# Analysis of a Periodic Impulsive Predator-Prey System with Disease in the Prey

Authors: Lianwen Wang; Zhijun Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We investigate a periodic predator-prey system subject to impulsive perturbations, in which a disease can be transmitted among the prey species only, in this paper. With the help of the theory of impulsive differential equations and Lyapunov functional method, sufficient conditions for the permanence, global attractivity, and partial extinction of system are established, respectively. It is shown that impulsive perturbations contribute to the above dynamics of the system. Numerical simulations are presented to substantiate the analytical results.


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## Body

## 1. Introduction

As a relatively new branch of study in theoretical biology, ecoepidemiology can be viewed as the coupling of ecology and epidemiology. Ecoepidemiological model is more appropriate than the ecological model (or epidemiological model) when species spreads the disease and is predated by other species. Following Anderson and May [1] who were the first to propose an ecoepidemiological model, a number of sophisticated predator-prey models with disease in prey population only are extensively investigated in the ecoepidemiological literature (see [2–5]).Notice that periodic phenomenon often occurs in many realistic ecoepidemiological models. The effect of a periodically varying environment is important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Nicholson [6] has suggested that any periodic change of climate tends to impose its period upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes. Thus, it is reasonable to assume that the coefficients in the systems are periodic functions. On the other hand, in reality, many evolution processes are characterized by the fact that they experience changes of state suddenly. These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, the abrupt changes can be well approximated as impulses. A natural description of the motion of impulsive processes can be expressed by impulsive differential equations. Some impulsive equations have been introduced in ecoepidemiological models in relation to chemotherapeutic [7] and vaccination [8, 9] and population disease control [10, 11].Considering the above facts, in this paper, we will consider a periodic predator-prey model subject to impulsive perturbations, in which a disease can be transmitted among the prey species only. Our motive comes from a delayed nonautonomous predator-prey system with disease in the prey in [5], and we consider the effect of impulsive perturbations on a corresponding undelayed periodic version in this paper. Here, we will establish the sufficient conditions for the permanence and partial extinction of the system by using the theory of impulsive differential equations and inequality analytical technique. By Lyapunov functional method, we will also establish sufficient conditions for the global attractivity of the system.


## 2. Assumptions and Formulation of Mathematical Model

The periodic predator-prey model with disease in the prey which is studied in this paper is the system of impulsive differential equations below. To formulate the mathematical model, we need to make the following assumptions which are the same as those in [5].(A1)
All newborns are susceptible in the model, in which only susceptible prey is capable of reproducing with logistic law while the infected species does not recover or become immune. The disease only spreads among the prey species and it is not genetically inherited.(A2)
The mortality terms for susceptible and infected prey are density dependent, and both contribute to population growth toward the environmental carrying capacity.(A3)
The predator species hunts on susceptible and infected prey with possibly different predation rates. For example, in some situations, the infected individuals may be caught easily, but the predators eat more fewer infected ones in other situations.Considering the above basic assumptions, we can now construct the following dynamic system:(1)S′(t)=S(t)[r(t)-k1(t)(S(t)+I(t))-a1(t)P(t)-β(t)I(t)],I′(t)=I(t)[β(t)S(t)-k2(t)(S(t)+I(t))-a2(t)P(t)],P′(t)=P(t)[-d(t)-b(t)P(t)+c1(t)S(t)+c2(t)I(t)],t≠τk,S(τk+)=(1+αk)S(τk),I(τk+)=I(τk)+p,P(τk+)=(1+γk)P(τk),t=τk,k∈ℕ.
Here, S(0)>0, I(0)>0, P(0)>0. r(t) is intrinsic birth rate function of the susceptible prey species S(t), d(t) is death rate function of the predator species P(t), k1(t), k2(t), and b(t) represent the self-inhibition rate functions of the susceptible prey species S(t), the infected prey species I(t), and the predator P(t), respectively. β(t) is the infection rate. a1(t), a2(t) and c1(t), c2(t) denote the capturing rates and conversion rates, respectively. The parameters r(t), d(t), k1(t), k2(t), b(t), a1(t), a2(t), c1(t), and c2(t) are positive continuous ω-periodic functions. The fixed impulsive points satisfy 0=τ0<τ1<τ2<⋯ and limk→∞τk=+∞ for k∈ℕ={1,2,…}. For two real number sequences {αk} and {γk}, there is an integer q>0 such that τk+q=τk+ω, αk+q=αk>-1, γk+q=γk>-1 for k∈ℕ. The jump conditions reflect the possibility of impulsive effects on the prey and predator. For biological reality, it is natural to assume αk>-1, γk>-1 and p>0. An obvious example of ecological situation giving rise to system (1) concerns the impulsive harvesting and stocking of the above species; that is, when αk>0, γk>0, p>0, the impulsive perturbations stand for stocking while αk<0, γk<0 mean harvesting.The organization of this paper is as follows. In the next section, we present some notations and preliminary lemmas. In Section4, we establish sufficient conditions for the permanence, global attractivity of positive solutions, and partial extinction for the above system, respectively. Several concrete examples and numerical simulations are also presented to substantiate the analytical results in the last section.


