Dynamic Analysis of a Predator-Prey (Pest) Model with Disease in Prey and Involving an Impulsive Control Strategy

Abstract

The dynamic behaviors of a predator-prey (pest) model with disease in prey and involving an impulsive control strategy to release infected prey at fixed times are investigated for the purpose of integrated pest management. Mathematical theoretical works have been pursuing the investigation of the local asymptotical stability and global attractivity for the semitrivial periodic solution and population persistent, which depicts the threshold expression of some critical parameters for carrying out integrated pest management. Numerical analysis indicates that the impulsive control strategy has a strong effect on the dynamical complexity and population persistent using bifurcation diagrams and power spectra diagrams. These results show that if the release amount of infective prey can satisfy some critical conditions, then all biological populations will coexist. All these results are expected to be of use in the study of the dynamic complexity of ecosystems.

1. Introduction

Predator-prey models with disease are a major concern and are now becoming a new field of study known as ecoepidemiology. The disease factor in predator-prey systems has been firstly considered by Anderson and May [1]. In subsequent years, many authors studied the dynamics of ecological models with infected prey, and their papers mainly focused on this issue [2–8]. The infection rate and the predation rate are the two primary factors, which can control the chaotic dynamics of an ecoepidemiological system [9]. Das et al. [10] studied the HP model [11] by introducing disease in prey populations, which can be described as follows: 𝑑𝑠𝑑𝑡=𝑟𝑠1−𝑘𝑠−𝛼𝑖𝑠−𝑐¹𝑎₁𝑝₁𝑠𝑑𝑖𝑏₁+𝑠,𝑑𝑡=𝛼𝑖𝑠−𝑎²𝑑𝑝₁𝑖𝑝₁−𝑑₁𝑖,𝑑𝑡=𝑎¹𝑝₁𝑠𝑏₁+𝑠+𝑐²𝑖𝑝₁−𝑎₃𝑝₁𝑝₂𝑏₂+𝑝₁𝑑𝑝₂−𝑑₂𝑝₁,𝑑𝑡=𝑐³𝑎₃𝑝₁𝑝₂𝑏₂+𝑝₁−𝑑₃𝑝₂,(1.1) where 𝑠,𝑖,𝑝₁, and 𝑝₂ are respectively the susceptible prey population, infected prey population, the intermediate predator population, and the top-predator population, 𝑎₁ and 𝑎₂ are the maximal predation rate of intermediate predator for susceptible and infected prey, respectively, 𝑎₃ is the maximal predation rate of top-predator for intermediate predator, 𝑏₁ and 𝑏₂ are the half saturation constant for functional response of intermediate and the top-predator respectively, 𝑐₁ is the conversion rate of susceptible prey to intermediate predator, 𝑐₂ is the conversion rate of infected prey to intermediate predator, and 𝑐₃ is the conversion rate of intermediate predator to top predator.

Through the dimensionless transformation (seeing [10]), the system can change into the following form: 𝑑𝑥𝑑𝑡=𝑥(1−𝑥)−𝑎𝑖𝑠−𝑏𝑝¹𝑠𝑑𝑖1+𝑐𝑠,𝑑𝑡=𝑎𝑖𝑠−𝑑𝑖𝑝¹𝑑𝑝₁−𝑒𝑖,𝑑𝑡=𝑓𝑝¹𝑠1+𝑐𝑠+𝑔𝑖𝑝¹−ℎ𝑝¹𝑝₂1+𝑚𝑝₁𝑑𝑝₂−𝑗𝑝₁,𝑑𝑡=𝑘𝑝¹𝑝₂1+𝑚𝑝₁−𝑙𝑝₂.(1.2)

In recent decades, technological revolutions have recently hit the industrial world; thus, infected population can now be controlled by many methods such as spraying pesticides and vaccination. It is well known that pest management involves using pesticides and releasing natural enemies, which have been focused by many researchers [12–14]. Control of an infected population can be achieved by chemical or biological control or both, which is called an impulsive control strategy in biomathematics. Systems with impulsive control strategies to describe time-varying processes are characterized by the fact that at certain moments, their states undergo abrupt change. Recently, impulsive control strategies have been recently introduced into population ecology [15–18], chemotherapeutic approaches to treat disease [19], and food webs [20–25].

