Dynamic Behaviors of Holling Type II Predator-Prey System with Mutual Interference and Impulses

Abstract

A class of Holling type II predator-prey systems with mutual interference and impulses is presented. Sufficient conditions for the permanence, extinction, and global attractivity of system are obtained. The existence and uniqueness of positive periodic solution are also established. Numerical simulations are carried out to illustrate the theoretical results. Meanwhile, they indicate that dynamics of species are very sensitive with the period matching between species’ intrinsic disciplinarians and the perturbations from the variable environment. If the periods between individual growth and impulse perturbations match well, then the dynamics of species periodically change. If they mismatch each other, the dynamics differ from period to period until there is chaos.

1. Introduction

As pointed out by Berryman [1], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both biology and mathematical biology due its universal existence and importance. Though predator-prey models with periodic or almost periodic coefficients have been studied extensively (see [2–14] and the references cited therein), few papers consider the mutual interference between the predators and prey, which was introduced by Hassell in 1971. During his research of the capturing behavior between hosts and parasites, he found that the hosts or parasites had the tendency to leave each other when they met, which interfered with hosts capture effects. Obviously, the mutual interference will be stronger, while the size of parasite becomes larger. From the observation, Hassell introduced the concept of mutual interference constant 𝑚  (0<𝑚≤1) and established a Lotka-Volterra predator-prey model with mutual interference as follows: 𝑥̇(𝑡)=𝑥𝑔(𝑥)−𝜑(𝑥)𝑦^𝑚,𝑦̇(𝑡)=𝑦−𝑑+𝛽𝜑(𝑥)𝑦^𝑚−1−𝑞𝑦, (1) where 𝑥 and 𝑦 are the sizes of the prey and predator populations, respectively, and 𝑚 is mutual interference constant; see [15, 16] for more details on the biological meaning of mutual interference constant 𝑚. Some results have appeared for this type of predator-prey system [17–20].

Recently, much more attention has paid on the functional responses. Based on experiments, Holling [21] investigated three different kinds of functional responses for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more consistent with the real ecosystem. The Lotka-Volterra system with Holling type II functional response is introduced by Wang and Zhu [22] in the following form: 𝑥̇(𝑡)=𝑥(𝑡)𝑟₁(𝑡)−𝑏₁(𝑡)𝑥(𝑡)−𝑐₁(𝑡)𝑥(𝑡)(𝑡)𝑦𝑘+𝑥^𝑚(𝑡),𝑦̇(𝑡)=𝑦(𝑡)−𝑟₂(𝑡)−𝑏₂(𝑡)𝑦(𝑡+𝑐)²(𝑡)𝑥(𝑡)(𝑡)𝑦𝑘+𝑥^𝑚(𝑡). (2)

However, ecological system is often deeply perturbed by nature and human exploit activities (e.g., fire, drought, flooding deforestation, hunting, harvesting, breeding, etc.), which are not suitable to be considered continually. To accurately describe the system, impulsive differential equations may be a better candidate than ordinary differential equation. The theory of impulsive differential equations not only is now being recognized to be richer than the corresponding theory of differential equations without impulse, but also represents a more natural framework for mathematical modelling of many real-world phenomena. In recent years, many important and interesting results on the permanence, extinction, global attractivity of systems, the existence and uniqueness of positive periodic solutions, bifurcation and dynamical complexity, and so forth have been proposed in [23–28] and references cited therein. Motivated by above studies, in this paper we will consider the following Holling II predator-prey system with mutual interference and impulses: 𝑥̇(𝑡)=𝑥(𝑡)𝑟₁(𝑡)−𝑏₁(𝑡)𝑥(𝑡)−𝑐₁(𝑡)𝑥(𝑡)(𝑡)𝑦𝑘+𝑥^𝑚(𝑡),𝑦̇(𝑡)=𝑦(𝑡)−𝑟₂(𝑡)−𝑏₂(𝑡)𝑦(𝑡+𝑐)²(𝑡)𝑥(𝑡)(𝑡)𝑦𝑘+𝑥^𝑚(𝑡),𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=1+ℎ_𝑘𝑥𝜏_𝑘,𝑦𝜏⁺_𝑘=1+𝑔_𝑘𝑦𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…, (3) where 𝑥 and 𝑦 are the sizes of the prey and predator populations, respectively. 0<𝑚<1 and 𝑘 is a positive constant. 𝑟_𝑖(𝑡),  𝑏_𝑖(𝑡),  and  𝑐_𝑖(𝑡)  (𝑖=1,2) are continuous periodic functions defined on [0,+∞) with a common period 𝑇>0. 𝑏_𝑖(𝑡),  𝑐_𝑖(𝑡),  (𝑖=1,2) are strictly positive. Also 𝜏_𝑘→+∞  (𝑡→+∞), 0<𝜏₁<𝜏₂<𝜏₃<⋯<𝜏_𝑘<𝜏_𝑘+1<⋯. Assume that ℎ_𝑘,  𝑔_𝑘  (𝑘=1,2…) are constants. 1+ℎ_𝑘>0 and 1+𝑔_𝑘>0, and there exists an integer 𝑞>0 such that ℎ_𝑘+𝑞=ℎ_𝑘, 𝑔_𝑘+𝑞=𝑔_𝑘, 𝜏_𝑘+𝑞=𝜏_𝑘+𝑇. The interpretations of parameters can be seen in Table 1.

Table 1 

Parameter values used in the simulations of system (3).

The organization of this paper is as follows. In Section 2, we give some definitions and state some lemmas which are essential to study the main theorems. In Section 3, sufficient conditions for the permanence, extinction of system (3), and the existence, uniqueness, and global attractivity of positive periodic solution are obtained. In Section 4, numerical examples are presented to illustrate our results. In the last section, we give a brief discussion.

2. Preliminaries

In this section, we introduce some notations and definitions and state some lemmas which will be useful in the subsequent sections.

Let 𝑅₊=[0,∞), 𝑅²₊={𝑋∈𝑅²∣𝑋≥0}. The map 𝑓=(𝑓₁,𝑓₂)^𝑇 is defined by the right hand of the first and second equations of system (3). Let 𝑉∶𝑅₊×𝑅²₊→𝑅₊; then 𝑉 is said to belong to class 𝑉₀ if(1)𝑉is continuous in (𝜏_𝑘,𝜏_𝑘+1]×𝑅²₊ and for each 𝑋∈𝑅²₊, 𝑘=1,2,…, there exists lim_{(𝑡,𝑌)→(𝜏}⁺_𝑘,𝑋)𝑉(𝑡,𝑌)=𝑉(𝜏⁺_𝑘,𝑋);(2)𝑉 is locally Lipschitzian in 𝑋.