## 3. Notations and Preliminary Lemmas

Before establishing our main results, we summarize several useful lemmas for the later sections.Let𝕁⊂ℝ=(-∞,+∞). Denote by PC(𝕁,ℝ) the set of functions v:𝕁→ℝ which are continuous for t∈𝕁, t≠τk, are continuous from the left for t∈𝕁, and have discontinuities of the first kind at the points τk∈𝕁. Denote by PC1(𝕁,ℝ) the set of functions v:𝕁→ℝ with a derivative dv/dt∈PC(𝕁,ℝ).Lemma 1 (see [12]).
Let the functionv∈PC1(𝕁,ℝ) satisfy the inequalities
(2)v′(t)≤g(t)v(t)+h(t),t≠τk,t≥τ0,v(τk+)≤dkv(τk)+βk,k∈ℕ,
where g(t), h(t)∈C(𝕁,ℝ), dk>0, and βk are constants for k∈ℕ. Then for t≥τ0,
(3)v(t)≤v(τ0)∏τ0<τk<t‍dkexp{∫τ0t‍g(s)ds}+∑τ0<τk<t‍(∏τ0<τj<t‍djexp{∫τkt‍g(s)ds})βk+∫τ0t‍∏s<τk<t‍dkexp{∫st‍g(r)dr}h(s)ds.
Analogously, one sees that
(4)v(t)≥v(τ0)∏τ0<τk<t‍dkexp{∫τ0t‍g(s)ds}+∑τ0<τk<t‍(∏τ0<τj<t‍djexp{∫τkt‍g(s)ds})βk+∫τ0t‍∏s<τk<t‍dkexp{∫st‍g(r)dr}h(s)ds,
for t≥τ0, when all the inequalities of (1) are reversed.Lemma 2 (see [13, 14]).
Consider the following single-species impulsive model:(5)x′(t)=m(t)x(t),t≠τk,k∈ℕ,x(τk+)=x(τk)+p,
where x(0)>0, the constant p>0, m(t) is a continuous ω-periodic function satisfying ∫0ω‍m(t)dt<0, and there is an integer q>0 such that τk+q=τk+ω, k∈ℕ. System (5) has a unique positive ω-periodic solution which is globally asymptotically stable.Lemma 3 (see [15]).
Consider the following impulsive system:(6)x′(t)=x(t)[m(t)-n(t)x(t)],t≠τk,k∈ℕ,x(τk+)=x(τk)+p,
where x(0)>0, n(t)>0, the constant p>0, and m(t), n(t) are continuous ω-periodic functions. Assume that there is an integer q>0 such that τk+q=τk+ω, k∈ℕ. Then(1)
speciesx is permanent;(2)
any two positive solutionsx(t) and x-(t) of (6) satisfy limt→+∞|x(t)-x-(t)|=0.Lemma 4 (see [13, 16]).
Consider the following single-species system with impulsive perturbations:(7)x′(t)=x(t)[m(t)-n(t)x(t)],t≠τk,k∈ℕ,x(τk+)=(1+αk)x(τk),
where x(0)>0, n(t)>0 for all t≥0, m(t), n(t) are continuous ω-periodic functions and {αk} is a real number sequence with αk>-1. Meanwhile, there exists an integer q>0 such that τk+q=τk+ω and αk+q=αk for k∈ℕ. Then(1)
any solutionx(t) of (7) satisfies limt→+∞x(t)=0 if ∏k=1q‍(1+αk)exp{∫0ω‍m(t)dt}<1;(2)
(7) has a unique positive ω-periodic solution which is globally asymptotically stable if ∏k=1q‍(1+αk)exp{∫0ω‍m(t)dt}>1.Lemma 5 (see [16]).
Suppose thatv(t) is a continuous ω-periodic function, where ω>0 and ∫0ω‍v(t)dt>0. Then the following inequality
(8)exp{c(t-s)}≤exp{1+dω+∫st‍v(r)dr},fort≥s,t⟶+∞
holds, where 0<c≤min{∫0ω‍v(t)dt/ω,1/ω}, and d=maxt∈[0,ω]{|v(t)|}.


## 4. Main Results

In this section, we will establish sufficient conditions for the permanence, global attractivity of positive solutions, and partial extinction of system (1), respectively. We first give the result on permanence.


### 4.1. Permanence

Theorem 6.
If(9)∏k=1q‍(1+αk)exp{∫0ω‍[r(t)-M2(k1(t)+β(t))-M3a1(t)(k1(t)+β(t))]dt∫0ω‍}>1,(10)∏k=1q‍(1+γk)exp{∫0ω‍[-d(t)+m1c1(t)+m2c2(t)]dt∫0ω‍}>1
hold, then system (1) is permanent, where M2, M3, m1, m2 are defined in (26), (29), (33), and (38), respectively.Proof.
For (1), it is easy to verify S(t)>0, I(t)>0, P(t)>0 for all t>0 if S(0+)=S(0)>0, I(0+)=I(0)>0, P(0+)=P(0)>0. To finish the proof of Theorem 6, we consider the following two steps (i.e., (I) and (II)).(I)
Consider the uniformly ultimately upper boundary (or UUUB) ofS(t), I(t), P(t).
First of all, we discuss the UUUB ofS(t). It follows from the first equation of (1) and impulsive condition that
(11)S′(t)≤r(t)S(t),t≠τk,k∈ℕ,S(τk+)=(1+αk)S(τk),(12)S′(t)≤S(t)[r(t)-k1(t)S(t)],t≠τk,k∈ℕ,S(τk+)=(1+αk)S(τk).
Now, we consider two cases to obtain the UUUB of S(t).
Case  1. Consider ∏k=1q‍(1+αk)exp{∫0ω‍r(t)dt}≤1.
For (11), applying Lemma 1, we have
(13)S(t)≤S(0)∏0<τk<t‍(1+αk)exp{∫0t‍r(s)ds}.
When t∈(nω,(n+1)ω], n∈ℕ∪{0}, we set
(14)𝒜=maxt∈[0,ω]∏0<τk<t‍(1+αk),ℛ=maxt∈(nω,(n+1)ω]exp{∫0ω‍r(s)ds}.
It follows from (13) that
(15)S(t)≤S(0)∏0<τk<nω‍(1+αk)exp{∫0nω‍r(s)ds}×∏nω≤τk<t‍(1+αk)exp{∫nωt‍r(s)ds}≤S(0)(∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds})n×∏0<τk<t-nω‍(1+αk)exp{∫nωt‍r(s)ds}≤S(0)(∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds})n𝒜ℛ.
If ∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds}=1, then
(16)S(t)≤S(0)𝒜ℛ.
If ∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds}<1, then
(17)limt→+∞S(0)(∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds})n𝒜ℛ=0.
This fact implies that limt→+∞S(t)=0. That is, there are a sufficient small constant σ1>0 and t1>0 such that
(18)S(t)≤σ1,fort≥t1.Case  2. Consider ∏k=1q‍(1+αk)exp{∫0ω‍r(t)dt}>1.
It follows from (2) in Lemma4 that the comparison system of (12)
(19)s1′(t)=s1(t)[r(t)-k1(t)s1(t)],t≠τk,k∈ℕ,s1(τk+)=(1+αk)s1(τk)
has a unique positive globally asymptotically stable ω-periodic solution denoted by s-1(t). Let s1(t) be the solution of (19) with s1(0)=S(0)>0. The asymptotic property of s-1(t) shows that there exist a sufficient small constant σ2>0 and t2>0 such that
(20)s1(t)≤s-1(t)+σ2,fort≥t2.
Applying the comparison theorem of impulsive differential equations, one has
(21)S(t)≤s-1(t)+σ2,fort≥t2.
Equations (18) and (21) show that there must be T1=max{t1,t2} and M1=maxt∈[0,ω]{S(0)𝒜ℛ, σ1, s-1(t)+σ2}>0 such that
(22)S(t)≤M1,fort≥T1.
Next, we discuss the UUUB ofI(t). From the second equation of (1) and impulsive condition, for t≥T1, one has
(23)I′(t)≤I(t)[M1β(t)-k2(t)I(t)],t≠τk,k∈ℕ,I(τk+)=I(τk)+p.
Using (2) in Lemma 3, we can see that the comparison system of (23)
(24)i1′(t)=i1(t)[M1β(t)-k2(t)i1(t)],t≠τk,k∈ℕ,i1(τk+)=i1(τk)+p
is permanent, which implies that there are M2>0 and T2≥T1 such that
(25)i1(t)≤M2,fort≥T2.
Using the comparison theorem of impulsive differential equations, we get
(26)I(t)≤M2,fort≥T2.
Finally, we verify the UUUB ofP(t). For t≥T2, from the third equation of (1) and impulsive condition, we obtain
(27)P′(t)≤P(t)[-d(t)+M1c1(t)+M2c2(t)],t≠τk,k∈ℕ,P(τk+)=(1+γk)P(τk)(28)P′(t)≤P(t)[-d(t)+M1c1(t)+M2c2(t)-b(t)P(t)],t≠τk,k∈ℕ,P(τk+)=(1+γk)P(τk).
Similar to the proof of the UUUB of S(t), it follows from (10) that there exist T3>0 and M3>0 such that
(29)P(t)≤M3,fort≥T3.(II)
Consider the uniformly ultimately lower boundary (or UULB) ofS(t), I(t), P(t).
Firstly, we prove the UULB ofS(t). From (I), for t≥T3, one has
(30)S′(t)≥S(t)[r(t)-M2(k1(t)+β(t))-M3a1(t)-k1(t)S(t)],t≠τk,k∈ℕ,S(τk+)=(1+αk)S(τk).
By (2) in Lemma 4 and the assumption in (9), we know easily that the comparison system of (30)
(31)s2′(t)=s2(t)[r(t)-M2(k1(t)+β(t))-M3a1(t)-k1(t)s2(t)],t≠τk,k∈ℕ,s2(τk+)=(1+αk)s2(τk)
has a unique positive globally asymptotically stable ω-periodic solution denoted by s-2(t). Let s2(t) be the solution of (31) satisfying s2(T3)=S(T3). The asymptotic property of s-2(t) implies that there exist a sufficient small constant ϵ>0 and T4≥T3 such that
(32)s2(t)≤s-2(t)-ϵ,fort≥T4,
where ϵ<(1/2)mint∈[0,ω]{s-2(t)}. It follows from the comparison theorem of impulsive differential equations that
(33)S(t)≥s2(t)-ϵ≥mint∈[0,ω]{s-2(t)}-ϵ≐m1,fort≥T4.
Secondly, we prove the UULB ofI(t). Recall the above results, and it follows from (1) that for t≥T4,
(34)I′(t)≥I(t)[m1β(t)-(M1+M2)k2(t)-M3a2(t)]≥I(t)[m1β(t)-(M1+M2)k2(t)-M3a2(t)-η],t≠τk,k∈ℕ,I(τk+)=I(τk)+p,
where the constant η>0 satisfies
(35)∫0ω‍[m1β(t)-(M1+M2)k2(t)-M3a2(t)-η]<0.
From Lemma 2, we know that the comparison system of (34)
(36)i2′(t)=i2(t)[m1β(t)-(M1+M2)k2(t)-M3a2(t)-ηM1+M2],t≠τk,k∈ℕ,i2(τk+)=i2(τk)+p
has a unique positive globally asymptotically stable ω-periodic solution denoted by i-2(t). The asymptotic property of i-2(t) implies that there are T5≥T4 and a sufficient small constant δ>0 such that
(37)i2(t)≥i-2(t)-δ,fort≥T5,
where δ<(1/2)mint∈[0,ω]{i-2(t)}. Let i2(t) be the solution of (36) with i2(T5)=I(T5). Using the comparison theorem of impulsive differential equations, we have
(38)I(t)≥i-2(t)-δ≥mint∈[0,ω]{i-2(t)}-δ≐m2,fort≥T5.
Finally, we prove the UULB of P(t). For t≥T5, (1), (33), and (38) reduce to
(39)P′(t)≥P(t)[-d(t)+m1c1(t)+m2c2(t)-b(t)P(t)m1],t≠τk,k∈ℕ,P(τk+)=(1+γk)P(τk).
Notice that the assumption in (10) is similar to the proof of the UULB of S(t); we obtain that there are T6≥T5 and m3>0 such that
(40)P(t)≥m3,fort≥T6.
(I) and (II) yield(41)m1≤S(t)≤M1,m2≤I(t)≤M2,m3≤P(t)≤M3,fort≥T6,
and hence system (1) is permanent. The proof is completed.


