Research Article Global Stability of an Eco-Epidemiological Model with Time Delay and Saturation Incidence

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1 Hindawi Publishing Copoation Discete Dynamics in Natue and Society Volume 2011, Aticle ID 7078, 22 pages doi: /2011/7078 Reseach Aticle Global Stability of an Eco-Epidemiological Model with Time Delay and Satuation Incidence Shuxue Mao, Rui Xu, Zhe Li, and Yunfei Li Institute of Applied Mathematics, Shijiazhuang Mechanical Engineeing College, Shijiazhuang 05000, China Coespondence should be addessed to Shuxue Mao, Received 0 June 2011; Accepted 2 August 2011 Academic Edito: He-Teng Yau Copyight q 2011 Shuxue Mao et al. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. We investigate a delayed eco-epidemiological model with disease in pedato and satuation incidence. Fist, by compaison aguments, the pemanence of the model is discussed. Then, we study the local stability of each equilibium of the model by analyzing the coesponding chaacteistic equations and find that Hopf bifucation occus when the delay τ passes though a sequence of citical values. Next, by means of an iteation technique, sufficient conditions ae deived fo the global stability of the disease-fee plana equilibium and the positive equilibium. Numeical examples ae caied out to illustate the analytical esults. 1. Intoduction Recently, moe attention has been paid to the eco-epidemiology model which consides both the ecological and epidemiological issues simultaneously due to the fact that most of the ecological populations suffe fom vaious infectious diseases which have a significant ole in egulating population sizes see, e.g., 1 6. Mukhejee 7 discussed a pedato-pey model with disease in pey. The citeia wee deived fo both local stability and instability involving system paametes. In addition, consideing the time equied by the susceptible individuals to become infective afte thei inteaction with the infectious individuals, Zhou et al. 8 fomulated a delayed eco-epidemiology model and found that the Hopf bifucation occus when the delay passes though a sequence of citical values. They also gave an estimation of the length of the time delay to peseve stability. On the othe hand, in the pedato-pey system, the disease not only can spead in pey but also can spead in pedato. Theefoe, Zhang et al. 9 studied an eco-epidemiological model with disease in pedato and showed that a Hopf bifucation can occu as the delay inceased. The above-mentioned woks all used bilinea incidence to model disease tansmission.

2 2 Discete Dynamics in Natue and Society Note that ecologically the assumption of standad incidence instead of the fome bilinea mass action incidence is meaningful fo lage populations and a low numbe of infected individuals, a vey good justification behind this assumption being found in 10. Han et al. 11 poposed fou modifications of a pedato-pey model with standad incidence to include an SIS o SIR paasitic infection. Thesholds wee identified, and global stability esults wee poved. When the disease pesists in the pey population and the pedatos have asufficient feeding efficiency to suvive, the disease also pesists in the pedato population. Hethcote et al. 12 consideed a pedato-pey model including an SIS paasitic infection in the pey with infected pey being moe vulneable to pedation. Thesholds wee identified which detemine when the pedato population suvives and when the disease emains endemic. Howeve, thee ae a vaiety of factos that emphasize the need fo a modification of the bilinea incidence and standad incidence. Fo example, the undelying assumption of homogeneous mixing may not always hold. Incidence ates that incease moe gadually than linealy in I and S may aise fom satuation effects. It has been stongly suggested by seveal authos that the disease tansmission pocess may follow satuation incidence. Afte studying the cholea epidemic spead in Bai in 197, Capasso and Seio 1 intoduced a satuated incidence ate g I S into epidemic models with g I βi/ 1 αi. A geneal satuation incidence ate g I S βi p S/ 1 αi p was poposed by Liu et al. 14 and used by a numbe of authos; see, fo example, Ruan and Wang 15 p 2, Bhattachayya and Mukhopadhyay 16 p 1, and so foth. βi p measues the infection foce of the disease, and 1/ 1 αi p measues the inhibition effect fom the behavioal change of the susceptible individuals when thei numbe inceases o fom the cowding effect of the infective individuals. This incidence ate seems moe easonable than the bilinea incidence ate βsi, because it includes the behavioal change and cowding effect of the infective individuals and pevents the unboundedness of the contact ate by choosing suitable paametes. Motivated by the woks of Zhang et al. 9 and Capasso and Seio 1, inthis pape, we ae concened with the effect of disease in pedato and satuated incidence on the dynamics of eco-epidemiological model. To this end, we conside the following delay diffeential equations: ẋ t x t 1 x t K ax t S t, Ṡ t bx t τ S t τ cs 2 t βs t I t 1 αi t, İ t βs t I t 1 αi t di t, 1.1 with initial conditions x θ φ 1 θ, S θ φ 2 θ, I θ φ θ, φ i θ 0, θ τ, 0, φ i 0 > 0 i 1, 2,, 1.2 whee φ 1 θ,φ 2 θ,φ θ C τ, 0, R, the Banach space of continuous functions mapping the inteval τ, 0 into R 0, hee R 0 { x 1,x 2,x : x i 0, i 1, 2, }.

