑚), there must exist 𝜏_𝑞^(𝑚)₃∈(𝜏_𝑞^(𝑚)₁,𝜏_𝑞^(𝑚)₂) such that𝑦𝜏_𝑞^(𝑚)₃,𝑧_𝑚=𝜀_𝑦,𝑦𝑡,𝑧_𝑚>𝜀_𝑦,for𝑡∈𝜏_𝑞^(𝑚)₃,𝜏_𝑞^(𝑚)₂. (3.43) Denote 𝑃^(𝑚) as the nonnegative integer, such that 𝜏_𝑞^(𝑚)₂∈(𝜏_𝑞^(𝑚)₃+𝑃^(𝑚)𝜔,𝜏_𝑞^(𝑚)₃+(𝑃^(𝑚)+1)𝜔]. By (3.15), we have 𝜀_𝑦exp(𝛼𝜔)<𝑦𝜏_𝑞^(𝑚)₂,𝑧_𝑚<𝑦𝜏_𝑞^(𝑚)₃,𝑧_𝑚exp^{𝜏𝑞2(𝑚)}_{𝜏𝑞3(𝑚)}𝛾(𝑡)𝑔𝑡,𝜀_𝑥𝜀_𝑥−𝜇(𝑡)−ℎ(𝑡)𝜀_𝑦𝑑𝑡=𝜀_𝑦exp^{𝜏𝑞3(𝑚)+𝑃(𝑚)𝜔}_{𝜏𝑞3(𝑚)}+^{𝜏𝑞2(𝑚)}_{𝜏𝑞3(𝑚)+𝑃(𝑚)𝜔}𝛾(𝑡)𝑔𝑡,𝜀_𝑥𝜀_𝑥−𝜇(𝑡)−ℎ(𝑡)𝜀_𝑦𝑑𝑡<𝜀_𝑦exp(𝛼𝜔). (3.44) This contradiction establishes that (3.41) is true, particularly (3.41) holds for 𝑡∈[𝑠_𝑞^(𝑚)+𝑃,𝑡_𝑞^(𝑚)].
By (3.31) and (3.38), we have𝜂_𝑥𝑚²=𝑥𝑡_𝑞^(𝑚),𝑧_𝑚≥𝑥𝑠_𝑞^(𝑚)+𝑃,𝑧_𝑚exp^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)+𝑃}𝑓𝑡,𝜀_𝑥−𝑀_𝑔𝜀_𝑦(𝑠exp𝛼𝜔)𝑑𝑡=𝑥_𝑞^(𝑚)+𝑃,𝑧_𝑚exp^{𝜏𝑞(𝑚)}_{𝑠𝑞(𝑚)+𝑃}𝜙_𝜀(𝑡)>𝜂𝑑𝑡_𝑥𝑚², (3.45) which is also a contradiction. This completes the proof of Proposition 3.6.

Proposition 3.7. Suppose 𝑓 satisfies (A1)–(A4), 𝑔 satisfies (A5)–(A7), and (2.2) holds. Then there exists a positive constant 𝜂_𝑦 such that lim_𝑡→∞sup𝑦(𝑡)>𝜂_𝑦 (3.46) for all solutions (𝑥(𝑡),𝑦(𝑡)) of (1.3) with positive initial values.

