Dynamics of a Predator-Prey System with Beddington-DeAngelis Functional Response and Delays

Abstract

We consider a predator-prey system with Beddington-DeAngelis functional response and delays, in which not only the stage structure on prey but also the delay due to digestion is considered. First, we give a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of a hyperbolic curve and a line. Then, by constructing an appropriate Lyapunov function, we prove that the positive equilibrium is locally asymptotically stable under a sufficient condition. Finally, by using comparison theorem and the 𝜔-limit set theory, we study the global asymptotic stability of the boundary equilibrium and the positive equilibrium, respectively. Also, we obtain a sufficient condition to assure the global asymptotic stability.

1. Introduction

The purpose of this paper is to consider the following predator-prey system with Beddington-DeAngelis functional response and delays: 𝑥^′_𝑖(𝑡)=𝑎𝑥(𝑡)−𝑑_𝑖𝑥_𝑖(𝑡)−𝑎𝑒^{−𝑑𝑖𝜏1}𝑥𝑡−𝜏₁,𝑥^′(𝑡)=𝑎𝑒^{−𝑑𝑖𝜏1}𝑥𝑡−𝜏₁−𝑏𝑥²(𝑡)−𝑐𝑥(𝑡)𝑦(𝑡)𝑚₁+𝑚₂𝑥(𝑡)+𝑚₃𝑦(𝑡),𝑦^′(𝑡)=−𝑑𝑦(𝑡)+𝑓𝑥𝑡−𝜏₂𝑦𝑡−𝜏₂𝑚₁+𝑚₂𝑥𝑡−𝜏₂+𝑚₃𝑦𝑡−𝜏₂, (1) where 𝑥_𝑖(𝑡) denotes the immature prey population density, 𝑥(𝑡) denotes the mature prey population density, and 𝑦(𝑡) denotes the mature predator population density, and all the parameters 𝜏₁, 𝜏₂, 𝑎, 𝑑_𝑖, 𝑏, 𝑐, 𝑑, 𝑓, 𝑚₁, 𝑚₂, and 𝑚₃ are positive. More precisely, the parameters 𝜏₁ and 𝜏₂ represent the time that the prey juveniles and the predator juveniles take to be mature, respectively. 𝑎 is the intrinsic growth rate of the prey and 𝑑_𝑖 is the death rate of immature preys. The term 𝑎𝑒^{−𝑑𝑖𝜏1} denotes the surviving rate of the immature prey during the delay period to be mature. The constant 𝑏 denotes the intensity of intraspecific action on the prey and 𝑑 is the death rate of the predator. The function 𝑐𝑥(𝑡)𝑦(𝑡)/(𝑚₁+𝑚₂𝑥(𝑡)+𝑚₃𝑦(𝑡)) represents the Beddington-DeAngelis functional response and the function 𝑓𝑥(𝑡−𝜏₂)𝑦(𝑡−𝜏₂)/(𝑚₁+𝑚₂𝑥(𝑡−𝜏₂)+𝑚₃𝑦(𝑡−𝜏₂)) represents the predator’s growth rate that comes from the Beddington-DeAngelis functional response during time period [𝑡−𝜏₂,𝑡].

Actually, the predator-prey system with Beddington-DeAngelis functional response has been extensively studied in the literature. In the original work, Beddington [1] and DeAngelis et al. [2] proposed the predator-prey system as 𝑥^′(𝑡)=𝑎𝑥(𝑡−𝜏)−𝑏𝑥²(𝑡)−𝑐𝑥(𝑡)𝑦(𝑡)𝑚₁+𝑚₂𝑥(𝑡)+𝑚₃𝑦(𝑡),𝑦^′(𝑡)(𝑡)+(𝑡)𝑦(𝑡)=−𝑑𝑦𝑓𝑥𝑚₁+𝑚₂𝑥(𝑡)+𝑚₃𝑦(𝑡). (2) Shaikhet [3] considered the stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations. Indeed, the author in [3] not only obtained sufficient conditions for asymptotic mean square stability of the considered systems, but also showed that the trivial solution is asymptotically mean square stable and the positive equilibrium is stable in probability by constructing appropriate Lyapunov functions. In particular, Shaikhet expounded a general method of construction of Lyapunov functions for stochastic functional differential equations and obtained the stability conditions for functional differential equations in [4]. See also [5–10] for the corresponding results about the predator’s functional response. Particularly, Huo [8] studied the global attractiveness of delay diffusive prey-predator systems with the Michaelis-Menten functional response.

In the real world, many species usually have a life history that takes them through two stages, immature stage and mature stage. Recently, the system with the stage structure on prey has been extensively studied. In particular, She and Li [11] analyzed the following Beddington-DeAngelis model with the stage structure on prey: 𝑥^′_𝑖(𝑡)=𝑎𝑥(𝑡)−𝑑_𝑖𝑥_𝑖(𝑡)−𝑎𝑒^{−𝑑𝑖𝜏}𝑥(),𝑥𝑡−𝜏_′(𝑡)=𝑎𝑒^{−𝑑𝑖𝜏}𝑥(𝑡−𝜏)−𝑏𝑥²(𝑡)−𝑐𝑥(𝑡)𝑦(𝑡)𝑚₁+𝑚₂𝑥(𝑡)+𝑚₃𝑦(𝑡),𝑦^′(𝑡)(𝑡)+(𝑡)𝑦(𝑡)=−𝑑𝑦𝑓𝑥𝑚₁+𝑚₂𝑥(𝑡)+𝑚₃𝑦(𝑡), (3) where the stage structure was introduced by a constant time delay. In fact, it is the case of 𝜏₂=0 in system (1). See also [12–15] for predator-prey models with Beddington-DeAngelis functional response and stage structure.

