Almost Periodic Solution of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response

Abstract

We consider a predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional response. By applying the comparison theorem of the differential equation and constructing a suitable Lyapunov function, sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained. Our results not only supplement but also improve some existing ones. One example is presented to verify our main results.

1. Introduction

Let 𝑓(𝑡) be a continuous bounded function on 𝑅, and we set 𝑓^𝑙=inf_𝑡∈𝑅𝑓(𝑡),𝑓^𝑢=sup_𝑡∈𝑅𝑓(𝑡). (1) Leslie [1] introduced the famous Leslie predator-prey system ̇()(𝑥)̇𝑦𝑥=𝑥𝑎−𝑏𝑥−𝑝𝑦,𝑦=𝑦𝑒−𝑓𝑥, (2) where 𝑥, 𝑦 stand for the population (the density) of the prey and the predator at time 𝑡, respectively, and 𝑝(𝑥) is the so-called predator functional response to prey. The term 𝑦/𝑥 is the Leslie-Gower term which measures the loss in the predator population due to rarity (per capita 𝑦/𝑥) of its favorite food. Leslie and Gower [2], Pielou [3] obtain some excellent results on the system (2) with the functional response 𝑝(𝑥)=𝑐𝑥 which is called Holling-type I. By applying the Dulac's criterion and constructing Lyapunov functions, Hsu and Huang [4] establish the global stability of system (2).

Recently, Aziz-Alaoui and Daher Okiye [5] pointed out that in the case of severe scarcity, 𝑦 can switch over to other populations, but its growth will be limited by the fact that its most favorite food 𝑥 is not available in abundance. To solve such problem, they suggested to add a positive constant 𝑑 to the denominator and proposed the following predator-prey model with modified Leslie-Gower and Holling-type II schemes: 𝑥̇=𝑟₁𝑎−𝑏𝑥−¹𝑦𝑥+𝑘₁̇𝑟𝑥,𝑦=₂−𝑎₂𝑦𝑥+𝑘₂𝑦, (3) with initial conditions 𝑥(0)>0 and 𝑦(0)>0, where 𝑥 and 𝑦 represent the population densities at time 𝑡. 𝑟₁, 𝑎₁, 𝑏, 𝑘₁, 𝑟₂, 𝑎₂, and 𝑘₂ are model parameters assuming only positive values. 𝑟₁ is the growth rate of prey 𝑥, 𝑏 measures the strength of competition among individuals of species 𝑥, 𝑎₁ is the maximum value of the per capita reduction rate of 𝑥 due to 𝑦, 𝑘₁ (resp., 𝑘₂) measures the extent to which the environment provides protection to prey 𝑥 (resp., to the predator 𝑦), 𝑟₂ describes the growth rate of 𝑦, and 𝑎₂ has a similar meaning to 𝑎₁. The authors studied the boundedness and global stability of positive equilibrium of the system (3). Since then, system (2) and its nonautonomous versions have been studied by incorporating delay, impulses, harvesting, and so on (see, e.g., [6–13]). In particular, Zhu and Wang [13] consider the following nonautonomous model: 𝑥̇(𝑡)=𝑥(𝑡)𝑟₁(𝑡)(𝑡)𝑥(𝑡)−𝑎−𝑏¹(𝑡)𝑦(𝑡)𝑥(𝑡)+𝑘₁,𝑦̇(𝑡)=𝑦(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦(𝑡)𝑥(𝑡)+𝑘₂. (4) Under the assumption that the coefficients of the system (4) are all continuous 𝑇-periodic functions, by utilizing the coincidence degree theorem and constructing a suitable Lyapunov function, they obtained sufficient conditions for the existence and global attractivity of positive periodic solutions of the system (4). More precisely, Zhu and Wang [13] obtained the following theorem (see [13, Theorems  3.1 and 3.2]).

Theorem 1. Suppose that(𝐶₁)𝑘₁𝑟₁−𝑎₁𝑘₂+𝑟₁𝑏^𝑢𝑟₂𝑎₂^𝑢𝑙>0 (5) holds, and further suppose that one of the following conditions:(𝐶₂)𝑘₁𝑟₁−𝑎₁𝑟₂𝑘₂+𝑒^𝐴5𝑎₂>0,𝑤ℎ𝑒𝑟𝑒𝐴₅=ln𝑟₁𝑏 (6)(𝐶₃)𝑘₁>𝑘₂ (7) holds; then system (4) has at least one positive 𝑇-periodic solution.

Corollary 6 given in Section 2 of this paper shows that when (𝐶₂) or (𝐶₃) does not satisfy, the conclusion of Theorem 1 also holds. Moreover, as we know, permanence is one of the most important topics in the study of population dynamics; however, Zhu and Wang [13] did not investigate this property of the system (4). One aim of this work is to obtain a set of sufficient conditions which guarantee the permanence of the system (4).

On the other hand, as Fan and Kuang [14] have mentioned, the Holling II type functional responses fail to model the interference among predators and have been facing challenges from the biology and physiology communities (see [15–17]). Some biologists have argued that in many situations, especially when predators have to search for food (and therefore have to share or compete for food), the functional response in a prey-predator model should be predator-dependent.

Stimulated by the previous reasons, in this paper we will incorporate the Beddington-DeAngelis functional response into model (4) and consider the following model which is the generalization of the model (4): 𝑥̇(𝑡)=𝑥(𝑡)𝑟₁(𝑡)(𝑡)𝑥(𝑡)−𝑐(𝑡)𝑦(𝑡)−𝑏𝛼(𝑡)(𝑡)𝑥(𝑡)(𝑡)𝑦(𝑡),𝑦̇(𝑡)(𝑡)𝑟+𝛽+𝛾=𝑦₂(𝑡)−𝑑(𝑡)𝑦(𝑡)𝑥(𝑡)(𝑡),+𝑘 (8) where 𝑥(𝑡) is the size of prey population and 𝑦(𝑡) is the size of predator population.

