Research Article Stability Analysis of a Ratio-Dependent Predator-Prey Model Incorporating a Prey Refuge

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1 Applied Mathematics Aticle ID pages Reseach Aticle Stability Analysis of a Ratio-Dependent Pedato-Pey Model Incopoating a Pey Refuge Lingshu Wang and Guanghui Feng School of Mathematics and Statistics Hebei Univesity of Economics & Business Shijiazhuang China Institute of Applied Mathematics Shijiazhuang Mechanical Engineeing College Shijiazhuang China Coespondence should be addessed to Lingshu Wang; wanglingshu@6com Received 4 Novembe 03; Revised Febuay 04; Accepted Febuay 04; Published 3 Mach 04 Academic Edito: Shan Zhao Copyight 04 L Wang and G Feng 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 wo is popely cited A atio-dependent pedato-pey model incopoating a pey efuge with disease in the pey population is fomulated and analyzed The effects of time delay due to the gestation of the pedato and stage stuctue fo the pedato on the dynamics of the system ae concened By analyzing the coesponding chaacteistic equations the local stability of a pedato-extinction equilibium and a coexistence equilibium of the system is discussed espectively Futhe it is poved that the system undegoes a Hopf bifucation at the coexistence equilibium when τ=τ 0 By compaison aguments sufficient conditions ae obtained fo the global stability of the pedato-extinction equilibium By using an iteation technique sufficient conditions ae deived fo the global attactivity of the coexistence equilibium of the poposed system Intoduction Since the pioneeing wo of Kemac-Mcendic on SIRS [] epidemiological models have eceived much attention fom scientists Mathematical models have become impotant tools in analyzing the spead and contol of infectious disease It is of moe biological significance to conside the effect of inteacting species when we study the dynamical behavios of epidemiological models Ecoepidemiology which is a elatively new banch of study in theoetical biology tacles such situations by dealing with both ecological and epidemiological issues It can be viewed as the coupling of an ecological pedato-pey model and an epidemiological SI SIS o SIRS model Following Andeso and May []who wee the fist to popose an ecoepidemiological model by meging the ecological pedato-pey model intoduced by Lota and Voltea the effect of disease in ecological system is an impotant issue fom mathematical and ecological point of view Many wos have been devoted to the study of the effects of a disease on a pedato-pey system [ 5] In [5] Xiao and Chen have consideed a atio-dependent pedatopey system with disease in the pey Conside =S( S+I K ) βsi =βsi biy ay + I dy = pbiy ay + I cy whee S(t) and I(t) epesent the densities of susceptible and infected pey population at time tespectivelyandy(t) epesents the density of the pedato population at time t The paametes K β d b a pandc aepositiveconstants epesenting the pey intinsic gowth ate caying capacity tansmission ate the infected pey death ate captuing ate half captuing satuation constant convesion ate and the pedato death ate espectively A peiodic solution can occu whethe the system () is pemanent o not; that is thee ae solutions which tend to disease-fee equilibium while bifucating peiodic solution exists Recently the qualitative analysis of pedato-pey models incopoating a pey efuge has been done by many authos see [3 4] In [3] Pal and Samanta incopoated a pey efuge ()

