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

Authors: Hongli Li; Long Zhang; Zhidong Teng; Yaolin Jiang
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## 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.


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## Body

## 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:
(1)x˙(t)=xg(x)-φ(x)ym,y˙(t)=y(-d+βφ(x)ym-1-q(y)),
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:
(2)x˙(t)=x(t)(r1(t)-b1(t)x(t))-c1(t)x(t)k+x(t)ym(t),y˙(t)=y(t)(-r2(t)-b2(t)y(t))+c2(t)x(t)k+x(t)ym(t).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:
(3)x˙(t)=x(t)(r1(t)-b1(t)x(t))-c1(t)x(t)k+x(t)ym(t),y˙(t)=y(t)(-r2(t)-b2(t)y(t))+c2(t)x(t)k+x(t)ym(t),hhhhhhhhht≠τk,x(τk+)=(1+hk)x(τk),y(τk+)=(1+gk)y(τk),t=τk,k=1,2,…,
where x and y are the sizes of the prey and predator populations, respectively. 0<m<1 and k is a positive constant. ri(t),  bi(t),  and  ci(t)  (i=1,2) are continuous periodic functions defined on [0,+∞) with a common period T>0. bi(t),  ci(t),  (i=1,2) are strictly positive. Also τk→+∞  (t→+∞), 0<τ1<τ2<τ3<⋯<τk<τk+1<⋯. Assume that hk,  gk  (k=1,2…) are constants. 1+hk>0 and 1+gk>0, and there exists an integer q>0 such that hk+q=hk, gk+q=gk, τk+q=τk+T. The interpretations of parameters can be seen in Table 1.Table 1
Parameter values used in the simulations of system (3).
Parameter
Interpretation
Value
r
1
(
t
)
Growth rate of the prey
1
+
sin
⁡
⁡
(
2
π
t
)
r
2
(
t
)
Death rate of the predator
0.03
+
0.01
sin
⁡
⁡
(
2
π
t
)
b
1
(
t
)
Intracompetition coefficient among the prey
1
-
0.1
sin
⁡
⁡
(2πt)
b
2
(
t
)
Intracompetition coefficient among the predators
0.4
-
0.1
sin
⁡
⁡
(
2
π
t
)
c
1
(
t
)
Capture rate of predator on prey
0.7
+
0.1
cos
⁡
(
2
π
t
)
c
2
(
t
)
Conversion rate of nutrition
0.2
+
0.1
sin
⁡
⁡
(
2
π
t
)
k
The environment carrying capacity
10
m
Mutual interference constant
1/3
h
k
Impulse perturbations on prey
—
g
k
Impulse perturbations on predator
—The organization of this paper is as follows. In Section2, 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.LetR+=[0,∞), R+2={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+×R+2→R+; then V is said to belong to class V0 if(1)
Vis continuous in (τk,τk+1]×R+2 and for each X∈R+2, k=1,2,…, there exists lim⁡(t,Y)→(τk+,X)V(t,Y)=V(τk+,X);(2)
V is locally Lipschitzian in X.Definition 1.
LetV∈V0; then, for (t,X)∈(τk,τk+1]×R+2, the upper right derivative of V(t,X) with respect to the impulsive differential system (3) is defined as
(4)D+V(t,x)=lim⁡h→0+sup⁡1h[V(t+h,X+hf(t,X))-V(t,X)].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
(5)m1≤liminf⁡t→+∞x(t)≤limsup⁡t→+∞x(t)≤M1,m2≤liminf⁡t→+∞y(t)≤limsup⁡t→+∞y(t)≤M2.Lemma 3 ([29]).