Definition 1. Let 𝑉∈𝑉₀; then, for (𝑡,𝑋)∈(𝜏_𝑘,𝜏_𝑘+1]×𝑅²₊, the upper right derivative of 𝑉(𝑡,𝑋) with respect to the impulsive differential system (3) is defined as 𝐷⁺𝑉(𝑡,𝑥)=lim_ℎ→0+1supℎ𝑉𝑡+ℎ,𝑋+ℎ𝑓(𝑡,𝑋)−𝑉(𝑡,𝑋). (4)

Definition 2. System (3) is permanent, if, for any positive solution (𝑥(𝑡),𝑦(𝑡))^𝑇 of system (3), there exist positive constants 𝑚₁,  𝑚₂,  𝑀₁, and 𝑀₂ such that 𝑚₁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀₁,𝑚₂≤liminf_𝑡→+∞𝑦(𝑡)≤limsup_𝑡→+∞𝑦(𝑡)≤𝑀₂. (5)

Lemma 3 ([29]). Let 𝑉∈𝑉₀ and 𝑋(𝑡) is a solution of (3) with initial value 𝑋(0⁺)=𝑋₀. Assume that 𝐷⁺𝑉(𝑡,𝑋(𝑡))≤𝑔(𝑡,𝑉(𝑡,𝑋(𝑡))),𝑡≠𝜏_𝑘𝑉𝑡,𝑘=1,2,…,𝑡,𝑋⁺≤𝜓_𝑘(𝑉(𝑡,𝑋(𝑡))),𝑡=𝜏_𝑘,𝑘=1,2,…, (6) where 𝑔∶𝑅₊×𝑅₊→𝑅 is continuous in (𝜏_𝑘,𝜏_𝑘+1]×𝑅₊ and for 𝑢∈𝑅₊, 𝑘=1,2,…, lim_{(𝑡,𝑣)→(𝜏}⁺_𝑘,𝑢)𝑔(𝑡,𝑣)=𝑔(𝜏⁺_𝑘,𝑢) exists and 𝜓_𝑘∶𝑅₊→𝑅₊ is nondecreasing. Let 𝑟(𝑡) be the maximal solution of scalar impulsive differential equation 𝑢̇(𝑡)=𝑔(𝑡,𝑢(𝑡)),𝑡≠𝜏_𝑘𝑢𝑡,𝑘=1,2,…,₊=𝜓_𝑘(𝑢(𝑡))≥0,𝑡=𝜏_𝑘,𝑡_𝑘𝑢0>0,𝑘=1,2,…,₊=𝑢₀ (7) existing on [0,∞). Then 𝑉(0⁺,𝑋₀)≤𝑢₀ implies that 𝑉(𝑡,𝑋(𝑡))≤𝑟(𝑡) for all 𝑡≥0.
Similarly, assume that inequality (6) is reversed. Let 𝑝(𝑡) be the minimal solution of (7) existing on [0,+∞). Then, 𝑉(0⁺,𝑋₀)≥𝑢₀ implies that 𝑉(𝑡,𝑋(𝑡))≥𝑝(𝑡) for all 𝑡≥0.

Let 𝑓(𝑡) be continuous 𝑇-periodic function defined on the 𝑅₊; we introduce the following notation: 𝑚𝑓(𝑡=1)𝑇^𝑇₀𝑓(𝑡)𝑑𝑡. (8)

Consider the following periodic logistic equation with impulses: 𝑥̇(𝑡)=𝑥(𝑡)𝑟₁(𝑡)−𝑏₁(𝑡)𝑥(𝑡),𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=1+ℎ_𝑘𝑥𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (9) We have the following results.

Lemma 4 (see [30]). (1) If ^𝑞_𝑘=1ln1+ℎ_𝑘+𝑇𝑚𝑟₁(𝑡)>0, A₁ then system (9) has a unique T-periodic solution 𝑥^∗(𝑡), which is globally asymptotically stable.
(2) If ^𝑞_𝑘=1ln1+ℎ_𝑘+𝑇𝑚𝑟₁(𝑡)≤0, A₂ then 𝑥^∗(𝑡)≡0.

Consider a periodic Logistic equation with impulses: 𝑥̇(𝑡)=𝑥(𝑡)𝑏(𝑡)−𝑎(𝑡)(𝑥(𝑡))^𝜃,𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=1+ℎ^′_𝑘𝑥𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…, (10) where 𝜃 is a positive constant, 𝑎(𝑡) and 𝑏(𝑡) are continuous 𝑇-periodic functions with 𝑎(𝑡)>0 and 𝑚[𝑏(𝑡)]>0, and  ℎ^′_𝑘+𝑞=ℎ^′_𝑘, 𝜏_𝑘+𝑞=𝜏_𝑘+𝑇. We have the following results.

Lemma 5 (see [31]). (1) If ^𝑞_𝑘=1ln1+ℎ^′_𝑘+𝑇𝑚[𝑏]>0, (11) then system (10) has a unique 𝑇-periodic solution 𝑥^∗(𝑡), which is globally asymptotically stable.
(2) If ^𝑞_𝑘=1ln1+ℎ^′_𝑘+𝑇𝑚[𝑏(𝑡)]<0, (12) then 𝑥^∗(𝑡)≡0.

Lemma 6 (see [31]). Let 𝑥^∗(𝑡) be the unique 𝑇-periodic positive solution of system (9) and 𝑥^∗_𝜀(𝑡) the unique 𝑇-periodic positive solution of the following system: 𝑥̇(𝑡)=𝑥(𝑡)𝑏_𝜀(𝑡)−𝑎(𝑡)(𝑥(𝑡))^𝜃,𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=1+ℎ^′_𝑘𝑥𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…, (13) where 𝑏_𝜀(𝑡) is the continuous T-periodic function with 𝑚[𝑏_𝜀(𝑡)]>0 and lim_𝜀→0𝑏_𝜀(𝑡)=𝑏(𝑡). Then lim_𝜀→0𝑥^∗_𝜀(𝑡)−𝑥^∗(𝑡)=0. (14)

Furthermore, we consider the following periodic logistic equation with impulses: 𝑣̇(𝑡)=𝑣(𝑡)𝑟₂(𝑡)−𝑐₂(𝑡)𝑥^∗(𝑡)𝑘+𝑥^∗(𝑡)𝑣^1−𝑚(𝑡),𝑡≠𝜏_𝑘,𝑣𝜏⁺_𝑘=11+𝑔_𝑘𝑣𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (15) By Lemma 5, we have the following Corollary.

Corollary 7. If −^𝑞_𝑘=1ln1+𝑔_𝑘+𝑇𝑚𝑟₂(𝑡)>0, H₁ then system (15) has a unique T-periodic solution 𝑣^∗(𝑡), which is globally asymptotically stable.

Next, one considers the following periodic logistic equation with impulses: 𝑥̇(𝑡)=𝑥(𝑡)𝑟₁(𝑡)−𝑐₁(𝑡)𝑘𝑦^∗(𝑡)^𝑚−𝑏₁(𝑡)𝑥(𝑡),𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=1+ℎ_𝑘𝑥𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (16)

By Lemma 5, we have the following Corollary.

