Permanence of Periodic Predator-Prey System with Functional Responses and Stage Structure for Prey

Abstract

A stage-structured three-species predator-prey model with Beddington-DeAngelis and Holling II functional response is introduced. Based on the comparison theorem, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. An example is also presented to illustrate our main results.

1. Introduction

The aim of this paper is to investigate the permanence of the following periodic stage-structure predator-prey system with Beddington-DeAngelis and Holling II functional response:𝑥^′₁(𝑡)=𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑝(𝑡)−¹(𝑡)𝑥₁(𝑡)𝑘₁(𝑡)+𝑚(𝑡)𝑦₁(𝑡)+𝑛(𝑡)𝑥₁𝑦(𝑡)¹𝑥(𝑡),_′2(𝑡)=𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂𝑝(𝑡)−²(𝑡)𝑥₂(𝑡)𝑘₂(𝑡)+𝑥₂𝑦(𝑡)²𝑦(𝑡),_′1(𝑡)=𝑦₁(𝑡)−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥₁(𝑡)𝑘₁(𝑡)+𝑚(𝑡)𝑦₁(𝑡)+𝑛(𝑡)𝑥₁(𝑡)−𝑞₁(𝑡)𝑦₁,𝑦(𝑡)_′2(𝑡)=𝑦₂(𝑡)−𝑔₂ℎ(𝑡)+²(𝑡)𝑥₂(𝑡)𝑘₂(𝑡)+𝑥₂(𝑡)−𝑞₂(𝑡)𝑦₂(𝑡), (1.1)where 𝑎(𝑡),𝑏(𝑡),𝑐(𝑡),𝑑(𝑡),𝑓(𝑡),𝑔_𝑖(𝑡),ℎ_𝑖(𝑡),𝑘_𝑖(𝑡),𝑚(𝑡),𝑛(𝑡),𝑝_𝑖(𝑡), and 𝑞_𝑖(𝑡)(𝑖=1,2) are all continuous positive 𝜔-periodic functions. Here, 𝑥₁(𝑡) and 𝑥₂(𝑡) denote the density of immature and mature prey species at time 𝑡, respectively, 𝑦₁(𝑡) represents the density of the predator that preys on immature prey, and 𝑦₂(𝑡) represents the density of the other predator that preys on mature prey at time 𝑡.

The birth rate into the immature population is given by 𝑎(𝑡)𝑥₂(𝑡), that is, it is assumed to be proportional to the existing mature population, with a proportionality coefficient 𝑎(𝑡). The death rate of the immature population is proportional to the existing immature population and to its square with coefficients 𝑏(𝑡) and 𝑑(𝑡), respectively. The death rate of the mature population is of a logistic nature, that is, it is proportional to the square of the population with a proportionality 𝑓(𝑡). The transition rate from the immature individuals to the mature individuals is assumed to be proportional to the existing immature population, with a proportionality coefficient 𝑐(𝑡). Similarly, −𝑔₁(𝑡)𝑦₁(𝑡)−𝑞₁(𝑡)𝑦²₁(𝑡) and −𝑔₂(𝑡)𝑦₂(𝑡)−𝑞₂(𝑡)𝑦²₂(𝑡) give the density dependent death rate of the two predators, respectively. 𝑝₁(𝑡) and 𝑝₂(𝑡) are the capturing rate of the two predators, respectively. ℎ₁(𝑡)/𝑝₁(𝑡) and ℎ₂(𝑡)/𝑝₂(𝑡) are the rate of conversion of nutrients into the reproduction of the two mature predators, respectively.

The functional response of predator species 𝑦₁(𝑡) to immature prey species takes the Beddington-DeAngelis form, that is, 𝑥₁(𝑡)/(𝑘₁(𝑡)+𝑚(𝑡)𝑦₁(𝑡)+𝑛(𝑡)𝑥₁(𝑡)). It was introduced by Beddington [1] and DeAngelis et al. [2] independently in 1975. It is similar to the well-known Holling type II functional response but has an extra term 𝑚(𝑡)𝑦₁(𝑡) in the denominator which models mutual interference between predators. The Beddington-DeAngelis form of functional response has some of the same qualitative features as the ratio-dependent models form but avoids some of the same behaviors of ratio-dependent models at low densities which have been the source of controversy. The function 𝑥₂(𝑡)/(𝑘₂(𝑡)+𝑥₂(𝑡)) represents the functional response of predator 𝑦₂(𝑡) to mature prey, which is called Holling type II function or Michaelis-Menten function. Holling type II is the second function that Holling proposed three kinds of functional response of the predator to prey based on numerous experiments for different species. The Holling type form of functional response is intituled prey-dependent model form. It is applied to almost invertebrate that is one of the most extensive applied functional responses.

Cui and Song [3] proposed the following predator-prey model with stage structure for prey:𝑥^′₁(𝑡)=𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁(𝑡)−𝑝(𝑡)𝑥₁𝑥(𝑡)𝑦(𝑡),_′2(𝑡)=𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂𝑦(𝑡),_′(𝑡)=𝑦(𝑡)−𝑔(𝑡)+ℎ(𝑡)𝑥₁(𝑡)−𝑞(𝑡)𝑦(𝑡). (1.2)They obtained a set of sufficient and necessary conditions which guarantee the permanence of the system. For more back ground and the relevant work on system (1.2), one could refer to [3–6] and the references cited therein. Recently, Chen [7, 8] and Yang [9] consider the functional response of the predator to immature prey species. Lin and Hong [10] consider a biological model for two predators and one prey with periodic delays.