Proof. By assumption (A5) and (2.2) we can choose a constant 𝜀₁>0 such that 𝜓_𝜀1(𝑡)>0, (3.47) where 𝜓_𝜀1(𝑡)=𝛾(𝑡)𝑔𝑡,𝑥^∗(𝑡)−𝜀₁𝑥^∗(𝑡)−𝜀₁−𝜇(𝑡)−ℎ(𝑡)𝜀₁. (3.48) Consider the following equation with positive parameter α: 𝑥̇=𝑥𝑓(𝑡,𝑥)−2𝑀_𝑔𝛼. (3.49) By Lemma 3.2, (3.49) has a unique positive 𝜔-periodic solution 𝑥^∗_𝛼(𝑡) which is globally asymptotically stable since 𝛼<𝜀₀/(2𝑀_𝑔). Let 𝑥_𝛼(𝑡) be the solution of (3.49) with initial condition 𝑥_𝛼(0)=𝑥^∗(0) in which 𝑥^∗(𝑡) is the unique periodic solution of (1.4) given by Lemma 2.1. Hence, for the above 𝜀₁, there exists sufficiently large 𝑇₂>𝑇₁, such that 𝑥_𝛼(𝑡)−𝑥^∗_𝛼(𝑡<𝜀)¹4,for𝑡≥𝑇₂. (3.50) By the continuity of the solution in the parameter, we have 𝑥_𝛼(𝑡)→𝑥^∗(𝑡) uniformly in [𝑇₂,𝑇₂+𝜔] as 𝛼→0. Hence, for 𝜀₁>0 there exists 𝛼₀>0, such that 𝑥_𝛼(𝑡)−𝑥^∗(𝑡<𝜀)¹4,for𝑡∈𝑇₂,𝑇₂+𝜔,0<𝛼<𝛼₀. (3.51) So, we have 𝑥^∗_𝛼(𝑡)−𝑥^∗(𝑡<𝜀)¹2,𝑡∈𝑇₂,𝑇₂+𝜔. (3.52) Notice that 𝑥_𝛼(𝑡) and 𝑥^∗(𝑡) are all 𝜔-periodic, hence 𝑥^∗_𝛼(𝑡)−𝑥^∗(𝑡<𝜀)¹2,𝑡≥0,0<𝛼<𝛼₀. (3.53) Choosing a constant 𝛼₁(0<𝛼₁<𝛼₀,2𝛼₁<𝜀₁), we have 𝑥^∗_𝛼1(𝑡)≥𝑥^∗(𝑡)−𝜀₁2,𝑡≥0. (3.54)
Suppose that (3.46) is not true. Then there exists 𝑧∈𝑅²₊, such thatlim_𝑡→∞sup𝑦(𝑡,𝑧)<𝛼₁, (3.55) where (𝑥(𝑡,𝑧),𝑦(𝑡,𝑧)) is the solution of (1.3) with (𝑥(0,𝑧),𝑦(0,𝑧))=𝑧. So, there exist 𝑇₃≥𝑇₂, such that 𝑦(𝑡,𝑧)<2𝛼₁<𝜀₁,𝑡≥𝑇₃ (3.56) and hence 𝑥̇≥𝑥(𝑡,𝑧)𝑓(𝑡,𝑥)−2𝑀_𝑔𝛼₁. (3.57) Let 𝑢(𝑡) be the solution of (3.49) with 𝛼=𝛼₁ and 𝑢(𝑇₃)=𝑥(𝑇₃,𝑧), then 𝑥(𝑡,𝑧)≥𝑢(𝑡),𝑡≥𝑇₃. (3.58) By the global asymptotic stability of 𝑥^∗_𝛼1(𝑡), for the given 𝜀=𝜀₁/2, there exists 𝑇₄≥𝑇₃, such that 𝑢(𝑡)−𝑥^∗_𝛼1(𝑡<𝜀)¹2,𝑡≥𝑇₄. (3.59) Hence 𝑥(𝑡,𝑧)≥𝑢(𝑡)>𝑥^∗_𝛼1(𝑡)−𝜀₁2,𝑡≥𝑇₄ (3.60) and hence 𝑥(𝑡,𝑧)>𝑥^∗(𝑡)−𝜀₁,𝑡≥𝑇₄ (3.61) from (3.54). This implies 𝑦̇(𝑡,𝑧)≥𝑦(𝑡,𝑧)𝛾(𝑡)𝑔𝑡,𝑥^∗(𝑡)−𝜀₁𝑥^∗−𝜀₁−𝜇(𝑡)−ℎ(𝑡)𝜀₁=𝜓_𝜀1𝑦(𝑡,𝑧),𝑡≥𝑇₄. (3.62) Integrating the above inequality from 𝑇₄ to 𝑡 yields 𝑦(𝑡,𝑧)≥𝑦𝑇₄,𝑧exp^𝑡_𝑇4𝜓_𝜀1(𝑡)𝑑𝑡. (3.63)𝑦(𝑡,𝑧)→∞ as 𝑡→∞ from (3.47) which is a contradiction. This completes the proof of Proposition 3.7.

Proposition 3.8. Suppose 𝑓 satisfies (A1)–(A4), 𝑔 satisfies (A5)–(A7), and (2.2) holds. Then there exists a positive constant 𝛾_𝑦, such that lim_𝑡→∞inf𝑦(𝑡)≥𝛾_𝑦 (3.64) for all solutions (𝑥(𝑡),𝑦(𝑡)) of (1.3) with positive initial values.