3 Discete Dynamics in Natue and Society We make the following assumptions fo ou model 1.1. A1 The pey population gows logistically with intinsic gowth ate and envionmental caying capacity K. A2 Thee is a spead of disease in pedatos which ae divided solely into susceptible and infectious population. a is the captuing ate of susceptible pedatos, b is the gowth ate of susceptible pedato due to pedation of pey. A Susceptible pedatos become infected when they come in contact with infected pedato, and this contact pocess is assumed to follow the satuation incidence ate βs t I t / 1 αi t, withβ measuing the foce of infection and α the inhibition effect. A4 c>0models death ate due to ovecowding, and τ is the time equied fo the gestation of susceptible pedato. d is the death ate of infected pedato. All the above-mentioned paametes ae assumed to be positive. The pape is oganized as follows. In the next section, the positivity of solutions and the pemanence of system ae discussed. By analyzing the coesponding chaacteistic equations, we find conditions fo local stability and bifucation esults in Section. In Section 4, sufficient conditions ae deived fo the global stability of the disease-fee plana equilibium and the positive equilibium of the system. Numeical examples ae caied out to illustate the validity of the main esults. The pape ends with a conclusion in the last section. 2. Pemanence To pove the pemanence of system 1.1, we need the following lemma, which is a diect application of Theoem in the study by Kuang 17. Lemma 2.1. Conside the following equation: ẋ t ax t τ bx t cx 2 t, 2.1 whee a, b, c, τ > 0 and x t > 0 fo all t τ, 0. 1 If a>b,thenlim sup x t a b /c. 2 If a<b,thenlim sup x t 0. Theoem 2.2. All the solutions of 1.1 with initial conditions 1.2 ae all nonnegative. Poof. Let x t,s t,i t be the solution of system 1.1 satisfying conditions 1.2.Fomthe fist and last equations of system 1.1, we have x t x 0 e t 0 1 x ξ /K as ξ dξ, I t I 0 e t 0 βs ξ / 1 αi ξ d dξ. 2.2 Hence, x t and I t ae positive.

4 4 Discete Dynamics in Natue and Society We now claim that S t > 0 fo all t>0. Othewise, thee exists a t 1 > 0 such that S t 1 0andS t > 0 fo all t 0,t 1. Then Ṡ t 1 0. Fom the second equation of 1.1, we have Ṡ t 1 bx t 1 τ S t 1 τ > 0, 2. which is a contadiction. Theoem 2.. All the solutions of 1.1 with initial conditions 1.2 ae ultimately bounded. Poof. Fom the fist equation of 1.1, we have ẋ t x t Hence, we get 1 x t K. 2.4 lim sup x t K M Fom the second equation of system 1.1,fot sufficiently lage, we have Ṡ t bx t τ S t τ cs 2 t βs t I t 1 αi t bks t τ cs 2 t. 2.6 Hence, by Lemma 2.1, one can get lim sup S t bk c M It follows fom the thid equation of 1.1 and the above inequality, that fo t sufficiently lage, we have İ t βs t I t 1 αi t di t βm 2I t 1 αi t di t. 2.8 Hence, one can see lim sup I t 1/ dα βm 2 d M. Now, we show that system 1.1 is pemanent. Theoem 2.4. Suppose that H 1 βm 2 >d, 2.9 whee m 2 is defined in 2.1, then system 1.1 is pemanent.