### 4.2. Global Attractivity

In this subsection, we will discuss the global attractivity of system (1) based on Theorem 6.Theorem 7.
If there exist constantsμ1>0, μ2>0, μ3>0 such that
(42)liminft→+∞{μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)}>0,liminft→+∞{μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)}>0,liminft→+∞{μ3b(t)-μ1a1(t)-μ2a2(t)}>0,m2+p=M2,
hold, then any positive solution {S(t),I(t),P(t)} of system  (1) is globally attractive, where M2, m2 are defined in (26) and (38).Proof.
Suppose that{S(t),I(t),P(t)} and {S*(t),I*(t),P*(t)} are any two positive solutions of system (1). It follows from Theorem 6 that there exists a large enough t* such that, for t>t*≥T6,
(43)m1≤S(t),S*(t)≤M1,m2≤I(t),I*(t)≤M2,m3≤P(t),P*(t)≤M3.
Consider the following Lyapunov function:
(44)V(t)=μ1|lnS(t)-lnS*(t)|+μ2|lnI(t)-lnI*(t)|+μ3|lnP(t)-lnP*(t)|,
where μ1, μ2, and μ3 are positive constants. Calculating and estimating the upper right derivative D+V(t) along the solutions of system (1), for t>t* and t≠τk, we obtain that
(45)D+V(t)=μ1sgn[S(t)-S*(t)][S′(t)S(t)-S*′(t)S*(t)]+μ2sgn[I(t)-I*(t)][I′(t)I(t)-I*′(t)I*(t)]+μ3sgn[P(t)-P*(t)][P′(t)P(t)-P*′(t)P*(t)]=μ1sgn[S(t)-S*(t)]×[-k1(t)(S(t)-S*(t))-(k1(t)+β(t))(I(t)-I*(t))-a1(t)(P(t)-P*(t))]+μ2sgn[I(t)-I*(t)]×[(β(t)-k2(t))(S(t)-S*(t))-k2(t)(I(t)-I*(t))-a2(t)(P(t)-P*(t))]+μ3sgn[P(t)-P*(t)]×[c1(t)(S(t)-S*(t))+c2(t)(I(t)-I*(t))-b(t)(P(t)-P*(t))]≤-[μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)]×|S(t)-S*(t)|-[μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)]×|I(t)-I*(t)|-[μ3b(t)-μ1a1(t)-μ2a2(t)]×|P(t)-P*(t)|.
Applying (43) and the differential mean value theorem, for any closed interval contained in t∈(τk,τk+1],k=h,h+1,…, and τh>t*, we thus have
(46)1M1|S(t)-S*(t)|≤|lnS(t)-lnS*(t)|≤1m1|S(t)-S*(t)|,1M2|I(t)-I*(t)|≤|lnI(t)-lnI*(t)|≤1m2|I(t)-I*(t)|,1M3|P(t)-P*(t)|≤|lnP(t)-lnP*(t)|≤1m3|P(t)-P*(t)|.
Hence, for t∈(τk,τk+1],k=h,h+1,…, and τh>t*, from (45)-(46), we get
(47)D+V(t)≤-[μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)]×|S(t)-S*(t)|-[μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)]×|I(t)-I*(t)|-[μ3b(t)-μ1a1(t)-μ2a2(t)]×|P(t)-P*(t)|≤-m1[μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)]×|lnS(t)-lnS*(t)|-m2[μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)]×|lnI(t)-lnI*(t)|-m3[μ3b(t)-μ1a1(t)-μ2a2(t)]×|lnP(t)-lnP*(t)|≤-ρ[|lnS(t)-lnS*(t)|+|lnI(t)-lnI*(t)|+|lnP(t)-lnP*(t)|]=-ρV(t),
where
(48)ρ=mint∈[0,ω]{m1[μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)],m2[μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)],m3[μ3b(t)-μ1a1(t)-μ2a2(t)]}>0.
For t=τk, k∈ℕ, one has
(49)V(τk+)=μ1|lnS(τk+)-lnS*(τk+)|+μ2|lnI(τk+)-lnI*(τk+)|+μ3|lnP(τk+)-lnP*(τk+)|=μ1|ln(1+αk)S(τk)-ln(1+αk)S*(τk)|+μ2|ln(I(τk)+p)-ln(I*(τk)+p)|+μ3|ln(1+γk)P(τk)-ln(1+γk)P*(τk)|=μ1|lnS(τk)-lnS*(τk)|+μ2g′(v)|v=ζk|I(τk)-I*(τk)|+μ3|lnP(τk)-lnP*(τk)|=μ1|lnS(τk)-lnS*(τk)|+μ2ζk+p|I(τk)-I*(τk)|+μ3|lnP(τk)-lnP*(τk)|,
where ζk is between I(τk) and I*(τk), g(v)=ln(v+p). It follows from (46) that
(50)μ2ζk+p|I(τk)-I*(τk)|≤μ2M2m2+p|lnI(τk)-lnI*(τk)|,
which, together with (49), leads to
(51)V(τk+)≤μ1|lnS(τk)-lnS*(τk)|+μ2M2m2+p|lnI(τk)-lnI*(τk)|+μ3|lnP(τk)-lnP*(τk)|≤λV(τk),
where λ=max{μ1,μ2M2/(m2+p),μ3}. Combining (47) and (51), we have
(52)D+V(t)≤-ρV(t),t≠τk,t>t*,V(τk+)≤λV(τk),τk>t*,k=h,h+1,….
From Lemma 1, for t>τh>t*, (52) yields
(53)V(t)≤λV(τh+)exp{-∫τhtρ‍ds}=λV(τh+)exp{-ρ(t-τh)}.
Since τh is an impulsive point, we have from (43) and (44) that,
(54)V(τh+)=μ1|lnS(τh+)-lnS*(τh+)|+μ2|lnI(τh+)-lnI*(τh+)|+μ3|lnP(τh+)-lnP*(τh+)|≤max{2μ1|lnM1|,2μ1|lnm1|}+max{2μ2|ln(M2+p)|,2μ2|ln(m2+p)|}+max{2μ3|lnM3|,2μ3|lnm3|},
which indicates that V(τh+) is bounded. As a consequence, we get