In reality, mature prey was also consumed by some predators. Different predator usually consumes prey in different stage structure. Some predators only prey on immature prey, and some predators only prey on mature prey. There is different functional response in different predator. So, we add a predator species which consumes mature prey to the model (1.2). By assuming that one predator consumes immature prey according to the Beddington-DeAngelis functional response while the other predator consumes mature prey according to Holling II functional response, we get model (1.1). In the resource limited environment, could the wild animals be coexistence for long-term under the animals' law of the jungle? To keep the biology's variety of the nature, the permanence of biotic population is a significant and comprehensive problem in biomathematics. So, it is meaningful to investigate the permanence of the model (1.1).

The aim of this paper is, by further developing the analysis technique of Cui [3, 11], to derive a set of sufficient and necessary conditions which ensure the permanence of the system (1.1). The rest of the paper is arranged as follows. In Section 2, we introduce some lemmas and then state the main result of this paper. The result is proved in Section 3. In Section 4, we give an example which shows the feasibility of our result. The last section is devoted to make some explanation on the biological meaning of our result.

Throughout this paper, for a continuous 𝜔-periodic function 𝑓(𝑡), we set𝐴_𝜔1(𝑓)=𝜔^𝜔₀𝑓(𝑡)𝑑𝑡. (1.3)

2. Main Result

In this section, we introduce a definition and some lemmas which will be useful in subsequent sections and state the main result.

Definition 2.1. System (1.1) is said to be permanent if there exist positive constants 𝑚,𝑀, and 𝑇₀, such that each positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of the system (1.1) with any positive initial value 𝜑 fulfills 𝑚≤𝑥_𝑖(𝑡)≤𝑀,𝑚≤𝑦_𝑖(𝑡)≤𝑀,𝑖=1,2 for all 𝑡≥𝑇₀, where 𝑇₀ may depend on 𝜑.

Lemma 2.2 (see [12]). If 𝑎(𝑡), 𝑏(𝑡),𝑐(𝑡), 𝑑(𝑡), and 𝑓(𝑡) are all 𝜔-periodic, then system 𝑥^′₁(𝑡)=𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥(𝑡),_′2(𝑡)=𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂(𝑡) (2.1) has a positive 𝜔-periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) which is globally asymptotically stable with respect to 𝑅²₊={(𝑥₁,𝑥₂)∶𝑥₁>0,𝑥₂>0}.

Lemma 2.3 (see [13]). If 𝑏(𝑡) and 𝑎(𝑡) are all 𝜔-periodic, and if 𝐴_𝜔(𝑏)>0 and 𝐴_𝜔(𝑎)>0 for all 𝑡∈𝑅, then the system 𝑥^′=𝑥𝑏(𝑡)−𝑎(𝑡)𝑥 (2.2)has a positive 𝜔-periodic solution which is globally asymptotically stable.

Now, we state the main result of this paper.

Theorem 2.4. System (1.1) is permanent if and only if 𝐴_𝜔−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥^∗₁(𝑡)𝑘₁(𝑡)+𝑛(𝑡)𝑥^∗₁(𝑡)>0,𝐴_𝜔−𝑔₂ℎ(𝑡)+²(𝑡)𝑥^∗₂(𝑡)𝑘₂(𝑡)+𝑥^∗₂(𝑡)>0, (2.3) where (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) is the unique positive periodic solution of system (2.1) given by Lemma 2.2.

3. Proof of the Main Result

We need the following propositions to prove Theorem 2.4. The hypothesis of the lemmas and theorem of the preceding section is assumed to hold in what follows.

Proposition 3.1. There exist positive constants 𝑀_𝑥 and 𝑀_𝑦 such that lim_𝑡→+∞sup𝑥_𝑖(𝑡)≤𝑀_𝑥,lim_𝑡→+∞sup𝑦_𝑖(𝑡)≤𝑀_𝑦,𝑖=1,2 (3.1) for all solution of system (1.1) with positive initial values.