Corollary 8. If ^𝑞_𝑘=1ln1+ℎ_𝑘+𝑇𝑚𝑟₁(𝑡)−𝑐₁(𝑡)𝑘𝑦^∗(𝑡)^𝑚>0, H₂ then system (16) has a unique T-periodic solution 𝑥_∗(𝑡), which is globally asymptotically stable.

3. Main Results

In this section, we will study the permanence, extinction, and global attractivity of system (3) and the existence and uniqueness of positive periodic solution.

Lemma 9. For any solution (𝑥(𝑡),𝑦(𝑡))^𝑇 of system (3), if 𝑥(0⁺)>0, 𝑦(0⁺)>0, then 𝑥(𝑡)>0 and 𝑦(𝑡)>0 for all 𝑡≥0.

Proof. From the first equation of (3), one has 𝑥̇(𝑡)=𝑝(𝑡)𝑥(𝑡),𝑡≠𝜏_𝑘, (17) where 𝑝(𝑡)=𝑟₁(𝑡)−𝑏₁(𝑡)𝑥(𝑡)−𝑐₁(𝑡)(𝑡)𝑦𝑘+𝑥^𝑚(𝑡); (18) thus, from 𝑥(0⁺)>0, we have 𝑥(𝑡)=_{0<𝜏𝑘<𝑡}1+ℎ_𝑘𝑥0⁺exp^𝑡₀𝑝(𝑠)𝑑𝑠>0. (19) Similarly, it follows from the second equation of system (3) that 𝑦(𝑡)>0. This completes the proof of Lemma 9.

Lemma 10. Assuming that A₁ and H₁ hold, for any positive solution (𝑥(𝑡),𝑦(𝑡))^𝑇 of system (3), there exist constants 𝑀_𝑖  (𝑖=1,2) such that limsup_𝑡→+∞𝑥(𝑡)≤𝑀₁,limsup_𝑡→+∞𝑦(𝑡)≤𝑀₂. (20)

Proof. First, we show that 𝑥(𝑡) is ultimate upper bounded. It follows from (3) that 𝑥̇(𝑡)≤𝑥(𝑡)𝑟₁(𝑡)−𝑏₁(𝑡)𝑥(𝑡),𝑡≠𝜏_𝑘,𝑥𝜏⁺_𝑘=1+ℎ_𝑘𝑥𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (21) Consider the following auxiliary system: 𝑧̇(𝑡)=𝑧(𝑡)𝑟₁(𝑡)−𝑏₁(𝑡)𝑧(𝑡),𝑡≠𝜏_𝑘,𝑧𝜏⁺_𝑘=1+ℎ_𝑘𝑧𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (22) By Lemma 3, we have 𝑥(𝑡)≤𝑧(𝑡), where 𝑧(𝑡) is the solution of (22) with 𝑧(0⁺)=𝑥(0⁺). By A₁, from Lemma 4 we get that system (22) has a unique 𝑇-periodic solution 𝑥^∗(𝑡), which is globally stable. Therefore, for any 𝜀>0 being small enough, there exists a 𝑇₁>0 such that 𝑧(𝑡)−𝑥^∗(𝑡)<𝜀, (23) for any 𝑡>𝑇₁. Let 𝑀₁=sup{𝑥^∗(𝑡)∶𝑡∈[0,𝑇]}; we have 𝑥(𝑡)≤𝑧(𝑡)<𝑥^∗(𝑡)+𝜀≤𝑀₁+𝜀 (24) for any 𝑡>𝑇₁. Let 𝜀→0; we obtain limsup_𝑡→+∞𝑥(𝑡)≤𝑀₁. (25)
Next, we prove that 𝑦(𝑡) is ultimately upper bounded. Define 𝑦(𝑡)=1/𝑢(𝑡). From system (3), we have 𝑢̇(𝑡)=𝑢(𝑡)𝑟₂(𝑡)+𝑏₂(𝑡)1𝑢(𝑡)−𝑐₂(𝑡)𝑥(𝑡)(𝑡)𝑢𝑘+𝑥^1−𝑚(𝑡),𝑡≠𝜏_𝑘,𝑢𝜏⁺_𝑘=11+𝑔_𝑘𝑢𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (26) For any 𝑡>𝑇₁ and 𝑡≠𝜏_𝑘(𝑘=1,2,…), we have 𝑢̇(𝑡)≥𝑢(𝑡)𝑟₂(𝑡)−𝑐₂(𝑡)𝑥(𝑡)(𝑡)𝑢𝑘+𝑥^1−𝑚(𝑡)≥𝑢(𝑡)𝑟₂(𝑡)−𝑐₂(𝑡)𝑥^∗(𝑡)+𝜀𝑘+𝑥^∗(𝑡)𝑢+𝜀^1−𝑚(𝑡). (27) Consider the following auxiliary system: 𝜓̇(𝑡)=𝜓(𝑡)𝑟₂(𝑡)−𝑐₂(𝑡)𝑥^∗(𝑡)+𝜀𝑘+𝑥^∗(𝑡)𝜓+𝜀^1−𝑚(𝑡),𝑡≠𝜏_𝑘,𝜓𝜏⁺_𝑘=11+𝑔_𝑘𝜓𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (28) By Lemma 3, for any 𝑡>𝑇₁, we have 𝑢(𝑡)≥𝜓(𝑡), where 𝜓(𝑡) is the solution of (28) with 𝜓(𝑇⁺)=𝑢(𝑇⁺). By H₁, from Lemma 5, system (28) has a unique 𝑇-periodic solution 𝑣^∗_𝜀(𝑡), which is globally stable. Therefore, letting 𝜀→0, from Lemma 6, we have 𝑣^∗_𝜀(𝑡)→𝑣^∗(𝑡). Letting 𝑀^∗₂=inf{𝑣^∗(𝑡)∶𝑡∈[0,𝑇]}, for any 𝜀₀>0 being small enough, there exists 𝑇₂>𝑇₁, and we have 𝑢(𝑡)=1𝑦(𝑡)≥𝜓(𝑡)>𝑣^∗(𝑡)−𝜀₀≥𝑀^∗₂−𝜀₀ (29) for any 𝑡>𝑇₂. Let 𝜀₀→0; we have limsup_𝑡→+∞𝑦(𝑡)≤1𝑀^∗₂≜𝑀₂. (30) This completes the proof of Lemma 10.

Theorem 11. Assume that H₁ and H₂ hold, and ^𝑞_𝑘=1ln1+𝑔_𝑘+𝑇𝑚−𝑟₂(𝑡)+𝑐₂(𝑡)𝑥_∗(𝑡)𝑘+𝑥_∗(𝑡)𝑦^∗(𝑡)^𝑚−1>0,𝑚𝑟₁(𝑡)−𝑐₁(𝑡)𝑘𝑦^∗(𝑡)^𝑚>0,𝑚−𝑟₂(𝑡)+𝑐₂(𝑡)𝑥_∗(𝑡)𝑘+𝑥_∗(𝑡)𝑦^∗(𝑡)^𝑚−1>0, H₃H₄H₅ where 𝑥_∗(𝑡) is the unique T-periodic solution of system (16) and 𝑦^∗(𝑡)≜1/𝑣^∗(𝑡). Then species 𝑥 and 𝑦 are permanent.