2 Applied Mathematics ( m)i into system () Sufficient conditions wee deived fo the stability of the equilibia of the system We note that it is assumed in system () that each individual pedato admits the same ability to feed on pey ThisassumptionseemsnottobeealisticfomanyanimalsIn the natual wold thee ae many species whose individuals pass though an immatue stage duing which they ae aised by thei paents and the ate at which they attac pey can be ignoed Moeove it can be assumed that thei epoductive ate duing this stage is zeo Stage-stuctue is a natual phenomenon and epesents fo example the division of a population into immatue and matue individuals Stagestuctued models have eceived geat attention in ecent yeas (see eg [6 9]) Time delays of one type o anothe have been incopoated into biological models by many eseaches (see eg [8 ]) In geneal delay diffeential equations exhibit much moe complicated dynamics than odinay diffeential equations since a time delay could cause the population to fluctuate Time delay due to gestation is a common example because geneally the consumption of pey by the pedato thoughoutitspasthistoygovensthepesentbithateof the pedato Theefoe moe ealistic models of population inteactions should tae into account the effect of time delays Basedontheabovediscussionsinthispapeweincopoate a pey efuge stage stuctue fo the pedato and time delay due to the gestation of pedato into the system () To this end we study the following diffeential equations: S (t) +I(t) =S(t) ( ) βs(t) I (t) K =βs(t) I (t) (t) b ( m) I (t) Y (t) ay (t) + ( m) I (t) dy = pb ( m) I (t τ) Y () (t τ) ay (t τ) + ( m) I (t τ) ( +d )Y (t) dy = Y (t) d Y (t) whee Y (t) and Y (t) epesent the densities of the immatue and the matue pedato population at time t espectively the paametes d d and ae positive constants in which d and d ae the death ates of the immatue and the matue pedato espectively denotes the ate of immatue pedato becoming matue pedato the constant popotion infectedpey efugeis ( m)iwheem [0)is a constant and τ 0 is a constant delay due to the gestation of the pedato The initial conditions fo system ()taethefom S (θ) =φ (θ) 0 I(θ) =φ (θ) 0 Y (θ) =φ (θ) 0 Y (θ) =φ (θ) 0 θ [ τ 0) φ (0) >0 φ (0) >0 φ (0) >0 φ (0) >0 (φ (θ) φ (θ) φ (θ) φ (θ)) C([ τ 0] R 4 +0 ) whee R 4 +0 ={(x x x 3 x 4 ):x i 0i=34} (3) It is well nown by the fundamental theoy of functional diffeential equations [] that system () has a unique solution (S(t) I(t) Y (t) Y (t)) satisfying initial conditions (3) The oganization of this pape is as follows In the next section we show the positivity and the boundedness of solutions of system() with initial conditions (3)In Section 3 we investigate the global stability of the pedato-extinction equilibium In Section 4 we establish the local stability and the global attactivity of the coexistence equilibium of system () Futhe we study the existence of Hopf bifucation fo system ()at the positive equilibiuma bief discussion is given in