1 Introduction

Predator-prey model is one of the basic models between different species in nature.These models have been studied extensively and many excellent results have been obtained(see[1,2]).On the other hand,the effect of disease in ecological system is an important topic from mathematical as well as ecological point of view.After the work of Kermack-McKendrick[3]on SIRS(susceptible-infected-removedsusceptible)systems,many authors have investigated the dynamical behavior of epidemiological models.Chattopadhyay and Arino[4]proposed a predator-prey epidemiological model with disease spreading in prey.They assumed that the sound prey population grows according to a logistic law involving the whole prey population,and discussed the positivity,uniqueness,boundedness of the solutions and the existence of supercritical Hopf bifurcation.Haque et al[5]investigated a Lotka-Volterra type predator-prey model with a transmissible disease in the predator species.They aussumed that the sound and infected predators can hunt the prey and studied the stability of system.

移栽机的核心部件是栽植机构，是保证秧苗栽植质量、提高工作效率的重要部件，它反映了移栽机的发展过程和进步水平。本文研究一种新型悬杯式移栽机的栽植机构作为栽植系统的研究基础，针对其在进行移栽作业时存在的一些问题，如移栽质量不稳定和钵苗直立度不能完全达到农业技术要求等，对栽植机构进行分析和改进，以寻求最优的结构形式和工作参数。

Sun and Yuan[6]proposed the following predator-prey model with disease in the predator

当前的药物治疗有一些局限性，主要在于起效缓慢和对不少抑郁症患者无效。解决难治性抑郁症的一个策略是研发和使用非典型抗抑郁药物。另外，NMDA受体的阻断剂——氯胺酮的快速抗抑郁效果的发现及相关研究，不仅提供了崭新的抗抑郁疗法，也极大的促进了抑郁症发作机制的病理学研究，靶向谷氨酸能的药物有望成为第三代抗抑郁药，解决当前药物在起效时间、有效性等方面存在的问题。

where x(t),S(t)and I(t)represent the densities of the prey,susceptible(sound)predator and the infected predator population at time t,respectively.They assumed that there is a spread of disease in predator and only the susceptible predators have ability to capture prey.They investigated the boundedness of solution and global asymptotical stability of the equilibriums.

On the other hand,prey species makes use of refuges to decrease predation risk,here refuge means a places or situations where predation risk is somehow reduced.Ma et al[7]studied the following predator-prey model with prey refuges and a class of functional responses

where the term φ(X)represents the functional response of the predator population.They obtained the local asymptotical stability of equilibrium point and showed that the refuges used by prey can increase the equilibrium density of prey population but decrease that of predator.Ma et al.[8]further studied the influence of prey refuge and density dependent of predator species on the traditional Lotka-Volterra model.Huang,Chen and Li[9]studied the influence of prey refuge on a predatorprey model with Holling type III response function.In[10],a global analysis of a Holling type II predator-prey model with a constant prey refuge was presented.Ma et al[11]and Chen,Chen and Wang[12]studied a Lotka-Volterra predatorprey model incorporating a prey refuge and predator mutual interference.For more details in this direction,please see[13,14].

However,there are still seldom scholars investigating the predator-prey model with prey refuge and disease in predator.More precisely,we study the global stability of the following model

where x(t),S(t)and I(t)represent the densities of the prey,susceptible(sound)predator and the infected predator population at time t,respectively with initial conditions x(0)＞0,S(0)＞0,I(0)＞0.All the parameters are positive constants and 0＜m＜1.The disease incidence follows the simple law of mass action incidence βS(t)I(t)with β being the transmission coefficient;d1 ≤ d2;mx(t)is the capacity of a refuge at time t.

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The organization of this paper is as follows.In Section 2,we study the stability of the equilibriums of system(1.3).In Section 3,numerical simulation is presented to illustrate the feasibility of our main results.In the last section,we give a brief discussion.

2 Main Results

In this section,we investigate the local and global stability of system(1.3).

Similar to the proof of[6],we have the following lemma.

Lemma 2.1 All solutions of system(1.3)with a positive initial data will remain positive and uniformly bounded.

The proofs of(ii)and(iii)are similar to that of(i)and we omit the detail here.This completes the proof of Proposition 2.1.