Li and Zhang [18] pointed out that in real world phenomenon, the environment varies due to the factors such as seasonal effects of weather, food supplies, mating habits, and harvesting. So it is usual to assume the periodicity of parameters in the system (8). However, if the various constituent components of the temporally nonuniform environment are with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions. For this reason, the assumption of almost periodicity is more realistic, more important, and more general when we consider the effects of the environmental factors. So it is assumed that the coefficients 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑘(𝑡), 𝛼(𝑡), 𝛽(𝑡), 𝛾(𝑡), 𝑟_𝑖(𝑡)(𝑖=1,2) are all continuous, almost periodic functions and satisfy min_𝑖=1,2𝑏^𝑙,𝑐^𝑙,𝑑^𝑙,𝑘^𝑙,𝛼^𝑙,𝛽^𝑙,𝛾^𝑙,𝑟^𝑙_𝑖>0,max_𝑖=1,2𝑏^𝑢,𝑐^𝑢,𝑑^𝑢,𝑘^𝑢,𝛼^𝑢,𝛽^𝑢,𝛾^𝑢,𝑟^𝑢_𝑖<+∞. (9)

We consider system (8) with the following initial conditions: 𝑥(0)>0,𝑦(0)>0. (10)

One can easily show that the solution of (8) with the initial condition (10) is defined and remains positive for all 𝑡≥0.

The aim of this paper is to obtain sufficient conditions for the existence of a unique globally attractive almost periodic solution of systems (8) and (10), by utilizing the comparison theorem of the differential equation and applying the analysis technique of papers [19–21].

The organization of this paper is as follows. In Section 2, by applying the theory of differential inequality, we present the permanence results for system (8). In Section 3, by constructing a suitable Lyapunov function, a set of sufficient conditions which ensure the existence and uniqueness of almost periodic solutions of system (8) are obtained. Then, in Section 4, a suitable example together with its numeric simulations is given to illustrate the feasibility of the main results. We end this paper by a brief discussion.

2. Permanence

Now let us state several definitions and lemmas which will be useful in the proving of the main result of this section.

Lemma 2 (see [22]). If 𝑎>0, 𝑏>0 and 𝑥̇≥𝑥(𝑏−𝑎𝑥), when 𝑡≥0 and 𝑥(0)>0, we have liminf_𝑡→+∞𝑥(𝑡)≥𝑏𝑎. (11) If 𝑎>0, 𝑏>0 and 𝑥̇≤𝑥(𝑏−𝑎𝑥), when 𝑡≥0 and 𝑥(0)>0, we have limsup_𝑡→+∞𝑥(𝑡)≤𝑏𝑎. (12)

Theorem 3. Suppose that system (8) with initial condition (10) satisfies the following conditions:(𝐻₁)𝑟^𝑙₁𝛼^𝑙−𝑟^𝑢₂𝑐^𝑢𝑀₁+𝑘^𝑢𝑑^𝑙>0,𝑤ℎ𝑒𝑟𝑒𝑀₁=𝑟^𝑢₁𝑏^𝑙. (13)Then system (8) is permanent, that is, any positive solution (𝑥(𝑡),𝑦(𝑡))^𝑇 of the system (8) satisfies 0<𝑚₁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀₁,0<𝑚₂≤liminf_𝑡→+∞𝑦(𝑡)≤limsup_𝑡→+∞𝑦(𝑡)≤𝑀₂, (14) where 𝑚₁=(𝑟^𝑙₁𝛼^𝑙−𝑐^𝑢𝑀₂)/𝑏^𝑢𝛼^𝑙, 𝑀₂=𝑟^𝑢₂(𝑀₁+𝑘^𝑢)/𝑑^𝑙, 𝑚₂=𝑟^𝑙₂(𝑚₁+𝑘^𝑙)/𝑑^𝑢.

Proof. From condition (𝐻₁), we can choose a small enough 𝜀 such that 𝑟^𝑙₁𝛼^𝑙>𝑟^𝑢₂𝑐^𝑢𝑀₁+𝜀+𝑘^𝑢𝑑^𝑙. (15) The first equation of (8) yields 𝑥̇(𝑡)≤𝑥(𝑡)𝑟^𝑢₁−𝑏^𝑙𝑥(𝑡). (16) Applying Lemma 2 to (16) leads to limsup_𝑡→+∞𝑥(𝑡)≤𝑟^𝑢₁𝑏^𝑙Δ=𝑀₁. (17) Equation (17) shows that there exists a large enough 𝑇₁>0 such that for all 𝑡≥𝑇₁, 𝑥(𝑡)≤𝑀₁+𝜀=^Δ𝑀_1𝜀. (18) It follows from (18) and the second equation of system (8) that, for 𝑡≥𝑇₁, 𝑦̇(𝑡)≤𝑦(𝑡)𝑟^𝑢₂−𝑑^𝑙𝑦(𝑡)𝑀_1𝜀+𝑘^𝑢. (19) According to Lemma 2 and (19), one has limsup_𝑡→+∞𝑦(𝑡)≤𝑟^𝑢₂𝑀_1𝜀+𝑘^𝑢𝑑^𝑙. (20) Thus, for previous 𝜀, there exists a 𝑇₂≥𝑇₁, such that 𝑦(𝑡)≤𝑟^𝑢₂𝑀_1𝜀+𝑘^𝑢𝑑^𝑙+𝜀=^Δ𝑀_2𝜀∀𝑡≥𝑇₂. (21) Equation (21) together with the first equation of (8) leads to 𝑥̇(𝑡)≥𝑥(𝑡)𝑟^𝑙₁−𝑏^𝑢𝑥(𝑡)−𝑐^𝑢𝑀_2𝜀𝛼^𝑙∀𝑡≥𝑇₂. (22) From (22), according to (15) and Lemma 2, we can obtain liminf_𝑡→+∞𝑥(𝑡)≥𝑟^𝑙₁𝛼^𝑙−𝑐^𝑢𝑀_2𝜀𝑏^𝑢𝛼^𝑙. (23) Hence, for previous 𝜀, there exists a 𝑇₃≥𝑇₂, such that 𝑥(𝑡)≥𝑟^𝑙₁𝛼^𝑙−𝑐^𝑢𝑀_2𝜀𝑏^𝑢𝛼^𝑙−𝜀=^Δ𝑚_1𝜀∀𝑡≥𝑇₃. (24) From (24) and the second equation of system (8), we know that for 𝑡≥𝑇₃, 𝑦̇(𝑡)≥𝑟^𝑙₂−𝑑^𝑢𝑦(𝑡)𝑚_1𝜀+𝑘^𝑙. (25) Applying Lemma 2 to (25) leads to liminf_𝑡→+∞𝑦(𝑡)≥𝑟^𝑙₂𝑚_1𝜀+𝑘^𝑙𝑑^𝑢. (26) Setting 𝜀→0, it follows from the previous discussion that limsup_𝑡→+∞𝑦(𝑡)≤𝑟^𝑢₂𝑀₁+𝑘^𝑢𝑑^𝑙Δ=𝑀₂,liminf_𝑡→+∞𝑥(𝑡)≥𝑟^𝑙₁𝛼^𝑙−𝑐^𝑢𝑀₂𝑏^𝑢𝛼^𝑙Δ=𝑚₁,liminf_𝑡→+∞𝑦(𝑡)≥𝑟^𝑙₂𝑚₁+𝑘^𝑙𝑑^𝑢Δ=𝑚₂. (27) Obviously, 𝑚_𝑖 and 𝑀_𝑖(𝑖=1,2) are independent of the solution of system (8); (17) and (27) show that the conclusion of Theorem 3 holds. The proof is completed.