Based on the two aspects discussed, the authors constructed a predator-prey model with disease in prey (a pest) and involving an impulsive control strategy for the purpose of integrated pest management. The impulsive control strategy was used to introduce infected prey (a pest) at a fixed time on the basis of system (1.2). The predator-prey model with disease in prey and involving an impulsive control strategy can be described by the following differential equations: 𝑑𝑥(𝑡)𝑑𝑡𝑑𝑦=(𝑡𝑥)(𝑡)(1−𝑥(𝑡))−𝑎𝑥(𝑡)𝑦(𝑡)−𝑏1𝑥+(𝑡)𝑐𝑧𝑥((𝑡𝑡)),𝑑𝑧(𝑡)𝑑𝑡=𝑎𝑥(𝑡)𝑦(𝑡)−𝑑𝑦(𝑡)𝑧(𝑡)−𝑒𝑦(𝑡),𝑑𝑡=𝑓1𝑥+(𝑑𝑡)𝑞𝑐𝑧𝑥(𝑡(()𝑡𝑡))+𝑔𝑦(𝑡)𝑧(𝑡)−ℎ1𝑞+(𝑡)𝑚𝑧𝑧((𝑡𝑡))−𝑗𝑧(𝑡),𝑑𝑡=𝐾ΔΔΔΔ𝑥𝑧𝑞𝑦1𝑞((((+(𝑡𝑡𝑡𝑡𝑡)))))𝑚====𝑧𝑧(00𝑝0(𝑡,𝑡,,,))−𝑡𝑙=𝑞(𝑛𝑡𝑇),,𝑡≠𝑛𝑇,(1.3) where 𝑥(𝑡),𝑦(𝑡),𝑧(𝑡), and 𝑞(𝑡) are respectively the densities of susceptible prey (a pest), infected prey (a pest), the intermediate predator (natural enemy), and the top predator at time 𝑡. Then, Δ𝑥(𝑡)=𝑥(𝑡⁺)−𝑥(𝑡), Δ𝑦(𝑡)=𝑦(𝑡⁺)−𝑦(𝑡), Δ𝑧(𝑡)=𝑧(𝑡⁺)−𝑧(𝑡), and Δ𝑞(𝑡)=𝑞(𝑡⁺)−𝑞(𝑡). We have 𝑎=𝛼𝑘𝑟,𝑏=𝑐¹𝑎₁𝑘𝑟𝑏₁,𝑐=𝑘𝑏₁,𝑑=𝑎²𝑘𝑟,𝑒=𝑑𝑟¹,𝑓=𝑎¹𝑘𝑔=𝑐₂𝑏₁𝑟,𝑎₂𝑘𝑟,ℎ=𝑎³𝑘𝑏₂𝑟,𝑚=𝑏𝑘₁,𝑗=𝑑²𝑟,𝐾=𝑐³𝑎₃𝑘𝑏₂𝑟,𝑙=𝑑𝑟³,(1.4) where 𝑎₁ and 𝑎₂ are the maximal predation rates of the intermediate predator on susceptible and infected prey respectively; 𝑎₃ is the maximal predation rate of the top predator on the intermediate predator; 𝑏₁ and 𝑏₂ are the half-saturation constants for functional response of the intermediate prey and the top predator respectively; 𝑐₁ is the conversion rate of susceptible prey to intermediate predators; 𝑐₂ and 𝑐₃ are, respectively, the conversion rate of infected prey to intermediate predators and the conversion rate of the intermediate predator to the top predator; 𝑑₁, 𝑑₂, 𝑑₃ are the death rates of infected prey, the intermediate predator, and the top predator, respectively; 𝛼 is the incidence rate; 𝑟 is the intrinsic growth rate; 𝑘 is the carrying capacity (see [10]); 𝑝>0 is the introduced amount of infective prey population at 𝑡=𝑛𝑇, 𝑛∈𝑁, 𝑁={0,1,2…}, where 𝑇 is the period of the impulsive control. It is known that pest outbreak will cause some serious ecological and economic problems, and we can directly gather infected prey to increase the amount of infected prey and indirectly carry out integrated pest management.

The paper is organized as follows: in the next section, a mathematical analysis of the model is carried out. Section 3 describes some numerical simulations, and the last section contains a brief discussion.

2. Mathematical Analysis

Some important notations, lemmas, and definitions will be provided, which are frequently used in subsequent proofs.

Let 𝑅₊=[0,+∞),𝑅⁴⁺={𝑋=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑞(𝑡))∈𝑅⁴|𝑋≥0}. Denote 𝑓=(𝑓₁,𝑓₂,𝑓₃,𝑓₄) as the map defined by the right-hand side of the first, second, third, and fourth equations of system (1.3). Let𝑉₀={𝑉∶𝑅₊×𝑅⁴⁺→𝑅₊}, then 𝑉 is said to belong to class 𝑉₀ if(1)𝑉 is continuous on (𝑛𝑇,(𝑛+1)𝑇]×𝑅⁴⁺,𝑛∈𝑁, and for each 𝑋∈𝑅⁴  lim_{(𝑡,𝜇)→(𝑛𝑇}⁺_,𝑋)𝑉(𝑡,𝜇)=𝑉(𝑛𝑇⁺,𝑋) exists;(2)𝑉 is locally Lipschitzian in 𝑋.

Definition 2.1 (see [26]). Let 𝑉∈𝑉₀, and then, for (𝑛𝑇,(𝑛+1)𝑇]×𝑅⁴⁺, the upper right derivative of 𝑉(𝑡,𝑋) with respect to the impulsive differential system (1.3) can be defined as 𝐷⁺𝑉(𝑡,𝑋)=lim_ℎ→0+sup1ℎ[𝑉(𝑡+ℎ,𝑋+ℎ𝑓(𝑡,𝑋))−𝑉(𝑡,𝑋)].(2.1)

The solution of system (1.3) is a piecewise continuous function 𝑋∶𝑅₊×𝑅⁴⁺, where 𝑋(𝑡) is continuous on(𝑛𝑇,(𝑛+1)𝑇],𝑛∈𝑁, and 𝑋(𝑛𝑇⁺)=lim_{𝑡→𝑛𝑇}⁺𝑋(𝑇) exists. Obviously the smoothness properties of 𝑓 can guarantee the global existence and uniqueness of the solution of system (1.3); for details see [26–28].