More recently, Xiang et al. [16] established a virus dynamics model with Beddington-DeAngelis functional response and delays. In [16], they not only incorporated the delay 𝜏₁, which described the time between viral entry into a target cell and the production of new virus particles, but also incorporated the delay 𝜏₂, which described the maturation time of the newly produced viruses. Additionally, it is well known that the growth of the predator is affected by the time delay due to digestion. This motivates us to resolve the issue.

Motivated by [16], in the present paper, we study the predator-prey system (1) with Beddington-DeAngelis functional response and delays, in which not only the stage structure on prey but also the delay due to digestion is considered.

More precisely, let 𝑎=𝑎𝑒^{−𝑑𝑖𝜏1} and assume that 𝑥_𝑖(0)=𝑎⁰_−𝜏1𝑒^{−𝑑𝑖𝑠}𝑥(𝑠)𝑑𝑠 (4) holds. Then, based on similar conclusions for Lemma 3.1 in [10], we obtain 𝑥_𝑖(𝑡)=𝑎⁰_−𝜏1𝑒^{−𝑑𝑖𝑠}𝑥(𝑡+𝑠)𝑑𝑠, (5) which means that 𝑥_𝑖(𝑡) depends on 𝑥(𝑡) completely. Thus, we can obtain the following system, which is separated from (1): 𝑥^′(𝑡)=𝑎𝑥𝑡−𝜏₁−𝑏𝑥²(𝑡)−𝑐𝑥(𝑡)𝑦(𝑡)𝑚₁+𝑚₂𝑥(𝑡)+𝑚₃𝑦(𝑡),𝑦^′(𝑡)=−𝑑𝑦(𝑡)+𝑓𝑥𝑡−𝜏₂𝑦𝑡−𝜏₂𝑚₁+𝑚₂𝑥𝑡−𝜏₂+𝑚₃𝑦𝑡−𝜏₂. (6)

It is obvious that, for studying the dynamics of the predator-prey system (1) with Beddington-DeAngelis functional response and delays, it is sufficient to study the dynamics of (6).

The rest of the organization of this paper is as follows. In Section 2, we first establish the local stability of the boundary equilibria of (6) by analyzing the characteristic equations, and then, we study the local asymptotic stability of the positive equilibria of (6) by constructing an appropriate Lyapunov function. In Section 3, we consider the global asymptotic stability of the boundary equilibrium and the positive equilibrium, respectively.

2. Local Stability of Equilibria

In this section, we study the local stability of the equilibria of (6). By direct calculations, we have that the system (6) has the origin equilibrium 𝐸₀(0,0) and the boundary equilibrium 𝐸₁((𝑎/𝑏),0).

In order to investigate the existence of the positive equilibrium, we consider the equations (𝑎−𝑏𝑥)𝑚₁+𝑚₂𝑥+𝑚₃𝑦𝑚−𝑐𝑦=0−𝑑₁+𝑚₂𝑥+𝑚₃𝑦+𝑓𝑥=0 (7) and obtain the following result.

Theorem 1. The system (6) has a unique positive equilibrium 𝐸^∗(𝑥^∗,𝑦^∗) if and only if the condition 𝑓−𝑑𝑚₂𝑎𝑏>𝑑𝑚₁ (8) holds, which is equivalent to the condition 𝑓−𝑑𝑚₂𝑎𝑒^{−𝑑𝑖𝜏1}𝑏>𝑑𝑚₁. (9)

Proof. We prove the existence of the positive equilibrium by analyzing (7). For the first equation of (7), it is easy to see that its corresponding curves go through the points (𝑎/𝑏,0), (−𝑚₁/𝑚₂,0), and (0,𝑎𝑚₁/(𝑐−𝑎𝑚₃)) when 𝑐−𝑎𝑚₃≠0. Furthermore, in case of ((𝑐−𝑎𝑚₃)/𝑏𝑚₃)≠𝑚₁/𝑚₂, the first equation is a hyperbolic equation and has two asymptotic lines 𝑥+((𝑐−𝑎𝑚₃)/𝑏𝑚₃)=0 and 𝑦+(𝑚₂/𝑚₃)𝑥+((𝑏𝑚₁𝑚₃−𝑐𝑚₂)/𝑏𝑚²₃)=0. If ((𝑐−𝑎𝑚₃)/𝑏𝑚₃)=𝑚₁/𝑚₂, the first equation can be simplified as (𝑚₁+𝑚₂𝑥)(𝑎𝑚₂−𝑏𝑚₂𝑥−𝑏𝑚₃𝑦)=0, which are two lines.
For the second equation of (7), it implies a line which goes through the points (0,−𝑚₁/𝑚₃) and (𝑑𝑚₁/(𝑓−𝑑𝑚₂),0) when 𝑑𝑚₁/(𝑓−𝑑𝑚₂)≠0. Their corresponding figures are as shown in Figures 1, 2, and 3.
Consequently, by analyzing Figures 1, 2, and 3 and discussions, we obtain that (6) has a unique positive equilibrium.

Figure 1 

−𝑚1/𝑚2<−(𝑐−𝑎𝑚3)/𝑏𝑚3<0.

Figure 2 

−(𝑐−𝑎𝑚3)/𝑏𝑚3<−𝑚1/𝑚2<0.

Figure 3 

−(𝑐−𝑎𝑚3)/𝑏𝑚3>0.