Proof. Obviously, 𝑅⁴₊={(𝑥₁,𝑥₂,𝑦₁,𝑦₂)∣𝑥_𝑖>0,𝑦_𝑖>0} is a positively invariant set of system (1.1). Given any solution (𝑥₁,𝑥₂,𝑦₁,𝑦₂) of system (1.1), we have𝑥^′₁(𝑡)≤𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥(𝑡),_′2(𝑡)≤𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂(𝑡). (3.2)By Lemma 2.2, the following auxiliary equation:𝑢^′₁(𝑡)=𝑎(𝑡)𝑢₂(𝑡)−𝑏(𝑡)𝑢₁(𝑡)−𝑑(𝑡)𝑢²₁𝑢(𝑡),_′2(𝑡)=𝑐(𝑡)𝑢₁(𝑡)−𝑓(𝑡)𝑢²₂(𝑡) (3.3)has a globally asymptotically stable positive 𝜔-periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)). Let (𝑢₁(𝑡),𝑢₂(𝑡)) be the solution of (3.3) with (𝑢₁(0),𝑢₂(0))=(𝑥₁(0),𝑥₂(0)). By comparison theorem, we then have𝑥_𝑖(𝑡)≤𝑢_𝑖(𝑡),𝑖=1,2, (3.4)for 𝑡⩾0. By (2.3), we can choose positive 𝜀>0 small enough such that𝐴_𝜔−𝑔_𝑖ℎ(𝑡)+^𝑖(𝑡)𝑥^∗_𝑖(𝑡)+𝜀𝑘_𝑖(𝑡)>0. (3.5)Thus, from the global attractivity of (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)), for the above given 𝜀>0, there exists a 𝑇₀>0, such that𝑢_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)<𝜀,𝑡≥𝑇₀. (3.6)Inequality (3.4) combined with (3.6) leads to𝑥_𝑖(𝑡)<𝑥^∗_𝑖(𝑡)+𝜀,𝑡>𝑇₀. (3.7)Let 𝑀_𝑥=max_{0≤𝑡≤𝜔}{𝑥^∗_𝑖(𝑡)+𝜀,𝑖=1,2}, we havelim_𝑡→+∞sup𝑥_𝑖(𝑡)≤𝑀_𝑥. (3.8)In addition, for 𝑡≥𝑇₀, from the third and fourth equations of (1.1) and (3.7) we get𝑦^′_𝑖(𝑡)≤𝑦_𝑖(𝑡)−𝑔_𝑖ℎ(𝑡)+^𝑖(𝑡)𝑥_𝑖(𝑡)𝑘_𝑖(𝑡)−𝑞_𝑖(𝑡)𝑦_𝑖(𝑡)≤𝑦_𝑖(𝑡)−𝑔_𝑖ℎ(𝑡)+^𝑖(𝑡)𝑥^∗_𝑖(𝑡)+𝜀𝑘_𝑖(𝑡)−𝑞_𝑖(𝑡)𝑦_𝑖(𝑡). (3.9)Consider the following auxiliary equation:𝑣^′_𝑖(𝑡)=𝑣_𝑖(𝑡)−𝑔_𝑖ℎ(𝑡)+^𝑖(𝑡)𝑥^∗_𝑖(𝑡)+𝜀𝑘_𝑖(𝑡)−𝑞_𝑖(𝑡)𝑣_𝑖(𝑡). (3.10)It follows from (3.5) and Lemma 2.3 that (3.10) has a unique positive 𝜔-periodic solution 𝑦^∗_𝑖(𝑡)>0 which is globally asymptotically stable. Similar to the above analysis, there exists a 𝑇₁>𝑇₀ such that for the above 𝜀, one has𝑦_𝑖(𝑡)<𝑦^∗_𝑖(𝑡)+𝜀,𝑡≥𝑇₁. (3.11)Let 𝑀_𝑦=max_{0≤𝑡≤𝜔}{𝑦^∗_𝑖(𝑡)+𝜀,𝑖=1,2}, then we havelim_𝑡→+∞sup𝑦_𝑖(𝑡)≤𝑀_𝑦,𝑖=1,2. (3.12)This completes the proof of Proposition 3.1.

Proposition 3.2. There exist positive constants 𝛿_𝑖𝑥<𝑀_𝑥, 𝑖=1,2, such that lim_𝑡→+∞inf𝑥_𝑖(𝑡)≥𝛿_𝑖𝑥,𝑖=1,2. (3.13)

Proof. By Proposition 3.1, there exists 𝑇₁>0 such that0<𝑥_𝑖(𝑡)≤𝑀_𝑥,0<𝑦_𝑖(𝑡)≤𝑀_𝑦,𝑡≥𝑇₁. (3.14)Hence, from the first and second equations of system (1.1), we have𝑥^′₁(𝑡)≥𝑎(𝑡)𝑥₂𝑝(𝑡)−𝑏(𝑡)+¹(𝑡)𝑘₁𝑀(𝑡)^𝑦𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥(𝑡),_′2(𝑡)≥𝑐(𝑡)𝑥₁𝑝(𝑡)−𝑓(𝑡)+²(𝑡)𝑘₂𝑀(𝑡)^𝑦𝑥²₂(𝑡), (3.15)for 𝑡≥𝑇₁. By Lemma 2.2, the following auxiliary equation:𝑢^′₁(𝑡)=𝑎(𝑡)𝑢₂𝑝(𝑡)−𝑏(𝑡)+¹(𝑡)𝑘₁𝑀(𝑡)^𝑦𝑢₁(𝑡)−𝑑(𝑡)𝑢²₁𝑢(𝑡),_′2(𝑡)=𝑐(𝑡)𝑢₁𝑝(𝑡)−𝑓(𝑡)+²(𝑡)𝑘₂𝑀(𝑡)^𝑦𝑢²₂(𝑡) (3.16)has a globally asymptotically stable positive 𝜔-periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)). Let (𝑢₁(𝑡),𝑢₂(𝑡)) be the solution of (3.16) with (𝑢₁(𝑇₁),𝑢₂(𝑇₂))=(𝑥₁(𝑇₁),𝑥₂(𝑇₂)), by comparison theorem, we have𝑥_𝑖(𝑡)≥𝑢_𝑖(𝑡)(𝑖=1,2),𝑡>𝑇_1. (3.17)Thus, from the global attractivity of (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)), there exists a 𝑇₂>𝑇₁, such that𝑢_𝑖(𝑡)−𝑥^∗_𝑖<𝑥(𝑡)^∗𝑖(𝑡)2(𝑖=1,2),𝑡>𝑇₂. (3.18)Inequality (3.18) combined with (3.17) leads to𝑥_𝑖(𝑡)>𝛿_𝑖𝑥=min_{0≤𝑡≤𝜔}𝑥^∗_𝑖(𝑡)2(𝑖=1,2),𝑡>𝑇₂. (3.19)And solim_𝑡→+∞inf𝑥_𝑖(𝑡)≥𝛿_𝑖𝑥,𝑖=1,2. (3.20)The proof of Proposition 3.2 is complete.