Definition 2.2 (see [21]). system (1.3) is said to be uniformly persistent if there is an 𝜔>0 (independent of the initial conditions) such that every solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑞(𝑡)) of system (1.3) satisfies the following: lim_𝑡→∞inf𝑥(𝑡)≥𝜔,lim_𝑡→∞inf𝑦(𝑡)≥𝜔,lim_𝑡→∞inf𝑧(𝑡)≥𝜔,lim_𝑡→∞inf𝑞(𝑡)≥𝜔.(2.2)

Definition 2.3 (see [24]). System (1.3) is said to be permanent if there exists a compact region Ω₀⊂int𝑅⁴⁺ such that every solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑞(𝑡)) of system (1.3) will eventually enter and remain in the region Ω₀.

Lemma 2.4 (see [24]). Suppose that 𝑋(𝑡) is a solution of system (1.3) with 𝑋(0⁺)≥0; then 𝑋(𝑡)≥0 for all 𝑡≥0. Furthermore, 𝑋(𝑡)>0, 𝑡>0 if 𝑋(0⁺)>0.

Lemma 2.5. There exists a constant 𝑀 such that 𝑥(𝑡)≤𝑀, 𝑦(𝑡)≤𝑀, 𝑧(𝑡)≤𝑀, and 𝑞(𝑡)≤𝑀 for each solution 𝑋=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑞(𝑡)) of system (1.3) for all sufficiently large 𝑡. Details can be found in Theorem 2.2 of [29].

Lemma 2.6 (see [26]). Let 𝑉∈𝑉₀, and assume that 𝑉𝐷𝑡,⁺𝑋𝑉(𝑡𝑡₊,𝑋)≤𝑔(𝑡,𝑉(𝑡,𝑋)),𝑡≠𝑛𝑇,≤Φ_𝑛(𝑉(𝑡,𝑋(𝑡))),𝑡=𝑛𝑇,(2.3) where 𝑔∶𝑅₊×𝑅₊→𝑅 is continuous in (𝑛𝑇,(𝑛+1)𝑇] for 𝑢∈𝑅²⁺,𝑛∈𝑁, lim_{(𝑡,𝑦)→(𝑛𝑇}⁺₎𝑔(𝑡,𝑣)=𝑔(𝑛𝑇⁺,𝑢) existing, and 𝜙^𝑖𝑛(𝑖=1,2)∶𝑅₊→𝑅₊ nondecreasing. Let 𝑟(𝑡) be a maximal solution of the scalar impulsive differential equation as follows: 𝑑𝑢(𝑡)𝑢𝑑𝑡𝑡=𝑔(𝑡,𝑢(𝑡)),𝑡≠𝑛𝑇,₊=𝑢Φ0_𝑛(𝑢(𝑡)),𝑡=𝑛𝑇,₊=𝑢₀,(2.4) existing on (0,+∞]. Then 𝑉(0⁺,𝑋₀)≤𝑢₀, implying that 𝑉(𝑡,𝑋(𝑡))≤𝑟(𝑡),𝑡≥0, where 𝑋(𝑡) is any solution of system (1.3). Note that if certain smoothness conditions on 𝑔 exist to guarantee the existence and uniqueness of solutions for (2.4), then 𝑟(𝑡) is the unique solution of (2.4).

For convenience, some basic properties of certain subsystems of system (1.3) are now provided as follows: 𝑑𝑦(𝑡)𝑦𝑑𝑡𝑡=−𝑒𝑦(𝑡),𝑡≠𝑛𝑇,₊=𝑦𝑦0(𝑡)+𝑝,𝑡=𝑛𝑇,₊=𝑦₀.(2.5) Therefore, the following lemma holds.

Lemma 2.7 (see [26]). For a positive periodic solution 𝑦^∗(𝑡) of system (2.5) and the solution 𝑦(𝑡) of system (2.5) with initial value 𝑦₀=𝑦(0⁺)≥0, |𝑦(𝑡)−𝑦^∗(𝑡)|→0, 𝑡→∞, where 𝑦^∗(𝑡)=𝑝exp(−𝑒(𝑡−𝑛𝑇))1−𝑦exp(−𝑒𝑇),𝑡∈(𝑛𝑇,(𝑛+1)𝑇],𝑛∈𝑁,_∗𝑦(𝑡)=𝑦0₊0⁺=1−exp𝑝(−𝑒𝑇),−1−exp𝑝(−𝑒𝑇)exp(−𝑒𝑇)+𝑦^∗(𝑡).(2.6) Next, the stability of susceptible prey and of predator-eradication periodic solutions will be studied.

Theorem 2.8. The solution (0,𝑦^∗(𝑡),0,0) is said to be locally asymptotically stable if 𝑇<(𝑝/𝑒).