Theorem 4. Suppose that system (8) with initial condition (10) satisfies the following conditions:(𝐻₂)𝑟^𝑙₁𝛾^𝑙−𝑐^𝑢>0. (28)Then system (8) is permanent, that is, any positive solution (𝑥(𝑡),𝑦(𝑡))^𝑇 of the system (8) satisfies 0<𝑚₁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀₁,0<𝑚₂≤liminf_𝑡→+∞𝑦(𝑡)≤limsup_𝑡→+∞𝑦(𝑡)≤𝑀₂, (29) where 𝑀₁, 𝑀₂ are defined in Theorem 3 and 𝑚₁=(𝑟^𝑙₁𝛾^𝑙−𝑐^𝑢)/𝑏^𝑢𝛾^𝑙, 𝑚₂=𝑟^𝑙₂(𝑚₁+𝑘^𝑙)/𝑑^𝑢.

Proof. From (17) and (20), one can derive limsup_𝑡→+∞𝑥(𝑡)≤𝑀₁,limsup_𝑡→+∞𝑦(𝑡)≤𝑀₂, (30) where 𝑀₁ and 𝑀₂ are defined in Theorem 3.
It follows from the first equation of system (8) that 𝑥(𝑡)≥𝑥(𝑡)𝑟^𝑙₁−𝑏^𝑢𝑥(𝑡)−𝑐^𝑢𝛾^𝑙. (31) According to (𝐻₂), by applying Lemma 2 to (31), we obtain liminf_𝑡→+∞𝑥(𝑡)≥𝑟^𝑙₁𝛾^𝑙−𝑐^𝑢𝑏^𝑢𝛾^𝑙Δ=𝑚₁. (32) Hence, for a small enough 𝜀, there exists a 𝑇>0, such that 𝑥(𝑡)≥𝑟^𝑙₁𝛾^𝑙−𝑐^𝑢𝑏^𝑢𝛾^𝑙−𝜀=^Δ𝑚_1𝜀∀𝑡≥𝑇. (33) From (33) and the second equation of system (8), we know that for 𝑡≥𝑇₃, 𝑦̇(𝑡)≥𝑟^𝑙₂−𝑑^𝑢𝑦(𝑡)𝑚_1𝜀+𝑘^𝑙. (34) Applying Lemma 2 to (34) leads to liminf_𝑡→+∞𝑦(𝑡)≥𝑟^𝑙₂𝑚_1𝜀+𝑘^𝑙𝑑^𝑢. (35) Setting 𝜀→0, it follows from the previous discussion that liminf_𝑡→+∞𝑦(𝑡)≥𝑟^𝑙₂𝑚₁+𝑘^𝑙𝑑^𝑢Δ=𝑚₂. (36) Obviously, 𝑚_𝑖 and 𝑀_𝑖(𝑖=1,2) are independent of the solution of system (8); (30), (32) and (36) show that the conclusion of Theorem 4 holds. The proof is completed.

As a direct corollary of Theorem 2 in [23], from Theorems 3 or 4, we have the following.

Corollary 5. Suppose that (𝐻₁) or (𝐻₂) holds, then system (8) admits at least one positive 𝑇-periodic solution if 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑘(𝑡), 𝛼(𝑡), 𝛽(𝑡), 𝛾(𝑡), 𝑟_𝑖(𝑡)(𝑖=1,2) are all continuous positive 𝑇-periodic functions.

When 𝛼(𝑡)=𝑘₁, 𝛽(𝑡)=1, 𝛾(𝑡)=0, 𝑘(𝑡)=𝑘₂, 𝑐(𝑡)=𝑎₁(𝑡), 𝑑(𝑡)=𝑎₂(𝑡), where 𝑘₁, 𝑘₂ are positive constants, (8) becomes (4) which was discussed in [13]. According to Theorem 3 and Corollary 5, we obtain the following.

Corollary 6. Suppose that(𝐻₁₁)𝑟^𝑙₁𝑘₁−𝑎^𝑢₁𝑟^𝑢₂𝑀₁+𝑘₂𝑎^𝑙₂>0,𝑤ℎ𝑒𝑟𝑒𝑀₁=𝑟^𝑢₁𝑏^𝑙 (37)holds; then system (4) is permanent and admits at least one positive 𝑇-periodic solution if 𝑎_𝑖(𝑡), 𝑏(𝑡), 𝑟_𝑖(𝑡)(𝑖=1,2) are all continuous positive 𝑇-periodic functions.

Remark 7. Comparing with Theorem 1, it follows from Corollary 6 that (𝐶₂) or (𝐶₃) is superfluous, so our results improve the main results in [13].