(55)V(t)≤λV(τh+)exp{-ρ(t-τh)}⟶0,ast⟶+∞,
and hence
(56)limt→+∞(|lnS(t)-lnS*(t)|+|lnI(t)-lnI*(t)|+|lnP(t)-lnP*(t)|)=0
or equivalently
(57)limt→+∞|lnS(t)-lnS*(t)|=0,limt→+∞|lnI(t)-lnI*(t)|=0,limt→+∞|lnP(t)-lnP*(t)|=0,
which, together with (46), yields
(58)limt→+∞1M1|S(t)-S*(t)|=0,limt→+∞1M2|I(t)-I*(t)|=0,limt→+∞1M3|P(t)-P*(t)|=0,
that is,
(59)limt→+∞|S(t)-S*(t)|=0,limt→+∞|I(t)-I*(t)|=0,limt→+∞|P(t)-P*(t)|=0.
This completes the proof.


### 4.3. Partial Extinction

This section concerns the partial extinction of system (1). We will see that species S, P tend to extinction while species I stabilizes at a positive solution of an impulsive system.Theorem 8.
If(60)∏k=1q‍(1+αk)exp{∫0ω‍r(t)dt}<1,(61)∏k=1q‍(1+γk)exp{∫0ω‍[-d(t)+M1*c1(t)+M2*c2(t)]dt}<1
hold, then any positive solution {S(t),I(t),P(t)} of system (1) satisfies limt→+∞S(t)=0, limt→+∞|I(t)-Γ(t)|=0, limt→+∞P(t)=0, where M1*, M2* are defined in (65) and (67), respectively, and Γ(t) is a positive solution of the impulsive system
(62)Γ′(t)=-k2Γ2(t),t≠τk,k∈ℕ,Γ(τk+)=Γ(τk)+p.Proof.
We first focus onS(t). We get from system (1) that
(63)S′(t)≤S(t)[r(t)-k1(t)S(t)],t≠τk,k∈ℕ,S(τk+)=(1+αk)S(τk).
It follows from (60) and (1) in Lemma 4 that any positive solution of the comparison system of (63)
(64)s′(t)=s(t)[r(t)-k1(t)s(t)],t≠τk,k∈ℕ,s(τk+)=(1+αk)s(τk)
satisfies limt→+∞s(t)=0. By the comparison theorem of impulsive differential equations we can obtain that limt→+∞S(t)=0. This fact implies that there exist a sufficient small constant M1*>0 and t^1>0 such that
(65)S(t)≤M1*,fort≥t^1.
Next, from system (1), we obtain that, for t≥t^1,
(66)I′(t)≤I(t)[M1*β(t)-k2(t)I(t)],t≠τk,k∈ℕ,I(τk+)=I(τk)+p.
Similar to (23)–(26), we get that there exist a constant M2*>0 and t^2≥t^1 such that
(67)I(t)≤M2*,fort≥t^2.
From system (1), (65), and (67), we have
(68)P′(t)≤P(t)[-d(t)+M1*c1(t)+M2*c2(t)-b(t)P(t)M1*],t≠τk,k∈ℕ,P(τk+)=(1+γk)P(τk).
Similar to S(t), when the assumption (61) holds, a simple proof can verify limt→+∞P(t)=0. This fact implies that there are a sufficient small constant M3*>0 and t^3≥t^2 such that
(69)P(t)≤M3*,fort≥t^3.
From system (1), (65), and (69), we obtain that for t≥t^3(70)I′(t)≥I(t)[-M1*k2(t)-M3*a2(t)-k2(t)I(t)],t≠τk,k∈ℕ,I(τk+)=I(τk)+p.
Similar to (23)–(26), we can obtain that there exist a constant m2*>0 and T*≥t^3 such that
(71)I(t)≥m2*,fort≥T*,
which, together with (67), leads to
(72)m2*≤I(t)≤M2*,fort≥T*.
Based on the above discussion, we will investigate the global attractivity of I(t). Let
(73)Z(t)=I(t)-Γ(t).
Then,
(74)Z′(t)=-k2(t)(I(t)+Γ(t))Z(t)+(β(t)-k2(t))S(t)I(t)-a2(t)I(t)P(t),t≠τk,k∈ℕ,Z(τk+)=(I(τk)+p)-(Γ(τk)+p)=Z(τk).
Clearly, Z(t) is continuous. Recalling (74) and using the formula of solutions of first-order linear differential equation, we get, for t≥T*,
(75)Z(t)=Z(T*)exp{∫T*t‍[-k2(v)(I(v)+Γ(v))]dv}+∫T*t‍[(β(v)-k2(v))S(v)I(v)-a2(v)I(v)P(v)]×exp{∫vt‍[-k2(s)(I(s)+Γ(s))]ds}dv,
Furthermore, applying Lemma 1, one obtains
(76)|Z(t)|≤|Z(T*)|exp{∫T*t‍[-k2(v)(I(v)+Γ(v))]dv}+∫T*t‍[|β(v)-k2(v)|S(v)I(v)+a2(v)I(v)P(v)]×exp{∫vt‍[-k2(s)(I(s)+Γ(s))]ds}dv,≤|Z(T*)|exp{-m2*∫T*t‍k2(v)dv}+M2*∫T*t‍[(βU+k2U)S(v)+a2UP(v)]×exp{-m2*∫vt‍k2(s)ds}dv,≤|Z(T*)|exp{1+m2*k2Uω-𝒦(t-T*)}+M2*exp{1+m2*k2Uω}×∫T*t‍[(βU+k2U)S(v)+a2UP(v)]×exp{𝒦(v-t)}dv,
where
(77)k2U=maxt∈[0,ω]{k2(t)},βU=maxt∈[0,ω]{β(t)},a2U=maxt∈[0,ω]{a2(t)},0<𝒦<min{∫0ω‍m2*k2(t)dtω,1ω}.
Notice that limt→+∞S(t)=0, limt→+∞P(t)=0, and thus a calculation shows that
(78)limt→+∞∫T*t‍S(v)exp{𝒦(v-t)}dv=0,limt→+∞∫T*t‍P(v)exp{𝒦(v-t)}dv=0.
Hence,
(79)limt→+∞∫T*t[(βU+k2U)S(v)+a2UP(v)]‍exp{𝒦(v-t)}dv=0.
Combining (76) and (79), we obtain that limt→+∞|Z(t)|=0, which implies limt→+∞|I(t)-Γ(t)|=0. This completes the proof.