Proof. The local stability of periodic solution (0,𝑦^∗(𝑡),0,0) may be determined by considering the behavior of small-amplitude perturbations of the solution. Define 𝑥(𝑡)=𝑢(𝑡),𝑦(𝑡)=𝑣(𝑡)+𝑦^∗(𝑡),𝑧(𝑡)=𝑤(𝑡),𝑞(𝑡)=ℎ(𝑡).(2.7) Substituting (2.7) into (1.3), a linearization of the system can be obtained as follows: 𝑑𝑢(𝑡)𝑑𝑡=1−𝑎𝑦^∗𝑑𝑣(𝑡)(𝑡)𝑢(𝑡),𝑑𝑡=𝑎𝑦^∗𝑑𝑤(𝑡()𝑡)𝑢(𝑡)−𝑒𝑣(𝑡)−𝑑𝑦^∗(𝑡)𝑤(𝑡),𝑑𝑡=𝑔𝑦^∗𝑑ℎ(𝑡)(𝑡)−𝑗𝑤(𝑡),𝑑ΔΔΔΔ𝑡𝑤ℎ𝑣𝑢((((=𝑡𝑡𝑡𝑡))))−====𝑙ℎ0𝑝00,,,,(𝑡),𝑡=𝑛𝑇.𝑡≠𝑛𝑇,(2.8) This can be rewritten as ⎛⎜⎜⎜⎜⎜⎜⎝𝑤ℎ𝑣𝑢((((𝑡𝑡𝑡𝑡))))⎞⎟⎟⎟⎟⎟⎟⎠=𝜙(𝑡)⎛⎜⎜⎜⎜⎜⎜⎝𝑤ℎ𝑣𝑢((((0000))))⎞⎟⎟⎟⎟⎟⎟⎠,0≤𝑡≤𝑇,(2.9) where 𝜙(𝑡) satisfies 𝑑𝜙(𝑡)𝑑𝑡=⎛⎜⎜⎜⎜⎜⎜⎝1−𝑎𝑦^∗𝑎𝑦_∗(𝑡)0000(𝑡)−0𝑒𝑔−𝑦_∗𝑑𝑦^∗(𝑡)000(0𝑡)−𝑗−0𝑙⎞⎟⎟⎟⎟⎟⎟⎠,(2.10) with 𝜙(0)=𝐼, where 𝐼 is the identity matrix, and ⎛⎜⎜⎜⎜⎜⎜⎝𝑣𝑢𝑛𝑛𝑇𝑇⁺₊𝑤𝑛𝑇₊ℎ𝑛𝑇₊⎞⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎝1000010000100001⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝𝑤ℎ𝑣𝑢((((𝑛𝑛𝑛𝑛𝑇𝑇𝑇𝑇))))⎞⎟⎟⎟⎟⎟⎟⎠.(2.11) Hence, the stability of the periodic solution (0,𝑦^∗(𝑡),0,0) is determined by the eigenvalues of 𝜃=⎛⎜⎜⎜⎜⎜⎜⎝1000010000100001⎞⎟⎟⎟⎟⎟⎟⎠𝜙(𝑡).(2.12) If the absolute values of all eigenvalues are less than one, the periodic solution (0,𝑦^∗(𝑡),0,0) is locally stable. Then all eigenvalues of 𝜙 can be denoted by 𝜆₁,𝜆₂,𝜆₃, and 𝜆₄, where 𝜆₁=exp∫^𝑇0(1−𝑎𝑦^∗(𝑡))𝑑𝑡, 𝜆₂=exp(−𝑒𝑇)<1, 𝜆₃=exp∫^𝑇0(𝑔𝑦^∗(𝑡)−𝑗)𝑑𝑡, 𝜆₄=exp(−𝑙𝑇)<1.
Clearly, |𝜆₃|=exp(−𝑔𝑝)<1 with |𝜆₁|<1 only if 𝑇<(𝑝/𝑒) according to the Floquet theory of impulsive differential equations, and the periodic solution (0,𝑦^∗(𝑡),0,0) is locally stable. This completes the proof.

Theorem 2.9. The solution (0,𝑦^∗(𝑡),0,0) is said to be globally attractive if 𝑔𝑀<𝑗 and 1𝑎<1𝑝−exexpp(−(−(𝑑(𝑑𝑀𝑀++𝑒)𝑒)𝑇𝑇)).(2.13)