Section 5 to conclude this wo Peinaies In this section we show the positivity and the boundedness of solutions of system () with initial conditions (3) Theoem Solutions of system () with initial conditions (3) ae positive fo all t 0 Poof Let (S(t) I(t) Y (t) Y (t)) be a solution of system () with initial conditions (3) It follows fom the fist and the second equations of system ()that t S (t) =S(0) exp { [ 0 K S (t) ( K +β)i(t)]}>0 (4) I (t) t b ( m) Y =I(0) exp { [βs (t) d (t) 0 ay (t)+( m) I (t) ]} >0 (5) Let us conside Y (t) and Y (t) fot [0τ]Since ψ (θ) 0 fo θ [ τ0] we deive fom the thid equation of system ()that dy ( +d ) Y (t) (6) Since ψ (0) > 0 a standad compaison agument shows that Y (t) Y (0) e ( +d )t >0; (7) that is Y (t) > 0 fo t [0τ]Fot [0τ]itfollowsfom the fouth equation of ()that dy d Y (t) (8) Since ψ (0) > 0 a standad compaison agument shows that Y (t) Y (0) e d t >0; (9) that is Y (t) > 0 fo t [0τ]Inasimilawayweteatthe intevals [ττ][nτ(n + )τ] n NThusS(t) > 0 I(t) > 0 Y (t) > 0andY (t) > 0foallt 0 This completes the poof

3 Applied Mathematics 3 Theoem Positive solutions of system () with initial conditions (3) ae ultimately bounded Poof Let (S(t) I(t) Y (t) Y (t)) be any positive solution of system () with initial conditions (3) Denote d = min{d d d }Define V (t) =ps(t τ) +pi(t τ) +Y (t) +Y (t) (0) Calculating the deivative of V(t) along positive solutions of system () it follows that dv = p S (t τ)[k S(t τ) I(t τ)] K p(t τ) d Y (t) d Y (t) dv (t) p K K(+ d) [S (t τ) ] + pk( + d) () which yields pk( + d) supv (t) t 4 d () If we choose M = (K( + d) )/4 d and M = (pk( + d) )/4 dthen sups (t) M t This completes the poof supi (t) M t supy i (t) M (i =) t 3 Pedato-Extinction Equilibium and Its Stability 4 (3) In this section we discuss the stability of the pedatoextinction equilibium It is easy to show that if Kβ > d system() admitsa pedato-extinction equilibium E (S I 00)whee S = d β I = (Kβ d) β(kβ+) (4) The chaacteistic equation of system () attheequilib- ium E is of the fom λ +g λ+g 0 +f 0 e λτ =0 (5) whee g =( +d +d ) g 0 =d ( +d ) f 0 = pb When τ=0ifd ( +d )>pb thene is locally asymptotically stable and if d ( +d )<pb thene is unstable It is easily seen that g g 0 =( +d ) +d >0 (6) g 0 f 0 =d ( +d )+pb >0 Hence if d ( +d )>pb bylemmabin[] it follows that the equilibium E is locally asymptotically stable fo all τ 0Ifd ( +d )<pb thene is unstable fo all τ 0 Theoem 3 Let Kβ > d hold; the pedato-extinction equilibium E is globally stable povided that d ( +d )>pb 0 < ( m) < a (Kβ d) (7) b Poof Basedonabovediscussionsweonlypovetheglobal attactivity of the equilibium E Let(S(t) I(t) Y (t) Y (t)) be any positive solution of system () with initial conditions (3) It follows fom the fist and the second equations of system ()that =S( S+I K ) βsi βsi (8) Conside the following auxiliay equations: dx dx =x ( x +x K ) βx x =βx x dx (9) If Kβ > d then by Theoem 3 in [4] it follows fom (9)that x (t) = d β By compaison we obtain that x (t) = (Kβ d) β(kβ+) (0) sups (t) d β supi (t) (Kβ d) β(kβ+) =I () Hence fo ε>0sufficiently small thee is a T >0such that if t>t theni(t) I +ε It follows fom the thid and the fouth equations of system ()thatfot>t +τ dy dy pb ( m) (I +ε)y (t τ) ay (t τ) + ( m) (I +ε) ( +d )Y (t) = Y (t) d Y (t) Conside the following auxiliay equations: du du = pb ( m) (I +ε)u (t τ) au (t τ) + ( m) (I +ε) ( +d )u (t) = u (t) d u (t) () (3) If d ( +d )> pb thenbylemma4in[9] it follows fom (3)that u (t) =0 By compaison we obtain that Y (t) =0 u (t) =0 (4) Y (t) =0 (5)