Proof. From Lemma 9, there exist constants 𝑀_𝑖(𝑖=1,2) such that limsup_𝑡→+∞𝑥(𝑡)≤𝑀₁,limsup_𝑡→+∞𝑦(𝑡)≤𝑀₂,𝑖=1,2. (31) From the proof of Lemma 10, for above 𝜀>0 being small enough, there exists 𝑇₂>0 such that 𝑦(𝑡)≤1𝑣^∗(𝑡)≜𝑦−𝜀^∗𝜀(𝑡) (32) for any 𝑡>𝑇₂. From the first equation of system (3), for 𝑡>𝑇₂ and 𝑡≠𝜏_𝑘, we have 𝑥̇(𝑡)≥𝑥(𝑡)𝑟₁(𝑡)−𝑐₁(𝑡)(𝑡)𝑦𝑘+𝑥^∗𝜀(𝑡)^𝑚−𝑏₁(𝑡)𝑥(𝑡),≥𝑥(𝑡)𝑟₁(𝑡)−𝑐₁(𝑡)𝑘𝑦^∗_𝜀(𝑡)^𝑚−𝑏₁(𝑡)𝑥(𝑡). (33) Consider the following auxiliary system: 𝜙̇(𝑡)=𝜙(𝑡)𝑟₁(𝑡)−𝑐₁(𝑡)𝑘𝑦^∗_𝜀(𝑡)^𝑚−𝑏₁(𝑡)𝜙(𝑡),𝑡≠𝜏_𝑘,𝜙𝜏⁺_𝑘=1+ℎ_𝑘𝜙𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…, (34) where 𝜙(𝑡) is the solution of (34) with 𝜙(𝑇⁺₂)=𝑥(𝑇⁺₂). By Lemma 3, for any 𝑡>𝑇₂, we have 𝑥(𝑡)≥𝜙(𝑡). From condition H₂, we have ^𝑞_𝑘=1ln1+ℎ_𝑘+𝑇𝑚𝑟₁(𝑡)−𝑐₁(𝑡)𝑘𝑦^∗_𝜀(𝑡)^𝑚>0. (35) From inequality (35) and Lemma 4, system (34) has a unique 𝑇-periodic solution 𝑥_∗𝜀(𝑡), which is globally stable. Since condition H₄ holds, for the above 𝜀>0 being small enough, we have 𝑚[𝑟₁(𝑡)−(𝑐₁(𝑡)/𝑘)(𝑦^∗_𝜀(𝑡))^𝑚]>0; letting 𝜀→0, from Lemma 6, we have 𝑥_∗𝜀(𝑡)→𝑥_∗(𝑡). Letting 𝑚₁=inf{𝑥_∗(𝑡)∶𝑡∈[0,𝑇]}, for the above 𝜀 being small enough, there exists 𝑇₃>𝑇₂ such that 𝑥(𝑡)≥𝜙(𝑡)≥𝑥_∗(𝑡)−𝜀>𝑚₁−𝜀, for any 𝑡>𝑇₃. Let 𝜀→0; we have lim_𝑡→+∞inf𝑥(𝑡)≥𝑚₁. From system (3), for 𝑡>𝑇₃ and 𝑡≠𝜏_𝑘, we have 𝑦̇(𝑡)≥𝑦(𝑡)−𝑟₂(𝑡)−𝑏₂(𝑡)𝑦(𝑡)+𝑐₂(𝑡)𝑥_∗(𝑡)−𝜀𝑘+𝑥_∗(𝑡)𝑦−𝜀^∗𝜀(𝑡)^𝑚−1. (36) Consider the following auxiliary system: 𝑤̇(𝑡)=𝑤(𝑡)−𝑟₂(𝑡)−𝑏₂(𝑡)𝑤(𝑡)+𝑐₂(𝑡)𝑥_∗(𝑡)−𝜀𝑘+𝑥_∗(𝑡)𝑦−𝜀^∗𝜀(𝑡)^𝑚−1,𝑡≠𝜏_𝑘,𝑤𝜏⁺_𝑘=1+𝑔_𝑘𝑤𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…, (37) where 𝑤(𝑡) is the solution of (37) with 𝑤(𝑇⁺₃)=𝑦(𝑇⁺₃). By Lemma 3, for all 𝑡>𝑇₃, we have 𝑦(𝑡)≥𝑤(𝑡), for the above 𝜀>0 being small enough. By condition H₃, we obtain ^𝑞_𝑘=1ln1+𝑔_𝑘+𝑇𝑚−𝑟₂+𝑐₂(𝑡)𝑥_∗(𝑡)−𝜀𝑘+𝑥_∗(𝑡)𝑦−𝜀^∗𝜀(𝑡)^𝑚−1>0. (38) From inequality (38), Lemma 4, system (37) has a unique 𝑇-periodic solution 𝑦_∗𝜀(𝑡), which is globally asymptotically stable. Since condition H₅ holds, for the above 𝜀>0 being small enough, such that 𝑚[−𝑟₂(𝑡)+((𝑐₂(𝑡)(𝑥_∗(𝑡)−𝜀))/(𝑘+(𝑥_∗(𝑡)−𝜀)))(𝑦^∗_𝜀(𝑡))^𝑚−1]>0, let 𝜀→0, and from Lemma 6, we have 𝑦_∗𝜀(𝑡)→𝑦_∗(𝑡), where 𝑦_∗(𝑡) is the unique 𝑇-periodic globally asymptotically stable solution of the following system: 𝑝̇(𝑡)=𝑝(𝑡)−𝑟₂(𝑡)−𝑏₂(𝑡)𝑝(𝑡)+𝑐₂(𝑡)𝑥_∗(𝑡)𝑘+𝑥_∗(𝑡)𝑦^∗(𝑡)^𝑚−1(𝑡),𝑡≠𝜏_𝑘,𝑝𝜏⁺_𝑘=1+𝑔_𝑘𝑝𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (39) Letting 𝑚₂=inf{𝑦_∗(𝑡)∶𝑡∈[0,𝑇]}, for the above 𝜀>0 being small enough, there exists 𝑇₄>𝑇₃ such that 𝑦(𝑡)≥𝑤(𝑡)>𝑦_∗(𝑡)−𝜀, for any 𝑡>𝑇₄. Let 𝜀→0; we have lim_𝑡→+∞𝑦(𝑡)≥𝑚₂.