3. Existence of a Unique Almost Periodic Solution

Now let us state several definitions and lemmas which will be useful in the proving of the main result of this section.

Definition 8 (see [24, 25]). A function 𝑓(𝑡,𝑥), where 𝑓 is an 𝑚-vector, 𝑡 is a real scalar, and 𝑥 is an 𝑛-vector, is said to be almost periodic in 𝑡 uniformly with respect to 𝑥∈𝑋⊂𝑅^𝑛, if 𝑓(𝑡,𝑥) is continuous in 𝑡∈𝑅 and 𝑥∈𝑋, and if for any 𝜀>0, there is a constant 𝑙(𝜀)>0, such that in any interval of length 𝑙(𝜀) there exists 𝜏 such that the inequality 𝑓(𝑡+𝜏)−𝑓(𝑡)=^𝑚_𝑖=1𝑓_𝑖(𝑡+𝜏,𝑥)−𝑓_𝑖(𝑡,𝑥)<𝜀 (38) is satisfied for all 𝑡∈(−∞,+∞), 𝑥∈𝑋. The number 𝜏 is called an 𝜀-translation number of 𝑓(𝑡,𝑥).

Definition 9 (see [24, 25]). A function 𝑓∶𝑅→𝑅 is said to be asymptotically almost periodic function if there exists an almost periodic function 𝑞(𝑡) and a continuous function 𝑟(𝑡) such that 𝑓(𝑡)=𝑞(𝑡)+𝑟(𝑡),𝑡∈𝑅,𝑟(𝑡)⟶0as𝑡⟶∞. (39)

We denote by 𝑆(𝐸) the set of all solutions 𝑧(𝑡)=(𝑥(𝑡),𝑦(𝑡))^𝑇 of system (8) satisfying 𝑚₁≤𝑥(𝑡)≤𝑀₁,𝑚₂≤𝑦(𝑡)≤𝑀₂ for all 𝑡∈𝑅.

Lemma 10. 𝑆(𝐸)≠∅.

Proof. Since 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑘(𝑡), 𝛼(𝑡), 𝛽(𝑡), 𝛾(𝑡), 𝑟_𝑖(𝑡)(𝑖=1,2) are almost periodic functions, there exists a sequence {𝑡_𝑛}, 𝑡_𝑛→∞ as 𝑛→∞ such that 𝑏𝑡+𝑡_𝑛⟶𝑏(𝑡),𝑐𝑡+𝑡_𝑛(𝑡),𝑑⟶𝑐𝑡+𝑡_𝑛⟶𝑑(𝑡),𝑘𝑡+𝑡_𝑛(𝑡),𝛼⟶𝑘𝑡+𝑡_𝑛⟶𝛼(𝑡),𝛽𝑡+𝑡_𝑛(𝑡),𝛾⟶𝛽𝑡+𝑡_𝑛⟶𝛾(𝑡),𝑟_𝑖𝑡+𝑡_𝑛⟶𝑟_𝑖(𝑡)(𝑖=1,2), (40) as 𝑛→∞ uniformly on 𝑅. Let 𝑧(𝑡)=(𝑥(𝑡),𝑦(𝑡))^𝑇 be solution of systems (8) and (10) satisfying 𝑚₁≤𝑥(𝑡)≤𝑀₁, 𝑚₂≤𝑦(𝑡)≤𝑀₂ for 𝑡>𝑇. Obviously, the sequence 𝑧(𝑡+𝑡_𝑛) is uniformly bounded and equicontinuous on each bounded subset of 𝑅. Therefore, by Ascoli-Arzela theorem, there exists a subsequence of {𝑡_𝑛}, and we still denote it as {𝑡_𝑛}, such that 𝑥(𝑡+𝑡_𝑛)→𝑝₁(𝑡),𝑦(𝑡+𝑡_𝑛)→𝑝₂(𝑡), as 𝑛→∞ uniformly on each bounded subset of 𝑅. For any 𝑇₁∈𝑅, we may assume that 𝑡_𝑛+𝑇₁≥𝑇 for all 𝑛. For 𝑡≥0, we have𝑥𝑡+𝑡_𝑛+𝑇₁−𝑥𝑡_𝑛+𝑇₁=^𝑡+𝑇1_𝑇1𝑥𝑠+𝑡_𝑛𝑟₁𝑠+𝑡_𝑛−𝑏𝑠+𝑡_𝑛𝑥𝑠+𝑡_𝑛−𝑐𝑠+𝑡_𝑛𝑦𝑠+𝑡_𝑛𝛼𝑠+𝑡_𝑛+𝛽𝑠+𝑡_𝑛𝑥𝑠+𝑡_𝑛+𝛾𝑠+𝑡_𝑛𝑦𝑠+𝑡_𝑛𝑦𝑑𝑠,𝑡+𝑡_𝑛+𝑇₁−𝑦𝑡_𝑛+𝑇₁=^𝑡+𝑇1_𝑇1𝑦𝑠+𝑡_𝑛𝑟₂𝑠+𝑡_𝑛−𝑑𝑠+𝑡_𝑛𝑦𝑠+𝑡_𝑛𝑥𝑠+𝑡_𝑛+𝑘𝑠+𝑡_𝑛𝑑𝑠. (41)Applying Lebesgue's dominated convergence theorem and letting 𝑛→∞ in the previous equations, we obtain 𝑝₁𝑡+𝑇₁−𝑝₁𝑇₁=^𝑡+𝑇1_𝑇1𝑝₁(𝑠)𝑟₁(𝑠)−𝑏(𝑠)𝑝₁(𝑠)−𝑐(𝑠)𝑝₂(𝑠)𝛼(𝑠)+𝛽(𝑠)𝑝₁(𝑠)+𝛾(𝑠)𝑝₂(𝑠)𝑝𝑑𝑠,₂𝑡+𝑇₁−𝑝₂𝑇₁=^𝑡+𝑇1_𝑇1𝑝₂(𝑠)𝑟₂(𝑠)−𝑑(𝑠)𝑝₂(𝑠)𝑝₁(𝑠)+𝑘(𝑠)𝑑𝑠, (42) for all 𝑡≥0. Since 𝑇₁∈𝑅 is arbitrarily given, (𝑝₁(𝑡),𝑝₂(𝑡))^𝑇 is a solution of system (8) on 𝑅. It is clear that 𝑚₁≤𝑝₁(𝑡)≤𝑀₁, 𝑚₂≤𝑝₂(𝑡)≤𝑀₂ for 𝑡∈𝑅. That is to say (𝑝₁(𝑡),𝑝₂(𝑡))^𝑇∈𝑆(𝐸). This completes the proof.