Proposition 3.3. Suppose that (2.3) holds, then there exist positive constants 𝛿_𝑖𝑦, 𝑖=1,2, such that any solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of system (1.1) with positive initial value satisfies lim_𝑡→+∞sup𝑦_𝑖(𝑡)≥𝛿_𝑖𝑦,𝑖=1,2. (3.21)

Proof. By Assumption (2.3), we can choose constant 𝜀₀>0 (without loss of generality, we may assume that 𝜀₀<(1/2)min_{0≤𝑡≤𝜔}{𝑥^∗_𝑖(𝑡)}, where (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) is the unique positive periodic solution of system (2.1)) such that𝐴_𝜔𝜑_𝜀0(𝑡)>0,𝐴_𝜔𝜓_𝜀0(𝑡)>0, (3.22)where𝜑_𝜀0(𝑡)=−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥^∗₁(𝑡)−𝜀₀𝑘₁(𝑡)+𝑚(𝑡)𝜀₀+𝑛(𝑡)𝑥^∗₁(𝑡)−𝜀₀−𝑞₁(𝑡)𝜀₀,𝜓_𝜀0(𝑡)=−𝑔₂ℎ(𝑡)+²(𝑡)𝑥^∗₂(𝑡)−𝜀₀𝑘₂(𝑡)+𝑥^∗₂(𝑡)−𝜀₀−𝑞₂(𝑡)𝜀₀. (3.23)Consider the following equations with a parameter 𝛽>0: 𝑥^′₁(𝑡)=𝑎(𝑡)𝑥₂𝑝(𝑡)−𝑏(𝑡)+2𝛽¹(𝑡)𝑘₁𝑥(𝑡)¹(𝑡)−𝑑(𝑡)𝑥²₁𝑥(𝑡),_′2(𝑡)=𝑐(𝑡)𝑥₁𝑝(𝑡)−𝑓(𝑡)+2𝛽²(𝑡)𝑘₂𝑥(𝑡)²²(𝑡). (3.24)By Lemma 2.2, the system (3.24) has a unique positive 𝜔-periodic solution (𝑥^∗_1𝛽(𝑡),𝑥^∗_2𝛽(𝑡)), which is globally attractive. Let (𝑥_1𝛽(𝑡),𝑥_2𝛽(𝑡)) be the solution of (3.24) with initial condition 𝑥_𝑖𝛽(0)=𝑥^∗_𝑖(0),𝑖=1,2. Hence, for above 𝜀₀, there exists a sufficiently large 𝑇₃>𝑇₂ such that𝑥_𝑖𝛽(𝑡)−𝑥^∗_𝑖𝛽<𝜀(𝑡)⁰4(𝑖=1,2),𝑡>𝑇₃. (3.25)By the continuity of the solution in the parameter, we have 𝑥_𝑖𝛽(𝑡)→𝑥^∗_𝑖(𝑡) uniformly in [𝑇₃,𝑇₃+𝜔] as 𝛽→0. Hence, for 𝜀₀>0, there exists a 𝛽₀=𝛽₀(𝜀₀)>0 such that𝑥_𝑖𝛽(𝑡)−𝑥^∗_𝑖<𝜀(𝑡)⁰4(𝑖=1,2),𝑡∈[𝑇₃,𝑇₃+𝜔],0<𝛽<𝛽₀. (3.26)So, we have𝑥^∗_𝑖𝛽(𝑡)−𝑥^∗_𝑖(𝑡)≤𝑥_𝑖𝛽(𝑡)−𝑥^∗_𝑖𝛽(𝑡)+𝑥_𝑖𝛽(𝑡)−𝑥^∗_𝑖<𝜀(𝑡)⁰2,𝑡∈[𝑇₃,𝑇₃+𝜔]. (3.27)Since 𝑥^∗_𝑖𝛽(𝑡) and 𝑥^∗_𝑖(𝑡) are all 𝜔-periodic, we have𝑥^∗_𝑖𝛽(𝑡)−𝑥^∗_𝑖<𝜀(𝑡)⁰2(𝑖=1,2),𝑡≥0,0<𝛽<𝛽₀. (3.28)Choosing a constant 𝛽₁(0<𝛽₁<𝛽₀,2𝛽₁<𝜀₀), we have𝑥^∗_𝑖𝛽1(𝑡)≥𝑥^∗_𝑖𝜀(𝑡)−⁰2(𝑖=1,2),𝑡≥0. (3.29)Suppose that Conclusion (3.21) is not true. Then, there exists 𝐹∈𝑅⁴₊ such that, for the positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of (1.1) with an initial condition (𝑥₁(0),𝑥₂(0),𝑦₁(0),𝑦₂(0))=𝐹, we havelim_𝑡→+∞sup𝑦_𝑖(𝑡)<𝛽₁,𝑖=1,2. (3.30)So, there exists 𝑇₄>𝑇₃ such that𝑦_𝑖(𝑡)<2𝛽₁<𝜀₀,𝑡≥𝑇₄. (3.31)By applying (3.31), from the first and second equations of system (1.1) it follows that for all 𝑡≥𝑇₄,𝑥^′₁(𝑡)≥𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)+2𝛽₁𝑝₁(𝑡)𝑘₁𝑥(𝑡)¹(𝑡)−𝑑(𝑡)𝑥²₁𝑥(𝑡),_′2(𝑡)≥𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)+2𝛽₁𝑝₂(𝑡)𝑘₂𝑥(𝑡)²²(𝑡). (3.32)Let (𝑢₁(𝑡),𝑢₂(𝑡)) be the solution of (3.24) with 𝛽=𝛽₁ and 𝑢_𝑖(𝑇₄)=𝑥_𝑖(𝑇₄),𝑖=1,2, then𝑥_𝑖(𝑡)≥𝑢_𝑖(𝑡)(𝑖=1,2),𝑡≥𝑇₄. (3.33)By the global asymptotic stability of (𝑥^∗_1𝛽1(𝑡),𝑥^∗_2𝛽2(𝑡)), for the given 𝜀=𝜀₀/2, there exists 𝑇₅≥𝑇₄, such that𝑢_𝑖(𝑡)−𝑥^∗_𝑖𝛽1<𝜀(𝑡)⁰2(𝑖=1,2),𝑡≥𝑇₅. (3.34)So,𝑥_𝑖(𝑡)≥𝑢_𝑖(𝑡)>𝑥^∗_𝑖𝛽1𝜀(𝑡)−⁰2(𝑖=1,2),𝑡≥𝑇₅, (3.35)and hence, by using (3.29), we get𝑥_𝑖(𝑡)>𝑥^∗_𝑖(𝑡)−𝜀₀(𝑖=1,2),𝑡≥𝑇₅. (3.36)Therefore, by (3.31) and (3.36), we have𝑦^′₁(𝑡)≥𝑦₁(𝑡)−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥^∗₁(𝑡)−𝜀₀𝑘₁(𝑡)+𝑚(𝑡)𝜀₀+𝑛(𝑡)𝑥^∗₁(𝑡)−𝜀₀−𝑞₁(𝑡)𝜀₀=𝜑_𝜀0(𝑡)𝑦₁𝑦(𝑡),_′2(𝑡)≥𝑦₂(𝑡)−𝑔₂ℎ(𝑡)+²(𝑡)𝑥^∗₂(𝑡)−𝜀₀𝑘₂(𝑡)+𝑥^∗₂(𝑡)−𝜀₀−𝑞₂(𝑡)𝜀₀=𝜓_𝜀0(𝑡)𝑦₂(𝑡), (3.37)for 𝑡≥𝑇₅. Integrating (3.37) from 𝑇₅ to 𝑡 yields𝑦₁(𝑡)≥𝑦₁𝑇₅exp^𝑡_𝑇5𝜑_𝜀0,𝑦(𝑡)𝑑𝑡₂(𝑡)≥𝑦₂𝑇₅exp^𝑡_𝑇5𝜓_𝜀0(𝑡)𝑑𝑡. (3.38)Thus, from (3.22) we know that 𝜑_𝜀0(𝑡)>0,𝜓_𝜀0(𝑡)>0. It follows that 𝑦₁(𝑡)→+∞,𝑦₂(𝑡)→+∞ as 𝑡→+∞. It is a contradiction. This completes the proof.