Thus V(t)is nonincreasing.From Lemma 2.1,(x−x1)2and I are bounded.On the other hand,it is easy to see that x′(t)and I′(t)are bounded.Therefore,(x − x1)2 and I are uniformly continuous on[0,+∞).Integrating both sides of(2.8)over the interval[0,+∞),we have

holds,then system(1.3)has a disease-free equilibrium E2(x1,S1,0),where x1=

Further,for the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)of system(1.3),x∗,S∗ and I∗ satisfy the following equations

Ifequation(2.5)has two positive rootsand z2=By calculation,we have0 and0.HenceThen,we have:

hold,then x∗,S∗ and I∗ are positive and satisfy

Letbe an arbitrary equilibrium of system(1.3).Then the Jacobian matrix about is given by

It is easy to prove that the equilibrium E0(0,0,0)is always unstable.

For the predator-extinction equilibrium E1(K,0,0),the Jacobian matrix is given by

采用SPSS 19.0统计学软件对数据进行处理，计量资料以“±s”表示，采用t检验。以P＜0.05为差异有统计学意义。

Hence,if≥1,then E1(K,0,0)is locally asymptotically stable for any 0＜m＜1.If＜1,then E1(K,0,0)is locally asymptotically stable if and only if 1−m＜

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For the disease-free equilibrium E2(x1,S1,0),the Jacobian matrix is given by

Then the characteristic equation is

By calculation,βS1−d2＜ 0 is equivalent to cd2(1−m)2−rβ(1−m)+＞0.Hence,the disease-free equilibrium E2(x1,S1,0)is locally asymptotically stable if and only if cd2(1−m)2−rβ(1−m)+＞0.

It follows from Lemma 2.2 that e(1−m)K−d1＜0.Then V′(t)＜0 for all(x,S,I)≠(K,0,0).By the Lyapunov-LaSalle invariance principle[15],E1(K,0,0)is globally asymptotically stable.This completes the proof of Lemma 2.3.

Then the characteristic equation is

whereNoting that 0,by Routh-Hurwitz criterion,the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is always locally asymptotically stable if and only if E∗exists.

Then we have the following result.

Lemma 2.2

(i)The trivial equilibrium E0(0,0,0)is unstable.

(ii)If≥1,then the predator-extinction equilibrium E1(K,0,0)is always locally asymptotically stable,but the disease-free equilibrium E2(x1,S1,0)and the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)do not exist.

(iii)If＜1,then the predator-extinction equilibrium E1(K,0,0)is locally asymptotically stable if 0＜1−m＜.The disease-free equilibrium E2(x1,S1,0)is locally asymptotically stable ifThe unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is locally asymptotically stable if 0＜1−m＜

By the definition of x∗,we have x∗ ＜ K.It follows from the second equation of(2.2)that 1−m＞,that is,the existence of E∗(x∗,S∗,I∗)implies that 1−m＞ holds.

De fine

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Let∆1=rβ(rβ −).

If rβ ＜ ,then H(z)＞0 for all＜z＜1.

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If rβ=and 0＜＜,then H(z)＞0 for

When＜1,then H(z)＞0 for all

From the above equation,if

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(i)If z2＜1,then H(z)＞0 for＜z＜z1and z2＜z＜1,H(z)＜0 for z1＜z＜z2.

(ii)If z1＜1≤z2,then H(z)＞0 for＜z＜z1,H(z)＜0 for z1＜z＜1.

(iii)If z1≥1,then H(z)＞0 for＜z＜1.

By analyse,we have the following results.

Proposition 2.1 If rβ ＞ and＜1,then

(i)z2 ＜ 1 if and only if

(ii)z1＜1≤z2if and only if one of the following conditions hold:

(ii.a)

(ii.b)=2.

(ii.c)

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(iii)z1 ≥ 1 if and only if

Proof (i)z2＜1 is equivalent to Hence,ifhold,then the conclusion(i)holds.