Proof. Otherwise, there exists a sequence {𝑧_𝑚}⊂𝑅²₊, such that lim_𝑡→∞inf𝑦𝑡,𝑧_𝑚<𝜂_𝑦(𝑚+1)²,𝑚=1,2,…. (3.65) However lim_𝑡→∞sup𝑦𝑡,𝑧_𝑚>𝜂_𝑦,𝑚=1,2,…, (3.66) from Proposition 3.7. Hence there are two time sequence {𝑠_𝑞^(𝑚)} and {𝑡_𝑞^(𝑚)} satisfying the following conditions: 0<𝑠₁^(𝑚)<𝑡₁^(𝑚)<𝑠₂^(𝑚)<𝑡₁^(𝑚)<⋯<𝑠_𝑞^(𝑚)<𝑡_𝑞^(𝑚)𝑠<⋯,_𝑞(𝑚)⟶∞,𝑡_𝑞^(𝑚)𝑦𝑡⟶∞,as𝑞⟶∞,_𝑞(𝑚),𝑧_𝑚=𝜂_𝑦(𝑚+1)²<𝑦𝑡,𝑧_𝑚<𝜂_𝑦𝑠𝑚+1=𝑦_𝑞^(𝑚),𝑧_𝑚,𝑡∈𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚). (3.67) By Proposition 3.4, for a given integer 𝑚>0 there is a 𝑇₁^(𝑚)>0, such that 𝑦𝑡,𝑧_𝑚≤𝑀_𝑦,for𝑡≥𝑇₁^(𝑚). (3.68) Because of 𝑠_𝑞^(𝑚)→∞ as 𝑞→∞, there is a positive integer 𝐾^(𝑚), such that 𝑠_𝑞^(𝑚)>𝑇₁^(𝑚) as 𝑞≥𝐾^(𝑚), and hence 𝑦̇𝑡,𝑧_𝑚≥𝑦𝑡,𝑧_𝑚−𝜇(𝑡)−ℎ(𝑡)𝑀_𝑦 (3.69) for 𝑞≥𝐾^(𝑚) and 𝑡∈[𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚)]. Integrating the above inequality from 𝑠^(𝑚) to 𝑡_𝑞^(𝑚), we have 𝑦𝑡_𝑞^(𝑚),𝑧_𝑚≥𝑦𝑠_𝑞^(𝑚),𝑧_𝑚exp^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)}−𝜇(𝑡)−ℎ(𝑡)𝑀_𝑦𝑑𝑡 (3.70) or ^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)}𝜇(𝑡)+ℎ(𝑡)𝑀_𝑦𝑑𝑡≥ln(𝑚+1),𝑞≥𝐾^(𝑚). (3.71)
Because of the boundedness of the function 𝜇(𝑡)+ℎ(𝑡)𝑀_𝑦, we know that𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)⟶∞,as𝑚⟶∞,𝑞≥𝐾^(𝑚). (3.72) By (3.47), there is a constant 𝑃>0 and a positive integer 𝑁₀ such that 𝜂_𝑦𝑚+1<𝛼₁<𝜀₁,𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)>2𝑃,_𝑞0𝜓_𝜀1(𝑡)𝑑𝑡>0 (3.73)(3.74) for 𝑚≥𝑁₀, 𝑞≥𝐾^(𝑚) and 𝑎≥𝑃. Further, 𝑦𝑡,𝑧_𝑚<𝛼₁<𝜀₁,𝑡∈𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚),𝑚≥𝑁₀,𝑞≥𝐾^(𝑚). (3.75) In addition, 𝑥̇𝑡,𝑧_𝑚≥𝑥𝑡,𝑧_𝑚𝑓𝑡,𝑥𝑡,𝑧_𝑚−𝑀_𝑔𝛼₁. (3.76) Let 𝑢(𝑡) be the solution of (3.49) with 𝛼=𝛼₁ and 𝑢(𝑠_𝑞^(𝑚))=𝑥(𝑠_𝑞^(𝑚),𝑧_𝑚). Then 𝑥𝑡,𝑧_𝑚≥𝑢(𝑡),𝑡∈𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚) (3.77) by Lemma 3.3. Further, by Propositions 3.4 and 3.6, we