LetV∈V0 and X(t) is a solution of (3) with initial value X(0+)=X0. Assume that
(6)D+V(t,X(t))≤g(t,V(t,X(t))),t≠τk,k=1,2,…,V(t,X(t+))≤ψk(V(t,X(t))),t=τk,k=1,2,…,
where g:R+×R+→R is continuous in (τk,τk+1]×R+ and for u∈R+, k=1,2,…, lim⁡(t,v)→(τk+,u)g(t,v)=g(τk+,u) exists and ψk:R+→R+ is nondecreasing. Let r(t) be the maximal solution of scalar impulsive differential equation
(7)u˙(t)=g(t,u(t)),t≠τk,k=1,2,…,u(t+)=ψk(u(t))≥0,t=τk,tk>0,k=1,2,…,u(0+)=u0
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.Letf(t) be continuous T-periodic function defined on the R+; we introduce the following notation:
(8)m[f(t)]=1T∫0T‍f(t)dt.Consider the following periodic logistic equation with impulses:(9)x˙(t)=x(t)(r1(t)-b1(t)x(t)),t≠τk,x(τk+)=(1+hk)x(τk),t=τk,k=1,2,….
We have the following results.Lemma 4 (see [30]).
(
1) If
(A1)∑k=1q‍ln⁡(1+hk)+Tm[r1(t)]>0,
then system (9) has a unique T-periodic solution x*(t), which is globally asymptotically stable.
(2) If
(A2)∑k=1q‍ln⁡(1+hk)+Tm[r1(t)]≤0,
then x*(t)≡0.Consider a periodic Logistic equation with impulses:(10)x˙(t)=x(t)(b(t)-a(t)(x(t))θ),t≠τk,x(τk+)=(1+hk′)x(τk),t=τk,k=1,2,…,
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  hk+q′=hk′, τk+q=τk+T. We have the following results.Lemma 5 (see [31]).
(
1) If
(11)∑k=1q‍ln⁡(1+hk′)+Tm[b]>0,
then system (10) has a unique T-periodic solution x*~(t), which is globally asymptotically stable.
(2) If
(12)∑k=1q‍ln⁡(1+hk′)+Tm[b(t)]<0,
then x*~(t)≡0.Lemma 6 (see [31]).
Letx*(t) be the unique T-periodic positive solution of system (9) and xɛ*~(t) the unique T-periodic positive solution of the following system:
(13)x˙(t)=x(t)(bɛ(t)-a(t)(x(t))θ),t≠τk,x(τk+)=(1+hk′)x(τk),t=τk,k=1,2,…,
where bɛ(t) is the continuous T-periodic function with m[bɛ(t)]>0 and lim⁡ɛ→0bɛ(t)=b(t). Then
(14)lim⁡ɛ→0|xɛ*~(t)-x*(t)|=0.Furthermore, we consider the following periodic logistic equation with impulses:(15)v˙(t)=v(t)(r2(t)-c2(t)x*(t)k+x*(t)v1-m(t)),t≠τk,v(τk+)=11+gkv(τk),t=τk,k=1,2,….
By Lemma 5, we have the following Corollary.Corollary 7.
If(H1)-∑k=1q‍ln⁡(1+gk)+Tm[r2(t)]>0,
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:(16)x˙(t)=x(t)(r1(t)-c1(t)ky*~(t)m-b1(t)x(t)),t≠τk,x(τk+)=(1+hk)x(τk),t=τk,k=1,2,….By Lemma5, we have the following Corollary.Corollary 8.
If(H2)∑k=1q‍ln⁡(1+hk)+Tm[r1(t)-c1(t)ky*~(t)m]>0,
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
(17)x˙(t)=p(t)x(t),t≠τk,
where
(18)p(t)=r1(t)-b1(t)x(t)-c1(t)k+x(t)ym(t);
thus, from x(0+)>0, we have
(19)x(t)=∏0<τk<t‍(1+hk)x(0+)exp⁡⁡(∫0t‍p(s)ds)>0.
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 Mi  (i=1,2) such that
(20)limsup⁡t→+∞x(t)≤M1,limsup⁡t→+∞y(t)≤M2.Proof.
First, we show thatx(t) is ultimate upper bounded. It follows from (3) that
(21)x˙(t)≤x(t)(r1(t)-b1(t)x(t)),t≠τk,x(τk+)=(1+hk)x(τk),t=τk,k=1,2,….
Consider the following auxiliary system:
(22)z˙(t)=z(t)(r1(t)-b1(t)z(t)),t≠τk,z(τk+)=(1+hk)z(τk),t=τk,k=1,2,….