Theorem 12. Assume that ^𝑞_𝑘=1ln1+𝑔_𝑘+𝑇𝑚−𝑟₂(𝑡)−𝑐₂(𝑡)𝑥^∗(𝑡)𝑘+𝑥^∗(𝑡)𝑦_∗(𝑡)^𝑚−1<0,_𝑞𝑘=1ln1+ℎ_𝑘+𝑇𝑚𝑟₁(𝑡)>0. H₆H₇ Then, for any positive solution (𝑥(𝑡),𝑦(𝑡))^𝑇 of system (3), one has lim_𝑡→+∞𝑦(𝑡)=0,lim_𝑡→+∞𝑥(𝑡)−𝑥^∗(𝑡)=0, (40) where 𝑥^∗(𝑡) and 𝑦_∗(𝑡) are unique T-periodic solutions of systems (9) and (39), respectively.

Proof. From system (3), for 𝑡>𝑇₄ and 𝑡≠𝜏_𝑘, we have 𝑦̇(𝑡)≤𝑦(𝑡)−𝑟₂(𝑡)−𝑏₂(𝑡)𝑦(𝑡)+𝑐₂(𝑡)𝑥^∗(𝑡)𝑘+𝑥^∗(𝑡)𝑦_∗(𝑡)^𝑚−1. (41) Consider the following auxiliary system: 𝜌̇(𝑡)=𝑦̇(𝑡)≤𝜌(𝑡)−𝑟₂(𝑡)−𝑏₂(𝑡)𝜌(𝑡)+𝑐₂(𝑡)𝑥^∗(𝑡)𝑘+𝑥^∗(𝑡)𝜌_∗(𝑡)^𝑚−1,𝑡≠𝜏_𝑘,𝜌𝜏⁺_𝑘=1+𝑔_𝑘𝜌𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (42) From H₆ and Lemma 4, we have lim_𝑡→+∞𝑦(𝑡)=0. (43) Therefore, for 𝜀>0 being small enough, there exists 𝑇₅>0 such that 𝑦(𝑡)<𝜀 for any 𝑡>𝑇₅. Since H₇ holds, we obtain ^𝑞_𝑘=1ln1+ℎ_𝑘+𝑇𝑚𝑟₁(𝑡)−𝑐₁(𝑡)𝜀^𝑚>0. (44) Thus, from the first equation of system (3), we have 𝑥(𝑡)𝑟₁(𝑡)−𝑏₁(𝑡)𝑥(𝑡)−𝑐₁(𝑡)𝜀^𝑚≤𝑥̇(𝑡)≤𝑥(𝑡)𝑟₁(𝑡)−𝑏₁(𝑡)𝑥(𝑡) (45) for any 𝑡>𝑇₅. Consider the following auxiliary system 𝜑̇(𝑡)=𝜑(𝑡)𝑟₁(𝑡)−𝑏₁(𝑡)𝜑(𝑡)−𝑐₁(𝑡)𝜀^𝑚,𝑡≠𝑡_𝑘,𝜑𝜏⁺_𝑘=1+ℎ_𝑘𝜑𝜏_𝑘,𝑡=𝜏_𝑘𝜎̇(𝑡)(𝑡)𝑟,𝑘=1,2,…,=𝜎₁(𝑡)−𝑏₁(𝑡)𝜎(𝑡),𝑡≠𝑡_𝑘,𝜎𝜏⁺_𝑘=1+ℎ_𝑘𝜎𝜏_𝑘,𝑡=𝜏_𝑘,𝑘=1,2,…. (46) From H₇ and Lemma 4, system (46) has unique 𝑇-periodic solutions 𝜑^∗_𝜀(𝑡) and 𝜎^∗(𝑡), respectively, which are globally asymptotically stable. By Lemma 3, for any 𝜀₁>0 being small enough, there exists 𝑇₆>0 such that 𝜑^∗_𝜀(𝑡)−𝜀₁≤𝑥(𝑡)≤𝜎^∗(𝑡)+𝜀₁ for any 𝑡>𝑇₆; letting 𝜀→0, we have 𝑥^∗(𝑡)−𝜀₁≤𝑥(𝑡)≤𝑥^∗(𝑡)+𝜀₁; since 𝜀₁ is arbitrarily small, we have lim_𝑡→+∞|𝑥(𝑡)−𝑥^∗(𝑡)|=0.

Theorem 13. Suppose that H₆ holds and ^𝑞_𝑘=1ln1+ℎ_𝑘+𝑇𝑚𝑟₁(𝑡)<0. H₈ Then species 𝑥 and 𝑦 are extinct; that is, lim_𝑡→∞𝑥(𝑡)=0,lim_𝑡→∞𝑦(𝑡)=0. (47)

Proof. From H₈ and Lemma 4, system (9) has a unique 𝑇-periodic solution 𝑥^∗(𝑡), and 𝑥^∗(𝑡)=0. By Lemma 3, we have lim_𝑡→∞𝑥(𝑡)≡0; therefore, lim_𝑡→∞𝑦(𝑡)=0.

Theorem 14. Suppose that all the conditions of Theorem 11 hold and there exist positive constants 𝛼 and 𝛿 such that min_{𝑡∈[0,𝑇]}𝛼𝑏₁(𝑡)−𝛼𝑐₁(𝑡)𝑀₂^𝑚𝑘+𝑚₁²−𝛿𝑐₂(𝑡)𝑘𝑚₂^𝑚−1𝑘+𝑚₁²>0,min_{𝑡∈[0,𝑇]}𝛿𝑏₂(𝑡)−𝛼𝑐₁(𝑡)𝑚𝑚₂^𝑚−1𝑘+𝑚₁−𝛿𝑐₂(𝑡)𝑀₁(𝑚−1)𝑚₂^𝑚−2𝑘+𝑚₁>0. H₉H₁₀ Then system (3) has a unique positive T-periodic solution which is globally asymptotically stable.