can choose 𝐾₁^(𝑚)>𝐾^(𝑚) such that 𝛾_𝑥≤𝑥𝑠_𝑞^(𝑚),𝑧_𝑚≤𝑀_𝑥 (3.78) for 𝑞≥𝐾₁^(𝑚). For 𝛼=𝛼₁, (3.49) has a unique positive 𝜔-periodic solution 𝑥^∗_𝛼1(𝑡) which is globally asymptotically stable. In addition, by the periodicity of (3.49), the periodic solution 𝑥_𝛼1(𝑡) is uniformly asymptotically stable with respect to the compact set Ω={𝑥∶𝛾_𝑥≤𝑥≤𝑀_𝑥}. Hence, for the given 𝜀₁ in Proposition 3.7, there exists 𝑇₀>𝑃 which is independent of 𝑚 and 𝑞, such that 𝑢(𝑡)≥𝑥^∗_𝛼1(𝑡)−𝜀₁2,𝑡≥𝑇₀+𝑠_𝑞^(𝑚). (3.79) Thus 𝑢(𝑡)≥𝑥^∗(𝑡)−𝜀₁,𝑡≥𝑇₀+𝑠_𝑞^(𝑚) (3.80) from (3.54). By (3.72), there exists a positive integer 𝑁₁≥𝑁₀ such that 𝑡_𝑞^(𝑚)>𝑠_𝑞^(𝑚)+2𝑇₀>𝑠_𝑞^(𝑚)+2𝑃 for 𝑚≥𝐾₁^(𝑚) and 𝑞≥𝐾₁^(𝑚). So we have 𝑥𝑡,𝑧_𝑚≥𝑥^∗(𝑡)−𝜀₁,𝑡∈𝑠_𝑞^(𝑚)+𝑇₀,𝑡_𝑞^(𝑚) (3.81) since 𝑚≥𝑁₁ and 𝑞≥𝐾₁^(𝑚). Hence, 𝑦̇𝑡,𝑧_𝑚≥𝜓_𝜀1(𝑡)𝑦𝑡,𝑧_𝑚,𝑡∈𝑠_𝑞^(𝑚)+𝑇₀,𝑡_𝑞^(𝑚) (3.82) from (3.75) and (3.81). Integrationg the above inequality from 𝑠_𝑞^(𝑚)+𝑇₀ to 𝑡_𝑞^(𝑚) yields 𝑦𝑡_𝑞^(𝑚),𝑧_𝑚≥𝑦𝑠_𝑞^(𝑚)+𝑇₀,𝑧_𝑚exp^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)+𝑇0}𝜓_𝜀1(𝑡)𝑑𝑡, (3.83) and hence 𝜂_𝑦(𝑚+1)²≥𝜂_𝑦(𝑚+1)²exp^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)+𝑇0}𝜓_𝜀1(𝑡)𝜂𝑑𝑡>^𝑦(𝑚+1)² (3.84) from (3.74). This is a contradiction. This completes the proof of Proposition 3.8. The result of Theorem 2.2 follows from Propositions 3.4–3.8.

Proof of Theorem 2.3. Suppose 𝑥^∗(𝑡) is the periodic solution of (1.4). For any solution (𝑥(𝑡),𝑦(𝑡)) of (1.3) with a positive initial value, there are two possible cases:(i)for all 𝑡≥0, 𝑥(𝑡)>𝑥^∗(𝑡);(ii)there is a 𝑇>0, such that 𝑥(𝑇)≤𝑥^∗(𝑡). Now, in the above two cases, we prove lim_𝑡→0𝑦(𝑡)=0, respectively.