Proof. Let 𝑉(𝑡)=𝑓𝐾𝑥(𝑡)+𝑏𝐾𝑧(𝑡)+𝑏ℎ𝑞(𝑡); then 𝑉^||_(1.1)=𝑓𝐾(1−𝑎𝑦(𝑡))𝑥(𝑡)+𝑏𝐾(𝑔𝑦(𝑡)−𝑗)𝑧(𝑡)−𝑓𝐾𝑥²(𝑡)−𝑏ℎ𝑙𝑞(𝑡).(2.14) By Lemma 2.5, there exists a constant 𝑀>0 such that 𝑥(𝑡)≤𝑀, 𝑦(𝑡)≤𝑀, 𝑧(𝑡)≤𝑀, 𝑞(𝑡)≤𝑀 for each solution 𝑋=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑞(𝑡)) of system (1.3) with sufficiently large 𝑡.
Then, 𝑑𝑦(𝑡)𝑉||𝑑𝑡=𝑎𝑦(𝑡)𝑥(𝑡)−𝑑𝑦(𝑡Δ)𝑧𝑦(=𝑡)𝑝−,𝑒𝑦𝑡(=𝑡)𝑛≥𝑇−(𝑑𝑀+𝑒)𝑦(𝑡),𝑡≠𝑛𝑇(2.15)_(1.1)≤=𝑓𝑓𝐾𝐾((11−−𝑎𝑎𝑦𝑦((𝑡𝑡))))𝑥𝑥((𝑡𝑡))++𝑏𝑏𝐾𝐾((𝑔𝑔𝑀𝑦(𝑡−)−𝑗)𝑗𝑧)𝑧(𝑡()𝑡−)−𝑓𝐾𝑓𝐾𝑥𝑥(𝑡²)(−𝑡)𝑏−ℎ𝑏𝑙𝑞ℎ(𝑙𝑡𝑞)(.𝑡)(2.16) By Lemmas 2.6 and 2.7, there exists a 𝑡₁>0, and an 𝜀>0 can be selected to be small enough so that 𝑦(𝑡)≥𝑦^∗1(𝑡)−𝜀 for all 𝑡≥𝑡₁. By (2.15), 𝑦(𝑡)≥𝑦^∗1(𝑡)−𝜀=𝑝exp(−(𝑑𝑀+𝑒)𝑇)𝜆1−exp(−(𝑑𝑀+𝑒)𝑇)−𝜀,_Δ=𝑝exp(−(𝑑𝑀+𝑒)𝑇)1−exp(−(𝑑𝑀+𝑒)𝑇)−𝜀.(2.17) Let 1−𝑎𝜆<0 and 𝑔𝑀−𝑗<0. Therefore, when 𝑡≥𝑡₁, by (2.16), 𝑉^|_(1.1)<0. So 𝑉(𝑡)→0 and 𝑥(𝑡)→0, 𝑧(𝑡)→0, 𝑞(𝑡)→0 as 𝑡→∞. It is known from the fact that the limiting state of system (1.3) is exactly system (2.5) and from Lemma 2.7 that (0,𝑦^∗(𝑡),0,0) is globally attractive. This completes the proof.

Theorem 2.10. System (1.3) is permanent if 𝑇>𝑝/𝑒, 𝑔𝑀>𝑗, 1𝑎>1𝑝−exexpp(−(−(𝑑(𝑑𝑀𝑀++𝑒)𝑒)𝑇𝑇)),𝑏ℎ𝑙𝑀>1𝑝−exexpp(((𝑎(𝑎𝑀𝑀−−𝑒)𝑒)𝑇𝑇)).(2.18)