Lemma 11 (see [26]). Let 𝑓 be a nonnegative function defined on [0,+∞) such that 𝑓 is integrable on [0,+∞) and is uniformly continuous on [0,+∞). Then lim_𝑡→+∞𝑓(𝑡)=0.

Theorem 12. In addition to (𝐻₁) or (𝐻₂), further suppose that(𝐻₃)𝑏(𝑡)−𝑐(𝑡)𝛽(𝑡)𝑀₂Δ²𝑡,𝑚₁,𝑚₂−𝑑(𝑡)𝑀₂𝑚₁+𝑘(𝑡)^2𝑙>0, (43)(𝐻₄)𝑑(𝑡)𝑀₁(𝑡)−𝑐(𝑡)+𝑘Δ𝑡,𝑚₁,𝑚₂−𝑐(𝑡)𝛾(𝑡)𝑀₂Δ²𝑡,𝑚₁,𝑚₂^𝑙>0, (44)where 𝑚_𝑖 and 𝑀_𝑖(𝑖=1,2) are defined in the proof of Theorem 3 (or Theorem 4) and Δ𝑡,𝑥(𝑡),𝑦(𝑡)=𝛼(𝑡)+𝛽(𝑡)𝑥(𝑡)+𝛾(𝑡)𝑦(𝑡). (45) Then system (8) with initial conditions (10) is globally attractive.