5 Discete Dynamics in Natue and Society 5 Poof. Fom the fist equation of system 1.1, we have ẋ t x t 1 x t K am It then follows that [ lim inf x t K 1 am 2 m Using the second equation of system 1.1,fot sufficiently lage, we have Ṡ t bm 1 S t τ cs 2 t βs t M 1 αm Hence, by Lemma 2.1 and H 1, one can deive that lim inf S t 1 [ bm 1 βm c 1 αm m Fom the thid equation of system 1.1 and, above inequality, we have İ t βm 2I t di t αi t Since H 1 holds, then lim inf I t 1 [ βm2 d m dα Theefoe, the above calculations and Theoem 2.2 imply that thee exist M i,m i i 1, 2, such that 0 <m 1 lim inf 0 <m 2 lim inf 0 <m lim inf x t lim sup x t M 1, S t lim sup S t M 2, I t lim sup I t M Local Stability System 1.1 possesses the following equilibia. 1 The tivial equilibium E 0 0, 0, 0. 2 The axial equilibium E 1 K, 0, 0. The disease-fee plana equilibium E 2 x 2,S 2, 0, whee x 2 Kc Kab c, S 2 Kb Kab c..1

6 6 Discete Dynamics in Natue and Society 4 The unique positive equilibium E x,s,i exists if βs >d, whee S Kbα 2 Kbα β β 4dα Kab/ c, 2α Kab/ c I βs d dα, x K KaS..2 In the following, we discuss the local stability of each equilibium of system 1.1 by analyzing the coesponding chaacteistic equations, espectively..1. Stability of Equilibium E 0 The chaacteistic equation of system 1.1 at the tivial equilibium E 0 is of the fom λ λ λ d 0.. It is easy to see that. always has a positive oot. Hence, E 0 is always unstable..2. Stability of Equilibium E 1 The chaacteistic equation of system 1.1 at the axial equilibium E 1 is of the fom λ K λ bke λτ λ d 0..4 Thee ae two chaacteistic oots λ 1 K, λ 2 d, and anothe chaacteistic oot is given by the oot of λ bke λτ..5 It is clea that Re λ>0. Hence, E 1 is always unstable... Stability of Equilibium E 2 Theoem.1. The disease-fee plana equilibium E 2 is locally asymptotically stable if βs 2 <d, and the equilibium E 2 is unstable if βs 2 >d. Poof. The chaacteistic equation of system 1.1 at the disease-fee plana equilibium E 2 is of the fom λ x 2 λ 2cS 2 cs 2 e λτ λ d βs K Clealy, λ 1 x 2 /K is a negative eigenvalue. The second eigenvalue is given by the oot of λ 2 cs 2 e λ2τ 2..7

7 Discete Dynamics in Natue and Society 7 Suppose that Re λ 2 0, then Re λ 2 cs 2 e Re λ 2τ cos τ Im λ 2 2 < 0. It is a contadiction, so Re λ 2 < 0. The last eigenvalue is λ βs 2 d. The equilibium E 2 is locally asymptotically stable if βs 2 <d, and the equilibium E 2 is unstable if βs 2 >d..4. Stability of Equilibium E The chaacteistic equation of system 1.1 at the positive equilibium E is of the fom λ A 1 λ 2 A 2 λ A e λτ B 1 λ 2 B 2 λ B 0,.8 whee A 1 x K 2cS βi dαi, 1 αi 1 αi A 2 x K dαi 1 αi 2cS βi 1 αi A x [ K dβi 2cS βi dαi, 1 αi 1 αi 1 αi B 1 bx, B 2 bx dαi x 1 αi K as, B bx dαi x 1 αi K as. dαi 1 αi x K βi 1 αi 2,.9 Fo τ 0, the tanscendental.8 educes to the following equation: λ A 1 B 1 λ 2 A 2 B 2 λ A B We can easily get A 1 B 1 x K cs dαi > 0, 1 αi x A 2 B 2 K dαi cs dαi x 1 αi 1 αi K βi 1 αi bx as 2 > 0, A B x [ K dβi cs dαi bx dαi as > 0, 1 αi 1 αi 1 αi.11 A 1 B 1 A 2 B 2 A B > 0. Theefoe, the Routh-Huwitz citeion implies that all the oots of.8 have negative eal pats and we can conclude that the positive equilibium E is asymptotically stable in the absence of delay.