Proof. From Lemma 2.5, there exists a constant 𝑀>0 such that 𝑥(𝑡)≤𝑀, 𝑦(𝑡)≤𝑀, 𝑧(𝑡)≤𝑀, 𝑞(𝑡)≤𝑀 for each solution 𝑋=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑞(𝑡)) of system (1.3) with 𝑡 sufficiently large. From (2.15), it is known that 𝑦(𝑡)≥𝑦^∗1(𝑡)−𝜀=(𝑝exp(−(𝑑𝑀+𝑒)𝑇))/(1−exp(−(𝑑𝑀+𝑒)𝑇))−𝜀=^Δ𝛿₁ for large enough 𝑡.
Therefore, it is only necessary to find a 𝛿₂ that satisfies 𝑥(𝑡)>𝛿₂, 𝑧(𝑡)>𝛿₂, 𝑞(𝑡)>𝛿₂. This will be achieved in the following two steps.
Let 𝛿₃>0,𝛿₄>0,𝛾=𝑒−𝑎𝛿₃, and 𝑉(𝑡)=𝑓𝐾𝑥(𝑡)+𝑏𝐾𝑧(𝑡)+𝑏ℎ𝑞(𝑡).
Then 𝑉||_(1.1)≥=𝑓𝑓𝐾𝐾((11−−𝑎𝑎𝑦𝑦((𝑡𝑡)))−𝑥𝑀(𝑡))+𝑥𝑏(𝑡𝐾)+(𝑔𝑏𝑦𝐾(𝑡)(𝑔−𝑦𝑗(𝑡))𝑧−(𝑡𝑗))−𝑧𝑓(𝑡𝐾)−𝑥²𝑓(𝑡𝐾)𝑥−(𝑏𝑡)ℎ−𝑙𝑞𝑏(ℎ𝑡)𝑙𝑞(𝑡).(2.19) First, it will be proved that there exists a 𝑡₂∈(0,+∞) such that 𝑥(𝑡₂)>𝛿₄, 𝑧(𝑡₂)>𝛿₄, and 𝑞(𝑡₂)>𝛿₄ because 𝑉(𝑡) is ultimately bounded.
Next, it will be proved that 𝑥(𝑡)<𝛿₃, 𝑧(𝑡)<𝛿₃, 𝑞(𝑡)<𝛿₃ cannot hold for all 𝑡∈(0,+∞).
Otherwise, 𝑑𝑦(𝑡)𝑑𝑡=𝑎𝑦(𝑡)𝑥(𝑡)−𝑑𝑦(𝑡)𝑧(𝑡)−𝑒𝑦(𝑡)≤𝑎𝛿³Δ𝑦=𝑝,𝑡=𝑛𝑇.−𝑒𝑦(𝑡),𝑡≠𝑛𝑇(2.20) Then let 𝑣₁(𝑡) be the solution of 𝑑𝑣₁(𝑡)𝑑𝑡=𝑎𝛿³Δ𝑣₁−𝑒𝑣₁(𝑡),𝑡≠𝑛𝑇(𝑡)=𝑝,𝑡=𝑛𝑇.(2.21) It follows that 𝑦(𝑡)<𝑣₁(𝑡) and 𝑣₁(𝑡)→𝑣^∗1(𝑡)(𝑡→∞) where 𝑣^∗1(𝑡)=(𝑝exp(−𝛾(𝑡−𝑛)𝑇))/(1−exp(−𝛾𝑇)).
So there exists a 𝑡₃>0 such that 𝑦(𝑡)<𝑣₁(𝑡)<𝑣^∗1(𝑡)+𝜀₁=𝑝exp(−𝛾(𝑡−𝑛)𝑇)1−exp(−𝛾𝑇)+𝜀¹<𝑝1−exp(−𝛾𝑇)+𝜀¹.(2.22) Then 𝑉||_(1.1)≥≥=𝑓𝑓𝑓𝐾𝐾𝐾((111−−−𝑎𝑎𝑎𝑦𝑦((𝑡𝑡)))−𝑥𝑀(𝑡𝑝))+𝑥𝑏(𝑡𝐾)+(𝑔𝑏𝑦𝐾(𝑡)(𝑔−𝑦𝑗()𝑡)𝑧−(𝑡𝑗))−𝑧𝑓(𝑡𝐾)𝑥−²𝑓(𝑡𝐾)𝑥−(𝑏𝑡)ℎ−𝑙𝑞𝑏(ℎ𝑡)𝑙𝑞(𝑡)1−exp(−𝛾𝑇)+𝜀¹−𝑀𝑥(𝑡)+𝑏𝐾𝑔𝑝1−exp(−𝛾𝑇)+𝜀¹−𝑗𝑧(𝑡).(2.23) According to the above conditions, 𝑉^|_(1.1)>0; then 𝑉(𝑡)→∞ and 𝑥(𝑡)→∞, 𝑧(𝑡)→∞, 𝑞(𝑡)→∞ as 𝑡→∞; however, this is a contradiction. Therefore, 𝑉(𝑡) is ultimately bounded.
Second, if 𝑥(𝑡)>𝛿₃, 𝑧(𝑡)>𝛿₃, 𝑞(𝑡)>𝛿₃ for all 𝑡≥𝑡₂, then the objective has been attained. To show this, let 𝑡^∗=inf_𝑡≥𝑡2{𝑉(𝑡)<𝛿₅}, and it follows that 𝑉(𝑡)≥𝛿₅ for 𝑡∈[𝑡₂,𝑡^∗) and that 𝑉(𝑡^∗)=𝛿₅. Suppose that 𝑡^∗∈(𝑛₁𝑇,(𝑛₁+1)𝑇],𝑛₁∈𝑁. Select 𝑛₂,𝑛₃∈𝑁 such that 𝑛₂𝑇 > (ln(𝜀₁/(𝑀+𝑝))/−𝛾), exp(𝑛₃𝛼₁𝑇)exp(𝛼₂(𝑛₂+1)𝑇)>1, where 𝛼₁=𝑓𝐾1−𝑎𝑝1−exp(−𝛾𝑇)+𝜀¹−𝑀𝑥(𝑡)+𝑏𝐾𝑔−𝑝1−exp(−𝛾𝑇)+𝜀¹𝛼−𝑗𝑧(𝑡)>0,₂=−𝑎𝑝exp((𝑎𝑀−𝑒)𝑇)1−exp((𝑎𝑀−𝑒)𝑇)𝑥(𝑡)−𝑗𝑧(𝑡)<0.(2.24) Let 𝑇₂=𝑛₂𝑇+𝑛₃𝑇. It is claimed that there must be a 𝑡₃∈((𝑛₁+1)𝑇,(𝑛₁+1)𝑇+𝑇₂] such that 𝑉(𝑡)≥𝛿₅. Otherwise, consider (2.21) with 𝑣₁(𝑡^∗+)=𝑦(𝑡^∗+). Then 𝑣₁(𝑡)=𝑣₁𝑛₁+1𝑇⁺−𝑝1−exp(−𝛾𝑇)exp−𝛾𝑡−𝑛¹+1𝑇+𝑣^∗1(𝑡).(2.25) For 𝑡∈((𝑛+1)𝑇,(𝑛+1)𝑇], 𝑛₁+1<𝑛<𝑛₁+𝑛₂+𝑛₃+1, it can be shown that |𝑣₁(𝑡)−𝑣^∗1(𝑡)|<(𝑀+𝑝)exp(−𝛾𝑛₁𝑇)<𝜀₁, 𝑦(𝑡)≤𝑣₁(𝑡)≤𝑣^∗1(𝑡)+𝜀₁ for 𝑡∈((𝑛₁+𝑛₂+1)𝑇,(𝑛₁+1)𝑇+𝑇₂], 𝑉^||_(1.1)≥𝑓𝐾1−𝑎𝑝1−exp(−𝛾𝑇)+𝜀¹−𝑀𝑥(𝑡)+𝑏𝐾𝑔𝑝1−exp(−𝛾𝑇)+𝜀¹=𝛼₁−𝑗𝑧(𝑡)>0,(2.26) and 𝑉((𝑛₁+1)𝑇+𝑇₂) ≥ 𝑉((𝑛₁+𝑛₂+1)𝑇)exp(𝛼₁𝑛₃𝑇). For 𝑡∈[𝑡^∗,(𝑛₁+𝑛₂+1)𝑇], it can be shown that 𝑉|_(1.1) ≥ −𝛼₂𝑉(𝑡)>0; then, 𝑉((𝑛₁+𝑛₂+1)𝑇) ≥ 𝑉^∗(𝑡)exp(−𝛼₂(𝑛₂+1)𝑇), so 𝑉((𝑛₁+1)𝑇+𝑇₂) ≥ 𝑉^∗(𝑡)exp(−𝛼₂(𝑛₂+1)𝑇+𝛼₁𝑛₃𝑇) > 𝛿₅, which is a contradiction. Therefore, there exists a 𝑡₃∈((𝑛₁+1)𝑇,(𝑛₁+1)𝑇+𝑇₂] such that 𝑉(𝑡)≥𝛿₅, resulting in 𝑉(𝑡) ≥ 𝑉^∗(𝑡)exp(−𝛼₂(𝑛₁ + 𝑛₂ + 𝑛₃ + 1)𝑇)=^Δ𝛿₆.
When 𝑡≥𝑡₃, the same procedure can be performed. According to the above discussion, if Ω₀={(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡),𝑞(𝑡))∶𝑉(𝑡)=𝑓𝐾𝑥(𝑡)+𝑏𝐾𝑧(𝑡)+𝑏ℎ𝑙𝑞(𝑡),𝛿≤𝑉(𝑡)≤𝑀₁}⊂int𝑅³⁺, every solution of system (1.3) will eventually enter and remain in the region Ω₀. This completes the proof.