最后借用何显斌教授的一句话:“只有当刑罚以正义和人道为基本价值取向的时候，刑法的价值—秩序和自由才能实现并确保和谐的共生关系。”［8］我们设立刑法是为了很好的维护正常的自由、秩序，而不是滥用刑罚。

根据试验安排和结果，建立RS3质量分数的回归模型，并对模型方差分析。结果分析表明，此模型的决定系数R2为0.767 6，响应面回归模型达到高度显著性水平(p=0.039 0)(如表2所示)。回归方程模型失拟p=0.057 3>0.05，不显著，说明该二次模型能够拟合真实的试验结果(如表3所示)。则

From Proposition 2.1,we have the following proposition.

Poposition 2.2 Proposition 2.1 is equivalent to the following statements.

Proof Note that for 0＜ x＜ 1.It follows from Proposition 2.1 thatThen

(i)It follows fromthatIt follows from(i)of Proposition 2.1 thatHence,ifthen z2＜1.The conditions(ii.a),(ii.b)and(ii.c)of Proposition 2.1 are equivalent toTherefore,ifthen z1＜1≤z2.Note that condition(iii)of Proposition 2.1 is impossible.

The proofs of(ii)and(iii)are similar to that of(i)and we omit the detail here.This completes the proof of Proposition 2.2.

By the above discussion and Propositions 2.1 and 2.2,we have the following result.

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Proposition 2.3 Equation(2.5)has the following result.

Next we show the global stability of equilibrium of system(1.3),and obtain the following lemmas.

由表3可以看出，增设磁选机后，生产系统介耗大幅下降，平均降到了1.8 0kg/t左右(2012年1～3月统计)。

Lemma 2.3 If the predator-extinction equilibrium E1(K,0,0)is locally asymptotically stable,then E1(K,0,0)is globally asymptotically stable.

Proof Consider the following Lyapunov function

where

Calculating the derivative of V along the solution(x(t),S(t),I(t))of system(1.3),we have

Again,for the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗),the Jacobian matrix is given by

Lemma 2.4 If the disease-free equilibrium E2(x1,S1,0)is locally asymptotically stable,then E2(x1,S1,0)is globally asymptotically stable.

Proof De fine a Lyapunov function as follows

where ηi=,i=1,2.

Calculating the derivative of V along the solution(x(t),S(t),I(t))of system(1.3),we have

It follows from Lemma 2.2 that βS1− d2 ＜ 0.LetThen it follows from(2.7)that

We can easily calculate that system(1.3)always has a trivial equilibrium E0(0,0,0)and a predator-extinction equilibrium E (K,0,0).If

Therefore,V(t)is bounded on[0,+∞)and satisfies

From the first equation of system(1.3)and the above equality,we have

so

It follows from the above inequality that(x(t)−x1)2∈ L1[0,+∞)and I(t)∈L1[0,+∞).By Barbalat’s lemma[16],we conclude that

Hence,

that is

Then,from(2.9)-(2.11),E2(x1,S1,0)is globally asymptotically stable.This completes the proof of Lemma 2.4.

Lemma 2.5 If the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is locally asymptotically stable,then E∗(x∗,S∗,I∗)is globally asymptotically stable.

Proof De fine a Lyapunov function as follows

where ηi=,i=1,2.

Calculating the derivative of V along the solution(x(t),S(t),I(t))of system(1.3),we obtain

Thus V(t)is nonincreasing.By Lemma 2.1,(x − x∗)2is bounded.On the other hand,it is easy to see that x′(t)is bounded.Therefore,(x − x∗)2is uniformly continuous on[0,+∞).Integrating both sides of(2.12)over the interval[0,+∞),we have

Hence,V(t)is bounded on[0,+∞)and satisfies∫

The above inequality implies that(x(t)− x∗)2 ∈ L1[0,+∞).By Barbalat’s lemma[16],we have

so

It follows from the first equation of system(1.3)and the above equality that

Hence,

that is

By the definition of x∗,we haveThen,from(2.15),we obtain

From the second equation of system(1.3)and(2.16),we have

Hence,from(2.14)and(2.16),we obtain

that is

Therefore,by(2.14),(2.16)and(2.17),E∗(x∗,S∗,I∗)is globally asymptotically stable.This completes the proof of Lemma 2.5.