## 4.1. Permanence

Theorem 6.
If(9)∏k=1q‍(1+αk)exp{∫0ω‍[r(t)-M2(k1(t)+β(t))-M3a1(t)(k1(t)+β(t))]dt∫0ω‍}>1,(10)∏k=1q‍(1+γk)exp{∫0ω‍[-d(t)+m1c1(t)+m2c2(t)]dt∫0ω‍}>1
hold, then system (1) is permanent, where M2, M3, m1, m2 are defined in (26), (29), (33), and (38), respectively.Proof.
For (1), it is easy to verify S(t)>0, I(t)>0, P(t)>0 for all t>0 if S(0+)=S(0)>0, I(0+)=I(0)>0, P(0+)=P(0)>0. To finish the proof of Theorem 6, we consider the following two steps (i.e., (I) and (II)).(I)
Consider the uniformly ultimately upper boundary (or UUUB) ofS(t), I(t), P(t).
First of all, we discuss the UUUB ofS(t). It follows from the first equation of (1) and impulsive condition that
(11)S′(t)≤r(t)S(t),t≠τk,k∈ℕ,S(τk+)=(1+αk)S(τk),(12)S′(t)≤S(t)[r(t)-k1(t)S(t)],t≠τk,k∈ℕ,S(τk+)=(1+αk)S(τk).
Now, we consider two cases to obtain the UUUB of S(t).
Case  1. Consider ∏k=1q‍(1+αk)exp{∫0ω‍r(t)dt}≤1.
For (11), applying Lemma 1, we have
(13)S(t)≤S(0)∏0<τk<t‍(1+αk)exp{∫0t‍r(s)ds}.
When t∈(nω,(n+1)ω], n∈ℕ∪{0}, we set
(14)𝒜=maxt∈[0,ω]∏0<τk<t‍(1+αk),ℛ=maxt∈(nω,(n+1)ω]exp{∫0ω‍r(s)ds}.
It follows from (13) that
(15)S(t)≤S(0)∏0<τk<nω‍(1+αk)exp{∫0nω‍r(s)ds}×∏nω≤τk<t‍(1+αk)exp{∫nωt‍r(s)ds}≤S(0)(∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds})n×∏0<τk<t-nω‍(1+αk)exp{∫nωt‍r(s)ds}≤S(0)(∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds})n𝒜ℛ.
If ∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds}=1, then
(16)S(t)≤S(0)𝒜ℛ.
If ∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds}<1, then
(17)limt→+∞S(0)(∏k=1q‍(1+αk)exp{∫0ω‍r(s)ds})n𝒜ℛ=0.
This fact implies that limt→+∞S(t)=0. That is, there are a sufficient small constant σ1>0 and t1>0 such that
(18)S(t)≤σ1,fort≥t1.Case  2. Consider ∏k=1q‍(1+αk)exp{∫0ω‍r(t)dt}>1.
It follows from (2) in Lemma4 that the comparison system of (12)
(19)s1′(t)=s1(t)[r(t)-k1(t)s1(t)],t≠τk,k∈ℕ,s1(τk+)=(1+αk)s1(τk)
has a unique positive globally asymptotically stable ω-periodic solution denoted by s-1(t). Let s1(t) be the solution of (19) with s1(0)=S(0)>0. The asymptotic property of s-1(t) shows that there exist a sufficient small constant σ2>0 and t2>0 such that
(20)s1(t)≤s-1(t)+σ2,fort≥t2.
Applying the comparison theorem of impulsive differential equations, one has
(21)S(t)≤s-1(t)+σ2,fort≥t2.
Equations (18) and (21) show that there must be T1=max{t1,t2} and M1=maxt∈[0,ω]{S(0)𝒜ℛ, σ1, s-1(t)+σ2}>0 such that
(22)S(t)≤M1,fort≥T1.
Next, we discuss the UUUB ofI(t). From the second equation of (1) and impulsive condition, for t≥T1, one has
(23)I′(t)≤I(t)[M1β(t)-k2(t)I(t)],t≠τk,k∈ℕ,I(τk+)=I(τk)+p.
Using (2) in Lemma 3, we can see that the comparison system of (23)
(24)i1′(t)=i1(t)[M1β(t)-k2(t)i1(t)],t≠τk,k∈ℕ,i1(τk+)=i1(τk)+p
is permanent, which implies that there are M2>0 and T2≥T1 such that
(25)i1(t)≤M2,fort≥T2.
Using the comparison theorem of impulsive differential equations, we get
(26)I(t)≤M2,fort≥T2.
Finally, we verify the UUUB ofP(t). For t≥T2, from the third equation of (1) and impulsive condition, we obtain
(27)P′(t)≤P(t)[-d(t)+M1c1(t)+M2c2(t)],t≠τk,k∈ℕ,P(τk+)=(1+γk)P(τk)(28)P′(t)≤P(t)[-d(t)+M1c1(t)+M2c2(t)-b(t)P(t)],t≠τk,k∈ℕ,P(τk+)=(1+γk)P(τk).
Similar to the proof of the UUUB of S(t), it follows from (10) that there exist T3>0 and M3>0 such that
(29)P(t)≤M3,fort≥T3.(II)
Consider the uniformly ultimately lower boundary (or UULB) ofS(t), I(t), P(t).
Firstly, we prove the UULB ofS(t). From (I), for t≥T3, one has
(30)S′(t)≥S(t)[r(t)-M2(k1(t)+β(t))-M3a1(t)-k1(t)S(t)],t≠τk,k∈ℕ,S(τk+)=(1+αk)S(τk).
By (2) in Lemma 4 and the assumption in (9), we know easily that the comparison system of (30)
(31)s2′(t)=s2(t)[r(t)-M2(k1(t)+β(t))-M3a1(t)-k1(t)s2(t)],t≠τk,k∈ℕ,s2(τk+)=(1+αk)s2(τk)
has a unique positive globally asymptotically stable ω-periodic solution denoted by s-2(t). Let s2(t) be the solution of (31) satisfying s2(T3)=S(T3). The asymptotic property of s-2(t) implies that there exist a sufficient small constant ϵ>0 and T4≥T3 such that
(32)s2(t)≤s-2(t)-ϵ,fort≥T4,
where ϵ<(1/2)mint∈[0,ω]{s-2(t)}. It follows from the comparison theorem of impulsive differential equations that
(33)S(t)≥s2(t)-ϵ≥mint∈[0,ω]{s-2(t)}-ϵ≐m1,fort≥T4.
Secondly, we prove the UULB ofI(t). Recall the above results, and it follows from (1) that for t≥T4,
(34)I′(t)≥I(t)[m1β(t)-(M1+M2)k2(t)-M3a2(t)]≥I(t)[m1β(t)-(M1+M2)k2(t)-M3a2(t)-η],t≠τk,k∈ℕ,I(τk+)=I(τk)+p,
where the constant η>0 satisfies
(35)∫0ω‍[m1β(t)-(M1+M2)k2(t)-M3a2(t)-η]<0.
From Lemma 2, we know that the comparison system of (34)
(36)i2′(t)=i2(t)[m1β(t)-(M1+M2)k2(t)-M3a2(t)-ηM1+M2],t≠τk,k∈ℕ,i2(τk+)=i2(τk)+p
has a unique positive globally asymptotically stable ω-periodic solution denoted by i-2(t). The asymptotic property of i-2(t) implies that there are T5≥T4 and a sufficient small constant δ>0 such that
(37)i2(t)≥i-2(t)-δ,fort≥T5,
where δ<(1/2)mint∈[0,ω]{i-2(t)}. Let i2(t) be the solution of (36) with i2(T5)=I(T5). Using the comparison theorem of impulsive differential equations, we have
(38)I(t)≥i-2(t)-δ≥mint∈[0,ω]{i-2(t)}-δ≐m2,fort≥T5.
Finally, we prove the UULB of P(t). For t≥T5, (1), (33), and (38) reduce to
(39)P′(t)≥P(t)[-d(t)+m1c1(t)+m2c2(t)-b(t)P(t)m1],t≠τk,k∈ℕ,P(τk+)=(1+γk)P(τk).
Notice that the assumption in (10) is similar to the proof of the UULB of S(t); we obtain that there are T6≥T5 and m3>0 such that
(40)P(t)≥m3,fort≥T6.
(I) and (II) yield(41)m1≤S(t)≤M1,m2≤I(t)≤M2,m3≤P(t)≤M3,fort≥T6,
and hence system (1) is permanent. The proof is completed.