Proposition 3.4. Suppose that (2.3) holds, then there exist positive constants 𝜂_𝑖𝑦, 𝑖=1,2, such that any solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑦₁(𝑡),𝑦₂(𝑡)) of system (1.1) with positive initial value satisfies lim_𝑡→+∞inf𝑦_𝑖(𝑡)>𝜂_𝑖𝑦,𝑖=1,2. (3.39)

Proof. Suppose that (3.39) is not true, then there exists a sequence {𝜉_𝑚}∈𝑅⁴₊, such thatlim_𝑡→+∞inf𝑦_𝑖𝑡,𝜉_𝑚<𝛿_𝑖𝑦(𝑚+1)²,𝑚=1,2,…. (3.40)On the other hand, by Proposition 3.3, we havelim_𝑡→+∞sup𝑦_𝑖𝑡,𝜉_𝑚>𝛿_𝑖𝑦,𝑚=1,2,…. (3.41)Hence, there are time sequences {𝑠_𝑞^(𝑚)} and {𝑡_𝑞^(𝑚)} satisfying0<𝑠₁^(𝑚)<𝑡₁^(𝑚)<𝑠₂^(𝑚)<𝑡₂^(𝑚)<⋯<𝑠_𝑞^(𝑚)<𝑡_𝑞^(𝑚)𝑠<⋯,_𝑞(𝑚)⟶+∞,𝑡_𝑞^(𝑚)𝑦⟶+∞as𝑞⟶+∞,_𝑖𝑠_𝑞^(𝑚),𝜉_𝑚=𝛿_𝑖𝑦𝑚+1,𝑦_𝑖𝑡_𝑞^(𝑚),𝜉_𝑚=𝛿_𝑖𝑦(𝑚+1)²,𝛿_𝑖𝑦(𝑚+1)²<𝑦_𝑖𝑡,𝜉_𝑚<𝛿_𝑖𝑦𝑠𝑚+1,𝑡∈_𝑞^(𝑚),𝑡_𝑞^(𝑚). (3.42)By Proposition 3.1, for a given positive integer 𝑚, there is a 𝑇₁^(𝑚)>0, such that for all 𝑡>𝑇₁^(𝑚)𝑥_𝑖𝑡,𝜉_𝑚<𝑀_𝑥,𝑦_𝑖𝑡,𝜉_𝑚<𝑀_𝑦,𝑖=1,2. (3.43)Because of 𝑠_𝑞^(𝑚)→+∞ as 𝑞→+∞, there is a positive integer 𝑍^(𝑚), such that 𝑠_𝑞^(𝑚)>𝑇₁^(𝑚) as 𝑞≥𝑍^(𝑚), hence𝑦^′_𝑖𝑡,𝜉_𝑚≥𝑦_𝑖𝑡,𝜉_𝑚−𝑔_𝑖(𝑡)−𝑞_𝑖(𝑡)𝑀_𝑦 (3.44)for 𝑡∈[𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚)],𝑞≥𝑍^(𝑚). Integrating (3.44) from 𝑠_𝑞^(𝑚) to 𝑡_𝑞^(𝑚) yields𝑦_𝑖𝑡_𝑞^(𝑚),𝜉_𝑚≥𝑦_𝑖𝑠_𝑞^(𝑚),𝜉_𝑚exp^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)}−𝑔_𝑖(𝑡)−𝑞_𝑖(𝑡)𝑀_𝑦𝑑𝑡, (3.45)or^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)}𝑔_𝑖(𝑡)+𝑞_𝑖(𝑡)𝑀_𝑦𝑑𝑡≥ln(𝑚+1)for𝑞≥𝑍^(𝑚). (3.46)Thus, from the boundedness of 𝑔_𝑖(𝑡)+𝑞_𝑖(𝑡)𝑀_𝑦, we have𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)⟶+∞as𝑚⟶+∞,𝑞≥𝑍^(𝑚). (3.47)By (3.22) and (3.47), there are constants 𝑃>0 and 𝑁₀>0, such that𝛿_𝑖𝑦𝑚+1<𝛽₁<𝜀₀,𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)>2𝑃,_𝑎0𝜑_𝜀0(𝑡)𝑑𝑡>0,^𝑎₀𝜓_𝜀0(𝑡)𝑑𝑡>0, (3.48)(3.49)for 𝑚≥𝑁₀,𝑞≥𝑍^(𝑚), and 𝑎≥𝑃. Inequality (3.48) implies that𝑦_𝑖𝑡,𝜉_𝑚<𝛽₁<𝜀₀,𝑡∈𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚), (3.50)for 𝑚≥𝑁₀,𝑞≥𝑍^(𝑚). In addition, from (3.43) and (3.50) we