Proof. It follows from conditions (𝐻₃) and (𝐻₄) that there exists a small enough 𝜀>0 such that 𝐴₁(𝜀)=𝑏(𝑡)−𝑐(𝑡)𝛽(𝑡)𝑀₂+𝜀Δ²𝑡,𝑚₁−𝜀,𝑚₂−𝑑(𝑡𝑀−𝜀)₂+𝜀𝑚₁−𝜀+𝑘(𝑡)^2𝑙𝐴>𝜀,₂(𝜀)=𝑑(𝑡)𝑀₁(𝑡)−𝑐(𝑡)+𝜀+𝑘Δ𝑡,𝑚₁−𝜀,𝑚₂−𝑐(𝑡)𝛾(𝑡)𝑀−𝜀₂+𝜀Δ²𝑡,𝑚₁−𝜀,𝑚₂−𝜀^𝑙>𝜀. (46) Let 𝑧₁(𝑡)=(𝑥(𝑡),𝑦(𝑡))^𝑇, 𝑧₂(𝑡)=(𝑥_∗(𝑡),𝑦_∗(𝑡))^𝑇 be any two positive solutions of system (8) with initial conditions (10). For previous 𝜀, according to Lemma 10 and Theorem 3 (or Theorem 4), there exists a 𝑇>0, when 𝑡≥𝑇, 𝑚₁−𝜀≤𝑥(𝑡)≤𝑀₁+𝜀,𝑚₂−𝜀≤𝑦(𝑡)≤𝑀₂𝑚+𝜀,₁−𝜀≤𝑥_∗(𝑡)≤𝑀₁+𝜀,𝑚₂−𝜀≤𝑦_∗(𝑡)≤𝑀₂+𝜀. (47)
Let 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡), where 𝑉₁(𝑡)=ln𝑥(𝑡)−ln𝑥_∗(𝑡,𝑉₂)(𝑡)=ln𝑦(𝑡)−ln𝑦_∗(𝑡). (48) Calculating the upper right derivatives of 𝑉₁(𝑡) along the solution of (8) leads to𝐷⁺𝑉₁(𝑡)=sgn𝑥(𝑡)−𝑥_∗(𝑡)−𝑏(𝑡)𝑥(𝑡)−𝑥_∗(𝑡)(𝑡)𝑦(𝑡)−𝑐Δ(𝑡)(𝑡−𝑦𝑡,𝑥,𝑦)^∗(𝑡)Δ𝑡,𝑥_∗(𝑡),𝑦_∗(𝑡(𝑡𝑥(𝑡))=−𝑏)−𝑥_∗(𝑡)−sgn𝑥(𝑡)−𝑥_∗(𝑡)𝑐(𝑡)𝑦(𝑡)Δ(𝑡)(𝑡)−𝑦𝑡,𝑥,𝑦^∗(𝑡)Δ(𝑡)(𝑡)+𝑦𝑡,𝑥,𝑦_∗(𝑡)Δ(𝑡)(𝑡)−𝑦𝑡,𝑥,𝑦^∗(𝑡)Δ𝑡,𝑥_∗(𝑡),𝑦_∗(𝑡)=−𝑏(𝑡)𝑥(𝑡)−𝑥_∗(𝑡)−sgn𝑥(𝑡)−𝑥_∗(𝑡)(𝑡)𝑦(𝑡)×𝑐−𝑦_∗(𝑡)Δ(𝑡)(𝑡)+𝑦𝑡,𝑥,𝑦^∗(𝑡)𝛽(𝑡)𝑥_∗(𝑡)−𝑥(𝑡)+𝛾(𝑡)𝑦_∗(𝑡)−𝑦(𝑡)Δ𝑡,𝑥_∗(𝑡),𝑦_∗(𝑡)(𝑡)(𝑡),𝐷⋅Δ𝑡,𝑥,𝑦₊𝑉₁(𝑡)≤−𝑏(𝑡)𝑥(𝑡)−𝑥_∗(𝑡(𝑡)𝑦(𝑡))+𝑐−𝑦_∗(𝑡)Δ(𝑡)(𝑡+𝑦𝑡,𝑥,𝑦)^∗(𝑡)𝛽(𝑡)𝑥(𝑡)−𝑥_∗(𝑡)+𝛾(𝑡)𝑦(𝑡)−𝑦_∗(𝑡)Δ𝑡,𝑥_∗(𝑡),𝑦_∗(𝑡(𝑡)(𝑡.)⋅Δ𝑡,𝑥,𝑦) (49)
Calculating the upper right derivatives of 𝑉₂(𝑡) along the solution of (8), one has 𝐷⁺𝑉₂(𝑡)=sgn𝑦(𝑡)−𝑦_∗(𝑡𝑑(𝑡𝑦))^∗(𝑡)𝑥_∗(𝑡)(𝑡)−𝑦(𝑡)+𝑘𝑥(𝑡)(𝑡)𝑦(𝑡)+𝑘=sgn−𝑦_∗(𝑡𝑑(𝑡)×𝑦_∗)(𝑡)𝑥_∗(𝑡)(𝑡)−𝑦+𝑘^∗(𝑡)𝑥(𝑡)(𝑡)+𝑦+𝑘_∗(𝑡)𝑥(𝑡)(𝑡)−𝑦(𝑡)+𝑘𝑥(𝑡)(𝑡)𝑦(𝑡)+𝑘=sgn−𝑦_∗(𝑡𝑑(𝑡)×𝑦_∗)(𝑡)𝑥(𝑡)−𝑥_∗(𝑡)𝑥_∗(𝑡)(𝑡)(𝑥(𝑡)(𝑡+𝑦+𝑘+𝑘))^∗(𝑡)−𝑦(𝑡)𝑥(𝑡)(𝑡)≤𝑑(𝑡)𝑦+𝑘_∗(𝑡)𝑥(𝑡)−𝑥_∗(𝑡)𝑥_∗(𝑡)(𝑡)(𝑥(𝑡)(𝑡−𝑑(𝑡𝑦(𝑡)+𝑘+𝑘)))−𝑦_∗(𝑡)𝑥(𝑡)(𝑡).+𝑘 (50)
It follows from (49)-(50) that for 𝑡≥𝑇, 𝐷⁺𝑉(𝑡)≤(𝑡)+𝑐(𝑡)𝑦−𝑏^∗(𝑡)𝛽(𝑡)Δ𝑡,𝑥_∗(𝑡),𝑦_∗(𝑡(𝑡)(𝑡+𝑑(𝑡)𝑦)⋅Δ𝑡,𝑥,𝑦)_∗(𝑡)𝑥_∗(𝑡)(𝑡)𝑥(𝑡)(𝑡𝑥(𝑡)+𝑘(+𝑘))−𝑥_∗(𝑡+−𝑑(𝑡))𝑥(𝑡)(𝑡)+𝑐(𝑡)+𝑘Δ(𝑡)(𝑡)+𝑐(𝑡)𝑦𝑡,𝑥,𝑦_∗(𝑡)𝛾(𝑡)Δ𝑡,𝑥_∗(𝑡),𝑦_∗(𝑡)(𝑡)(𝑡)×𝑦(𝑡)⋅Δ𝑡,𝑥,𝑦−𝑦_∗(𝑡𝑏(𝑡)−𝑐(𝑡)𝛽(𝑡)𝑀)≤−²+𝜀Δ²𝑡,𝑚₁−𝜀,𝑚₂−𝑑(𝑡)𝑀−𝜀²+𝜀𝑚₁−𝜀+𝑘(𝑡)²×𝑥(𝑡)−𝑥_∗(𝑡−𝑑(𝑡))𝑀₁(𝑡)−𝑐(𝑡)+𝜀+𝑘Δ𝑡,𝑚₁−𝜀,𝑚₂−𝑐(𝑡)𝛾(𝑡)𝑀−𝜀₂+𝜀Δ²𝑡,𝑚₁−𝜀,𝑚₂𝑦(𝑡)−𝜀−𝑦_∗(𝑡). (51) It follows from (46) and (51) that for 𝑡≥𝑇, 𝐷⁺𝑉(𝑡)≤−𝜀𝑥(𝑡)−𝑥_∗(𝑡)−𝜀𝑦(𝑡)−𝑦_∗(𝑡), (52) which implies 𝑉(𝑡) is nonincreasing on [𝑇,+∞). Integrating the previous inequality from 𝑇 to 𝑡 leads to 𝑉(𝑡)+𝜀^𝑡_𝑇𝑥(𝑠)−𝑥_∗(𝑠×)𝑑𝑠+𝜀_𝑡𝑇𝑦(𝑠)−𝑦_∗(𝑠)𝑑𝑠<𝑉(𝑇)<+∞,𝑡≥𝑇. (53) Then for 𝑡≥𝑇, we obtain that ^𝑡_𝑇𝑥(𝑠)−𝑥_∗(𝑠𝑉(𝑇))𝑑𝑠<𝜀<+∞,_𝑡𝑇𝑦(𝑠)−𝑦_∗(𝑠𝑉(𝑇))𝑑𝑠<𝜀<+∞. (54) Hence, |𝑥(𝑡)−𝑥_∗(𝑡)|, |𝑦(𝑡)−𝑦_∗(𝑡)|∈𝐿¹([𝑇,+∞)). By system (8) and Theorem 3 (or Theorem 4), we get 𝑥(𝑡), 𝑥_∗(𝑡), 𝑦(𝑡), 𝑦_∗(𝑡), and their derivatives are bounded on [𝑇,+∞), which implies that both |𝑥(𝑡)−𝑥_∗(𝑡)| and |𝑦(𝑡)−𝑦(𝑡)| are uniformly continuous on [𝑇,+∞). By Lemma 11, we obtain lim_𝑡→+∞𝑥(𝑡)−𝑥_∗(𝑡)=0,lim_𝑡→+∞𝑦(𝑡)−𝑦_∗(𝑡)=0. (55) Then the solution of systems (8) and (10) is globally attractive.

Theorem 13. Suppose all conditions of Theorem 12 hold; then there exists a unique almost periodic solution of systems (8) and (10).