## 4.2. Global Attractivity

In this subsection, we will discuss the global attractivity of system (1) based on Theorem 6.Theorem 7.
If there exist constantsμ1>0, μ2>0, μ3>0 such that
(42)liminft→+∞{μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)}>0,liminft→+∞{μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)}>0,liminft→+∞{μ3b(t)-μ1a1(t)-μ2a2(t)}>0,m2+p=M2,
hold, then any positive solution {S(t),I(t),P(t)} of system  (1) is globally attractive, where M2, m2 are defined in (26) and (38).Proof.
Suppose that{S(t),I(t),P(t)} and {S*(t),I*(t),P*(t)} are any two positive solutions of system (1). It follows from Theorem 6 that there exists a large enough t* such that, for t>t*≥T6,
(43)m1≤S(t),S*(t)≤M1,m2≤I(t),I*(t)≤M2,m3≤P(t),P*(t)≤M3.
Consider the following Lyapunov function:
(44)V(t)=μ1|lnS(t)-lnS*(t)|+μ2|lnI(t)-lnI*(t)|+μ3|lnP(t)-lnP*(t)|,
where μ1, μ2, and μ3 are positive constants. Calculating and estimating the upper right derivative D+V(t) along the solutions of system (1), for t>t* and t≠τk, we obtain that
(45)D+V(t)=μ1sgn[S(t)-S*(t)][S′(t)S(t)-S*′(t)S*(t)]+μ2sgn[I(t)-I*(t)][I′(t)I(t)-I*′(t)I*(t)]+μ3sgn[P(t)-P*(t)][P′(t)P(t)-P*′(t)P*(t)]=μ1sgn[S(t)-S*(t)]×[-k1(t)(S(t)-S*(t))-(k1(t)+β(t))(I(t)-I*(t))-a1(t)(P(t)-P*(t))]+μ2sgn[I(t)-I*(t)]×[(β(t)-k2(t))(S(t)-S*(t))-k2(t)(I(t)-I*(t))-a2(t)(P(t)-P*(t))]+μ3sgn[P(t)-P*(t)]×[c1(t)(S(t)-S*(t))+c2(t)(I(t)-I*(t))-b(t)(P(t)-P*(t))]≤-[μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)]×|S(t)-S*(t)|-[μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)]×|I(t)-I*(t)|-[μ3b(t)-μ1a1(t)-μ2a2(t)]×|P(t)-P*(t)|.
Applying (43) and the differential mean value theorem, for any closed interval contained in t∈(τk,τk+1],k=h,h+1,…, and τh>t*, we thus have
(46)1M1|S(t)-S*(t)|≤|lnS(t)-lnS*(t)|≤1m1|S(t)-S*(t)|,1M2|I(t)-I*(t)|≤|lnI(t)-lnI*(t)|≤1m2|I(t)-I*(t)|,1M3|P(t)-P*(t)|≤|lnP(t)-lnP*(t)|≤1m3|P(t)-P*(t)|.
Hence, for t∈(τk,τk+1],k=h,h+1,…, and τh>t*, from (45)-(46), we get
(47)D+V(t)≤-[μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)]×|S(t)-S*(t)|-[μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)]×|I(t)-I*(t)|-[μ3b(t)-μ1a1(t)-μ2a2(t)]×|P(t)-P*(t)|≤-m1[μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)]×|lnS(t)-lnS*(t)|-m2[μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)]×|lnI(t)-lnI*(t)|-m3[μ3b(t)-μ1a1(t)-μ2a2(t)]×|lnP(t)-lnP*(t)|≤-ρ[|lnS(t)-lnS*(t)|+|lnI(t)-lnI*(t)|+|lnP(t)-lnP*(t)|]=-ρV(t),
where
(48)ρ=mint∈[0,ω]{m1[μ1k1(t)-μ2|β(t)-k2(t)|-μ3c1(t)],m2[μ2k2(t)-μ1(k1(t)+β(t))-μ3c2(t)],m3[μ3b(t)-μ1a1(t)-μ2a2(t)]}>0.
For t=τk, k∈ℕ, one has
(49)V(τk+)=μ1|lnS(τk+)-lnS*(τk+)|+μ2|lnI(τk+)-lnI*(τk+)|+μ3|lnP(τk+)-lnP*(τk+)|=μ1|ln(1+αk)S(τk)-ln(1+αk)S*(τk)|+μ2|ln(I(τk)+p)-ln(I*(τk)+p)|+μ3|ln(1+γk)P(τk)-ln(1+γk)P*(τk)|=μ1|lnS(τk)-lnS*(τk)|+μ2g′(v)|v=ζk|I(τk)-I*(τk)|+μ3|lnP(τk)-lnP*(τk)|=μ1|lnS(τk)-lnS*(τk)|+μ2ζk+p|I(τk)-I*(τk)|+μ3|lnP(τk)-lnP*(τk)|,
where ζk is between I(τk) and I*(τk), g(v)=ln(v+p). It follows from (46) that
(50)μ2ζk+p|I(τk)-I*(τk)|≤μ2M2m2+p|lnI(τk)-lnI*(τk)|,
which, together with (49), leads to
(51)V(τk+)≤μ1|lnS(τk)-lnS*(τk)|+μ2M2m2+p|lnI(τk)-lnI*(τk)|+μ3|lnP(τk)-lnP*(τk)|≤λV(τk),
where λ=max{μ1,μ2M2/(m2+p),μ3}. Combining (47) and (51), we have
(52)D+V(t)≤-ρV(t),t≠τk,t>t*,V(τk+)≤λV(τk),τk>t*,k=h,h+1,….
From Lemma 1, for t>τh>t*, (52) yields
(53)V(t)≤λV(τh+)exp{-∫τhtρ‍ds}=λV(τh+)exp{-ρ(t-τh)}.
Since τh is an impulsive point, we have from (43) and (44) that,
(54)V(τh+)=μ1|lnS(τh+)-lnS*(τh+)|+μ2|lnI(τh+)-lnI*(τh+)|+μ3|lnP(τh+)-lnP*(τh+)|≤max{2μ1|lnM1|,2μ1|lnm1|}+max{2μ2|ln(M2+p)|,2μ2|ln(m2+p)|}+max{2μ3|lnM3|,2μ3|lnm3|},
which indicates that V(τh+) is bounded. As a consequence, we get
(55)V(t)≤λV(τh+)exp{-ρ(t-τh)}⟶0,ast⟶+∞,
and hence
(56)limt→+∞(|lnS(t)-lnS*(t)|+|lnI(t)-lnI*(t)|+|lnP(t)-lnP*(t)|)=0
or equivalently
(57)limt→+∞|lnS(t)-lnS*(t)|=0,limt→+∞|lnI(t)-lnI*(t)|=0,limt→+∞|lnP(t)-lnP*(t)|=0,
which, together with (46), yields
(58)limt→+∞1M1|S(t)-S*(t)|=0,limt→+∞1M2|I(t)-I*(t)|=0,limt→+∞1M3|P(t)-P*(t)|=0,
that is,
(59)limt→+∞|S(t)-S*(t)|=0,limt→+∞|I(t)-I*(t)|=0,limt→+∞|P(t)-P*(t)|=0.
This completes the proof.