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 ɛ>0 being small enough, there exists a T1>0 such that
(23)|z(t)-x*(t)|<ɛ,
for any t>T1. Let M1=sup⁡{x*(t):t∈[0,T]}; we have
(24)x(t)≤z(t)<x*(t)+ɛ≤M1+ɛ
for any t>T1. Let ɛ→0; we obtain
(25)limsup⁡t→+∞x(t)≤M1.
Next, we prove thaty(t) is ultimately upper bounded. Define y(t)=1/u(t). From system (3), we have
(26)u˙(t)=u(t)(r2(t)+b2(t)1u(t)hhhhh-c2(t)x(t)k+x(t)u1-m(t)),t≠τk,u(τk+)=11+gku(τk),t=τk,k=1,2,….
For any t>T1 and t≠τk(k=1,2,…), we have
(27)u˙(t)≥u(t)(r2(t)-c2(t)x(t)k+x(t)u1-m(t))≥u(t)(r2(t)-c2(t)x*(t)+ɛk+x*(t)+ɛu1-m(t)).
Consider the following auxiliary system:
(28)ψ˙(t)=ψ(t)(r2(t)-c2(t)x*(t)+ɛk+x*(t)+ɛψ1-m(t)),hhhhhhhhhhhhhhhhhhhhhhhht≠τk,ψ(τk+)=11+gkψ(τk),t=τk,k=1,2,….
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 vɛ*~(t), which is globally stable. Therefore, letting ɛ→0, from Lemma 6, we have vɛ*~(t)→v*~(t). Letting M2*=inf⁡{v*(t):t∈[0,T]}, for any ɛ0>0 being small enough, there exists T2>T1, and we have
(29)u(t)=1y(t)≥ψ(t)>v*~(t)-ɛ0≥M2*-ɛ0
for any t>T2. Let ɛ0→0; we have
(30)limsup⁡t→+∞y(t)≤1M2*≜M2.
This completes the proof of Lemma 10.Theorem 11.
Assume that(H1) and (H2) hold, and
(H3)∑k=1q‍ln⁡(1+gk)+Tm[-r2(t)+c2(t)x*(t)k+x*(t)y*~(t)m-1]>0,(H4)m[r1(t)-c1(t)k(y*~(t))m]>0,(H5)m[-r2(t)+c2(t)x*(t)k+x*(t)(y*~(t))m-1]>0,
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 Lemma9, there exist constants Mi(i=1,2) such that
(31)limsup⁡t→+∞x(t)≤M1,limsup⁡t→+∞y(t)≤M2,i=1,2.
From the proof of Lemma 10, for above ɛ>0 being small enough, there exists T2>0 such that
(32)y(t)≤1v*~(t)-ɛ≜yɛ*~(t)
for any t>T2. From the first equation of system (3), for t>T2 and t≠τk, we have
(33)x˙(t)≥x(t)(r1(t)-c1(t)k+x(t)yɛ*~(t)m-b1(t)x(t)),≥x(t)(r1(t)-c1(t)kyɛ*~(t)m-b1(t)x(t)).
Consider the following auxiliary system:
(34)ϕ˙(t)=ϕ(t)(r1(t)-c1(t)kyɛ*~(t)m-b1(t)ϕ(t)),t≠τk,ϕ(τk+)=(1+hk)ϕ(τk),t=τk,k=1,2,…,
where ϕ(t) is the solution of (34) with ϕ(T2+)=x(T2+). By Lemma 3, for any t>T2, we have x(t)≥ϕ(t). From condition (H2), we have
(35)∑k=1q‍ln⁡(1+hk)+Tm[r1(t)-c1(t)k(yɛ*~(t))m]>0.
From inequality (35) and Lemma 4, system (34) has a unique T-periodic solution x*ɛ(t), which is globally stable. Since condition (H4) holds, for the above ɛ>0 being small enough, we have m[r1(t)-(c1(t)/k)(yɛ*~(t))m]>0; letting ɛ→0, from Lemma 6, we have x*ɛ(t)→x*(t). Letting m1=inf⁡{x*(t):t∈[0,T]}, for the above ɛ being small enough, there exists T3>T2 such that x(t)≥ϕ(t)≥x*(t)-ɛ>m1-ɛ, for any t>T3. Let ɛ→0; we have lim⁡t→+∞inf⁡x(t)≥m1. From system (3), for t>T3 and t≠τk, we have
(36)y˙(t)≥y(t)(c2(t)(x*(t)-ɛ)k+(x*(t)-ɛ)(yɛ*~(t))m-1-r2(t)-b2(t)y(t)h+c2(t)(x*(t)-ɛ)k+(x*(t)-ɛ)(yɛ*~(t))m-1).