Proof. Let (𝑥(𝑡),𝑦(𝑡))^𝑇 and (𝑥(𝑡),𝑦(𝑡))^𝑇 be any two positive solutions of system (3); for any sufficiently small 𝜀>0, from Theorem 11, there is a large enough 𝑇₄>0 such that for any 𝑡>𝑇₄, we have 𝑚₁−𝜀≤𝑥(𝑡),𝑥(𝑡)≤𝑀₁𝑚+𝜀,₂−𝜀≤𝑦(𝑡),𝑦(𝑡)≤𝑀₂+𝜀. (48) Define the Lyapunov function as follows: 𝑉(𝑡)=𝛼|ln𝑥(𝑡)−ln𝑥(𝑡)|+𝛿ln𝑦(𝑡)−ln𝑦(𝑡). (49) By the mean value theorem and (48), for any 𝑡∈(𝜏_𝑘,𝜏_𝑘+1], 𝑘=𝑝,𝑝+1,…, and 𝜏_𝑝>𝑇₄, we have 1𝑀₁|𝑥(𝑡)−+𝜀𝑥(𝑡)|≤|ln𝑥(𝑡)−ln𝑥(𝑡)|≤|𝑥(𝑡)−𝑥(𝑡,1)|𝑀₂𝑦(𝑡)−+𝜀𝑦(𝑡)≤ln𝑦(𝑡)−ln𝑦(𝑡)≤𝑦(𝑡)−𝑦(𝑡,1)𝑘+𝑥(𝑡)−1(𝑡)=1𝑘+𝑥𝑘+𝜉₁₁(𝑡)²(𝑥(𝑡)(𝑡,𝑥(𝑡)−𝑥))(𝑡)−𝑘+𝑥𝑥(𝑡)𝑘+𝑥(𝑡)=𝑘𝑘+𝜉₁₂(𝑡)²(𝑥(𝑡)−𝑥(𝑡)),𝑦(𝑡)^𝑚−𝑦(𝑡)^𝑚=𝑚𝜉₂₁(𝑡)^𝑚−1𝑦(𝑡)(𝑡),𝑦(𝑡)−𝑦_𝑚−1−𝑦(𝑡)^𝑚−1=(𝑚−1)𝜉₂₂(𝑡)^𝑚−2𝑦(𝑡)−𝑦(𝑡), (50) where 𝜉_1𝑖 is suited between 𝑥(𝑡) and 𝑥(𝑡), 𝜉_2𝑖 is suited between 𝑦(𝑡) and 𝑦(𝑡), and  𝑖=1,2. For 𝑡>𝑇₄,  𝑡≠𝜏_𝑘  (𝑘=1,2,…) and the above small enough 𝜀>0; from conditions H₉ and H₁₀, there exist positive constants 𝜂 and 𝜃 such that min_{𝑡∈[0,𝑇]}𝛼𝑏₁(𝑡)−𝛼𝑐₁(𝑡)𝑀₂+𝜀^𝑚𝑘+𝑚₁−𝜀²−𝛿𝑐₂(𝑡)𝑘𝑚₂−𝜀^𝑚−1𝑘+𝑚₁−𝜀²>𝜂,min_{𝑡∈[0,𝑇]}𝛿𝑏₂(𝑡)−𝛼𝑐₁(𝑡)(𝑚)𝑚₂−𝜀^𝑚−1𝑘+𝑚₁−−𝜀𝛿𝑐₂(𝑡)𝑀₁+𝜀(𝑚−1)𝑚₂−𝜀^𝑚−2𝑘+𝑚₁𝑚+𝜀>𝜂,𝜂min¹−𝜀𝛼,𝑚₂−𝜀𝛿>𝜃. (51) Calculating the upper right derivative of 𝑉(𝑡), we obtain 𝐷⁺𝑉(𝑡)=𝛼sgn(𝑥(𝑡)−𝑥(𝑡𝑥̇(𝑡)))𝑥(𝑡)−𝑥̇(𝑡)𝑥(𝑡)+𝛿sgn𝑦(𝑡)−𝑦(𝑡)𝑦̇(𝑡)𝑦₁(𝑡)−𝑦̇(𝑡)𝑦(𝑡)=𝛼sgn(𝑥(𝑡)−𝑥(𝑡×))−𝑏₁(𝑡)(𝑥(𝑡)−𝑥(𝑡))−𝑐₁(𝑡)𝑦(𝑡)^𝑚(𝑡)−𝑘+𝑥𝑦(𝑡)^𝑚𝑘+𝑥(𝑡)+𝛿sgn𝑦(𝑡)−𝑦(𝑡)×−𝑏₂(𝑡)𝑦(𝑡)−𝑦(𝑡)−𝑐₂(𝑡)𝑥(𝑡)𝑦(𝑡)^𝑚−1(𝑡)−𝑘+𝑥𝑥(𝑡)𝑦(𝑡)^𝑚−1𝑘+𝑥(𝑡)=−𝛼𝑏₁(𝑡)|𝑥(𝑡)−𝑥(𝑡)|−𝛿𝑏₂(𝑡)𝑦(𝑡)−𝑦(𝑡)+𝛼𝑐₁(𝑡)sgn(𝑥(𝑡)−𝑥(𝑡×))𝑦(𝑡)^𝑚𝑘+𝑥(𝑡)−𝑦(𝑡)^𝑚𝑘+𝑥(𝑡)+𝛿𝑐₂(𝑡)sgn𝑦(𝑡)−𝑦(𝑡)×𝑥(𝑡)𝑦(𝑡)^𝑚−1(𝑡)−𝑘+𝑥𝑥(𝑡)𝑦(𝑡)^𝑚−1𝑘+𝑥(𝑡). (52) In view of 𝑐₁(𝑡)sgn(𝑥(𝑡)−𝑥(𝑡))𝑦(𝑡)^𝑚𝑘+𝑥(𝑡)−𝑦(𝑡)^𝑚𝑘+𝑥(𝑡)=𝑐₁(𝑡)sgn(𝑥(𝑡)−𝑥(𝑡×))𝑦(𝑡)^𝑚𝑘+𝑥(𝑡)−𝑦(𝑡)^𝑚(𝑡)+𝑘+𝑥𝑦(𝑡)^𝑚(𝑡)+𝑦(𝑡)𝑘+𝑥𝑘+𝑥(𝑡^𝑚)=𝑐₁(𝑡)sgn(𝑥(𝑡)−𝑥(𝑡))𝑦(𝑡)^𝑚×1𝑘+𝜉₁₁²(𝑥(𝑡)(𝑡+𝑚𝜉−𝑥))₂₁(𝑡)^𝑚−1𝑘+𝑥(𝑡)𝑦(𝑡)−𝑦(𝑡),𝑐₂(𝑡)sgn𝑦(𝑡)−𝑦(𝑡)𝑥(𝑡)𝑦(𝑡)^𝑚−1(𝑡)−𝑘+𝑥𝑥(𝑡)𝑦(𝑡)^𝑚−1𝑘+𝑥(𝑡)=𝑐₂(𝑡)sgn𝑦(𝑡)−𝑦(𝑡)×𝑥(𝑡)𝑦(𝑡)^𝑚−1𝑘+𝑥₁(𝑡)−𝑥(𝑡)𝑦(𝑡)^𝑚−1(𝑡)+𝑥(𝑡𝑘+𝑥)𝑦(𝑡)^𝑚−1(𝑡)−𝑘+𝑥𝑥(𝑡)𝑦(𝑡)^𝑚−1𝑘+𝑥(𝑡)=𝑐₂(𝑡)sgn𝑦(𝑡)−𝑦(𝑡)×𝑥(𝑡)(𝑚−1)𝜉₂₂(𝑡)^𝑚−2(𝑡)𝑦(𝑡)−𝑘+𝑥𝑦(𝑡)+𝑘𝑘+𝜉₁₂(𝑡)²𝑦(𝑡)^𝑚−1(𝑥(𝑡)−𝑥(𝑡)), (53)(54) substituting (53) and (54) into (52), we have 𝐷⁺𝑉(𝑡)≤−𝛼𝑏₁(𝑡)−𝛼𝑐₁(𝑡)𝑦(𝑡)^𝑚𝑘+𝜉₁₁²(𝑡)−𝛿𝑐₂(𝑡)𝑘𝑦(𝑡)^𝑚−1𝑘+𝜉²₁₂×|𝑥(𝑡)−𝑥(𝑡−)|𝛿𝑏₂(𝑡)−𝛼𝑐₁(𝑡)𝑦(𝑡)^𝑚𝑚𝜉^𝑚−1₂₁(𝑡)−𝑘+𝑥𝛿𝑐₂(𝑡)𝑥(𝑡)(𝑚−1)𝜉₂₂^𝑚−2(𝑡)×𝑦(𝑡)−𝑘+𝑥𝑦(𝑡)≤−𝛼𝑏₁(𝑡)−𝛼𝑐₁(𝑡)𝑀₂+𝜀^𝑚𝑘+𝑚₁−𝜀²−𝛿𝑐₂(𝑡)𝑘𝑚₂−𝜀^𝑚−1𝑘+𝑚₁−𝜀²|𝑥(𝑡)−𝑥(𝑡−)|𝛿𝑏₂(𝑡)−𝛼𝑐₁(𝑡)𝑀₂+𝜀^𝑚𝑚𝑚₂−𝜀^𝑚−1𝑘+𝑚₁−−𝜀𝛿𝑐₂(𝑡)𝑀₁+𝜀(𝑚−1)𝑚₂−𝜀^𝑚−2𝑘+𝑚₁×𝑦(𝑡)−−𝜀𝑦(𝑡𝑚)≤−𝜂¹−𝜀𝛼𝛼|ln𝑥(𝑡)−ln𝑥(𝑡+𝑚)|₂−𝜀𝛿𝛿ln𝑦(𝑡)−ln𝑦(𝑡). (55) For all 𝑡∈𝑅₊ and 𝑡=𝜏⁺_𝑘  (𝑘=1,2,…), we have 𝑉̇𝜏⁺_𝑘=𝛼ln𝑥𝜏⁺_𝑘−ln𝑥𝜏⁺_𝑘+𝛿ln𝑦𝜏⁺_𝑘−ln𝑦𝜏⁺_𝑘=𝛼ln1+ℎ_𝑘𝑥𝜏_𝑘−ln1+ℎ_𝑘𝑥𝜏_𝑘+𝛿ln1+𝑔_𝑘𝑦𝜏_𝑘−ln1+𝑔_𝑘𝑦𝜏_𝑘=𝑉𝜏_𝑘. (56) Hence, 𝑉(𝑡) is continuous for all 𝑡>𝜏_𝑝>𝑇₄. The above analysis shows that, for all 𝑡>𝜏_𝑝>𝑇₄, 𝐷⁺𝑉(𝑡)≤−𝜃𝑉(𝑡), (57) and we further have 𝑉(𝑡)≤𝑉𝜏_𝑝exp−𝜃𝑡−𝜏_𝑝. (58) It is obvious that 𝑉(𝑡)→0 as 𝑡→+∞; that is, lim_𝑡→+∞|ln𝑥(𝑡)−ln𝑥(𝑡)|=0,lim_𝑡→+∞ln𝑦(𝑡)−ln𝑦(𝑡)=0. (59) We further have lim_𝑡→+∞|𝑥(𝑡)−𝑥(𝑡)|=0,lim_𝑡→+∞𝑦(𝑡)−𝑦(𝑡)=0. (60) Choose constants 𝐴=min{𝑚₁,𝑚₂} and 𝐵=max{𝑀₁,𝑀₂}. By Theorem 11, we have 𝐴≤𝑥(𝑡),𝑦(𝑡),𝑥(𝑡),𝑦(𝑡)≤𝐵,∀𝑡≥𝑇₄. (61) Now let us consider the (𝑥(𝑚𝑇,𝑧₀),𝑦(𝑚𝑇,𝑧₀))=𝑧(𝑚𝑇,𝑧₀), 𝑚=1,2,…, and 𝑧₀=(𝑥(0),𝑦(0)). It is compact in the domain [𝐴,𝐵]², since 𝐴≤𝑥(𝑡) and 𝑦(𝑡)≤𝐵 for all 𝑡≥𝑇₄. Let 𝑧 be a limit point of this sequence 𝑧=lim_𝑛→∞𝑧(𝑚_𝑛𝑇,𝑧₀); then we have 𝑧(𝑇,𝑧)=𝑧. In fact, since 𝑧(𝑇,𝑧(𝑚_𝑛𝑇,𝑧₀))=𝑧(𝑚_𝑛𝑇,𝑧(𝑇,𝑧₀)) and 𝑧(𝑚_𝑛𝑇,𝑧(𝑇,𝑧₀))−𝑧(𝑚_𝑛𝑇,𝑧₀)→0 as 𝑚_𝑛→∞, we have ‖𝑧(𝑇,𝑧)−𝑧‖_[𝐴,𝐵]²≤𝑧(𝑇,𝑧)−𝑧𝑇,𝑧𝑚_𝑛𝑇,𝑧_{0[𝐴,𝐵]2}+𝑧𝑇,𝑧𝑚_𝑛𝑇,𝑧₀−𝑧𝑚_𝑛𝑇,𝑧_{0[𝐴,𝐵]2}+𝑧𝑚_𝑛𝑇,𝑧₀−𝑧_[𝐴,𝐵]2⟶0. (62) The sequence 𝑧(𝑚𝑇,𝑧₀)(𝑚=1,2,…) has a unique limit point; otherwise, let the sequence have two limit points 𝑧=lim_𝑛→∞𝑧(𝑚_𝑛𝑇,𝑧₀) and 𝑧=lim_𝑛→∞𝑧(𝑚_𝑛𝑇,𝑧₀). Then, taking into account (60) and 𝑧=𝑧(𝑚_𝑛𝑇,𝑧), we have ‖𝑧−𝑧‖_[𝐴,𝐵]²≤𝑧−𝑍𝑚_𝑛𝑇,𝑧_{0[𝐴,𝐵]2}+𝑍𝑚_𝑛𝑇,𝑧₀−𝑧_[𝐴,𝐵]2⟶0,𝑛⟶∞, (63) and hence 𝑧=𝑧; the solution (𝑥(𝑡,𝑥₀),𝑦(𝑡,𝑦₀)) is the unique periodic solution of system (3). By (60), it is globally asymptotically stable. This completes the proof of Theorem 14.