4 4 Applied Mathematics Hence fo ε>0sufficiently small thee is a T >0such that if t>t theny (t) ε It follows fom the fist and the second equations of system ()thatfot>t : =S( S+I K ) βsi βsi b ( m) εi aε + ( m) I Conside the following auxiliay equations: dx dx =x ( x +x K ) βx x =βx x dx b ( m) εx aε + ( m) x (6) (7) If Kβ > dand( m) < (a/b)(kβ d) then by Theoem 3 in [4] it follows fom (7)that x ( m) (Kβ d) εaβ(kβ+) (t) = ( m) β(kβ+) +(([( m) (Kβ d) εaβ(kβ+)] +4ε( m) β (Kβ + ) [a(kβ d) b( m)]) / then system () has a unique coexistence equilibium E (S I Y Y )whee S = da p+( m) [ pb d ( +d )] βa p I = +Kβ (K S ) Y = ( m)[ pb d ( +d )] I a ( +d ) Y = ( m)[ pb d ( +d )] I ad ( +d ) (3) The chaacteistic equation of system () at the equilibium E taes the fom λ 4 +p 3 λ 3 +p λ +p λ+p 0 + (q λ +q λ+q 0 ) e λτ =0 (3) whee p 3 = +d +d + K S α p =d ( +d )+( +d +d )( K S α ) +S [β ( K +β)i K α ] p =S ( +d +d )[β( K +β)i K α ] (( m) β(kβ+)) ):=x x (t) =K Kβ + x By compaison fo ε sufficiently small we obtain that (8) +d ( +d )( K S α ) p 0 =d ( +d )S [β ( K +β)i K α ] q = p α q = p α ( K S α )+p α α 3 inf S (t) d β inf which togethe with () yields S (t) = d β I (t) (Kβ d) β(kβ+) (9) (Kβ d) I (t) = β(kβ+) (30) Hence if Kβ > d d ( +d )>pb ( m) < (a/b) (Kβ d) hold then the equilibium E (S I 00)is globally stable 4 Coexistence Equilibium and Its Stability In this section we discuss the stability of the coexistence equilibium and the existence of a Hopf bifucation It is easy to show that if the following holds: (H) d ( +d ) < pb 0 < ( m) < (a p(kβ d))/( pb d ( +d )) q 0 α = = p α S [β ( K +β)i K α ]+p α α 3 K S b( m) I Y [ay + ( m) I ] α = b( m) (I ) [ay + ( m) I ] α 3 = ab ( m) (Y ) [ay + ( m) I ] (33) When τ=0(3)becomes λ 4 +p 3 λ 3 + (p +q ) λ+p 0 +q 0 =0 (34) If the following holds: (H) (/K)S α >0 β((/k) + β)i (/K)α >0 then it is easy to show that p 3 >0 p 0 +q 0 >0 p +q >0 p +q >0 (35)

5 Applied Mathematics 5 If (p +q )[p 3 (p +q ) (p +q )] > p 3 (p 0 +q 0 )then by the Routh-Huwitz theoem when τ=0 the coexistence equilibium E of system () is locally asymptotically stable and E is unstable if (p +q )[p 3 (p +q ) (p +q )] < p 3 (p 0 +q 0 ) If iω(ω > 0) is a solution of (34) sepaating eal and imaginay pats we have (q ω q 0 ) sin ωτ + q ω cos ωτ = p 3 ω 3 p ω (q ω q 0 ) cos ωτ q ω sin ωτ = ω 4 p ω +p 0 (36) By squaing and adding the two equations of (36) it follows that whee ω 8 +h 3 ω 6 +h ω 4 +h ω +p 0 q 0 =0 (37) h 3 =p 3 p h =p +p 0 q p p 3 h =p +q 0q q p 0p (38) If h 3 > 0 h > 0 h > 0 and p 0 q 0 > 0bythe geneal theoy on chaacteistic equation of delay diffeential equation fom [3] (Theoem 4) E emains stable fo all τ>0 If h i > 0 (i = 3) and p 0 q 0 <0then(37) hasa unique positive oot ω 0 ;thatis(34) admitsapaiofpuely imaginay oots of the fom ±ω 0 Fom(36) we see that τ n = nπ + accos ω 0 ω 0 ((q ω 0 q 0)(ω 4 0 p ω 0 +p 0) +q ω 0 (p 3 ω 3 0 p ω 0 )) ((q ω 0 ) +(q ω 0 q 0) ) n = 0 (39) By