## 4.3. Partial Extinction

This section concerns the partial extinction of system (1). We will see that species S, P tend to extinction while species I stabilizes at a positive solution of an impulsive system.Theorem 8.
If(60)∏k=1q‍(1+αk)exp{∫0ω‍r(t)dt}<1,(61)∏k=1q‍(1+γk)exp{∫0ω‍[-d(t)+M1*c1(t)+M2*c2(t)]dt}<1
hold, then any positive solution {S(t),I(t),P(t)} of system (1) satisfies limt→+∞S(t)=0, limt→+∞|I(t)-Γ(t)|=0, limt→+∞P(t)=0, where M1*, M2* are defined in (65) and (67), respectively, and Γ(t) is a positive solution of the impulsive system
(62)Γ′(t)=-k2Γ2(t),t≠τk,k∈ℕ,Γ(τk+)=Γ(τk)+p.Proof.
We first focus onS(t). We get from system (1) that
(63)S′(t)≤S(t)[r(t)-k1(t)S(t)],t≠τk,k∈ℕ,S(τk+)=(1+αk)S(τk).
It follows from (60) and (1) in Lemma 4 that any positive solution of the comparison system of (63)
(64)s′(t)=s(t)[r(t)-k1(t)s(t)],t≠τk,k∈ℕ,s(τk+)=(1+αk)s(τk)
satisfies limt→+∞s(t)=0. By the comparison theorem of impulsive differential equations we can obtain that limt→+∞S(t)=0. This fact implies that there exist a sufficient small constant M1*>0 and t^1>0 such that
(65)S(t)≤M1*,fort≥t^1.
Next, from system (1), we obtain that, for t≥t^1,
(66)I′(t)≤I(t)[M1*β(t)-k2(t)I(t)],t≠τk,k∈ℕ,I(τk+)=I(τk)+p.
Similar to (23)–(26), we get that there exist a constant M2*>0 and t^2≥t^1 such that
(67)I(t)≤M2*,fort≥t^2.
From system (1), (65), and (67), we have
(68)P′(t)≤P(t)[-d(t)+M1*c1(t)+M2*c2(t)-b(t)P(t)M1*],t≠τk,k∈ℕ,P(τk+)=(1+γk)P(τk).
Similar to S(t), when the assumption (61) holds, a simple proof can verify limt→+∞P(t)=0. This fact implies that there are a sufficient small constant M3*>0 and t^3≥t^2 such that
(69)P(t)≤M3*,fort≥t^3.
From system (1), (65), and (69), we obtain that for t≥t^3(70)I′(t)≥I(t)[-M1*k2(t)-M3*a2(t)-k2(t)I(t)],t≠τk,k∈ℕ,I(τk+)=I(τk)+p.
Similar to (23)–(26), we can obtain that there exist a constant m2*>0 and T*≥t^3 such that
(71)I(t)≥m2*,fort≥T*,
which, together with (67), leads to
(72)m2*≤I(t)≤M2*,fort≥T*.
Based on the above discussion, we will investigate the global attractivity of I(t). Let
(73)Z(t)=I(t)-Γ(t).
Then,
(74)Z′(t)=-k2(t)(I(t)+Γ(t))Z(t)+(β(t)-k2(t))S(t)I(t)-a2(t)I(t)P(t),t≠τk,k∈ℕ,Z(τk+)=(I(τk)+p)-(Γ(τk)+p)=Z(τk).
Clearly, Z(t) is continuous. Recalling (74) and using the formula of solutions of first-order linear differential equation, we get, for t≥T*,
(75)Z(t)=Z(T*)exp{∫T*t‍[-k2(v)(I(v)+Γ(v))]dv}+∫T*t‍[(β(v)-k2(v))S(v)I(v)-a2(v)I(v)P(v)]×exp{∫vt‍[-k2(s)(I(s)+Γ(s))]ds}dv,
Furthermore, applying Lemma 1, one obtains
(76)|Z(t)|≤|Z(T*)|exp{∫T*t‍[-k2(v)(I(v)+Γ(v))]dv}+∫T*t‍[|β(v)-k2(v)|S(v)I(v)+a2(v)I(v)P(v)]×exp{∫vt‍[-k2(s)(I(s)+Γ(s))]ds}dv,≤|Z(T*)|exp{-m2*∫T*t‍k2(v)dv}+M2*∫T*t‍[(βU+k2U)S(v)+a2UP(v)]×exp{-m2*∫vt‍k2(s)ds}dv,≤|Z(T*)|exp{1+m2*k2Uω-𝒦(t-T*)}+M2*exp{1+m2*k2Uω}×∫T*t‍[(βU+k2U)S(v)+a2UP(v)]×exp{𝒦(v-t)}dv,
where
(77)k2U=maxt∈[0,ω]{k2(t)},βU=maxt∈[0,ω]{β(t)},a2U=maxt∈[0,ω]{a2(t)},0<𝒦<min{∫0ω‍m2*k2(t)dtω,1ω}.
Notice that limt→+∞S(t)=0, limt→+∞P(t)=0, and thus a calculation shows that
(78)limt→+∞∫T*t‍S(v)exp{𝒦(v-t)}dv=0,limt→+∞∫T*t‍P(v)exp{𝒦(v-t)}dv=0.
Hence,
(79)limt→+∞∫T*t[(βU+k2U)S(v)+a2UP(v)]‍exp{𝒦(v-t)}dv=0.
Combining (76) and (79), we obtain that limt→+∞|Z(t)|=0, which implies limt→+∞|I(t)-Γ(t)|=0. This completes the proof.


## 5. Examples and Numerical Simulations

In this paper, we investigate the dynamic behaviors of a periodic predator-prey system subject to impulsive perturbations, in which disease only spreads among the prey species. A good understanding of the permanence, global attractivity, and partial extinction of the above system is obtained. Our three main results (i.e., Theorems6, 7, and 8) show that the impulsive perturbations play important roles in shaping the dynamics. To substantiate the theoretical results, we consider the following system:
(80)S′(t)=S(t)[0.83+0.2sin2πt-(0.96+0.01cos2πt)×(S(t)+I(t))-(0.19+0.01sin2πt)P(t)-(0.17+0.02sin2πt)I(t)],I′(t)=I(t)[(0.17+0.02sin2πt)S(t)-(0.59+0.02sin2πt)(S(t)+I(t))-(0.26+0.02cos2πt)P(t)],P′(t)=P(t)[-(0.32+0.18sin2πt)-(1.78+0.05cos2πt)P(t)+(0.16+0.02sin2πt)S(t)+(0.12+0.02sin2πt)I(t)],t≠τk,S(τk+)=(1+αk)S(τk),I(τk+)=I(τk)+p,P(τk+)=(1+γk)P(τk),t=τk,k∈ℕ.
Obviously, the period ω=1. Let q=1, τk=k∈ℕ; then, τk+1=τk+1. When αk=0, p=0, γk=0, from Figures 1(a)–1(c), that species S of system (80) is permanent while I, P tend to extinction.Partial extinction of system (80) with αk=0, p=0, and γk=0. (a) Time series of S for t∈[0,30]. (b) Time series of I for t∈[0,30]. (c) Time series of P for t∈[0,30].
(a)
(b)
(c)Provided that all the other coefficients remain unchanged, we chooseαk=0.26, p=0.1, and γk=0.53, a calculation showing that the assumptions of Theorem 6 hold. So species S, I, P are permanent (see Figures 2(a)–2(c)), and moreover, we can prove that the assumptions of Theorem 7 are satisfied, so system (80) is globally attractive. From Figures 3(a)–3(c), it is true that the positive solution {S(t),I(t),P(t)} with {S(0),I(0),P(0)}={0.69,0.42,0.265} tends to the other positive solution {S*(t),I*(t),P(t)} with {S*(0),I*(0),P*(0)}={1.24,0.11,0.139}. To verify partial extinction we choose αk=-0.45, p=0.13, and γk=-0.04; system (62) with Γ(0)=0.68 has a positive solution Γ(t) (see Figure 4(b)). We can see that the assumptions of Theorem 8 hold, so species I stabilizes at a certain solution of a impulsive system (see Figures 4(b) and 4(c)) while species S, P tend towards extinction (see Figures 4(a) and 4(d)). The above facts demonstrate that the use of impulsive control strategy can change the dynamic behaviors of the system, and hence a suitable harvesting or stocking policy is important.Permanence of system (80) with αk=0.26, p=0.1, and γk=0.53. (a) Time series of S for t∈[0,30]. (b) Time series of I for t∈[0,30]. (c) Time series of P for t∈[0,30].
(a)
(b)
(c)Global attractivity of system (80) with αk=0.26, p=0.1, and γk=0.53. (a) Positive solution S(t) with S(0)=0.69 tends to the other positive solution S*(t) with S*(0)=1.24 for t∈[0,20]. (b) Positive solution I(t) with I(0)=0.42 tends to the other positive solution I(t) with I*(0)=0.11 for t∈[0,20]. (c) Positive solution P(t) with P(0)=0.265 tends to the other positive solution P*(t) with P*(0)=0.139 for t∈[0,20].
(a)
(b)
(c)Partial extinction of system (80) with αk=-0.45, p=0.13, and γk=-0.04. (a) Time series of S for t∈[0,20]. (b) Time series of Γ for t∈[0,20]. (c) Time series of Γ and I for t∈[0,20]. (d) Time series of P for t∈[0,20].
(a)
(b)
(c)
(d)



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*Source: 656920-2013-10-10.xml*