(i) Suppose 𝑣(𝑡) is a solution of (1.4) with 𝑣(0)=𝑥(0). Comparing (1.3) and (1.4) we have 𝑣(𝑡)≥𝑥(𝑡)≥𝑥^∗(𝑡) and lim_𝑡→0(𝑣(𝑡)−𝑥^∗(𝑡))=0. Hencelim_𝑡→∞𝑥(𝑡)−𝑥^∗(𝑡)=0. (3.85) From system (1.3), for all 𝑡₀∈[0,∞), we have 𝑥𝑡₀+𝜔−𝑥𝑡₀=^𝑡0_𝑡^+𝜔₀𝑥(𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠−^𝑡0_𝑡^+𝜔₀𝑥(𝑠)𝑔()𝑥𝑠,𝑥𝑦𝑑𝑠_∗𝑡₀+𝜔−𝑥^∗𝑡₀=^𝑡0_𝑡^+𝜔₀𝑥^∗(𝑠)𝑓𝑠,𝑥^∗𝑑𝑠=0. (3.86) So 𝑥𝑡₀+𝜔−𝑥𝑡₀=^𝑡0_𝑡^+𝜔₀𝑥𝑓(𝑠,𝑥)−𝑥^∗𝑓𝑠,𝑥^∗𝑑𝑠−^𝑡0_𝑡^+𝜔₀()=𝑥𝑔𝑠,𝑥𝑦𝑑𝑠_{𝑡0𝑡+𝜔0}𝑥−𝑥^∗𝑓(𝑠,𝑥)+𝑥^∗𝑓(𝑠,𝑥)−𝑓𝑠,𝑥^∗𝑑𝑠−^𝑡0_𝑡^+𝜔₀()=𝑥𝑔𝑠,𝑥𝑦𝑑𝑠_{𝑡0𝑡+𝜔0}𝑥−𝑥^∗𝑓(𝑠,𝑥)+𝑥^∗𝜕𝑓𝜕𝑥𝑠,𝑥^∗+𝜃𝑥−𝑥^∗𝑑𝑠−^𝑡0_𝑡^+𝜔₀𝑥𝑔(𝑠,𝑥)𝑦𝑑𝑠, (3.87) where 0<𝜃<1. By the boundedness of 𝑥, 𝑥^∗, and the continuity of 𝑓 and 𝜕𝑓/𝜕𝑥, there must exist a constant 𝐴>0, such that 𝑓(𝑠,𝑥)+𝑥^∗𝜕𝑓𝜕𝑥𝑠,𝑥^∗+𝜃𝑥−𝑥^∗≤𝐴. (3.88) For all 𝜀>0, by (3.85), there exists a 𝑇⁰>0, such that 0<𝑥(𝑡)−𝑥^∗(𝑡)<𝜀, for all 𝑡≥𝑇⁰. Hence 𝑥𝑔(^𝑡0_𝑡^+𝜔₀𝑠,𝑥)𝑦𝑑𝑠≤−𝑥𝑡₀+𝜔+𝑥𝑡₀𝑥𝑡+𝐴𝜔𝜀=−₀+𝜔−𝑥^∗𝑡₀+𝜔+𝑥𝑡₀−𝑥^∗𝑡₀=()𝜀+𝐴𝜔𝜀<0+𝜀+𝐴𝜔𝜀𝐴𝜔+1 (3.89) for 𝑡₀≥𝑇⁰. Denote 𝐺^𝐿=min_{0≤𝑡≤𝜔}𝑥^∗(𝑡)𝑔(𝑡,𝑥^∗(𝑡)), 𝑦^𝐿_𝑡0=min_{𝑡0≤𝑡≤𝑡0+𝜔}𝑦(𝑡). Then 0<𝐺^𝐿𝑡0_𝑡^+𝜔₀𝑦𝑑𝑠≤^𝑡0_𝑡^+𝜔₀𝑥𝑔(𝑠,𝑥)𝑦𝑑𝑠≤(𝐴𝜔+1)𝜀. (3.90) Hence 0<𝑦^𝐿_𝑡0≤1𝜔^𝑡0_𝑡^+𝜔₀𝑦(𝑠)𝐴𝑑𝑠≤𝐺^𝐿+1𝜔𝐺^𝐿𝜀. (3.91) This implies lim_𝑡0→∞𝑦^𝐿_𝑡0=lim_𝑡0→∞^𝑡0_𝑡^+𝜔₀𝑦(𝑠)𝑑𝑠=0. (3.92)
On the other hand,𝑦(𝑡)=𝑦𝑡₀exp^𝑡0_𝑡^+𝜔₀𝛾(𝑠)𝑥(𝑠)𝑔(𝑠,𝑥(𝑠))−𝜇(𝑠)−ℎ(𝑠)𝑦(𝑠)𝑑𝑠. (3.93) By the boundedness of 𝑥, 𝑦, 𝑔, and the continuity of 𝛾, μ, and ℎ, there exists a 𝐶>0 such that 𝛾(𝑡)𝑔(𝑡,𝑥)𝑥−𝜇(𝑡)−ℎ(𝑡)𝑦(𝑡)≤𝐶,𝑡≥0. (3.94) Hence 𝑦(𝑡)ln𝑦𝑡₀≤𝐶^𝑡0_𝑡^+𝜔₀𝑑𝑠=𝐶𝜔,for𝑡∈[0,𝜔]. (3.95) For all 𝑡, 𝑠∈[0,𝜔], we have 𝑦(𝑡)ln𝑦(𝑠)=𝑦(𝑡)ln𝑦𝑡₀𝑦(𝑠)−ln𝑦𝑡₀≤2𝐶𝜔. (3.96) It follows that ()≤𝑦(𝑡)exp−2𝐶𝜔𝑦(𝑠)≤exp(2𝐶𝜔),|𝑡−𝑠|≤𝜔. (3.97) Considering (3.92), we have lim_𝑡→∞𝑦(𝑡)=0. (3.98)
 (ii) Consider the second case. We claim that once there is a 𝑇>0 such that 𝑥(𝑇)≤𝑥^∗(𝑇), then𝑥(𝑡)<𝑥^∗(𝑡),∀𝑡>𝑇. (3.99)
Otherwise, suppose 𝑇=min_𝑡>𝑇{𝑡∶𝑥(𝑡)=𝑥^∗(𝑡)}. Then𝑥̇𝑇=𝑥^′𝑇−0=lim_𝑡→𝑇₋₀𝑥(𝑡)−𝑥𝑇𝑡−𝑇=lim_𝑡→𝑇₋₀𝑥(𝑡)−𝑥^∗𝑇𝑡−𝑇≥lim_𝑡→𝑇₋₀𝑥^∗(𝑡)−𝑥^∗𝑇𝑇=𝑥̇𝑡−^∗𝑇. (3.100) This is a contradiction since from (1.3) and (1.4) we know 𝑥̇(𝑡)<𝑥̇^∗(𝑡) at the same point (𝑡,𝑥)∈𝑅²₊. So (3.99) holds.