Theoem 34 in [3] we see that E emains stable fo τ<τ 0 In the following we claim that d(re(λ)) >0 (40) dτ τ=τ 0 This will show that thee exists at least one eigenvalue with apositiveealpatfoτ>τ 0 Moeove the conditions fo the existence of a Hopf bifucation (Theoem 9 in [3]) ae then satisfied yielding a peiodic solution To this end by diffeentiating equation (34)withespecttoτitfollowsthat ( dλ dτ ) 4λ 3 +3p = 3 λ +p λ+p λ (λ 4 +p 3 λ 3 +p λ +p λ+p 0 ) q + λ+q λ(q λ +q λ+q 0 ) τ λ (4) Hence a diect calculation shows that sgn { d(re λ) } dτ λ=iω 0 = sgn {Re ( dλ dτ ) } λ=iω 0 = sgn { ((3p 3 ω 0 p )(p 3 ω 0 p ) We deive fom (36)that Hence it follows that sgn { + (ω 0 p )(ω 4 0 p ω 0 +p 0)) (ω 0 (p p 3 ω 0 ) +(ω 4 0 p ω 0 +p 0) ) + q +q (q 0 q ω 0 ) (q ω 0 ) +(q ω 0 q 0) } (4) ω 0 (p p 3 ω 0 ) +(ω 4 0 p ω 0 +p 0) =(q ω 0 ) +(q ω 0 q 0) d (Re λ) } dτ λ=iω 0 = sgn { 4ω6 0 +3h 3ω 4 0 +h ω 0 +h (q ω 0 ) +(q ω 0 q 0) }>0 (43) (44) Theefoe the tansvesal condition holds and a Hopf bifucation occus at ω=ω 0 τ=τ 0 In conclusion we have the following esults Theoem 4 Fo system () let(h) and (H) hold; we have the following: (i) if (p +q )[p 3 (p +q ) (p +q )] > p 3 (p 0+q 0 ) h i >0 and p 0 q 0 >0 then the coexistence equilibium E is locally asymptotically stable fo all τ 0; (ii) if (p +q )[p 3 (p +q ) (p +q )] > p 3 (p 0+q 0 ) h i >0 and p 0 q 0 <0 then thee exists a positive numbe τ 0 such that the coexistence equilibium E is locally asymptotically stable if 0 τ<τ 0 and is unstable fo τ>τ 0 ;futhesystem() undegoes a Hopf bifucation at E when τ=τ n n=0; (iii) if (p +q )[p 3 (p +q ) (p +q )] < p 3 (p 0 +q 0 ) then the coexistence equilibium E is unstable fo all τ 0 We now give some examples to illustate the main esults above Example 5 In () we let a = p = 09 = 0 = 05 = β = d = d = d = 0 b = and

6 6 Applied Mathematics Solution S I Time t Figue : The tempoal solution found by numeical integation of system ()withτ= m = 05System() with the above coefficients has a unique coexistence equilibium E ( )It is easy to show that (/K)S α 3468 > 0 β((/k) + β)i (/K)S α > 0 thatisthecondition(h) holds We can get (p +q )[p 3 (p +q ) (p +q )] p 3 (p 0+q 0 ) 350 > 0 h > 0 h 6563 > 0 h 6535 > 0 andp 0 q > 0 ByTheoem4(i) the coexistence equilibium E is locally asymptotically stable fo all τ 0 Numeical simulation illustates ou esult (see Figue ) Example 6 In () we let a = 055 p = 095 = 0 = 05 = β = d = d = d = 0 b = and m = 05System() with the above coefficients has a unique coexistence equilibium E ( )It is easy to show that (/K)S α > 0andβ((/K)+ β)i (/K)S α 0083 > 0; that is the condition (H) holds We can get (p +q )[p 3 (p +q ) (p +q )] p 3 (p 0+q 0 ) 9949 > 0 h > 0 h 6975 > 0 h 33 > 0 andp 0 q < 0 ByTheoem4(ii) thee exists a positive numbe τ suchthatthe coexistence equilibium E is locally asymptotically stable if 0 τ<τ 0 and is unstable fo τ>τ 0 Numeicalsimulations illustate ou esults (see Figue ) Now we ae concened with the global attactiveness of the coexistence equilibium E Theoem 7 The coexistence equilibium E (S I Y Y ) of system () is globally attactive povided that the following conditions hold: Y Y That is the system () is pesistent if conditions (i) and (ii) hold Poof Let (S(t) I(t) Y (t) Y (t)) be any