have𝑥^′₁𝑡,𝜉_𝑚≥𝑎(𝑡)𝑥₂𝑡,𝜉_𝑚−𝑏(𝑡)+2𝑝₁(𝑡)𝛽₁𝑘₁𝑥(𝑡)¹𝑡,𝜉_𝑚−𝑑(𝑡)𝑥²₁𝑡,𝜉_𝑚,𝑥^′₂𝑡,𝜉_𝑚≥𝑐(𝑡)𝑥₁𝑡,𝜉_𝑚−𝑓(𝑡)+2𝑝₂(𝑡)𝛽₁𝑘₂𝑥(𝑡)²²𝑡,𝜉_𝑚, (3.51)for 𝑡∈[𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚)]. Let (𝑢₁(𝑡),𝑢₂(𝑡)) be the solution of (3.24) with 𝛽=𝛽₁ and 𝑢_𝑖(𝑠_𝑞^(𝑚))=𝑥_𝑖(𝑠_𝑞^(𝑚),𝜉_𝑚), then by applying comparison theorem, we have𝑥_𝑖𝑡,𝜉_𝑚≥𝑢_𝑖(𝑡),𝑡∈𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚). (3.52)Further, by using Propositions 3.1 and 3.2, there exists an enough large 𝑍₁^(𝑚)>𝑍^(𝑚) such that𝜂_𝑖𝑥<𝑥_𝑖𝑠_𝑞^(𝑚),𝜉_𝑚<𝑀_𝑥, (3.53)for 𝑞≥𝑍₁^(𝑚). For 𝛽=𝛽₁, (3.24) has a unique positive 𝜔-periodic solution (𝑥^∗_1𝛽1(𝑡),𝑥^∗_2𝛽1(𝑡)) which is globally asymptotically stable. In addition, by the periodicity of (3.24), the periodic solution (𝑥^∗_1𝛽1(𝑡),𝑥^∗_2𝛽1(𝑡)) is uniformly asymptotically stable with respect to the compact set Ω={𝑥∣𝜂_𝑖𝑥<𝑥<𝑀_𝑥}. Hence, for given 𝜀₀ in Proposition 3.3, there exists 𝑇₀>𝑃, which is independent of 𝑚 and 𝑞, such that𝑢_𝑖(𝑡)>𝑥^∗_𝑖𝛽1𝜀(𝑡)−⁰2,𝑖=1,2as𝑡>𝑇₀+𝑠_𝑞^(𝑚). (3.54)Thus, by using (3.29), we get𝑢_𝑖(𝑡)>𝑥^∗_𝑖(𝑡)−𝜀₀,𝑖=1,2as𝑡>𝑇₀+𝑠_𝑞^(𝑚). (3.55)By (3.47), there exists a positive integer 𝑁₁≥𝑁₀ such that 𝑡_𝑞^(𝑚)>𝑠_𝑞^(𝑚)+2𝑇₀>𝑠_𝑞^(𝑚)+2𝑃 for 𝑚≥𝑁₁ and 𝑞≥𝑍₁^(𝑚). So, we have𝑥_𝑖𝑡,𝜉_𝑚≥𝑥^∗_𝑖(𝑡)−𝜀₀,𝑖=1,2as𝑡∈𝑇₀+𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚), (3.56)where 𝑚≥𝑁₁ and 𝑞≥𝑍₁^(𝑚). Hence, by using (3.50) and (3.56), from the third and fourth equations of system (1.1), we have𝑦^′₁𝑡,𝜉_𝑚≥𝜑_𝜀0(𝑡)𝑦₁𝑡,𝜉_𝑚,𝑦^′₂𝑡,𝜉_𝑚≥𝜓_𝜀0(𝑡)𝑦₂𝑡,𝜉_𝑚,𝑡∈𝑇₀+𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚). (3.57)Integrating the above inequalities from 𝑇₀+𝑠_𝑞^(𝑚) to 𝑡_𝑞^(𝑚), we have𝑦₁𝑡_𝑞^(𝑚),𝜉_𝑚≥𝑦₁𝑇₀+𝑠_𝑞^(𝑚),𝜉_𝑚exp^{𝑡𝑞(𝑚)}_{𝑇0+𝑠𝑞(𝑚)}𝜑_𝜀0,𝑦(𝑡)𝑑𝑡₂𝑡_𝑞^(𝑚),𝜉_𝑚≥𝑦₂𝑇₀+𝑠_𝑞^(𝑚),𝜉_𝑚exp^{𝑡𝑞(𝑚)}_{𝑇0+𝑠𝑞(𝑚)}𝜓_𝜀0(𝑡)𝑑𝑡, (3.58)that is𝛿_1𝑦(𝑚+1)²≥𝛿_1𝑦(𝑚+1)²exp^{𝑡𝑞(𝑚)}_{𝑇0+𝑠𝑞(𝑚)}𝜑_𝜀0>𝛿(𝑡)𝑑𝑡^1𝑦(𝑚+1)²,𝛿_2𝑦(𝑚+1)²≥𝛿_2𝑦(𝑚+1)²exp^{𝑡𝑞(𝑚)}_{𝑇0+𝑠𝑞(𝑚)}𝜓_𝜀0>𝛿(𝑡)𝑑𝑡^2𝑦(𝑚+1)². (3.59)These are contradictions. This completes the proof of Proposition 3.4.