Now we show for all 𝜀>0, there is a 𝑇⁽¹⁾>𝑇≥0 such that𝑦𝑇⁽¹⁾<𝜀. (3.101) Otherwise, suppose for all 𝑡≥𝑇, 𝑦(𝑡)≥𝜀. Then 𝑥̇≤𝑥𝑓(𝑡,𝑥)−𝑔(𝑡,𝑥)𝜀,𝑡≥𝑇. (3.102) Suppose 𝑣(𝑡) is a solution of (1.4) with 𝑣(𝑇)=𝑥(𝑇). Then by Lemma 2.1 we have 𝑣(𝑡)→𝑥^∗(𝑡)(𝑡→∞), where 𝑥^∗(𝑡) denotes the periodic solution of (1.4). Since 𝑥̇(𝑇)<𝑣(̇𝑇), there exists a 𝜎>0 such that 𝑥(𝑡)<𝑣(𝑡) for 𝑡∈(𝑇,𝑇+𝜎]. Denote 𝛿(𝑡)=𝑣(𝑡)−𝑥(𝑡). From (3.99) we know 𝛿(𝑡)>0,for 𝑡≥𝑇. We show that there is an 𝜀₁>0 such that lim_𝑡→∞inf𝛿(𝑡)≥𝜀₁. In fact, ̇̇()−()()𝜀𝑓()()()()𝜀=𝑣𝑣−𝑥=𝑣𝑓𝑡,𝑣𝑥𝑓𝑡,𝑥−𝑔𝑡,𝑥=𝑣𝑡,𝑣−𝑓𝑡,𝑥+𝑓𝑡,𝑥)(𝑣−𝑥+𝑥𝑔𝑡,𝑥𝜕𝑓((()()𝜀𝜕𝑥𝑡,𝑥+𝜃𝑣−𝑥))+𝑓𝑡,𝑥)(𝑣−𝑥+𝑥𝑔𝑡,𝑥 (3.103) where 0<𝜃<1. By the boundedness of 𝑥, 𝑣, and the continuity of 𝑓 and 𝜕𝑓/𝜕𝑥, there exists an 𝐿>0, such that |𝑣(𝜕𝑓/𝜕𝑥)(𝑡,𝑥+𝜃(𝑣−𝑥))+𝑓(𝑡,𝑥)|<𝐿. Hence 𝛿̇>−𝐿𝛿+𝑥𝑔(𝑡,𝑥)𝜀. (3.104) Choosing 𝑇⁽⁰⁾>𝑇, such that 𝑣(𝑡)≥𝑥^∗(𝑡)/2for 𝑡≥𝑇⁽⁰⁾, and letting 𝜀₁=minmax_{0≤𝑡≤𝜎}𝛿(𝑡),min_{0≤𝑡≤;𝜔}𝑥^∗(𝑡)4,min_{0≤𝑡≤𝜔}𝑔𝑡,𝑥^∗(𝑡)/4𝑥^∗(𝑡)𝜀4𝐿. (3.105) Then whenever 𝛿(𝑡)≤𝜀₁ for 𝑡≥𝑇⁽⁰⁾, we have 𝑥(𝑡)=𝑣(𝑡)−𝛿(𝑡)≥𝑥^∗(𝑡)/2−𝑥^∗(𝑡)/4=𝑥^∗(𝑡)/4. By assumption (A6), 𝛿̇(𝑡)>−𝐿𝜀₁+𝑔(𝑡,𝑥^∗(𝑡)/4)𝑥^∗(𝑡)/4≥0, and this implies lim_𝑡→∞inf𝛿(𝑡)>𝜀₁. Choosing 𝑇^′>𝑇, such that 𝑣(𝑡)<𝑥^∗(𝑡)+𝜀₁/2and 𝛿(𝑡)≥𝜀₁ for 𝑡≥𝑇′, we have 𝑥(𝑡)≤𝑣(𝑡)−𝜀₁≤𝑥^∗(𝑡)+𝜀₁2−𝜀₁=𝑥^∗(𝑡)−𝜀₁2,for𝑡≥𝑇^′. (3.106) Hence, there exists an 𝜀₀>0 such that 𝛾(𝑡)𝑔(𝑡,𝑥(𝑡))𝑥(𝑡)−𝜇(𝑡)−ℎ(𝑡)𝜀≤𝛾(𝑡)𝑔𝑡,𝑥^∗(𝑡)−𝜀₁2𝑥^∗(𝑡)−𝜀₁2−𝜇(𝑡)≤−𝜀₀<0 (3.107) for 𝑡≥𝑇^′, by (2.3) and assumption (A5). So 𝜀≤𝑦(𝑡)≤𝑦𝑇^′exp^𝑡_𝑇′𝛾(𝑠)𝑔(𝑠,𝑥(𝑠))𝑥(𝑠)−𝜇(𝑠)−ℎ(𝑠)𝜀𝑑𝑠⟶0(𝑡⟶∞). (3.108) This is a contradiction and it implies that (3.101) holds.