positive solution of system () with initial conditions (3) Let U S = sups (t) U I = supi (t) L S = infs (t) L I = infi (t) (45) U Yi = supy i (t) L Yi = inf Y i (t) We now claim that U S =L S =S U I =L I =I U Yi =L Yi = Y i (i = ) The stategy of the poof is to use an iteation technique We deive fom the fist and the second equations of the system ()that =S( S+I K ) βsi βsi (46) Conside the following auxiliay equations: dx dx =x ( x +x K ) βx x =βx x dx (47) If Kβ > d then by Theoem 3 in [4] it follows fom (47) that x (t) = d β By compaison we obtain that x (t) = (Kβ d) β(kβ+) (48) U S = sups (t) d β := MS U I = supi (t) (Kβ d) β(kβ+) := MI (49) Hence fo ε>0sufficiently small thee is a T >0such that if t>t theni(t) M I +ε It follows fom the thid and the fouth equations of system ()thatfot>t +τ dy dy pb ( m) (MI +ε)y (t τ) ay (t τ) + ( m) (M I +ε) ( +d )Y (t) = Y (t) d Y (t) Conside the following auxiliay equations: du = pb ( m) (MI +ε)u (t τ) au (t τ) + ( m) (M I +ε) ( +d )u (t) (50) (i) 0 < ( m) < (a/b)(kβ d); (ii) d ( +d )< pb < d ( +d ) du = u (t) d u (t) (5)

7 Applied Mathematics Solution Solution Time t Time t S I Y Y S I Y Y (a) (b) Figue : The tempoal solution found by numeical integation of system () with(a) τ=and (b) τ=5 If pb > d ( +d )thenbylemma4in[9] it follows fom (5)that Since ( m) < (a/b)(kβ d) holds by Theoem 3 in [4] it follows fom (54) and compaison agument that u (t) = ( m)(mi +ε)[ pb d ( +d )] a ( +d ) u (t) = ( m)(mi +ε)[ pb d ( +d )] ad ( +d ) (5) L S = inf d b ( m) + := N S β aβ L I = inf +βk (55) ad + b ( m) (K ):=N I aβ By compaison we obtain that U Y U Y = supy (t) ( m)[ pb d ( +d )] M I a ( +d ) := MY = supy (t) = ( m)[ pb d ( +d )] M I ad ( +d ) := MY (53) Hence fo ε>0sufficiently small thee is a T 3 >T such that if t>t 3 theni(t) N I ε We deive fom the thid and the fouth equations of system ()thatfot>t 3 +τ dy dy pb ( m) (NI ε)y (t τ) ( m) (N I ε)+ay (t τ) (d + )Y (t) = Y (t) d Y (t) (56) Hence fo ε>0sufficiently small thee is a T >T +τsuch that if t>t theny (t) M Y +ε Wedeivefomthefistanhesecondequationsof system ()that =S( S+I K ) βsi βsi b ( m) I a (54) Since pb > d ( +d ) holds by Lemma 4 of [9] it follows fom (56) and compaison agument that L Y L Y = inf Y (t) ( m)[ pb d ( +d )] (N I a ( +d ) ε) = inf Y (t) ( m) [ pb d ( +d )] ad ( +d ) (N I ε) (57)

8 8 Applied Mathematics Since these two inequalities hold fo abitay ε > 0 sufficiently small we conclude that L Y N Y L Y N Y whee N Y = ( m)[ pb d ( +d )] N I a ( +d ) (58) = ( m)[ pb d ( +d )] N I ad ( +d ) N Y Hence fo ε>0sufficiently small thee is a T 4 T 3 +τsuch that if t>t 4 Y (t) N Y ε Fo ε>0sufficiently small we deive fom the fist and the second equations of system ()thatfot>t 4 =S( S+I K ) βsi βsi Y b ( m)(n ε) a(n Y ε)+( m) (M I +ε)i By compaison and Theoem 3 in [4] it follows that U S = sups (t) U I d β + b ( m)(n Y ε) β[a(n Y ε)+( m) (M I +ε)] := MS = supi (t) (59) +Kβ (K MS ) (60) Since these two inequalities hold fo abitay ε > 0 sufficiently small we conclude that U S M S U I M I whee M S = d β + b ( m) N Y β[an Y + ( m) M I] M I = +Kβ (K MS ) (6) Theefoe fo ε>0sufficiently small thee is a T 5 T 4 such that if t>t 5 I(t) M I +ε Fo ε>0sufficiently small we deive fom the thid and the