8 8 Discete Dynamics in Natue and Society Theoem.2. Fo system 1.1, if the condition H 2 A <B holds, the positive equilibium E is conditionally stable. Poof. Substituting λ iω into.8 and sepaating the eal and imaginay pats, one can get A 1 ω 2 A B B 1 ω 2 cos ωτ B 2 ω sin ωτ, ω A 2 ω B 2 ω cos ωτ B B 1 ω 2 sin ωτ..12 Squaing and adding.12 we get ω 6 D 1 ω 4 D 2 ω 2 D 0,.1 whee D 1 A 2 1 2A 2 B 2 1, D 2 A 2 2 B2 2 2A 1A 2B 1 B, D A 2 B2..14 We know that D < 0 povided that the condition H 2 holds. Thee is at least a positive ω 0 satisfying.1, that is, the chaacteistic equation.8 has a pai of puely imaginay oots of the fom ±iω 0.Fom.12, we can get the coesponding τ k > 0 such that the chaacteistic.8 has a pai of puely imaginay oots τ k 1 accos ω 0 A1 ω 2 0 A B B 1 ω0 2 ω 0 A 2ω 0 B2 ω 0 B B 1 ω B2 ω 0 2 2kπ, ω 0 k 0, 1, 2, Let λ τ ν τ iω τ be the oots of.8 such that τ τ k satisfying ν τ k 0andω τ k ω 0.Diffeentiating the two sides of.8 with espect to τ, weget dλ 1 2λ A 1 λ 2 A 2 dτ λ 2 λ A 1 λ 2 A 2 λ A B 1 λ 2 B λ 2 B 1 λ 2 B 2 λ B τ λ..16 Theefoe, [ dreλ sign dτ τ τ k [ dλ 1 sign Re dτ λ iω 0 1 [ A A 1 ω 2 0 sign Re i2ω 0 ω 2 0 A 1 ω 2 0 A i B 1 ω 2 0 ω 0 A B 2ω 0 B 1 ω 2 0 ib 2ω 0 B 1 sign 2ω6 0 A 2 1 2A 2 B 2 1 ω 4 0 B2 A2 ω 2 0 B B 1 ω B2 ω

9 Discete Dynamics in Natue and Society 9 If the conditions H 2 and H A 2 1 2A 2 >B 2 1 hold, one can see [ dreλ sign > dτ τ τ k Theefoe, the tansvesality condition holds, hence, the Hopf bifucation occus at ω ω 0 and τ τ k. Theoem.. Suppose that the conditions H 2 and H ae satisfied. 1 The positive equilibium E of system 1.1 is asymptotically stable fo all τ 0,τ 0 and unstable fo τ>τ 0. 2 System 1.1 undegoes a Hopf Bifucation at the positive equilibium E when τ τ k k 0, 1, Global Stability In this section, we study the global stability of equilibiums E 2 and E. The stategy of poofs is to use an iteation technique and compaison aguments, espectively. Theoem 4.1. If H 4 βbk < cd, Kab < c holds, then the disease-fee plana equilibium E 2 is globally asymptotically stable. Poof. Let x t,s t,i t be any positive solution of system 1.1 with initial conditions 1.2. Let the following hold: U 1 lim sup x t, V 1 lim inf x t, U 2 lim sup S t, V 2 lim inf S t, U lim sup I t, V lim inf I t. 4.1 In the following we shall claim that U 1 V 1 x 2,U 2 V 2 S 2,U V 0. It follows fom the fist equation of system 1.1 that By compaison, we obtain that ẋ t x t 1 x t K. 4.2 U 1 lim sup x t K ε. 4. Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that U 1 M x 1, whee M x 1 K. 4.4 Hence, fo ε>0sufficiently small, thee is a T 1 > 0 such that, if t>t 1, x t M x 1 ε.