Consider the following auxiliary system:
(37)w˙(t)=w(t)(-r2(t)-b2(t)w(t)c2(t)(x*(t)-ɛ)k+(x*(t)-ɛ)(y~ɛ*(t))m-1hhhhh+c2(t)(x*(t)-ɛ)k+(x*(t)-ɛ)(yɛ*~(t))m-1),t≠τk,w(τk+)=(1+gk)w(τk),t=τk,k=1,2,…,
where w(t) is the solution of (37) with w(T3+)=y(T3+). By Lemma 3, for all t>T3, we have y(t)≥w(t), for the above ɛ>0 being small enough. By condition (H3), we obtain
(38)∑k=1q‍ln⁡(1+gk)+Tm[-r2+c2(t)(x*(t)-ɛ)k+(x*(t)-ɛ)(yɛ*~(t))m-1]>0.
From inequality (38), Lemma 4, system (37) has a unique T-periodic solution y*ɛ(t), which is globally asymptotically stable. Since condition (H5) holds, for the above ɛ>0 being small enough, such that m[-r2(t)+((c2(t)(x*(t)-ɛ))/(k+(x*(t)-ɛ)))(yɛ*~(t))m-1]>0, let ɛ→0, and from Lemma 6, we have y*ɛ(t)→y*(t), where y*(t) is the unique T-periodic globally asymptotically stable solution of the following system:
(39)p˙(t)=p(t)(-r2(t)-b2(t)p(t)c2(t)x*(t)k+x*(t)(y~*(t))m-1hhhhh+c2(t)x*(t)k+x*(t)(y*~(t))m-1(t)),t≠τk,p(τk+)=(1+gk)p(τk),t=τk,k=1,2,….
Letting m2=inf⁡{y*(t):t∈[0,T]}, for the above ɛ>0 being small enough, there exists T4>T3 such that y(t)≥w(t)>y*(t)-ɛ, for any t>T4. Let ɛ→0; we have lim⁡t→+∞y(t)≥m2.Theorem 12.
Assume that(H6)∑k=1q‍ln⁡(1+gk)+Tm[-r2(t)-c2(t)x*(t)k+x*(t)(y*(t))m-1]<0,(H7)∑k=1q‍ln⁡(1+hk)+Tm[r1(t)]>0.
Then, for any positive solution (x(t),y(t))T of system (3), one has
(40)lim⁡t→+∞y(t)=0,lim⁡t→+∞|x(t)-x*(t)|=0,
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≠τk, we have
(41)y˙(t)≤y(t)(-r2(t)-b2(t)y(t)+c2(t)x*(t)k+x*(t)(y*(t))m-1).
Consider the following auxiliary system:
(42)ρ˙(t)=y˙(t)≤ρ(t)(-r2(t)-b2(t)ρ(t)c2(t)x*(t)k+x*(t)(ρ*(t))m-1hhhhh+c2(t)x*(t)k+x*(t)(ρ*(t))m-1),t≠τk,ρ(τk+)=(1+gk)ρ(τk),t=τk,k=1,2,….
From (H6) and Lemma 4, we have
(43)lim⁡t→+∞y(t)=0.
Therefore, for ɛ>0 being small enough, there exists T5>0 such that y(t)<ɛ for any t>T5. Since (H7) holds, we obtain
(44)∑k=1qln⁡(1+hk)+Tm[r1(t)-c1(t)εm]>0.