fouth equations of system ()thatfot>t 5 +τ dy dy pb ( m) (MI +ε)y (t τ) ( m) (M I +ε)+ay (t τ) (d + )Y (t) = Y (t) d Y (t) (6) Since pb >d ( +d ) holds by Lemma 4 of [9] it follows fom (6)that U Y U Y = supy (t) ( m)[ pb d (d + )] (M I a (d + ) +ε) = supy (t) ( m)[ pb d (d + )] (M I ad (d + ) +ε) (63) Since these two inequalities hold fo abitay ε > 0 sufficiently small we conclude that U Y M Y U Y M Y whee = ( m)[ pb d (d + )] M I a (d + ) (64) = ( m)[ pb d (d + )] M I ad (d + ) M Y M Y Theefoe fo ε>0sufficiently small thee is a T 6 T 5 +τ such that if t>t 6 y (t) M Y +ε Fo ε>0sufficiently small it follows fom the fist and the second equations of system ()thatfot>t 6 =S( S+I K ) βsi βsi Y b ( m)(m +ε) a(m Y +ε)+( m) (NI ε)i (t) (65) By Theoem 3 in [4] and compaison agument we can obtain L S = infs (t) d β + b ( m) (M Y +ε) β[a(m Y +ε)+( m) (NI ε)] L I = infi (t) +βk [ K d β b ( m)(m Y +ε) ] β[a(m Y [ +ε)+( m) (NI ε)] ] (66) Since these two inequalities hold fo abitay ε > 0 sufficiently small we conclude that L S N S L I N I whee N S = d β + b ( m) M Y β[am Y + ( m) NI ] N I = +βk (K NS ) (67)

9 Applied Mathematics 9 Hence fo ε>0sufficiently small thee is a T 7 T 6 such that if t>t 7 I(t) N I ε We theefoe obtain fom the thid and the fouth equations of system ()thatfot>t 7 +τ dy dy pb ( m) (NI ε)y (t τ) ( m) (N I ε)+ay (t τ) (d + )Y (t) = Y (t) d Y (t) (68) Since pb > d ( +d ) holds by Lemma 4 in [9] and compaison agument we deive that L Y L Y = inf Y (t) ( m)[ pb d ( +d )] a ( +d ) = inf Y (t) ( m) [ pb d ( +d )] ad ( +d ) (N I ε) (N I ε) (69) Since these inequalities hold fo abitay ε>0sufficiently small we conclude that L Y N Y L Y N Y whee N Y = ( m)[ pb d ( +d )] N I a ( +d ) (70) N Y = ( m)[ pb d ( +d )] N I ad ( +d ) Continuing this pocess we deive eight sequences M S MI MY MY NS NI NY and NY ( = ) such that fo M S = β [d + b ( m) N Y an Y + ( m) ] MI M I = +Kβ (K MS ) M Y = ( m)[ pb d ( +d )] M I a ( +d ) M Y = ( m)[ pb d ( +d )] M I ad ( +d ) N S = β [d + b ( m) M Y am Y + ( m) ] NI N I = +Kβ (K NS ) N Y = ( m)[ pb d ( +d )] N I a ( +d ) N Y = ( m)[ pb d ( +d )] N I ad ( +d ) (7) It is eadily seen that N S L S U S M S NI L I U I M I N Y i L Y i U Yi M Y i (i =) (7) Noting that the sequences M S MI MY MY ae noninceasing and the sequences N S NI NY NY ae nondeceasing Hence the it of each sequence in M S MI MY MY NS NI NY andny exists Denote + MS = S + MI = I + MY i = Y i (i =) + NS =S Fom (7) we can obtain + NI =I + NY i =Y i (i =) S= β [d + b ( m) Y ay + ( m) I ] I= (K S) +Kβ Y = ( m)[ pb d ( +d )] I a ( +d ) Y = ( m)[ pb d ( +d )] I ad ( +d ) S = β [d + b ( m) Y ay + ( m) I ] I= (K S) +Kβ Y = ( m)[ pb d ( +d )] I a ( +d ) Y = ( m)[ pb d ( +d )] I ad ( +d ) It follows fom (74)that αβ ( + Kβ) d ( +d )(I) +αβ(+kβ)[ pb d ( +d )] II =(Kβ d)ad ( +d ) I (73) (74) +[a(kβ d) b( m)][ pb d ( +d )] I (75) αβ ( + Kβ) d ( +d )(I) +αβ(+kβ)[ pb d ( +d )] II =(Kβ d)ad ( +d )I +[a(kβ d) b( m)][ pb d ( +d )] I (76)