3. Numerical Analysis

3.1. Bifurcation

To study the dynamics of system (1.3), the period 𝑇 and the impulsive control parameter 𝑝 are used as the bifurcation parameter. The bifurcation diagram provides a summary of the basic dynamic behavior of the system [30, 31].

First, the influence of the period 𝑇 is studied using the time series shown in Figure 1. The bifurcation diagrams are shown in Figures 2 and 3. Next, the influence of the impulsive control parameter 𝑝 is investigated. The bifurcation diagrams for this are shown in Figure 4.

(a)

(b)

(c)

(d)

Figure 1 

Dynamics of system (1.3) with 𝑏=8, 𝑑=3, 𝑐=3, 𝑒=0.2, 𝑓=3, 𝑔=2.5, ℎ=0.3, 𝑚=1.2, 𝑗=0.2, 𝑘=0.6, 𝑙∶=0.008, 𝑎∶=3, 𝑝=1.03, 𝑇=14. Time series of (a) susceptible prey, (b) infected prey, (c) the intermediate predator, and (d) the top predator.

(a)

(b)

(c)

(d)

Figure 2 

Bifurcation diagram of system (1.3) with initial conditions 𝑥(0)=0.1,𝑦(0)=0.2,𝑧(0)=0.3,𝑞(0)=0.4, 7≤𝑇≤37, 𝑏=6, 𝑑=3, 𝑐=3, 𝑒=0.2, 𝑓=2, 𝑔=2.5, ℎ=0.1, 𝑚=2; 𝑗=0.1, 𝑘=0.6, 𝑙=0.01, 𝑎=3, and 𝑝=0.6.

(a)

(b)

(c)

(d)

Figure 3 

Bifurcation diagram of system (1.3) with initial conditions 𝑥(0)=0.1,𝑦(0)=0.2,𝑧(0)=0.3,𝑞(0)=0.4, 7≤𝑇≤27, 𝑏=10, 𝑑=3, 𝑐=3, 𝑒=0.5, 𝑓=5, 𝑔=2.5, ℎ=0.1, 𝑚=2; 𝑗=0.2, 𝑘=0.3, 𝑙=0.01, 𝑎=1.3, and 𝑝=0.6.

(a)

(b)

(c)

(d)

Figure 4 

Bifurcation diagram of system (1.3) with initial conditions 𝑥(0)=0.1, 𝑦(0)=0.2, 𝑧(0)=0.3, 𝑞(0)=0.4, 0.1≤𝑝≤2.4, 𝑏=8, 𝑑=3, 𝑐=3, 𝑒=0.2, and 𝑓=3 s, 𝑔=2.5, ℎ=0.1, 𝑚=0.2, 𝑗=0.1, 𝑘=0.6, 𝑙=0.008, 𝑎=3, 𝑝=0.6.

To clearly see the dynamics of system (1.3), it is necessary to examine the phase diagrams at different value of the period 𝑇 and parameter 𝑝 corresponding to the bifurcation diagrams in Figures 2 and 4; the results of this analysis are shown in Figures 5 and 6.

(a)

(b)

(c)

(d)

Figure 5 

Periodic and chaotic behavior corresponding to Figure 2 as shown by phase diagrams: (a) 𝑇=10, (b) 𝑇=17, (c) 𝑇=15, and (d) 𝑇=27.7.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 6 

Periodic and chaotic behavior corresponding to Figure 3, as shown in phase diagrams: (a) 𝑝=0.13, (b) 𝑝=0.25, (c) 𝑝=0.3, (d) 𝑝=0.35, (e) 𝑝=0.45, (f) 𝑝=0.7, (g) 𝑝=0.9, and (h) 𝑝=1.0009.