10 10 Discete Dynamics in Natue and Society We, theefoe, deive fom the second equation of system 1.1 that, fo t>t 1 τ, Ṡ t b M x 1 ε S t τ cs 2 t. 4.5 Hence, by Lemma 2.1, one can get U 2 lim sup S t b Mx 1 ε c M S Hence, fo ε>0sufficiently small, thee is a T 2 > 0 such that, if T 2 >T 1 τ, S t M S 1 ε. It follows fom the thid equation of system 1.1 that, fo t>t 2, İ t β M S 1 ε I t 1 αi t di t. 4.7 Since H 4 holds, one can see U lim sup I t Accoding to Theoem 2.2, we can get lim I t U V 0. We deive fom the fist equation of system 1.1 that, fo t>t 2 τ, ẋ t x t 1 x t K a M S 1 ε. 4.9 By compaison we deive that V 1 lim inf [1 x t K a M S 1 ε Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that V 1 N x 1, whee [ N x 1 K 1 ams Hence, fo ε>0sufficiently small, thee is a T > 0 such that, if T >T 2 τ, x t N x 1 ε. We deive fom the second equation of system 1.1 that, fo t>t, Hence, by Lemma 2.1, one can get Ṡ t b N x 1 ε S t τ cs 2 t βs t ε 1 αε. 4.12

11 Discete Dynamics in Natue and Society 11 V 2 lim inf [ b N x 1 c βε. 1 αε 4.1 Since this is tue fo abitay ε>0sufficiently small, we conclude that V 2 N S 1, whee N S 1 bnx 1 c Hence, fo ε>0sufficiently small, thee is a T 4 > 0 such that, if T 4 >T τ, S t N S 1 ε. Again, it follows fom the fist equation of system 1.2 that, fo t>t 4, ẋ t x t 1 x t K a N S 1 ε A compaison agument yields U 1 lim sup x t K [1 a N S 1 ε Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that U 1 M x 2, whee [ M x 2 K 1 ans Hence, fo ε>0sufficiently small, thee is a T 5 > 0 such that, if T 5 >T 4 τ, x t M x 2 ε. It follows fom the second equation of system 1.1 that, fo t>t 5, Ṡ t b M x 2 ε S t τ cs 2 t βs t ε 1 αε By Lemma 2.1, one can deive that V 2 lim sup S t 1 c [ b M x 2 ε βε αε Since this is tue fo abitay ε>0sufficiently small, we conclude that U 2 M S 2, whee M S 2 bmx 2 c Hence, fo ε>0sufficiently small, thee is a T 6 > 0 such that, if T 6 >T 5 τ, S t M S 2 ε. We deive fom the fist equation of system 1.1 that, fo t>t 6,

12 12 Discete Dynamics in Natue and Society ẋ t x t 1 x t K a M S 2 ε By compaison it follows that V 1 lim inf [1 x t K a M S 2 ε Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that V 1 N x 2, whee [ N x 2 K 1 ams Hence, fo ε>0sufficiently small, thee is a T 7 > 0 such that, if T 7 >T 6 τ, x t N x 2 ε. We deive fom the second equation of system 1.1 that, fo t>t 7, Hence, by Lemma 2.1, one can get Ṡ t b N x 2 ε S t τ cs 2 t βs t ε 1 αε V 2 lim inf S t 1 c [ b N x 2 ε βε αε Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that V 2 N S 2, whee N S 2 bnx 2 c Hence, fo ε>0sufficiently small, thee is a T 7 > 0 such that, if T 8 >T 7 τ, S t N S 2 ε. Continuing this pocess, we get fou sequences M x n,m S n,n x n,n S n n 1, 2,... such that, fo n 2, [ Mn x K 1 ans n 1, [ Nn x K 1 ams n, Mn S bmx n, c N S n bnx n c. 4.27

13 Discete Dynamics in Natue and Society 1 Clealy, we have N x n V 1 U 1 M x n, N S n V 2 U 2 M S n It follows fom 4.27 that [ M x n 1 K 1 Kab M x K 2 a 2 b 2 n c 2 c 2 Noting that M x n S 2 and Kab < c, we deive fom 4.29 that [ M x n 1 K 1 Kab c [ K 1 Kab c [ Kab Mn x Kc Kab c [ Kab c 1 c 1 [ [ Kab Kab c 1 c Thus, the sequence M x n is monotonically noninceasing. Theefoe, it follows that lim n M x n exists. Taking n, weobtainfom 4.29 that [ lim n Mx n 1 K 1 Kab c K 2 a 2 b 2 lim n Mx n c 2 Noting that lim n Mx n 1 lim n Mx n, 4.2 it follows fom 4.1 that lim n Mx n 1 lim n Mx n x We deive fom 4. and the thid equation of 4.27 that lim n MS n 1 lim n MS n S Similaly, one can deive fom 4.27 and 4.4 that lim n Nx n x 2, lim n NS n S It follows fom 4.28, 4.,and 4.5 that V 1 U 1 x 2, V 2 U 2 S