Thus, from the first equation of system (3), we have
(45)x(t)(r1(t)-b1(t)x(t)-c1(t)εm)≤x˙(t)≤x(t)(r1(t)-b1(t)x(t))
for any t>T5. Consider the following auxiliary system
(46)φ˙(t)=φ(t)(r1(t)-b1(t)φ(t)-c1(t)εm),t≠tk,φ(τk+)=(1+hk)φ(τk),t=τk,k=1,2,…,σ˙(t)=σ(t)(r1(t)-b1(t)σ(t)),t≠tk,σ(τk+)=(1+hk)σ(τk),t=τk,k=1,2,….
From (H7) and Lemma 4, system (46) has unique T-periodic solutions φε*(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 φε*(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
(H8)∑k=1q‍ln⁡(1+hk)+Tm[r1(t)]<0.
Then species x and y are extinct; that is,
(47)lim⁡t→∞x(t)=0,lim⁡t→∞y(t)=0.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 Theorem11 hold and there exist positive constants α and δ such that
(H9)min⁡t∈[0,T]{αb1(t)-αc1(t)(M2)m(k+m1)2-δc2(t)k(m2)m-1(k+m1)2}>0,(H10)min⁡t∈[0,T]{δb2(t)-αc1(t)m(m2)m-1k+m1hhhhhh-δc2(t)M1(m-1)(m2)m-2k+m1}>0.
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
(48)m1-ε≤x(t),x-(t)≤M1+ε,m2-ε≤y(t),y-(t)≤M2+ε.
Define the Lyapunov function as follows:
(49)V(t)=α|ln⁡x(t)-ln⁡x-(t)|+δ|ln⁡y(t)-ln⁡y-(t)|.
By the mean value theorem and (48), for any t∈(τk,τk+1], k=p,p+1,…, and τp>T4, we have
(50)1M1+ε|x(t)-x-(t)|≤|ln⁡x(t)-ln⁡x-(t)|≤|x(t)-x-(t)|,1M2+ε|y(t)-y-(t)|≤|ln⁡y(t)-ln⁡y-(t)|≤|y(t)-y-(t)|,1k+x-(t)-1k+x(t)=1(k+ξ11(t))2(x-(t)-x(t)),x(t)k+x(t)-x-(t)k+x-(t)=k(k+ξ12(t))2(x(t)-x-(t)),(y-(t))m-(y(t))m=m(ξ21(t))m-1(y-(t)-y(t)),(y(t))m-1-(y-(t))m-1=(m-1)(ξ22(t))m-2(y(t)-y-(t)),
where ξ1i is suited between x(t) and x-(t), ξ2i is suited between y(t) and y-(t), and  i=1,2. For t>T4,  t≠τk  (k=1,2,…) and the above small enough ε>0; from conditions (H9) and (H10), there exist positive constants η and θ such that
(51)min⁡t∈[0,T]{αb1(t)-αc1(t)(M2+ε)m(k+m1-ε)2hhhhhh-δc2(t)k(m2-ε)m-1(k+m1-ε)2}>η,min⁡t∈[0,T]{δb2(t)-αc1(t)(m)(m2-ε)m-1k+m1-εhhhhhh-δc2(t)(M1+ε)(m-1)(m2-ε)m-2k+m1+ε}>η,ηmin⁡{m1-εα,m2-εδ}>θ.
Calculating the upper right derivative of V(t), we obtain
(52)D+V(t)=αsgn⁡⁡(x(t)-x-(t))(x˙(t)x(t)-x-˙(t)x-(t))+δsgn⁡⁡(y(t)-y-(t))(y˙(t)y1(t)-y-˙(t)y-(t))=αsgn⁡⁡(x(t)-x-(t))×[-b1(t)(x(t)-x-(t))((y(t))mk+x(t))-(y-(t))mk+x-(t)hhh-c1(t)((y(t))mk+x(t)-(y-(t))mk+x-(t))]+δsgn⁡⁡(y(t)-y-(t))×[-b2(t)(y(t)-y-(t))(x(t)(y(t))m-1k+x(t)-x-(t)(y-(t))m-1k+x-(t))hhh-c2(t)(x(t)(y(t))m-1k+x(t)hhhhhhhhhhh-x-(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)sgn⁡⁡(y(t)-y-(t))×(x(t)(y(t))m-1k+x(t)-x-(t)(y-(t))m-1k+x-(t)).