Figures 2, 3, and 4 reveal the complex dynamics of system (1.3), including period-doubling cascades, symmetry-breaking pitchfork bifurcation, chaos, and nonunique dynamics. Because every bifurcation diagram is similar, only one needs to be explained. Take Figure 4(a) as an example. When 𝑝∈[0,0.124], the dynamics of the system are not obvious, but with increasing 𝑝, the dynamics become more obvious. The system enters into a chaotic band with periodic windows. When 𝑝 is between 0.124 and 0.153, the chaotic behavior is intense, as can be seen in Figure 6(a). When 𝑝 moves beyond 0.153, the chaotic behavior disappears. When 𝑝∈[0.203,0.219], the chaotic attractor gains in strength, and the chaotic behavior appears again. When 𝑝 becomes greater than 0.219, periodic windows appear, as can be seen in Figures 6(b) and 6(c). When 𝑝 is in the interval between 0.328 and 0.35, chaotic behavior ensues, as can be seen in Figure 6(d). As the value of 𝑝 increases further, the system enters a stable state, as is shown in Figures 6(e), 6(f), and 6(g). When 𝑝 moves beyond 1.001, an unexpectedly chaotic phase appears, as is shown in Figure 6(h). It is clear that seasonal disturbances have little effect on the maximum density of all species; however, serious periodic oscillations are generated, and weak periodic solutions lose their stability and move into chaos. In summary, the key factor in the long-term dynamic behavior of system (1.3) is impulse perturbations, but seasonal disturbances can aggravate periodic oscillations and promote the emergence of chaos. Based on the above numerical simulation analysis, it is clear that impulsive control strategy has an important effect on the dynamical behaviors of the system, and weak periodic solutions lose their stability and move into chaos. In summary, the key factor in the long-term dynamic behavior of system (1.3) is impulsive control strategy, but disease disturbances can aggravate periodic oscillations and promote the emergence of chaos.

3.2. The Iargest Lyapunov Exponent

To detect whether the system exhibits chaotic behavior, one of the commonest methods is to calculate the largest Lyapunov exponent. The largest Lyapunov exponent takes into account the average exponential rates of divergence or convergence of nearby orbits in phase space [32]. A positive largest Lyapunov exponent indicates that the system is chaotic. If the largest Lyapunov exponent is negative, there must be periodic windows or a stable state. Through the largest Lyapunov exponent, it is possible to judge that at what time the system is chaotic, and at what time the system is stable. The largest Lyapunov exponents corresponding to Figures 2, 3, and 4 can be calculated and are shown in Figures 7, 8, and 9, which shows the accuracy and effectiveness of numerical simulation. Moreover, using the simulation of the largest Lyapunov exponents, the existence of chaotic behavior in system (1.3) can be further confirmed.

Figure 7 

The largest Lyapunov exponents (LLE) corresponding to Figure 2.

Figure 8 

The largest Lyapunov exponents (LLE) corresponding to Figure 3.

Figure 9 

The largest Lyapunov exponents (LLE) corresponding to Figure 4.

3.3. Strange Attractors and Power Spectra

To understand the qualitative nature of strange attractors, power spectra are used [33]. From Section 3.2, it is known that the largest Lyapunov exponent for strange attractor (a) is 0.0413, and for strange attractor (b) is 0.124. Therefore, they are both chaotic attractors, and the exponent of (b) is larger than that of (a), which means that the chaotic dynamics of (b) are more extreme than those of (a). The power spectrum of strange attractor (a) is composed of strong broadband components and sharp peaks, as are shown in Figure 10(c). On the contrary, in the spectrum of strong chaotic attractor (b), it is difficult to distinguish any sharp peaks, as can be seen in Figure 10(d). These power spectra can be interpreted as meaning that (a) comes from a strong limit cycle, but that (b) experiences some weak limit cycles. Hence, it is obvious that the impulsive control strategy has a strong effect on the dynamical behaviors of system (1.3) with 𝑡 the period of the impulsive control 𝑇 varying but that (b) experiences some weak limit cycles.

(a)

(b)

(c)

(d)

Figure 10 

Strange attractors and power spectra: (a) strange attractor when 𝑇=32, (b) strange attractor when 𝑇=21, (c) power spectrum of attractor (a), and (d) power spectrum of attractor (b).

4. Conclusions and Remarks

In the paper, the dynamic behaviors of a predator-prey (pest) model with disease in prey and involving an impulsive control strategy are presented analytically and numerically. The critical conditions are obtained to ensure the local asymptotical stability and global attractivity of semitrivial periodic solution as well as population permanence. Numerical analysis indicates that the impulsive control strategy has a strong effect on the dynamical complexity and population persistent using bifurcation diagrams and power spectra diagrams. In addition, the largest Lyapunov exponents are computed. This computation further confirms the existence of chaotic behavior and the accuracy of numerical simulation. These results revealed that the introduction of disease and the use of an impulsive control strategy can change the dynamic behaviors of the system. The same results also have been observed in continuous-time models of predator-prey or three-species food-chain models [34–37] and other systems [38]. In a word, it should be stressed that the impulsive control strategy is an effective method to control complex dynamics of predator-prey (pest) model.

Acknowledgment

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions on this paper. This work was supported by the National Natural Science Foundation of China (NSFC no. 31170338 and no. 30970305) and also by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001).