14 14 Discete Dynamics in Natue and Society We, theefoe, have lim x t x 2, lim S t S 2, limi t Hence, the disease-fee plana equilibium E 2 is globally asymptotically stable. The poof is complete. Theoem 4.2. If H 5 βbk > cd and Kab < c, β > Kbα holds, then the positive equilibium E is globally asymptotically stable. Poof. Let x t,s t,i t be any positive solution of system 1.1 with initial conditions 1.2. Let the following hold: x lim sup x t, x lim inf x t, S lim sup S t, S lim inf S t, I lim sup I t, I lim inf I t. 4.8 In the following we claim that x x x, S S S, I I I. It follows fom the fist equation of system 1.1 that ẋ t x t 1 x t K. 4.9 By compaison we obtain x lim sup x t K ε Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that x M x 1, whee M x 1 K Hence, fo ε>0sufficiently small, thee is a T 1 > 0 such that, if t>t 1, x t M x 1 ε.we obtain fom the second equation of system 1.1 that, fo t>t 1 τ, Ṡ t b M x 1 ε S t τ cs 2 t Hence, by Lemma 2.1, we deive that S lim sup S t b Mx 1 ε. 4.4 c

15 Discete Dynamics in Natue and Society 15 Since it is tue fo abitay ε>0sufficiently small, we conclude that S M S 1, whee M S 1 bmx 1 c Hence, fo ε>0sufficiently small, thee is a T 2 > 0 such that, if T 2 >T 1 τ, S t M S 1 ε. It follows fom the thid equation of system 1.1 that İ t β M S 1 ε I t 1 αi t di t Since H 5 holds, one can see I lim sup I t β M S 1 ε d dα Since it is tue fo abitay ε>0sufficiently small, we conclude that I M I 1, whee M I 1 βms 1 d dα We deive fom the fist equation of system 1.1 that, fo t>t 2, ẋ t x t 1 x t K a M S 1 ε By compaison we deive that x lim inf [1 x t K a M S 1 ε Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that x N x 1, whee [ N x 1 K 1 ams Hence, fo ε>0sufficiently small, thee is a T > 0 such that, if T >T 2 τ, x t N x 1 ε. We deive fom the second equation of system 1.1 that, fo t>t, Ṡ t b N x 1 ε S t τ cs 2 t βs t MI αm I 1

16 16 Discete Dynamics in Natue and Society Hence, by Lemma 2.1 and H 5, one can get S lim inf S t 1 c [ b N x 1 ε βmi 1 1 αm I Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that S N S 1, whee N S 1 1 [ bn x 1 c βmi αm I 1 Hence, fo ε>0sufficiently small, we get S t N S 1 ε. It follows fom the thid equation of system 1.1 that İ t β N S 1 ε I t 1 αi t di t Povided that βn S 1 >d, one can see I lim inf I t β N S 1 ε d dα Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that I N I 1, whee It follows fom the fist equation of system 1.1 that ẋ t x t 1 x t K N I 1 βns 1 d dα a N S 1 ε By compaison we deive that x lim sup x t K [1 a N S 1 ε Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that x M x 2, whee [ M x 2 K 1 ans Hence, fo ε>0sufficiently small, thee is a T 4 > 0 such that, if t>t 4, x t M x 2 ε.

17 Discete Dynamics in Natue and Society 17 We obtain fom the second equation of system 1.1 that, fo t>t 4 τ, Ṡ t b M x 2 ε S t τ cs 2 t βs t NI αn I 1 Hence, by Lemma 2.1, one can get S lim sup S t 1 c [ b M x 2 ε βni 1 1 αn I Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that S M S 2, whee M S 2 1 [ bm x 2 c βni αn I 1 Hence, fo ε>0sufficiently small, thee is a T 5 > 0 such that, if T 5 >T 4 τ, S t M S 2 ε. It follows fom the thid equation of system 1.1 that İ t β M S 2 ε I t 1 αi t di t. 4.6 Hence, by H 5, one can see I lim sup I t β M S 2 ε d dα Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that I M I 2, whee M I 2 βms 2 d dα We deive fom the fist equation of system 1.1 that, fo t>t 5, ẋ t x t 1 x t K a M S 2 ε By compaison we deive that x lim inf [1 x t K a M S 2 ε. 4.67