Condition (9) indicates that the positive equilibrium 𝐸^∗ exists for all 𝜏₁ in the interval 𝐼=(0,𝜏^∗), where 𝜏^∗=1𝑑_𝑖𝑎ln𝑓−𝑑𝑚₂𝑏𝑑𝑚₁. (10) Furthermore, 𝐸^∗ will coincide with 𝐸₁ when 𝜏₁=𝜏^∗ and there exists no positive equilibrium for 𝜏₁>𝜏^∗.

Now, we study the stability of equilibria 𝐸₀(0,0),𝐸₁((𝑎/𝑏),0) and 𝐸^∗(𝑥^∗,𝑦^∗), respectively.

At the beginning, we rewrite system (6) as 𝑍^′(𝑡)=𝐹(𝑍(𝑡),𝑍(𝑡−𝜏₁),𝑍(𝑡−𝜏₂)), where 𝑍(𝑡)=(𝑥(𝑡),𝑦(𝑡)), 𝑍(𝑡−𝜏₁)=(𝑥(𝑡−𝜏₁),𝑦(𝑡−𝜏₁)), 𝑍(𝑡−𝜏₂)=(𝑥(𝑡−𝜏₂),𝑦(𝑡−𝜏₂)).

Then, for an arbitrary fixed point 𝑍^∗=(𝑥,𝑦), by letting 𝑃=(𝜕𝐹/𝜕𝑍(𝑡))_𝑍^∗, 𝑄=(𝜕𝐹/𝜕𝑍(𝑡−𝜏₁))_𝑍^∗, and 𝑅=(𝜕𝐹/𝜕𝑍(𝑡−𝜏₂))_𝑍^∗, we have 𝑃=−2𝑏𝑥−𝑐𝑞^′_𝑥−𝑐𝑞^′_𝑦0−𝑑_𝑋∗,𝑄=𝑎000_𝑋∗,𝑅=00𝑓𝑞^′_𝑥𝑓𝑞^′_𝑦𝑋∗, (11) where 𝑞𝑥,𝑦=𝑥𝑦𝑚₁+𝑚₂𝑥+𝑚₃𝑦,𝑞^′_𝑥=𝑦𝑚₁+𝑚₃𝑦𝑚₁+𝑚₂𝑥+𝑚₃𝑦²,𝑞^′_𝑦=𝑥𝑚₁+𝑚₂𝑥𝑚₁+𝑚₂𝑥+𝑚₃𝑦². (12) Therefore, the characteristic equation of the system (6) at the point 𝑍^∗ is as follows:𝑃+𝑄𝑒^−𝜆𝜏1+𝑅𝑒^−𝜆𝜏2−𝜆𝐼=−2𝑏𝑥−𝑐𝑞^′_𝑥+𝑎𝑒^−𝜆𝜏1−𝜆−𝑐𝑞^′_𝑦𝑓𝑞^′_𝑥𝑒^−𝜆𝜏2−𝑑+𝑓𝑞^′_𝑦𝑒^−𝜆𝜏2=𝑚−𝜆−2𝑏𝑥−𝑐𝑦₁+𝑚₃𝑦𝑚₁+𝑚₂𝑥+𝑚₃𝑦²+𝑎𝑒^−𝜆𝜏1𝑚−𝜆−𝑐𝑥₁+𝑚₂𝑥𝑚₁+𝑚₂𝑥+𝑚₃𝑦²𝑓𝑦𝑚₁+𝑚₃𝑦𝑚₁+𝑚₂𝑥+𝑚₃𝑦²𝑒^−𝜆𝜏2𝑚−𝑑+𝑓𝑥₁+𝑚₂𝑥𝑚₁+𝑚₂𝑥+𝑚₃𝑦²𝑒^−𝜆𝜏2−𝜆=0. (13)

Thus, according to the characteristic equation at the point 𝐸₀, we obtain the following result.

Theorem 2. The equilibrium 𝐸₀(0,0) is unstable.

Proof. The characteristic equation of the system (6) at the point 𝐸₀ is 𝑃+𝑄𝑒^−𝜆𝜏1+𝑅𝑒^−𝜆𝜏2−𝜆𝐼_(0,0)=𝑎𝑒^−𝜆𝜏1=()−𝜆00−𝑑−𝜆𝜆+𝑑𝜆−𝑎𝑒^−𝜆𝜏1=0. (14)
Obviously, 𝜆=−𝑑 is a negative eigenvalue and the other eigenvalue depends on the solution of 𝜆−𝑎𝑒^−𝜆𝜏1=0.
Noticing that the line 𝑦=𝜆 and the curve 𝑦=𝑎𝑒^−𝜆𝜏1 must intersect at a unique point (𝜆,𝑦), where 𝜆 is a positive value, we can obtain that 𝐸₀ is a saddle. Thus, 𝐸₀ is unstable.

According to the characteristic equation of the point 𝐸₁, we have the following theorem.

Theorem 3. The equilibrium 𝐸₁((𝑎/𝑏),0) is unstable if (𝑓−𝑑𝑚₂)(𝑎/𝑏)>𝑑𝑚₁ and is locally asymptotically stable if (𝑓−𝑑𝑚₂)(𝑎/𝑏)<𝑑𝑚₁.