Proof of Theorem 2.4. The sufficiency of Theorem 2.4 now follows from Propositions 3.1–3.4. We thus only need to prove the necessity of Theorem 2.4. Suppose that𝐴_𝜔−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥^∗₁(𝑡)𝑘₁(𝑡)+𝑛(𝑡)𝑥^∗₁(𝑡)≤0,𝐴_𝜔−𝑔₂ℎ(𝑡)+²(𝑡)𝑥^∗₂(𝑡)𝑘₂(𝑡)+𝑥^∗₂(𝑡)≤0. (3.60)We will show thatlim_𝑡→+∞𝑦_𝑖(𝑡)=0,𝑖=1,2. (3.61)In fact, by (3.60), we know that, for any given positive constant 0<𝜀<1, there exist 𝜀₁>0, (0<𝜀₁<𝜀), 𝜀₀>0 such that𝐴_𝜔−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥^∗₁(𝑡)+𝜀₁𝑘₁(𝑡)+𝑛(𝑡)𝑥^∗₁(𝑡)+𝜀₁−𝑞₁𝜀(𝑡)𝜀≤−2𝐴_𝜔𝑞₁(𝑡)<−𝜀₀,𝐴_𝜔−𝑔₂ℎ(𝑡)+²(𝑡)𝑥^∗₂(𝑡)+𝜀₁𝑘₂(𝑡)+𝑥^∗₂(𝑡)+𝜀₁−𝑞₂𝜀(𝑡)𝜀≤−2𝐴_𝜔𝑞₂(𝑡)<−𝜀₀. (3.62)Since𝑥^′₁(𝑡)≤𝑎(𝑡)𝑥₂(𝑡)−𝑏(𝑡)𝑥₁(𝑡)−𝑑(𝑡)𝑥²₁𝑥(𝑡),_′2(𝑡)≤𝑐(𝑡)𝑥₁(𝑡)−𝑓(𝑡)𝑥²₂(𝑡). (3.63)We know that, for above 𝜀₁ there exists a 𝑇⁽¹⁾>0 such that𝑥_𝑖(𝑡)<𝑥^∗_𝑖(𝑡)+𝜀,𝑡≥𝑇⁽¹⁾. (3.64)It follows from (3.62) and (3.64) that for 𝑡≥𝑇⁽¹⁾,𝐴_𝜔−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥₁(𝑡)𝑘₁(𝑡)+𝑛(𝑡)𝑥₁(𝑡)−𝑞₁(𝑡)𝜀<−𝜀₀,𝐴_𝜔−𝑔₂ℎ(𝑡)+²(𝑡)𝑥₂(𝑡)𝑘₂(𝑡)+𝑥₂(𝑡)−𝑞₂(𝑡)𝜀<−𝜀₀. (3.65)
First, we show that there exists a 𝑇⁽²⁾>𝑇⁽¹⁾ such that 𝑦_𝑖(𝑇⁽²⁾)<𝜀,𝑖=1,2. Otherwise, by (3.65), we have𝜀≤𝑦₁(𝑡)≤𝑦₁𝑇⁽¹⁾exp^𝑡_𝑇(1)−𝑔₁ℎ(𝑠)+¹(𝑠)𝑥₁(𝑠)𝑘₁(𝑠)+𝑛(𝑠)𝑥₁(𝑠)−𝑞₁(𝑠)𝜀𝑑𝑠≤𝑦₁𝑇⁽¹⁾exp−𝜀₀𝑡−𝑇⁽¹⁾⟶0 (3.66)as 𝑡→+∞. Similarly, we have𝜀≤𝑦₂(𝑡)≤𝑦₂𝑇⁽¹⁾exp−𝜀₀𝑡−𝑇⁽¹⁾⟶0,𝑡⟶+∞, (3.67)which are contradictions.
Second, we now show that𝑦_𝑖(𝑡)≤𝜀exp𝑀(𝜀)𝜔,𝑖=1,2,for𝑡≥𝑇⁽²⁾, (3.68)where𝑀(𝜀)=max_{0≤𝑡≤𝜔}𝑔₁ℎ(𝑡)+¹(𝑡)𝑥^∗₁(𝑡)+𝜀𝑘₁(𝑡)+𝑛(𝑡)𝑥^∗₁(𝑡)+𝜀+𝑞₁(𝑡)𝜀,𝑔₂ℎ(𝑡)+²(𝑡)𝑥^∗₂(𝑡)+𝜀𝑘₂(𝑡)+𝑥^∗₂(𝑡)+𝜀+𝑞₂(𝑡)𝜀 (3.69)is a bounded constant for 0<𝜀<1. Otherwise, there exists a 𝑇⁽³⁾>𝑇⁽²⁾ such that𝑦_𝑖𝑇⁽³⁾>𝜀exp𝑀(𝜀)𝜔,𝑖=1,2. (3.70)By the continuity of 𝑦_𝑖(𝑡), there must exist 𝑇⁽⁴⁾∈(𝑇⁽²⁾,𝑇⁽³⁾) such that 𝑦_𝑖(𝑇⁽⁴⁾)=𝜀 and 𝑦_𝑖(𝑡)>𝜀 for 𝑡∈(𝑇⁽⁴⁾,𝑇⁽³⁾]. Let 𝑃₁ be the nonnegative integer such that 𝑇⁽³⁾∈(𝑇⁽⁴⁾+𝑃₁𝜔,𝑇⁽⁴⁾+(𝑃₁+1)𝜔]. By the first inequality of (3.65), we have𝜀exp𝑀(𝜀)𝜔<𝑦₁𝑇⁽³⁾<𝑦₁𝑇⁽⁴⁾exp^𝑇(3)_𝑇(4)−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥₁(𝑡)𝑘₁(𝑡)+𝑛(𝑡)𝑥₁(𝑡)−𝑞₁(𝑡)𝜀𝑑𝑡=𝜀exp^{𝑇(4)+𝑃1𝜔}_𝑇(4)+^𝑇(3)_{𝑇(4)+𝑃1𝜔}−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥₁(𝑡)𝑘₁(𝑡)+𝑛(𝑡)𝑥₁(𝑡)−𝑞₁(𝑡)𝜀𝑑𝑡<𝜀exp^𝑇(3)_{𝑇(4)+𝑃1𝜔}𝑔₁ℎ(𝑡)+¹(𝑡)𝑥₁(𝑡)𝑘₁(𝑡)+𝑛(𝑡)𝑥₁(𝑡)+𝑞₁(𝑡)𝜀𝑑𝑡<𝜀exp^𝑇(3)_{𝑇(4)+𝑃1𝜔}𝑔₁ℎ(𝑡)+¹(𝑡)𝑥^∗₁(𝑡)+𝜀𝑘₁(𝑡)+𝑛(𝑡)𝑥^∗₁(𝑡)+𝜀+𝑞₁.(𝑡)𝜀𝑑𝑡≤𝜀exp𝑀(𝜀)𝜔 (3.71)Similarly, by the second inequality of (3.65), we have𝜀exp𝑀(𝜀)𝜔<𝑦₂𝑇⁽³⁾≤𝜀exp𝑀(𝜀)𝜔, (3.72)which are contradictions. These imply that (3.68) holds. By the arbitrariness of 𝜀, it immediately follows that 𝑦_𝑖(𝑡)→0 as 𝑡→+∞. This completes the proof of Theorem 2.4.