18 18 Discete Dynamics in Natue and Society Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that x N x 2, whee [ N x 2 K 1 ams Hence, fo ε>0sufficiently small, thee is a T 6 > 0 such that, if T 6 >T 5 τ, x t N x 2 ε. We deive fom the second equation of system 1.1 that, fo t>t 6, Ṡ t b N x 2 ε S t τ cs 2 t βs t MI αm I 2 By Lemma 2.1, one can get S lim inf S t 1 c [ b N x 2 ε βmi 2 1 αm I Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that S N S 1, whee N S 2 1 [ bn x 2 c βmi αm I 2 Hence, fo ε>0sufficiently small, we get S t N S 2 ε. It follows fom the thid equation of system 1.1 that İ t β N S 2 ε I t 1 αi t di t Since H 5 holds, one can see I lim inf I t β N S 2 ε d. 4.7 dα Since this inequality holds tue fo abitay ε>0sufficiently small, we conclude that I N I 2, whee N I 2 βns 2 d dα

19 Discete Dynamics in Natue and Society 19 Continuing this pocess, we obtain six sequences Mn,M x n,m S n,n I n,n x n S,Nn I n 1, 2,... such that, fo n 2, [ Mn x K 1 ans n 1, Mn S 1 [ bmn x βni n 1, c 1 αn I n 1 Mn I βms n d, dα [ Nn x K 1 ams n, Nn S 1 [ bnn x βmi n, c 1 αmn I 4.75 Nn I βns n d. dα A diect calculation shows that M x 2 Mx 1 KaNS 1 < 0, M S 2 MS 1 1 [ b M x 2 c βn I Mx 1 1 < 0, 1 αn I 1 M I 2 MI 1 β M S 2 MS 1 dα < 0, N x 2 Nx 1 Ka M S 1 Ms 2 > 0, N S 2 NS 1 1 [ b N x 2 c βm I Nx αm I 1 N I 2 NI 1 β N S 2 NS 1 dα > 0. βmi 2 1 αm I 2 > 0, 4.76 By induction, we can show that M S n 1 < MS n, N S n 1 > NS n. Theefoe, the sequence M S n is deceasing, and the sequence N S n is inceasing. Clealy, we have N x n x x M x n, N S n S S M S n, N I n I I M I n Hence, the limits of the sequences M S n and N S n exist. Denote S lim MS n, S lim NS n. 4.78

20 20 Discete Dynamics in Natue and Society Solution Time t x s i Figue 1: The tempoal solution found by numeical integation of system 1.1 with 2, K 2, a 2, c 2,β 2, α 1,d 0.8, b 0., τ 1, and x 0,S 0,I 0 1, 1, 1. We deive fom 4.75 that [ S S [ β Kbα Kba d S S Since H 5 holds, β Kbα Kba/d S S > 0. It, theefoe, follows fom 4.79 that S S. Accodingly, we deive fom 4.75 that S 1 [ bk 1 as β c α d S By a simple calculation, we obtain S S S It follows fom 4.75 and 4.81 that I I I, x x x. Hence, the unique positive equilibium E is globally asymptotically stable. The poof is complete. In the following we will pesent two examples to veify ou esults obtained ealie. Example 4.. In system 1.1, welet 2, K 2, a 1, c 2, β 2, α 1, d 0.8, b 0., τ 1. It is easy to show that Kbβ cd 0.4 < 0,c Kab.4 > 0. By Theoem 4.1 we see that the equilibium E , , 0 of system 1.1 is globally stable see Figue 1.