In view of
(53)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))mk+x(t)+(y-(t))mk+x(t)+(y(t))mk+x(t)]=c1(t)sgn⁡(x(t)-x-(t))(y-(t))m×[1(k+ξ11)2(x-(t)-x(t))m(ξ21(t))m-1k+x(t)hhh+m(ξ21(t))m-1k+x(t)(y-(t)-y(t))],(54)c2(t)sgn⁡(y(t)-y-(t))(x(t)(y(t))m-1k+x(t)-x-(t)(y-(t))m-1k+x-(t))=c2(t)sgn⁡(y(t)-y-(t))×[x(t)(y(t))m-1k+x1(t)-x(t)(y-(t))m-1k+x(t)hhh+x(t)(y-(t))m-1k+x(t)-x-(t)(y-(t))m-1k+x-(t)]=c2(t)sgn⁡(y(t)-y-(t))×[x(t)(m-1)(ξ22(t))m-2k+x(t)(y(t)-y-(t))hhh+k(k+ξ12(t))2(y-(t))m-1(x(t)-x-(t))],
substituting (53) and (54) into (52), we have
(55)D+V(t)≤-(αb1(t)-αc1(t)(y-(t))m(k+ξ11)2(t)hhhh-δc2(t)k(y-(t))m-1(k+ξ122))×|x(t)-x-(t)|-[δb2(t)-αc1(t)(y-(t))mm(ξ21m-1(t))k+xhhhhhhhhhαc1(t)(y-(t))mm(ξ21m-1(t))k+x-δc2(t)x(t)(m-1)(ξ22)m-2(t)k+x]×|y(t)-y-(t)|≤-[αb1(t)-αc1(t)(M2+ε)m(k+m1-ε)2hhhhhhhhh-δc2(t)k(m2-ε)m-1(k+m1-ε)2]|x(t)-x-(t)|-[δb2(t)-αc1(t)(M2+ε)mm(m2-ε)m-1k+m1-εhhhhhhhhh-δc2(t)(M1+ε)(m-1)(m2-ε)m-2k+m1-ε]×|y(t)-y-(t)|≤-η(m1-εαα|ln⁡x(t)-ln⁡x-(t)|hhhhhh+m2-εδδ|ln⁡y(t)-ln⁡y-(t)|).
For all t∈R+ and t=τk+  (k=1,2,…), we have
(56)V˙(τk+)=α|ln⁡x(τk+)-ln⁡x-(τk+)|+δ|ln⁡y(τk+)-ln⁡y-(τk+)|=α|ln⁡(1+hk)x(τk)-ln⁡(1+hk)x-(τk)|+δ|ln⁡(1+gk)y(τk)-ln⁡(1+gk)y-(τk)|=V(τk).
Hence, V(t) is continuous for all t>τp>T4. The above analysis shows that, for all t>τp>T4,
(57)D+V(t)≤-θV(t),
and we further have
(58)V(t)≤V(τp)exp⁡(-θ(t-τp)).
It is obvious that V(t)→0 as t→+∞; that is,
(59)lim⁡t→+∞|ln⁡x(t)-ln⁡x-(t)|=0,lim⁡t→+∞|ln⁡y(t)-ln⁡y-(t)|=0.
We further have
(60)lim⁡t→+∞|x(t)-x-(t)|=0,lim⁡t→+∞|y(t)-y-(t)|=0.
Choose constants A=min⁡{m1,m2} and B=max⁡{M1,M2}. By Theorem 11, we have
(61)A≤x(t),y(t),x-(t),y-(t)≤B,∀t≥T4.
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(mnT,z0); then we have z(T,z-)=z-. In fact, since z(T,z(mnT,z0))=z(mnT,z(T,z0)) and z(mnT,z(T,z0))-z(mnT,z0)→0 as mn→∞, we have
(62)∥z(T,z-)-z-∥[A,B]2≤∥z(T,z-)-z(T,z(mnT,z0))∥[A,B]2+∥z(T,z(mnT,z0))-z(mnT,z0)∥[A,B]2+∥z(mnT,z0)-z-∥[A,B]2⟶0.