Now,we give the main result of this section.Let z=1−m in(2.5).According to the above analysis and summarizing Propositions 2.1-2.3 and Lemmas 2.2-2.5,we obtain the following theorem.

Theorem 2.1

(i)If≥1,then the predator-extinction equilibrium E1(K,0,0)is globally asymptotically stable.

(ii)Ifwe obtain:

(ii.a)Ifthen the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for alland the predator-extinction equilibrium E1(K,0,0)is globally asymptotically stable for all

(ii.b)Ifthen the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is globally asymptotically stable for all 0＜ m ＜ 1−z1,the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for alland the predator-extinction equilibrium E1(K,0,0)is globally asymptotically stable for all 1−＜m＜1.

(iii)Ifwe obtain:

(iii.a)Ifthen the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for all 0＜m＜0.5,and the predator-extinction equilibrium E1(K,0,0)is globally asymptotically stable for all 0.5＜m＜1.

(iii.b)Ifthen the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is globally asymptotically stable for all 0＜m＜1−z1,the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for all 1−z1＜m＜0.5,and the predator-extinction equilibrium E1(K,0,0)is globally asymptotically stable for all 0.5＜m＜1.

(iv)Ifwe obtain:

(iv.a)Ifthen the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for all and the predator-extinction equilibrium E1(K,0,0)is globally asymptotically stable for all

(iv.b)Ifthen the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for alland the predator-extinction equilibrium E1(K,0,0)is globally asymptotically stable for all

(iv.c)Ifthen the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for all 0＜m＜1−z2or the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is globally asymptotically stable for all 1−z2＜m＜1−z1,and the predator-extinction equilibrium E1(K,0,0)is globally asymptotically stable for all

(iv.d)If,then the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is globally asymptotically stable for all 0 ＜ m ＜ 1− z1,the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for all and the predator-extinction equilibrium E1(K,0,0)is globally asymptotically stable for all

3 Numerical Simulations

In this section we give some numerical simulations of systems(1.3).We consider the following system

where r=1.6;K=2;c=0.8;e=1;d2=1.2.

When d1=1.5,0.5＜=0.75 ＜ 1.Let β =3,thenIt follows from Theorem 2.1(ii.b)that the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is globally asymptotically stable for all 0 ＜ m ＜ 0.081(see Figure 1(b)),the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for all 0.081＜m＜0.25(See Figure 1(c)),and the predator-extinction equilibrium E1(2,0,0)is globally asymptotically stable for all 0.25＜m＜1(See Figure 1(d)).

When d1=1,Let β =1.5,thenIt follows from Theorem 2.1(iii.b)that the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is globally asymptotically stable for all 0＜m＜0.309(see Figure 2(b)),the diseasefree equilibrium E2(x1,S1,0)is globally asymptotically stable for all 0.309＜m＜0.5(see Figure 2(c)),and the predator-extinction equilibrium E1(2,0,0)is globally asymptotically stable for all 0.5＜m＜1(see Figure 2(d)).

When d1=0.4,=0.2 ＜ 0.5.Let β =0.6,then=0.8＜ =1＜=1.25.It follows from Theorem 2.1(iv.c)that the disease-free equilibrium E2(x1,S1,0)is globally asymptotically stable for all 0＜m＜0.276 or 0.724＜m＜0.8(see Figures 3(b)and 3(d)),the unique endemic-coexistence equilibrium E∗(x∗,S∗,I∗)is globally asymptotically stable for all 0.276 ＜ m ＜ 0.724(see Figure 3(c)),and the predator-extinction equilibrium E1(2,0,0)is globally asymptotically stable for all 0.8＜m＜1(see Figure 3(e)).

4 Discussion

In this paper,we study the global dynamics of a predator-prey model with prey refuge and disease in the predator.We show that prey refuge palys an important role in the dynamics of a predator-prey system(1.3).Form the above results,if the refuge m is sufficiently small,then the dynamics behavior of system is accordant with the corresponding system without prey refuges.By decreasing the value of natural death rate of the susceptible predator d1,the dynamics behavior of system becomes complicated,that is the global stable equilibrium may be changed by increasing the value of refuges m.This shows that prey refuge palys an important role in the dynamics behavior of system(1.3).

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