21 Discete Dynamics in Natue and Society Solution x s i Time t Figue 2: The tempoal solution found by numeical integation of system 1.1 with 2, K 2, a 1, c 2,β 2, α 1,d 0.4, b 0., τ 1, and x 0,S 0,I 0 1, 1, 1. Example 4.4. In system 1.1,welet 2,K 2,a 1,c 2,β 2,α 1,d 0.4,b 0.,τ 1. It is easy to show that Kbβ cd 0.4 > 0,Kab c.4 < 0,β Kbα > 0. By Theoem 4.2 we see that the equilibium E , , of system 1.1 is globally stable, as depicted in Figue Conclusion In this pape, we have incopoated the disease fo the pedato and the time delay into an eco-epidemiology model. A satuation incidence function was used to model the behavioal change of the susceptible pedato when thei numbe inceases o due to the cowding effect of the infected pedato. Fist, by compaison aguments, the pemanence of system 1.1 was studied. Then, by analyzing the coesponding chaacteistic equations, sufficient conditions wee deived fo the local stability of each equilibium of system 1.1. FomTheoem., we showed that system 1.1 undegoes a Hopf bifucation when the delay passes though a sequence of citical values. Next, by using the iteation technique and compaison aguments, we deived sufficient conditions fo the global stability of the disease-fee plane equilibium and positive equilibium of system 1.1. By Theoems 4.1 and 4.2, we showed that 1 if H 4 holds, the infected pedato population becomes extinct and the disease will be eliminated; that is, only sound pedato and pey coexist; 2 if H 5 holds, the pey, the sound pedato and the infected pedato coexist. The disease will not be eliminated, and the system is pemanent. Acknowledgment This wok was suppoted by the National Natual Science Foundation of China no

22 22 Discete Dynamics in Natue and Society Refeences 1 K. Kundu and J. Chattopadhyay, A atio-dependent eco-epidemiological model of the Salton Sea, Mathematical Methods in the Applied Sciences, vol. 29, no. 2, pp , K. P. Das, S. Roy, and J. Chattopadhyay, Effect of disease-selective pedation on pey infected by contact and extenal souces, BioSystems, vol. 95, no., pp , X. Zhou and J. Cui, Stability and Hopf bifucation analysis of an eco-epidemiological model with delay, Jounal of the Fanklin Institute, vol. 47, no. 9, pp , B. Mukhopadhyay and R. Bhattachayya, Role of pedato switching in an eco-epidemiological model with disease in the pey, Ecological Modelling, vol. 220, no. 7, pp , X. Zhou, X. Shi, and X. Song, The dynamics of an eco-epidemiological model with distibuted delay, Nonlinea Analysis: Hybid Systems, vol., no. 4, pp , N. Baiagi, R. R. Saka, and J. Chattopadhyay, Impacts of incubation delay on the dynamics of an eco-epidemiological system a theoetical study, Bulletin of Mathematical Biology, vol. 70, no. 7, pp , D. Mukhejee, Hopf bifucation in an eco-epidemic model, Applied Mathematics and Computation, vol. 217, no. 5, pp , X. Zhou, X. Shi, and X. Song, Analysis of a delay pey-pedato model with disease in the pey species only, Jounal of the Koean Mathematical Society, vol. 46, no. 4, pp , J.-F. Zhang, W.-T. Li, and X.-P. Yan, Hopf bifucation and stability of peiodic solutions in a delayed eco-epidemiological system, Applied Mathematics and Computation, vol. 198, no. 2, pp , M. Haque and D. Geenhalgh, A pedato-pey model with disease in the pey species only, Mathematical Methods in the Applied Sciences, vol. 0, no. 8, pp , L. Han, Z. Ma, and H. W. Hethcote, Fou pedato pey models with infectious diseases, Mathematical and Compute Modelling, vol. 4, no. 7-8, pp , H. W. Hethcote, W. Wang, L. Han, and Z. Ma, A pedato pey model with infected pey, Theoetical Population Biology, vol. 66, no., pp , V. Capasso and G. Seio, A genealization of the Kemack-McKendick deteministic epidemic model, Mathematical Biosciences, vol. 42, no. 1-2, pp. 4 61, W. M. Liu, S. A. Levin, and Y. Iwasa, Influence of nonlinea incidence ates upon the behavio of SIRS epidemiological models, Jounal of Mathematical Biology, vol. 2, no. 2, pp , S. Ruan and W. Wang, Dynamical behavio of an epidemic model with a nonlinea incidence ate, Jounal of Diffeential Equations, vol. 188, no. 1, pp , R. Bhattachayya and B. Mukhopadhyay, On an eco-epidemiological model with pey havesting and pedato switching: local and global pespectives, Nonlinea Analysis: Real Wold Applications, vol. 11, no. 5, pp , Y. Kuang, Delay Diffeential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineeing, Academic Pess, London, UK, 199.

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