Proof. The characteristic equation of the system (6) at the point 𝐸₁ is 𝑃+𝑄𝑒^−𝜆𝜏1+𝑅𝑒^−𝜆𝜏2−𝜆𝐼_{((𝑎/𝑏),0)}=−2𝑎+𝑎𝑒^−𝜆𝜏1−𝜆−𝑎𝑐𝑏𝑚₁+𝑎𝑚₂0−𝑑+𝑎𝑓𝑏𝑚₁+𝑎𝑚₂𝑒^−𝜆𝜏2=−𝜆𝜆+2𝑎−𝑎𝑒^−𝜆𝜏1𝜆+𝑑−𝑎𝑓𝑏𝑚₁+𝑎𝑚₂𝑒^−𝜆𝜏2=0. (15)
For the equation 𝜆+2𝑎−𝑎𝑒^−𝜆𝜏1=0, it is clear that the line 𝑦=𝜆+2𝑎 and the curve 𝑦=𝑎𝑒^−𝜆𝜏1 must intersect at a unique point (𝜆,𝑦), where 𝜆 is a negative value. And for the equation 𝜆+𝑑−(𝑎𝑓/(𝑏𝑚₁+𝑎𝑚₂))𝑒^−𝜆𝜏2=0, noticing that the line 𝑦=𝜆+𝑑 and the curve 𝑦=(𝑎𝑓/(𝑏𝑚₁+𝑎𝑚₂))𝑒^−𝜆𝜏2 must intersect at a unique point (𝜆,𝑦), we have that 𝜆 is positive when (𝑓−𝑑𝑚₂)(𝑎/𝑏)>𝑑𝑚₁ and is negative when (𝑓−𝑑𝑚₂)(𝑎/𝑏)<𝑑𝑚₁. Consequently, 𝐸₁ is unstable if (𝑓−𝑑𝑚₂)(𝑎/𝑏)>𝑑𝑚₁ and locally asymptotically stable if (𝑓−𝑑𝑚₂)(𝑎/𝑏)<𝑑𝑚₁.

In the rest of this section, by constructing an appropriate Lyapunov function, we study the stability of the positive equilibrium 𝐸^∗(𝑥^∗,𝑦^∗) under condition (8). Let 𝑥(𝑡)=𝑥^∗(𝑡),𝑦(𝑡)+𝑢=𝑦^∗+𝑣(𝑡); (16) then, the linearization of the system (6) is 𝑢^′(𝑡)=𝐴𝑢𝑡−𝜏₁(𝑡)(𝑡),𝑣−𝐵𝑢−𝐶𝑣_′(𝑡)=−𝐷𝑣(𝑡)+𝐸𝑣𝑡−𝜏₂+𝐹𝑢𝑡−𝜏₂, (17) where 𝐴=𝑎,𝐵=2𝑏𝑥^∗+𝑐𝑦^∗𝑚₁+𝑚₃𝑦^∗𝑚₁+𝑚₂𝑥^∗+𝑚₃𝑦^∗2,𝐶=𝑐𝑥^∗𝑚₁+𝑚₂𝑥^∗𝑚₁+𝑚₂𝑥^∗+𝑚₃𝑦^∗2,𝐷=𝑑,𝐸=𝑓𝑥^∗𝑚₁+𝑚₂𝑥^∗𝑚₁+𝑚₂𝑥^∗+𝑚₃𝑦^∗2,𝐹=𝑓𝑦^∗𝑚₁+𝑚₃𝑦^∗𝑚₁+𝑚₂𝑥^∗+𝑚₃𝑦^∗2. (18) Obviously, all the constants defined above are positive, and the system (17) can be recast as 𝑢^′(𝑡)=(𝐴−𝐵)𝑢(𝑡)−𝐶𝑣(𝑡)−𝐴^𝑡_𝑡−𝜏1𝑢^′(𝑠)=()𝑢(𝑡)(𝑡)𝑑𝑠𝐴−𝐵−𝐶𝑣−𝐴^𝑡_𝑡−𝜏1𝐴𝑢𝑠−𝜏₁(𝑠)(𝑠𝑣−𝐵𝑢−𝐶𝑣)𝑑𝑠_′(𝑡)=(𝐸−𝐷)𝑣(𝑡)−𝐸^𝑡_𝑡−𝜏2𝑣^′(𝑠)𝑑𝑠+𝐹𝑢(𝑡)−𝐹^𝑡_𝑡−𝜏2𝑢^′(𝑠)=()𝑣(𝑡)(𝑡)𝑑𝑠𝐸−𝐷+𝐹𝑢−𝐹^𝑡_𝑡−𝜏2𝐴𝑢𝑠−𝜏₁−𝐵𝑢(𝑠)−𝐶𝑣(𝑠)𝑑𝑠−𝐸^𝑡_𝑡−𝜏2−𝐷𝑣(𝑠)+𝐸𝑣𝑠−𝜏₂+𝐹𝑢𝑠−𝜏₂𝑑𝑠. (19) By constructing an appropriate Lyapunov function, we have the following theorem.

Theorem 4. Let 𝜏₁∈(0,𝜏^∗) with 𝜏^∗ defined in (10), and assume that 𝜏₁,𝜏₂∈0,𝜏_∗,|𝐹−𝐶|<min{2(𝐷−𝐸),2(𝐵−𝐴)}, (20) also holds, where 𝜏_∗2()−||=min𝐷−𝐸𝐹−𝐶𝐴𝐶+𝐹(𝐴+𝐵+𝐸)+2𝐶𝐹+𝐷𝐸+𝐸²,2(𝐵−𝐴)−|𝐹−𝐶|𝐴()().2𝐴+2𝐵+𝐶+𝐹𝐴+𝐵+𝐸 (21) Then, the equilibrium (0,0) of the system (17) is locally asymptotically stable, which implies that the positive equilibrium 𝐸^∗(𝑥^∗,𝑦^∗) of the system (6) is locally asymptotically stable.