Second, we show that𝑦(𝑡)≤𝜀exp(𝑀(𝜀)𝜔),for𝑡≥𝑇⁽¹⁾, (3.109) where 𝑀(𝜀)=max_{0≤𝑡≤𝜔}𝛾(𝑡)𝑔𝑡,𝑥^∗(𝑡)𝑥^∗(𝑡)+𝜇(𝑡)+ℎ(𝑡)𝜀. (3.110) Otherwise, there exists 𝑇⁽²⁾>𝑇⁽¹⁾, such that 𝑦𝑇⁽²⁾>𝜀exp(𝑀(𝜀)𝜔). (3.111) By the continuity of 𝑦(𝑡), there must exist 𝑇⁽³⁾∈(𝑇⁽¹⁾,𝑇⁽²⁾) such that 𝑦(𝑇⁽³⁾)=𝜀 and 𝑦(𝑡)>𝜀 for 𝑡∈(𝑇⁽³⁾,𝑇⁽²⁾]. Let 𝑃₁ be the nonnegative integer, such that 𝑇⁽²⁾∈(𝑇⁽³⁾+𝑃₁𝜔,𝑇⁽³⁾+(𝑃₁+1)𝜔], and by (2.3), (A6) and (3.99), we have 𝜀exp(𝑀(𝜀)𝜔)<𝑦𝑇⁽²⁾<𝑦𝑇⁽³⁾exp^𝑇(2)_𝑇(3)𝛾(𝑠)𝑔(𝑠,𝑥(𝑠))𝑥(𝑠)−𝜇(𝑠)−ℎ(𝑠)𝜀𝑑𝑠=𝜀exp^{𝑇(3)+𝑃1𝜔}_𝑇(3)+_𝑇^𝑇(2)_{(3)+𝑃1𝜔}𝛾(𝑠)𝑔(𝑠,𝑥)𝑥−𝜇(𝑠)−ℎ(𝑠)𝜀𝑑𝑠<𝜀exp^{𝑇(3)+𝑃1𝜔}_𝑇(3)+_𝑇^𝑇(2)_{(3)+𝑃1𝜔}𝛾(𝑠)𝑔𝑠,𝑥^∗𝑥^∗−𝜇(𝑠)−ℎ(𝑠)𝜀𝑑𝑠≤𝜀exp^𝑇(2)_{𝑇(3)+𝑃1𝜔}𝛾(𝑠)𝑔𝑠,𝑥^∗𝑥^∗−𝜇(𝑠)−ℎ(𝑠)𝜀𝑑𝑠≤𝜀exp^𝑇(2)_{𝑇(3)+𝑃1𝜔}𝛾(𝑠)𝑔𝑠,𝑥^∗𝑥^∗+𝜇(𝑠(𝑠)𝜀(𝑀(𝜀)𝜔),)+ℎ𝑑𝑠<𝜀exp (3.112) which is a contradiction. This completes the proof of conclusion (i) of Theorem 2.3.
Now we prove the second conclusion. Since (i) holds and 𝑔(𝑡,𝑥) is bounded, there is a 𝑡₁>0 and an 𝜀>0, such that 𝑔(𝑡,𝑥)𝑦<𝜀≤𝜀₀ for 𝑡≥𝑡₁. Then we have𝑥̇≥𝑥𝑓(𝑡,𝑥)−𝜀,for𝑡≥𝑡₁. (3.113) The following auxiliary equation, 𝑣̇=𝑣𝑓(𝑡,𝑣)−𝜀 (3.114) has a globally asymptotically stable positive 𝜔-periodic solution 𝑥^∗_𝜀(𝑡) by Lemma 3.2. So by Lemma 3.3, we have lim_𝑡→∞inf𝑥(𝑡)≥𝑥^∗_𝜀(𝑡). (3.115) This completes the proof of conclusion (ii) of Theorem 2.3.