Proof. According to Lemma 10, there exists a bounded positive solution 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡))^𝑇 of (8) with initial condition (10). Then there exists a sequence {𝑡^′_𝑘}, {𝑡^′_𝑘}→∞ as 𝑘→∞, such that (𝑢₁(𝑡+𝑡^′_𝑘),𝑢₂(𝑡+𝑡^′_𝑘))^𝑇 is a solution of the following system: 𝑥̇(𝑡)=𝑥(𝑡)𝑟₁𝑡+𝑡^′_𝑘−𝑏𝑡+𝑡^′_𝑘𝑥(𝑡)−𝑐𝑡+𝑡^′_𝑘𝑦(𝑡)𝛼𝑡+𝑡^′_𝑘+𝛽𝑡+𝑡^′_𝑘𝑥(𝑡)+𝛾𝑡+𝑡^′_𝑘𝑦(𝑡),𝑦̇(𝑡)=𝑦(𝑡)𝑟₂𝑡+𝑡^′_𝑘−𝑑𝑡+𝑡^′_𝑘𝑦(𝑡)𝑥(𝑡)+𝑘𝑡+𝑡^′_𝑘. (56) According to Theorem 3 (or Theorem 4) and the fact that 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑘(𝑡), 𝛼(𝑡), 𝛽(𝑡), 𝛾(𝑡), 𝑟_𝑖(𝑡)(𝑖=1,2) are all continuous, positive almost periodic functions, we know that both {𝑢_𝑖(𝑡+𝑡^′_𝑘)}(𝑖=1,2) and their derivative function {𝑢̇(𝑡+𝑡^′_𝑘)}(𝑖=1,2) are uniformly bounded; thus {𝑢_𝑖(𝑡+𝑡^′_𝑘)}(𝑖=1,2) are uniformly bounded and equicontinuous. By Ascoli's theorem, there exists a uniformly convergent subsequence {𝑢_𝑖(𝑡+𝑡_𝑘)}⊆{𝑢_𝑖(𝑡+𝑡^′_𝑘)} such that for any 𝜀>0, there exists a 𝐾(𝜀)>0 with the property that if 𝑚,𝑘≥𝐾(𝜀), then 𝑢_𝑖𝑡+𝑡_𝑚−𝑢_𝑖𝑡+𝑡_𝑘<𝜀,𝑖=1,2. (57) That is to say 𝑢_𝑖(𝑡)(𝑖=1,2) are asymptotically almost periodic functions; hence there exists two almost periodic functions 𝑝_𝑖(𝑡+𝑡_𝑘)(𝑖=1,2) and two continuous functions 𝑞_𝑖(𝑡+𝑡_𝑘)(𝑖=1,2) such that 𝑢_𝑖𝑡+𝑡_𝑘=𝑝_𝑖𝑡+𝑡_𝑘+𝑞_𝑖𝑡+𝑡_𝑘,𝑖=1,2, (58) where lim_𝑘→+∞𝑝_𝑖𝑡+𝑡_𝑘=𝑝_𝑖(𝑡),lim_𝑘→+∞𝑞_𝑖𝑡+𝑡_𝑘=0,𝑖=1,2, (59)𝑝_𝑖(𝑡)(𝑖=1,2) are also almost periodic functions.
Therefore, lim_𝑘→+∞𝑢_𝑖𝑡+𝑡_𝑘=𝑝_𝑖(𝑡),𝑖=1,2. (60) On the other hand, lim_𝑘→+∞𝑢̇_𝑖𝑡+𝑡_𝑘=lim_𝑘→+∞lim_ℎ→0𝑢_𝑖𝑡+𝑡_𝑘+ℎ−𝑢_𝑖𝑡+𝑡_𝑘ℎ=lim_ℎ→0lim_𝑘→+∞𝑢_𝑖𝑡+𝑡_𝑘+ℎ−𝑢_𝑖𝑡+𝑡_𝑘ℎ=lim_ℎ→0𝑝_𝑖(𝑡+ℎ)−𝑝_𝑖(𝑡)ℎ,𝑖=1,2. (61) So 𝑝̇_𝑖(𝑡)(𝑖=1,2) exist. Now we will prove that (𝑝₁(𝑡),𝑝₂(𝑡))^𝑇 is an almost solution of system (8).
From properties of almost periodic function, there exists an sequence {𝑡_𝑛}, {𝑡_𝑛}→∞ as 𝑛→∞, such that 𝑏𝑡+𝑡_𝑛⟶𝑏(𝑡),𝑐𝑡+𝑡_𝑛(𝑡),𝑑⟶𝑐𝑡+𝑡_𝑛⟶𝑑(𝑡),𝑘𝑡+𝑡_𝑛(𝑡),𝛼⟶𝑘𝑡+𝑡_𝑛⟶𝛼(𝑡),𝛽𝑡+𝑡_𝑛(𝑡),𝛾⟶𝛽𝑡+𝑡_𝑛⟶𝛾(𝑡),𝑟_𝑖𝑡+𝑡_𝑛⟶𝑟_𝑖(𝑡),(𝑖=1,2) (62) as 𝑛→∞ uniformly on 𝑅.
It is easy to know that 𝑢_𝑖(𝑡+𝑡_𝑛)→𝑝_𝑖(𝑡)(𝑖=1,2) as 𝑛→∞; then we have𝑝̇₁(𝑡)=lim_𝑛→+∞𝑢̇₁𝑡+𝑡_𝑛=lim_𝑛→+∞𝑢₁𝑡+𝑡_𝑛𝑟₁𝑡+𝑡_𝑛−𝑏𝑡+𝑡_𝑛𝑢₁𝑡+𝑡_𝑛−𝑐𝑡+𝑡_𝑛𝑢₂𝑡+𝑡_𝑛𝛼𝑡+𝑡_𝑛+𝛽𝑡+𝑡_𝑛𝑢₁𝑡+𝑡_𝑛+𝛾𝑡+𝑡_𝑛𝑢₂𝑡+𝑡_𝑛=𝑝₁(𝑡)𝑟₁(𝑡)−𝑏(𝑡)𝑝₁(𝑡)−𝑐(𝑡)𝑝₂(𝑡)𝛼(𝑡)+𝛽(𝑡)𝑝₁(𝑡)+𝛾(𝑡)𝑝₂(𝑡),𝑝̇₂(𝑡)=lim_𝑛→+∞𝑢̇₂𝑡+𝑡_𝑛=lim_𝑛→+∞𝑢₂𝑡+𝑡_𝑛𝑟₂𝑡+𝑡_𝑛−𝑑𝑡+𝑡_𝑛𝑢₂𝑡+𝑡_𝑛𝑢₁𝑡+𝑡_𝑛+𝑘𝑡+𝑡_𝑛=𝑝₂(𝑡)𝑟₂(𝑡)−𝑑(𝑡)𝑝₂(𝑡)𝑝₁(𝑡)(𝑡).+𝑘 (63)
These show that (𝑝₁(𝑡),𝑝₂(𝑡))^𝑇 satisfies system (8). Hence, (𝑝₁(𝑡),𝑝₂(𝑡))^𝑇 is a positive almost periodic solution of (8). Then, it follows from Theorem 12 that system (8) has a unique positive almost periodic solution. The proof is completed.