Proof. To illustrate the equilibrium (0,0) of the system (17) is locally asymptotically stable, it is sufficient to study the existence of a strict Lyapunov function. By letting 𝑊₁₁(𝑡)=𝑢²(𝑡) and differentiating 𝑊₁₁(𝑡) with respect to 𝑡, we have 𝑊^′₁₁(𝑡)=2(𝐴−𝐵)𝑢²(𝑡)(𝑡)𝑣(𝑡)(𝑡)−2𝐶𝑢−2𝐴𝑢^𝑡_𝑡−𝜏1𝐴𝑢𝑠−𝜏₁(𝑠)(𝑠()𝑢−𝐵𝑢−𝐶𝑣)𝑑𝑠≤2𝐴−𝐵²(𝑡)(𝑡)𝑣(𝑡)()𝜏−2𝐶𝑢+𝐴𝐴+𝐵+𝐶₁𝑢²(𝑡)+𝐴^𝑡_𝑡−𝜏1𝐴𝑢²𝑠−𝜏₁+𝐵𝑢²(𝑠)+𝐶𝑣²(𝑠)𝑑𝑠. (22) Letting 𝑊₁₂(𝑡)=𝐴∫^𝑡_𝑡−𝜏1∫^𝑡_𝑧[𝐴𝑢²(𝑠−𝜏₁)+𝐵𝑢²(𝑠)+𝐶𝑣²(𝑠)]𝑑𝑠𝑑𝑧 and differentiating 𝑊₁₂(𝑡) with respect to 𝑡, we have 𝑊^′₁₁(𝑡)+𝑊^′₁₂(𝑡)≤2(𝐴−𝐵)𝑢²(𝑡)(𝑡)𝑣(𝑡)()𝜏−2𝐶𝑢+𝐴𝐴+𝐵+𝐶₁𝑢²(𝑡)+𝐴𝜏₁𝐴𝑢²𝑡−𝜏₁+𝐵𝑢²(𝑡)+𝐶𝑣²(𝑡). (23) Furthermore, by letting 𝑊₁₃(𝑡)=𝐴²𝜏₁∫^𝑡_𝑡−𝜏1𝑢²(𝑠)𝑑𝑠 and 𝑊₁(𝑡)=𝑊₁₁(𝑡)+𝑊₁₂(𝑡)+𝑊₁₃(𝑡), we have 𝑊^′₁(𝑡)≤2(𝐴−𝐵)+𝐴(2𝐴+2𝐵+𝐶)𝜏₁𝑢²(𝑡)+𝐴𝐶𝜏₁𝑣²(𝑡)−2𝐶𝑢(𝑡)𝑣(𝑡). (24)
Next, by letting 𝑊₂₁(𝑡)=𝑣²(𝑡) and differentiating 𝑊₂₁(𝑡) with respect to 𝑡, we obtain 𝑊^′₂₁(𝑡)=2(𝐸−𝐷)𝑣²(𝑡)(𝑡)𝑣(𝑡)(𝑡)×+2𝐹𝑢−2𝑣_{𝑡𝑡−𝜏2}𝐹𝐴𝑢𝑠−𝜏₁(𝑠)(𝑠(𝑠)−𝐵𝑢−𝐶𝑣)+𝐸−𝐷𝑣+𝐸𝑣𝑠−𝜏₂+𝐹𝑢𝑠−𝜏₂()𝑣𝑑𝑠≤2𝐸−𝐷²(𝑡)(𝑡)𝑣(𝑡)+[𝐹()(𝜏+2𝐹𝑢𝐴+𝐵+𝐶+𝐸𝐷+𝐸+𝐹)]₂𝑣²(𝑡)+^𝑡_𝑡−𝜏2𝐹𝐴𝑢²𝑠−𝜏₁+𝐵𝑢²(𝑠)+𝐶𝑣²(𝑠)+𝐸𝐷𝑣²(𝑠)+𝐸𝑣²𝑠−𝜏₂+𝐹𝑢²𝑠−𝜏₂𝑑𝑠. (25) Letting 𝑊₂₂(𝑡)=∫^𝑡_𝑡−𝜏2∫^𝑡_𝑧{𝐹[𝐴𝑢²(𝑠−𝜏₁)+𝐵𝑢²(𝑠)+𝐶𝑣²(𝑠)]+𝐸[𝐷𝑣²(𝑠)+𝐸𝑣²(𝑠−𝜏₂)+𝐹𝑢²(𝑠−𝜏₂)]}𝑑𝑠𝑑𝑧 and differentiating 𝑊₂₂(𝑡) with respect to 𝑡, we have 𝑊^′₂₁(𝑡)+𝑊^′₂₂(𝑡)≤2(𝐸−𝐷)𝑣²(𝑡)(𝑡)𝑣(𝑡)+[𝐹()(𝜏+2𝐹𝑢𝐴+𝐵+𝐶+𝐸𝐷+𝐸+𝐹)]₂𝑣²(𝑡)+𝐹𝐴𝑢²𝑡−𝜏₁+𝐵𝑢²(𝑡)+𝐶𝑣²(𝑡)+𝐸𝐷𝑣²(𝑡)+𝐸𝑣²𝑡−𝜏₂+𝐹𝑢²𝑡−𝜏₂𝜏₂. (26) By letting 𝑊₂₃(𝑡)=𝐴𝐹𝜏₂∫^𝑡_𝑡−𝜏1𝑢²(𝑠)𝑑𝑠, 𝑊₂₄(𝑡)=𝐸²𝜏₂∫^𝑡_𝑡−𝜏1𝑣²(𝑠)𝑑𝑠, and 𝑊₂₅(𝑡)=𝐸𝐹𝜏₂∫^𝑡_𝑡−𝜏2𝑢²(𝑠)𝑑𝑠, we have 𝑊^′₂₃(𝑡)=𝐴𝐹𝜏₂𝑢²(𝑡)−𝑢²𝑡−𝜏₁,𝑊^′₂₄(𝑡)=𝐸²𝜏₂𝑣²(𝑡)−𝑣²𝑡−𝜏₂,𝑊^′₂₅(𝑡)=𝐸𝐹𝜏₂𝑢²(𝑡)−𝑢²𝑡−𝜏₂. (27) By letting 𝑊₂(𝑡)=𝑊₂₁(𝑡)+𝑊₂₂(𝑡)+𝑊₂₃(𝑡)+𝑊₂₄(𝑡)+𝑊₂₅(𝑡), we have 𝑊^′₂(𝑡)≤2(𝐸−𝐷)+𝜏₂𝐹(𝐴+𝐵+𝐸)+2𝐶𝐹+𝐷𝐸+𝐸²×𝑣²(𝑡)+𝐹(𝐴+𝐵+𝐸)𝜏₂𝑢²(𝑡)+2𝐹𝑢(𝑡)𝑣(𝑡). (28) Finally, we let 𝑊(𝑡)=𝑊₁(𝑡)+𝑊₂(𝑡). Obviously, 𝑊(𝑡)≥0 and 𝑊(𝑡)=0 if and only if 𝑢(𝑡)=𝑣(𝑡)=0. Besides, when 𝑢(𝑡)=𝑣(𝑡)=0, 𝑊^′(𝑡)=0. Furthermore, 𝑊^′(𝑡)≤2(𝐸−𝐷)+𝐴𝐶𝜏₁+𝜏₂𝐹(𝐴+𝐵+𝐸)+2𝐶𝐹+𝐷𝐸+𝐸²×𝑣²(𝑡)()𝑢(𝑡)𝑣(𝑡)+2()()𝜏+2𝐹−𝐶𝐴−𝐵+𝐴2𝐴+2𝐵+𝐶₁+𝐹(𝐴+𝐵+𝐸)𝜏₂×𝑢²(𝑡)≤2(𝐸−𝐷)+𝐴𝐶𝜏₁+𝜏₂𝐹(𝐴+𝐵+𝐸)+2𝐶𝐹+𝐷𝐸+𝐸²×𝑣²(𝑡)|𝑢(𝑡𝑣(𝑡+2()()𝜏+2𝐹−𝐶||)||)|𝐴−𝐵+𝐴2𝐴+2𝐵+𝐶₁+𝐹(𝐴+𝐵+𝐸)𝜏₂×𝑢²(𝑡)≤2(𝐸−𝐷)+|𝐹−𝐶|+𝐴𝐶𝜏₁+𝜏₂𝐹(𝐴+𝐵+𝐸)+2𝐶𝐹+𝐷𝐸+𝐸²𝑣²(𝑡)+2(𝐴−𝐵)+|𝐹−𝐶|+𝐴(2𝐴+2𝐵+𝐶)𝜏₁+𝐹(𝐴+𝐵+𝐸)𝜏₂𝑢²(𝑡). (29) Based on the conditions in Theorem 4, we have 𝑊^′(𝑡)<0 if 𝑢²(𝑡)+𝑣²(𝑡)>0. Thus, 𝑊(𝑡) is a strict Lyapunov function. According to the Lyapunov stability theorem, the equilibrium (0,0) of the system (17) is locally asymptotically stable, which implies that the positive equilibrium 𝐸^∗(𝑥^∗,𝑦^∗) of the system (6) is locally asymptotically stable.