4. Example

Consider the following predator-prey system:𝑥^′₁(𝑡)=5𝑥₂(𝑡)−2𝑥₁(𝑡)−𝑥²₁𝑥(𝑡)−2+sin(𝑡)/200₁(𝑡)5+𝑦₁(𝑡)+𝑥₁𝑦(𝑡)¹𝑥(𝑡),_′2(𝑡)=3𝑥₁(𝑡)−𝑥²₂𝑥(𝑡)−2+sin(𝑡)/100₂(𝑡)4+𝑥₂,𝑦(𝑡)_′1(𝑡)=𝑦₁−1(𝑡)3−sin(𝑡)+𝑥1002+sin(𝑡)/200₁(𝑡)5+𝑦₁(𝑡)+𝑥₁−𝑦(𝑡)4+cos(𝑡)₁,𝑦(𝑡)_′2(𝑡)=𝑦₂−1(𝑡)2−sin(𝑡)+𝑥1002+sin(𝑡)/100₂(𝑡)4+𝑥₂−𝑦(𝑡)3+cos(𝑡)₂(𝑡). (4.1)

In this case, corresponding to system (1.1), one has 𝑎(𝑡)=5, 𝑏(𝑡)=2, 𝑐(𝑡)=3, 𝑑(𝑡)=1, 𝑓(𝑡)=1, 𝑔₁(𝑡)=1/3+sin(𝑡)/100, 𝑔₂(𝑡)=1/2+sin(𝑡)/100, ℎ₁(𝑡)=𝑝₁(𝑡)=2+sin(𝑡)/200, ℎ₂(𝑡)=𝑝₂(𝑡)=2+sin(𝑡)/100, 𝑘₁(𝑡)=5, 𝑘₂(𝑡)=4, 𝑚(𝑡)=𝑛(𝑡)=1, 𝑞₁(𝑡)=4+cos(𝑡), 𝑞₂(𝑡)=3+cos(𝑡).

One could easily see that 𝑥^′₁(𝑡)=5𝑥₂(𝑡)−2𝑥₁(𝑡)−𝑥²₁𝑥(𝑡),_′2(𝑡)=3𝑥₁(𝑡)−𝑥²₂(𝑡) (4.2)has a unique positive periodic solution (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡))=(3,3), that is, in this case, the positive periodic solution is the positive equilibrium. By simple computation, one has𝐴_𝜔−𝑔₁ℎ(𝑡)+¹(𝑡)𝑥^∗₁(𝑡)𝑘₁(𝑡)+𝑛(𝑡)𝑥^∗₁=5(𝑡)𝐴12>0,_𝜔−𝑔₂ℎ(𝑡)+²(𝑡)𝑥^∗₂(𝑡)𝑘₂(𝑡)+𝑥^∗₂=5(𝑡)14>0. (4.3)Hence, corresponding to Theorem 2.4, we know that system (4.1) is permanent.

5. Conclusion

In this paper, a model which describes the nonautonomous periodic predator-prey system with Beddington-DeAngelis and Holling II functional response and stage structure for prey is proposed. Under Assumption (2.3), sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

The results of this paper suggest the following biological implication. Note that (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡)) is the globally asymptotically stable periodic solution of system (1.1) without predation, which, as showed by Lemma 2.2, always exists. Hence, condition (2.3) implies that if the death rate of the two predator species is all small enough and the growth by foraging minus the death for the second predator is sufficiently high, the system is permanent.

Acknowledgment

This work is supported by the Natural Science Foundation of Gansu Province (3ZS062-B25-019).