4. Examples and Numeric Simulations

Consider the following example: 𝑥̇(𝑡)=𝑥(𝑡)10+sin√(𝑡)−√5𝑡−11𝑥0.3+0.2sin3𝑡𝑦(𝑡)8+cos√11𝑡+10+sin√𝑥(𝑡)(𝑡),𝑦̇(𝑡)(𝑡)×√3𝑡+5𝑦=𝑦0.5+0.3sin√2𝑡−12+0.2sin13𝑡𝑦(𝑡)𝑥(𝑡).+2 (64) In this case, we have 𝑟^𝑢₁=11, 𝑟^𝑙₁=9, 𝑏^𝑢=𝑏^𝑙=11, 𝑐^𝑢=0.5, 𝑐^𝑙=0.1, 𝛼^𝑢=9, 𝛼^𝑙=7, 𝛽^𝑢=11, 𝛽^𝑙=9, 𝛾^𝑢=𝛾^𝑙=5, 𝑟^𝑢₂=0.8, 𝑟^𝑙₂=0.2, 𝑑^𝑢=12.2,𝑑^𝑙=11.8, 𝑘^𝑢=𝑘^𝑙=2. According to Theorem 3 (or Theorem 4), we have 𝑚₁(),𝑚≈0.81686or0.80909₂(),𝑀≈0.04618or0.04605₁=1,𝑀₂≈0.20339. (65)

Considering (𝐻₃) and (𝐻₄), we choose 𝑚₁=0.80909, 𝑚₂=0.04605. Hence, 𝑟^𝑙₁𝛼^𝑙−𝑟^𝑢₂𝑐^𝑢𝑀₁+𝑘^𝑢𝑑^𝑙𝑟≈62.8983>0,_𝑙1𝛾^𝑙−𝑐^𝑢𝑏(𝑡)−𝑐(𝑡)𝛽(𝑡)𝑀≈44.5>0,₂Δ²𝑡,𝑚₁,𝑚₂−𝑑(𝑡)𝑀₂𝑚₁+𝑘(𝑡)^2𝑙≥𝑏^𝑙−𝑐^𝑢𝛽^𝑢𝑀₂𝛼^𝑙+𝛽^𝑙𝑚₁+𝛾^𝑙𝑚₂²−𝑑^𝑢𝑀₂𝑚₁+𝑘^𝑙2𝑑(𝑡)≈10.6802>0,𝑀₁(𝑡)−𝑐(𝑡)+𝐾Δ𝑡,𝑚₁,𝑚₂−𝑐(𝑡)𝛾(𝑡)𝑀₂Δ²𝑡,𝑚₁,𝑚₂^𝑙≥𝑑^𝑙𝑀₁+𝑘^𝑢−𝑐^𝑢𝛼^𝑙+𝛽^𝑙𝑚₁+𝛾^𝑙𝑚₂−𝑐^𝑢𝛾^𝑢𝑀₂𝛼^𝑙+𝛽^𝑙𝑚₁+𝛾^𝑙𝑚₂²≈3.89646>0. (66) Equation (66) means that all conditions of Theorem 13 are satisfied in system (64). Thus, it admits a unique, globally attractive, positive, almost periodic solution. Figure 1 shows the dynamic behaviors of the solution (𝑥(𝑡),𝑦(𝑡))^𝑇 with the four group initial values (𝑥(0),𝑦(0))=(0.5,1)^𝑇, (1,1.2)^𝑇, (0,2,0.05)^𝑇, and (0.1,1.3)^𝑇. From the figure, we could easily see that the solution (𝑥(𝑡),𝑦(𝑡))^𝑇 is asymptotic to the unique, almost periodic solution of the system (64).

Figure 1 

Dynamic behavior of the system (64) with the initial condition (𝑥(0),𝑦(0))=(0.5,1)𝑇, (1,1.2)𝑇, (0,2,0.05)𝑇, and (0.1,1.3)𝑇, respectively.

5. Conclusion

In this paper, we consider a predator-prey with modified Leslie-Gower model and Beddington-DeAngelis functional response. When 𝛼(𝑡)=𝑘₁, 𝛽(𝑡)=1, 𝛾(𝑡)=0, 𝑘(𝑡)=𝑘₂, 𝑐(𝑡)=𝑎₁(𝑡), 𝑑(𝑡)=𝑎₂(𝑡), (8) we discussed reduces to (4) which was studied by Zhu and Wang [13]. By utilizing the coincidence degree theorem and constructing a suitable Lyapunov function, the authors in [13] investigated the existence and global attractivity of positive periodic solutions of (4) and obtained Theorem 1. More precisely, comparing Theorem 1 with Corollary 6, we find that conditions (𝐶₂) or (𝐶₃) of Theorem 1 are redundant, which implies that our results improve those of [13]. Example together with numeric simulation shows the feasibility of our main results.