3. Global Stability of Equilibria

In this section, we study the global asymptotic stability of the boundary equilibrium 𝐸₁((𝑎/𝑏),0) and the positive equilibrium 𝐸^∗(𝑥^∗,𝑦^∗), respectively. We first give two lemmas which will be used later.

Lemma 5 (see [10, Lemma 3.2]). For a given system 𝑥^′(𝑡)=𝑏₁𝑥(𝑡−𝜏)−𝑎₁𝑥(𝑡)−𝑎₂𝑥²(𝑡), (30) where 𝑏₁>0, 𝑎₁≥0, 𝑎₂>0, 𝜏≥0, and 𝑥(𝑡)>0 for all −𝜏≤𝑡≤0, we have (i)lim_𝑡→+∞𝑥(𝑡)=(𝑏₁−𝑎₁)/𝑎₂ if and only if 𝑏₁>𝑎₁;(ii)lim_𝑡→+∞𝑥(𝑡)=0 if and only if 𝑏₁≤𝑎₁.

Lemma 6 (see [11, Lemma 3.2]). Let 𝑆={(𝑥,𝑦)∶𝑥>0,𝑦>0} and 𝑆={(𝑥,𝑦)∶𝑥≥0,𝑦≥0}. Then, the sets 𝑆 and 𝑆 are both invariant sets.

Based on Lemmas 5 and 6, we prove that the boundary equilibrium 𝐸₁ is globally attractive.

Theorem 7. Let (𝑥(𝑡),𝑦(𝑡)) be the solution of (6) with positive initial condition in 𝑆.(i)If (𝑓−𝑑𝑚₂)(𝑎/𝑏)<𝑑𝑚₁, then 𝐸₁((𝑎/𝑏),0) is globally attractive.(ii)If (𝑓−𝑑𝑚₂)(𝑎/𝑏)>𝑑𝑚₁, then 𝐸₁((𝑎/𝑏),0) is unstable.