Proof of Corollary 2.6. We claim that once 𝑔(𝑡,𝑥) satisfies a local Lipschitz condition with respect to 𝑥, then the function 𝐹=𝑥𝑓()()𝑦𝑦𝛾(𝑡)𝑔()(𝑡)(𝑡)𝑦𝑡,𝑥,𝑦𝑡,𝑥−𝑔𝑡,𝑥𝑡,𝑥𝑥−𝜇−ℎ (3.116) satisfies a local Lipschitz condition with respect to 𝑥 and 𝑦. Considering assumptions (A1)–(A7), this is clearly the case. So the uniqueness of solutions of system (1.3) is guaranteed, and by Lemma 2.5 we know that there exsits an 𝜔-periodic solution 𝑍^∗(𝑡)=(𝑥^∗(𝑡),𝑦^∗(𝑡)). From the proof of Theorem 15.5 in [37], the initial value of the periodic solution is the fix point of the mapping 𝑇∶𝑧₀→𝑧(𝜔;0,𝑧₀) which is the limit point of a subsequence of the sequence {𝑇^𝑛𝑧₀}^∞₁, where 𝑧₀=(𝑥₀,𝑦₀) is the initial value of a bounded solution of system (1.3). Taking any positive 𝑥₀ and 𝑦₀, by Theorem 2.2 we know the solution started from this point is bounded and the limit of any subsequence of the sequence {𝑇^𝑛𝑧₀}^∞₁ is positive. So the periodic solution (𝑥^∗(𝑡),𝑦^∗(𝑡)) is positive.

4. Examples

Example 4.1. Suppose 𝑓(𝑡,𝑥)=1−(2+cos𝑡)𝑥,𝑔(𝑡,𝑥)=2𝑥𝑦/(2+sin𝑡+𝑥²), 𝛾(𝑡)=0.5, 𝜇(𝑡)=(1/10)−(1/20)sin𝑡, ℎ(𝑡)=1. The corresponding system is 𝑥̇=𝑥1−(2+cos𝑡)𝑥−2𝑥𝑦2+sin𝑡+𝑥²,̇𝑥𝑦=𝑦²2+sin𝑡+𝑥²−1−110.20sin𝑡 (4.1) We know that the periodic solution of system (3.1) is 𝑥^∗(𝑡)=1−exp−∫^𝜔₀𝑎(𝑠)𝑑𝑠∫^𝜔₀𝑏(𝑡−𝑠)exp−∫^𝜔₀𝑎(𝑡−𝜏)𝑑𝜏𝑑𝑠. (4.2) Using this formula we can compute the periodic solution of (4.1) in absence of predators 𝑥^∗(𝑡)=21−e^−2𝜋4e^2𝜋+cos(𝑡)e^2𝜋+sin(𝑡)e^2𝜋−4−cos(𝑡)−sin(𝑡)e^−2𝜋≈2((𝑡)(𝑡,4+cos+sin))𝛾(𝑡)𝑔(𝑡,𝑥^∗(𝑡))𝑥^∗1(𝑡)−𝜇(𝑡)=2𝜋₀^2𝜋𝑥^∗22+sin𝑡+(𝑥^∗)²−1−11020sin𝑡𝑑𝑡≈0.05313>0. (4.3) System (4.1) is permanent.

Example 4.2. Taking 𝑓(𝑡,𝑥), 𝑔(𝑡,𝑥) and 𝛾(𝑡) the same as in Example 4.1, and 𝜇(𝑡)=(1/5)−(1/10)sin𝑡, we can compute 𝛾(𝑡)𝑔(𝑡,𝑥^∗(𝑡))𝑥^∗1(𝑡)−𝜇(𝑡)=2𝜋₀^2𝜋𝑥^∗22+sin𝑡+(𝑥^∗)²−15−110sin𝑡𝑑𝑡≈−0.04687<0. (4.4)
In this circumstance the predator population tends to extinction and the prey population keeps oscillating.
Simulation results of the two examples are shown in Figure 1.

(a)

(b)

Figure 1 

(a) Simulation results of Example 4.1 with initial values 𝑥(0)=0.3, 𝑦(0)=0.5. (b) Simulation results of Example 4.2 with initial values 𝑥(0)=0.3, 𝑦(0)=0.6.