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(mnT,z0) and z~=lim⁡n→∞z(mnT,z0). Then, taking into account (60) and z~=z(mnT,z~), we have
(63)∥z--z~∥[A,B]2≤∥z--Z(mnT,z0)∥[A,B]2+∥Z(mnT,z0)-z~∥[A,B]2hhhh⟶0,n⟶∞,
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:
(64)-∑k=1q‍ln⁡(1+gk)+Tm[r2(t)]=0.0004>0,∑k=1q‍ln⁡(1+hk)+Tm[r1(t)-c1(t)(y*~(t))m]>1.143>0,∑k=1q‍ln⁡(1+gk)+Tm[-r2(t)+c2(t)x*(t)k+x*(t)y*~(t)m-1]>0.0536>0.m[r1(t)-c1(t)(y*~(t))m]>0.961>0,m[-r2(t)+c2(t)x*(t)k+x*(t)y*~(t)m-1]>0.024>0.
Taking α=1.5, δ=1, conditions (H9) and (H10) are, respectively, equivalent to the following inequalities:
(65)min⁡t∈[0,T]{αb1(t)-αc1(t)(M2)m(k+m1)2-δc2(t)k(m2)m-1(k+m1)2}⋍1.221>0,min⁡t∈[0,T]{δb2(t)-αc1(t)(m)(m2)m-1k+m1h-δc2(t)M1(m-1)(m2)m-2k+m1}⋍0.5498>0.
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 ofx(t) and y(t).Figure1 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:
(66)∑k=1q‍ln⁡(1+gk)+Tm[-r2(t)-c2(t)x*(t)k+x*(t)(y*(t))m-1]=-0.9463<0,∑k=1q‍ln⁡(1+hk)+Tm[r1(t)]=0.7769>0.
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 ofx(t) and y(t).Figure2 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 ofhk and gk on the dynamical behavior of system (3).
Case
h
k
g
k
x
y
Figure
1
0.2
0.03
Permanent
Permanent
Figure1
2
−0.2
−0.6
Permanent
Extinct
Figure2Furthermore, we investigate the effects of impulse coefficientshk and gk on the dynamical behavior of system (3); we keep the parameter gk fixed at gk=0.3. From Table 3, it is found that the impulse coefficient hk has an important influence on the sizes of prey and predator populations. Therefore, we plot the bifurcation diagram of system (3) when hk∈[-0.4,0.8]; see Figure 3(a); it is very clear that the number of prey populations increases with increasing parameter hk.Table 3
The effects ofhk on the dynamical behavior of system (3).
Case
h
k
g
k
Figure
—
Bifurcation parameter
0.3
Figure3(a)
1
−0.4
0.3
Figure3(b)
2
0
0.3
Figure3(c)
3
0.6
0.3
Figure3(d)(a) Bifurcation diagram ofhk when gk is fixed at 0.3; (b)–(d) phase portraits.
(a)
(b)
(c)
(d)(a) Bifurcation ofgk when hk is fixed at 0.4; (b)–(d) phase portraits.
(a)
(b)
(c)
(d)Dynamical behavior of system (3); here, we take initial value (x(0),y(0))=(0.721,0.156).
(a)
(b)
(c)
(d)On the other hand, we keep the parameterhk fixed at hk=0.4. For 0.26≤gk≤0.36, the effects of gk on the dynamical behavior of system (3) have been also investigated. From Table 4, it is found that the impulse coefficient gk has an important influence on the sizes of prey and predator populations.Table 4
The effects ofgk on the dynamical behavior of system (3).
Case
h
k
g
k
Figure
—
0.4
Bifurcation parameter
Figure4(a)
1
0.4
0.1
Figure4(b)
2
0.4
0.3
Figure4(c)
3
0.4
0.4
Figure4(d)Moreover, on increasing the period of impulse perturbations, some interesting phenomena happen; the details are given in Table5.Table 5
The effects of impulse periodT on the dynamical behavior of system (3).
Case
T
x
y
Figure
1
1
Periodic
Periodic
Figures5(a) and 5(b)
2
1.002
Almost periodic
Almost periodic
Figure5(c)
3
1.014
Chaotic
Chaotic
Figure5(d)


## 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 Theorem13 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.



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