Proof. If (𝑓−𝑑𝑚₂)(𝑎/𝑏)<𝑑𝑚₁, we first prove that lim_𝑡→+∞𝑦(𝑡)=0.
Due to the first equation (6), there is 𝑥^′(𝑡)≤𝑎𝑥𝑡−𝜏₁−𝑏𝑥²(𝑡). (31) In view of Lemma 6 and by considering the following comparison equation 𝑝^′(𝑡)=𝑎𝑝𝑡−𝜏₁−𝑏𝑝²(𝑡),𝑝(0)=𝑥(0)>0, (32) we have 𝑥(𝑡)≤𝑝(𝑡) for all 𝑡≥0. In addition, by Lemma 5, we obtain that lim_𝑡→+∞𝑝(𝑡)=(𝑎/𝑏). Thus, there exists a sufficiently small positive constant 𝜀 with (𝑓−𝑑𝑚₂)((𝑎/𝑏)+𝜀)<𝑑𝑚₁ such that, for this 𝜀, there exists a 𝑇_𝜀>0 such that 𝑥(𝑡)<(𝑎/𝑏)+𝜀 for all 𝑡>𝑇_𝜀. Substituting it into the second equation of (6), we have 𝑦^′(𝑡)(𝑡)+𝑓))≤−𝑑𝑦((𝑎/𝑏+𝜀𝑚₁+𝑚₂))𝑦((𝑎/𝑏+𝜀𝑡−𝜏₂, (33) for all 𝑡>𝑇_𝜀. Considering the following comparison equation: 𝑞^′(𝑡)(𝑡)+𝑓))=−𝑑𝑞((𝑎/𝑏+𝜀𝑚₁+𝑚₂))𝑞((𝑎/𝑏+𝜀𝑡−𝜏₂,𝑞𝑇_𝜀+𝜏₂=𝑦𝑇_𝜀+𝜏₂>0, (34) and letting 𝑞(𝑡)=𝑒^𝜆𝑡, we have 𝑒^𝜆𝑡𝑓))𝜆+𝑑−((𝑎/𝑏+𝜀𝑚₁+𝑚₂))𝑒((𝑎/𝑏+𝜀^−𝜆𝜏2=0. (35) Then the line 𝑦=𝜆+𝑑 and the curve 𝑦=(𝑓((𝑎/𝑏)+𝜀)/(𝑚₁+𝑚₂((𝑎/𝑏)+𝜀)))𝑒^−𝜆𝜏2 must intersect at a unique point (𝜆,𝑦). Since (𝑓−𝑑𝑚₂)((𝑎/𝑏)+𝜀)<𝑑𝑚₁, we can obtain that 𝑑>𝑓((𝑎/𝑏)+𝜀)/(𝑚₁+𝑚₂((𝑎/𝑏)+𝜀)). Thus, we deduce that 𝜆<0. Therefore, lim_𝑡→+∞𝑞(𝑡)=lim_𝑡→+∞𝑒^𝜆𝑡=0.
Obviously, based on the comparison theorem, we get that 𝑦(𝑡)≤𝑞(𝑡) for all 𝑡≥𝑇+𝜏₂. Thus, lim_𝑡→+∞𝑦(𝑡)=0.
Next, we prove that 𝑥(𝑡)→(𝑎/𝑏) as 𝑡→+∞.
Since lim_𝑡→+∞𝑦(𝑡)=0, for any given 𝜀₁∈(0,(𝑎/𝑏)), there exists a 𝑇₁>0 such that 0<𝑦(𝑡)<(𝑏𝑚₁/2𝑐)𝜀₁ for all 𝑡≥𝑇₁. Thus, we have 𝑎𝑥𝑡−𝜏₁−𝑏𝑥²(𝑡)≥𝑥^′(𝑡)=𝑎𝑥𝑡−𝜏₁−𝑏𝑥²(𝑡)−𝑐𝑥(𝑡)𝑦(𝑡)𝑚₁+𝑚₂𝑥(𝑡)+𝑚₃𝑦(𝑡)≥𝑎𝑥𝑡−𝜏₁−𝑏𝑥²(𝑡)−𝑐𝑥(𝑡)𝑦(𝑡)𝑚₁≥𝑎𝑥𝑡−𝜏₁−𝑏𝑥²(𝑡)−𝑏𝜀₁2𝑥(𝑡), (36) for all 𝑡≥𝑇₁. That is, 𝑎𝑥𝑡−𝜏₁−𝑏𝑥²(𝑡)≥𝑥^′(𝑡)≥𝑎𝑥𝑡−𝜏₁−𝑏𝑥²(𝑡)−𝑏𝜀₁2𝑥(𝑡). (37) Then, we consider the following comparison equation: 𝑝^′(𝑡)=𝑎𝑝𝑡−𝜏₁−𝑏𝑝²(𝑡)−𝑏𝜀₁2𝑝(𝑡),𝑝𝑇₁=𝑥𝑇₁>0. (38) Since 𝑎>𝑏𝜀₁, based on Lemma 5, we have that lim_𝑡→+∞𝑝(𝑡)=(𝑎/𝑏)−(𝜀₁/2). Additionally, we have that 𝑝(𝑡)≤𝑥(𝑡)≤𝑝(𝑡) for all 𝑡≥𝑇₁.
Since lim_𝑡→+∞𝑝(𝑡)=(𝑎/𝑏), for the above 𝜀₁, there exists a 𝑇₂>0 such that 𝑝(𝑡)≤(𝑎/𝑏)+𝜀₁ for all 𝑡>𝑇₂. Similarly, since lim_𝑡→+∞𝑝(𝑡)=(𝑎/𝑏)−(𝜀₁/2) for the above 𝜀₁, we also get that there exists a 𝑇₃>0 such that 𝑝(𝑡)−(𝑎/𝑏)+(𝜀₁/2)>−(𝜀₁/2) for all 𝑡>𝑇₃. Therefore, if we let 𝑇₀=max{𝑇₁,𝑇₂,𝑇₃}, there is −𝜀₁<𝑥(𝑡)−(𝑎/𝑏)<𝜀₁ for all 𝑡>𝑇₀. Thus, we have lim_𝑡→+∞𝑥(𝑡)=(𝑎/𝑏).
At last, we prove the second part of this theorem. We can assure that (𝑓−𝑑𝑚₂)(𝑎/𝑏)>𝑑𝑚₁ and try to derive a contradiction. If (𝑓−𝑑𝑚₂)(𝑎/𝑏)>𝑑𝑚₁, then (6) has a unique positive equilibrium (𝑥^∗,𝑦^∗) by Theorem 1, which is also a solution of (6); that is, (𝑥^∗,𝑦^∗)=(𝑥^∗(𝑡),𝑦^∗(𝑡)) satisfies (6) and satisfies lim_𝑡→+∞(𝑥^∗(𝑡),𝑦^∗(𝑡))=((𝑎/𝑏),0) by the above arguments, which contradicts the fact that (𝑥^∗,𝑦^∗) is a positive equilibrium. Thus, conclusion (ii) holds. This completes the proofs.