10 0 Applied Mathematics (75)minus(76) αβ (Kβ + ) d ( +d )[(I) (I) ] =(Kβ d)ad ( +d )(I I) [a(kβ d) b( m)] [ pb d ( +d )] (I I) Assume that I =I Then we deive fom (77)that αβ (Kβ + ) d ( +d )(I+I) =(Kβ d)ad ( +d ) (75)plus(76) (77) [a(kβ d) b( m)][ pb d ( +d )] (78) αβ (Kβ + ) d ( +d )(I+I) +αβ(kβ+)[ pb d ( +d )] II =[(Kβ d)ad ( +d ) + (a (Kβ d) b ( m)) ( pb d ( +d ))] (I+I) On substituting (78)into(79) it follows that αβ (Kβ + ) [ pb d ( +d )] II =[a(kβ d) b( m)] [ pb d ( +d )] (I+I) (79) (80) Note that I>0and I >0Ifd ( +d )< pb < d ( +d ) we deive that m > a(kβ d)/b This is a contadiction Hence we have S=S It theefoe follows fom (74)thatI= I Y =Y and Y =Y We theefoe conclude that E is globally attactive The poof is complete 5 Conclusion In this pape we have incopoated a pey efuge stage stuctuefothepedatoanimedelayduetothegestationofthe pedato into a pedato-pey system Incopoating a efuge into system () povides a moe ealistic model A efuge can be impotant fo the biological contol of a pest; howeve inceasing the amount of efuge can incease pey densities andleaopopulationoutbeasbyusingtheiteation technique and compaison aguments espectively we have established sufficient conditions fo the global stability of the pedato-extinction equilibium and the globally attactivity fo the coexistence equilibium As a esult we have shown the theshold fo the pemanence and extinction of the system By Theoem 3 we see that the pedato population go to extinction if 0 < ( m) < (a/b)(kβ d) and β > d d ( +d )>pb ByTheoem7weseethatif0<( m)< (a/b)(kβ d) and d ( +d )< pb < d ( +d )thenboth the pey and pedato species of system ()aepemanent Conflict of Inteests The authos declae that thee is no conflict of inteests egading the publication of this pape Acnowledgment This wo was suppoted by the National Natual Science Foundation of China (07) Refeences [] W O Kemac and A G Mcendic A contibution on the mathematical theoy of epidemics Poceedings of the Royal Society A vol 5 pp []RMAndesoandRMMayIntections Disease of Humans Dynamics and Contol OxfodUnivesityPessOxfodUK 99 [3]AKPalandGPSamanta Stabilityanalysisofanecoepidemiological model incopoating a pey efuge Nonlinea Analysis: Modelling and Contolvol5no4pp [4] S Wang The eseach of eco-epidemiological of models incopoating pey efuges [PhD thesis] Lanzhou Univesity 0 [5] Y Xiao and L Chen A atio-dependent pedato-pey model with disease in the pey Applied Mathematics and Computation vol 3 no -3 pp [6] W Wang and L Chen A pedato-pey system with stagestuctue fo pedato Computes & Mathematics with Applicationsvol33no8pp [7] YNXiaoandLSChen Globalstabilityofapedato-pey system with stage stuctue fo the pedato Acta Mathematica Sinicavol0nopp [8] R Xu and Z Ma Stability and Hopf bifucation in a pedatopey model with stage stuctue fo the pedato Nonlinea Analysis: Real Wold Applications vol9no4pp [9]RXuandZMa StabilityandHopfbifucationinaatiodependent pedato-pey system with stage stuctue Chaos Solitons & Factalsvol38no3pp [0] W G Aiello and H I Feedman A time-delay model of singlespecies gowth with stage stuctue Mathematical Biosciences vol 0 no pp [] Y Kuang and J W-H So Analysis of a delayed two-stage population model with space-ited ecuitment SIAM Jounal on Applied Mathematicsvol55no6pp [] J Hale Theoy of Functional Diffeential Equations Spinge New Yo NY USA nd edition 977 [3] Y Kuang Delay Diffeential Equations with Applications in Population Dynamics vol 9 Academic Pess New Yo NY USA 993

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