4. Numerical Simulations

In system (3), we take 𝑇=1, 𝑞=1, which means that system (3) is a one-period system with one impulse perturbation during one period.

In case 1, H₁–H₅ are, respectively, equivalent to the following inequalities: −^𝑞_𝑘=1ln1+𝑔_𝑘+𝑇𝑚𝑟₂(𝑡)=0.0004>0,_𝑞𝑘=1ln1+ℎ_𝑘+𝑇𝑚𝑟₁(𝑡)−𝑐₁(𝑡)𝑦^∗(𝑡)^𝑚>1.143>0,_𝑞𝑘=1ln1+𝑔_𝑘+𝑇𝑚−𝑟₂(𝑡)+𝑐₂(𝑡)𝑥_∗(𝑡)𝑘+𝑥_∗(𝑡)𝑦^∗(𝑡)^𝑚−1𝑚𝑟>0.0536>0.₁(𝑡)−𝑐₁(𝑡)𝑦^∗(𝑡)^𝑚𝑚>0.961>0,−𝑟₂(𝑡)+𝑐₂(𝑡)𝑥_∗(𝑡)𝑘+𝑥_∗(𝑡)𝑦^∗(𝑡)^𝑚−1>0.024>0. (64) Taking 𝛼=1.5, 𝛿=1, conditions H₉ and H₁₀ are, respectively, equivalent to the following inequalities: min_{𝑡∈[0,𝑇]}𝛼𝑏₁(𝑡)−𝛼𝑐₁(𝑡)𝑀₂^𝑚𝑘+𝑚₁²−𝛿𝑐₂(𝑡)𝑘𝑚₂^𝑚−1𝑘+𝑚₁²⋍1.221>0,min_{𝑡∈[0,𝑇]}𝛿𝑏₂(𝑡)−𝛼𝑐₁(𝑡)(𝑚)𝑚₂^𝑚−1𝑘+𝑚₁−𝛿𝑐₂(𝑡)𝑀₁(𝑚−1)𝑚₂^𝑚−2𝑘+𝑚₁⋍0.5498>0. (65) By calculation, conditions H₁–H₅, H₉, and H₁₀ hold; from Theorem 11 and Theorem 14, system (3) is globally asymptotically stable. Numerical simulations of these results can be seen in Figure 1.

Figure 1 

Time series of 𝑥(𝑡) and 𝑦(𝑡).

Figure 1 shows the dynamical behavior of the solutions (𝑥(𝑡),𝑦(𝑡)) with seven group initial values (𝑥(0),𝑦(0))=(0.5,0.1), (0.6,0.2), (0.7,0.3), (0.8,0.4), (0.9,0.5), (1.0,0.6), and (1.1,0.7), respectively. From Figure 1, we can obtain that there exists a unique positive periodic solution of system (3) such that any solution of system (3) with initial value (𝑥(0),𝑦(0)) tends to the positive periodic solution as 𝑡→∞.

In case 2, conditions H₆ and H₇ are, respectively, equivalent to the following inequalities: ^𝑞_𝑘=1ln1+𝑔_𝑘+𝑇𝑚−𝑟₂(𝑡)−𝑐₂(𝑡)𝑥^∗(𝑡)𝑘+𝑥^∗(𝑡)𝑦_∗(𝑡)^𝑚−1=−0.9463<0,_𝑞𝑘=1ln1+ℎ_𝑘+𝑇𝑚𝑟₁(𝑡)=0.7769>0. (66) By calculation, conditions H₆ and H₇ hold. From Theorem 12, we have that species 𝑦 will be extinct and species 𝑥 will be globally asymptotically stable. Numerical simulations of these results are plotted in Figure 2.

Figure 2 

Time series of 𝑥(𝑡) and 𝑦(𝑡).

Figure 2 shows the dynamical behavior of the solutions (𝑥(𝑡),𝑦(𝑡)) with seven group initial values (𝑥(0),𝑦(0))=(0.2,0.2), (0.3,0.5), (0.4,0.8), (0.5,1.1), (0.6,1.4), (0.7,1.7), and (0.8,2.0), respectively. From Figure 2, we can see that species 𝑦 will be extinct and species 𝑥 will be globally asymptotically stable (Table 2).

Table 2 

The effects of ℎ_𝑘 and 𝑔_𝑘 on the dynamical behavior of system (3).

Furthermore, we investigate the effects of impulse coefficients ℎ_𝑘 and 𝑔_𝑘 on the dynamical behavior of system (3); we keep the parameter 𝑔_𝑘 fixed at 𝑔_𝑘=0.3. From Table 3, it is found that the impulse coefficient ℎ_𝑘 has an important influence on the sizes of prey and predator populations. Therefore, we plot the bifurcation diagram of system (3) when ℎ_𝑘∈[−0.4,0.8]; see Figure 3(a); it is very clear that the number of prey populations increases with increasing parameter ℎ_𝑘.

Table 3 

The effects of ℎ_𝑘 on the dynamical behavior of system (3).

(a)

(b)

(c)

(d)

Figure 3 

(a) Bifurcation diagram of ℎ𝑘 when 𝑔𝑘 is fixed at 0.3; (b)–(d) phase portraits.

(a)

(b)

(c)

(d)

Figure 4 

(a) Bifurcation of 𝑔𝑘 when ℎ𝑘 is fixed at 0.4; (b)–(d) phase portraits.

(a)

(b)

(c)

(d)

Figure 5 

Dynamical behavior of system (3); here, we take initial value (𝑥(0),𝑦(0))=(0.721,0.156).

On the other hand, we keep the parameter ℎ_𝑘 fixed at ℎ_𝑘=0.4. For 0.26≤𝑔_𝑘≤0.36, the effects of 𝑔_𝑘 on the dynamical behavior of system (3) have been also investigated. From Table 4, it is found that the impulse coefficient 𝑔_𝑘 has an important influence on the sizes of prey and predator populations.

Table 4 

The effects of 𝑔_𝑘 on the dynamical behavior of system (3).

Moreover, on increasing the period of impulse perturbations, some interesting phenomena happen; the details are given in Table 5.

Table 5 

The effects of impulse period 𝑇 on the dynamical behavior of system (3).

5. Conclusion

In this paper, a class of Holling type II predator-prey systems with mutual interference and impulses is studied. Some sufficient conditions for the permanence and extinction of the system and existence and uniqueness of positive periodic solution are obtained. These obtained results are new and the complement to previously known results. From the view point of biology, the mathematical results are full of biological meanings and can be used to provide reliable foundation for making control strategy. The conditions of Theorem 13 show that human activities may save the extinct species living in a week environment.

From the numerical simulations, we find that the behavior of impulse perturbations of species has an important influence on the system. Besides, the dynamics of species are very sensitive with the period matching between species' intrinsic disciplinarians and the perturbations from the variable environment. If the periods between individual growth and impulse perturbations match well, the dynamics behavior of species periodically changes. If they mismatch each other, the dynamic behaviors would change from period to period until there is chaos. It is very consistent with real ecosystems, which helps us interpret the diversity and many phenomena of ecosystems.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 11361059, 10901130, 11271312, and 11371287), the Natural Science Foundation of Xinjiang (Grant no. 2012211B07), and the Scientific Research Programmes of Colleges in Xinjiang (XJEDU2013I03).