Remark 8. In the above theorem, we derive the global attractiveness of the boundary equilibrium 𝐸₁((𝑎/𝑏),0). Unfortunately, we just obtain a sufficient condition that (𝑓−𝑑𝑚₂)(𝑎/𝑏)<𝑑𝑚₁ to ensure that 𝐸₁ is globally asymptotically stable. If (𝑓−𝑑𝑚₂)(𝑎/𝑏)=𝑑𝑚₁, since the second equation of (6) is a delay diffusion equation, we cannot derive that lim_𝑡→+∞𝑦(𝑡)=0 by using comparison theorem. We leave it for a further study.

Furthermore, for the first equation of (1), we can solve explicitly as lim_𝑡→+∞𝑥_𝑖(𝑡)=𝑎²𝑒^{𝑑𝑖𝜏1}−1𝑏𝑑_𝑖. (39) Thus, by Theorem 7, we can obtain the following corollary directly.

Corollary 9. Let (𝑥_𝑖(𝑡),𝑥(𝑡),𝑦(𝑡)) be the solution of (1) with the positive initial condition. If 𝑓≤𝑑𝑚₂, then lim_𝑡→+∞𝑥_𝑖(𝑡)(𝑡)(𝑡)=𝑎,𝑥,𝑦²𝑏𝑑_𝑖𝑒^{𝑑𝑖𝜏1},𝑎−1𝑏,0. (40)

In the rest of this section, we will study the global attractiveness of the positive equilibrium 𝐸^∗ and obtain the following result.

Theorem 10. Assume that (8) and the following conditions 𝑎𝑚₃𝑓>𝑐,𝑎𝑚₃−𝑐𝑏𝑚₁𝑚₃+𝑚₂𝑎𝑚₃−𝑐>𝑑, (41) hold, and let (𝑥(𝑡),𝑦(𝑡)) be the solution of (6) with the positive initial condition in 𝑆 such that lim_𝑡→+∞(𝑥(𝑡),𝑦(𝑡))=𝐸^∗. Then, the positive equilibrium 𝐸^∗ of (6) is globally attractive.

Proof. Provided that conditions (8) and (41) hold, based on comparison theorem and by similar arguments in [11], we can obtain that, for any solution (𝑥(𝑡),𝑦(𝑡)) of (6) with the positive initial condition in 𝑆, there exists a point 𝐸^∗(𝑥^∗,𝑦^∗) such that lim_𝑡→+∞(𝑥(𝑡),𝑦(𝑡))=(𝑥^∗,𝑦^∗). Indeed, just by modifying the coefficients of (6), the discussion in [11] also holds. We omit the proof here and refer to [11] for details.
Then, in view of the property of the 𝜔-limit set in [17] and by the uniqueness result of the positive equilibrium in Theorem 1, we have 𝐸^∗=𝐸^∗. Thus, we derive that 𝐸^∗ is globally attractive.

4. Conclusion

In the present paper, we considered the stability of the equilibria of a predator-prey system with Beddington-DeAngelis functional response and delays. More precisely, we derived a sufficient and necessary condition for the existence of a unique positive equilibrium and proved that the positive equilibrium is locally asymptotically stable under a sufficient condition. Also, we established the local stability of the boundary equilibria by analyzing the characteristic equations. Finally, by using comparison theorem and the 𝜔-limit set theory, we studied the global asymptotic stability for both the boundary equilibrium and the positive equilibrium.

In particular, in Theorem 7, we derived the global attractiveness of the boundary equilibrium 𝐸₁((𝑎/𝑏),0). But unfortunately, we just obtained a sufficient condition that (𝑓−𝑑𝑚₂)(𝑎/𝑏)<𝑑𝑚₁ to ensure that 𝐸₁ was globally asymptotically stable. If (𝑓−𝑑𝑚₂)(𝑎/𝑏)=𝑑𝑚₁, since the second equation of (6) was a delay diffusion equation, we could not derive that lim_𝑡→+∞𝑦(𝑡)=0 by using comparison theorem.

Moreover, as pointed out in [11], there are two interesting problems about system (1), which are related to the phenomena of bifurcations and the existence of periodic orbits of (1). We leave them for a further study.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author was partially supported by NSFC (11201402, 11371179) and NSF of Shandong Province of China (ZR2010AQ006). The second author was partially supported by Graduate Innovation Foundation of Yantai University (GIFYTU).