Impulsive Integrated Pest Management Model with a Holling Type II Functional Response

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1 ISSN: Volume 1, Issue 10, Impulsive Inegraed Pes Managemen Model wih a Holling Type II Funcional Response Bhanu Gupa 1, Ami Sharma 2 1 P.G. Deparmen of Mahemaics, Jagdish Chandra DAV College, Hoshiarpur, India 2 P.G. Deparmen of Mahemaics, Jagdish Chandra DAV College, Hoshiarpur, India 2 IKG PTU, PhD Scholar, Kapurhala, India Absrac: In his paper, a prey predaor impulsive mahemaical model of inegraed pes managemen is esablished, in which infeced prey and predaor (naural enemy) are released impulsively. By using he Floque s heory for impulsive differenial equaions, small-ampliude perurbaion mehods and comparison echniques, we invesigae he local and global sabiliy of he suscepible pes-eradicaion periodic soluion. Keywords: Suscepible prey; Infeced prey; Predaor; Impulse. I. INTRODUCTION Mahemaical modeling is a echnique in which various naural processes and phenomena are convered ino mahemaical erms (expressions) and hen sudied for heir soluions, and coninually refined over a period of ime o ge more and more accurae and efficien resuls. During las few years mahemaical modeling has played an imporan role in sudying he biological, pharmaceuical and agriculural processes. From housands of years, pes conrol in agriculure has been a major concern for farmers. People in differen era used heir own mehodologies and echnologies o conrol he pess which desroy he plans and ulimaely reduce crop producion. However wih he passage of ime, a number of pes conrol sraegies are available o farmers such as physical conrol, biological conrol and chemical conrol and remoe sensing. In biological conrol echnique, he pess are conrolled via manipulaing he naure in which some organisms (predaors or infeced pess) are released, which consume or infec he pes populaion and ulimaely improve he crop producion. In chemical conrol, he pesicides are used o conrol he pess and are very effecive as hese can kill he pess rapidly bu he exensive use of hese pesicides are creaing major healh problems. So he ime demands he use of more and more biological conrol o conrol he pess desroying crops. A number of auhors sudied he managemen of pes conrol using biological echnique [1, 2, 8, 9, and 10]. Many evoluion processes are characerized by he fac ha a cerain momens of ime, hey experience a change of sae abruply. The impulsive sysems of differenial equaion are an adequae apparaus for he mahemaical modeling of numerous processes and phenomena sudied in biology, economics and echnology ec. As in pes conrol sraegies discussed above (chemical and biological), he pesicide, he infeced pess or predaors are released impulsively, so he pes managemen can be very efficienly sudied hrough modeling by impulsive differenial equaions. The sudy of pes managemen using impulsive differenial equaions was sared in 2005 due o Zhang e al [7]. They modeled he sysem as: , IJAERA - All Righs Reserved 424

2 x () = x()(1 x()) a 1x() y() 1+b 1 x() y () = a 1x() y() a 2 y()z() d 1+b 1 x() 1+b 2 y() 1 y() z () = a 2 y()z() d 1+b 2 y() 2 z() x(+) = x() y( +) = y() = nt z(+) = z() + p nt where x(), y() and z() are he populaion densiy of prey, predaor and op predaor a ime respecively and p is he amoun of he predaor ha is inroduced ino he populaion a periodic inervals of lengh T. The auhors esablished he condiions for global asympoic sabiliy of pes exincion periodic soluion (1,0, z ) as well as for he permanency of he sysem. They proved ha periodic soluion (1,0, z ()) is locally asympoically sable when T < a 2 p(1+b 1 ). and sysem is globally asympoically sable. d 2 (a 1 b 1 d 1 d 1 ) Shi and Chan [6] sudied an impulsive prey predaor model wih disease in prey for purpose of inegraed pes managemen. Here he auhors derived a sufficien condiion for he global sabiliy of suscepible pes eradicaion periodic soluion. The aim of his paper is o sudy a Holling II funcional responses prey predaor impulsive mahemaical model of inegraed pes managemen in which boh infeced prey and predaor (naural enemy) are released impulsively. The local sabiliy of pes exincion periodic soluion is sudied using Floque heory and global sabiliy of pes exincion periodic soluion is sudied by using comparison principle of impulsive differenial equaions. II. MATHEMATICAL MODEL Assumpions: A1 : Due o disease in pes populaion he oal pes populaion is divided ino wo classes, suscepible pes populaion and infeced pes populaion. A2 : The incidence rae among suscepible and infeced pes populaion is Holling ype II i.e., αs() Z() 1+xS() suscepible pes. where α is he conac number per uni ime of infeced pes wih A3 : A4 : The predaor aacks suscepible pess only and he predaion funcional response is again Holling ype II i.e., βs()i() where β is he conac number per uni ime of 1+yS() predaor wih suscepible pes. A ime = nt, n Z + = {1,2,3,, } he infeced pes and predaor are released periodically wih amoun u and v respecively, u > 0, v > 0. In his paper, we sudy he following model for inegraed pes managemen , IJAERA - All Righs Reserved 425

3 S () = S()(1 S()) αs() Z() 1+xS() βs()i() 1+yS(), I () = βs()i() d 1 +ys() 1 I(), Z αs() Z() () = d 1+xS() 2 Z(), S() = 0, = nt, n Z + = {1,2,3,, } I() = u, Z() = v. (1) Where S(): Densiy of suscepible pes a ime. I(): Densiy of infeced pes a ime. Z(): Densiy of predaor (naural enemy) a ime. d 1 : Naural deah rae of infeced pes. d 2 : Naural deah rae of predaor (naural enemy). α : The conac number per uni ime of predaor wih suscepible pes. β : The conac number per uni ime of infeced pes wih suscepible pes. u : The amoun of infeced pes released periodically. v: The amoun of predaor released periodically. S() = S( + ) S(), I() = I( + ) I(), Z() = Z( + ) Z(). T is period of impulsive effec. III. PRELIMINARIES nt, R + = [0, ) and R 3 + = {x = (x 1,x 2, x 3 ) R 3 + : x 1,x 2, x 3 > 0}. Le V: R + R 3 + R +. Then V is said o belong o class V0 if 3 (i) V is coninuous in (nt, (n + 1)T] R + and for each x R 3 +,n Z + = {1,2,3,, } and he lim V(,y) = (,y) (nt +,x) V(nT+, x) exiss and is finie. (ii) V is locally Lipschizian in x. Definiion 3.1 For V V 0 and (, x) (nt, (n + 1)T] R 3 +, he upper righ Dini derivaive of V(, x) wih respec o he impulsive differenial sysem (1) is defined as D + V(, x) = lim sup 1 [V( + h, x + hf(,x)) V(, x)]. h 0 + h 3 Definiion 3.2 Sysem (1) is said o be permanen if here exiss a compac region D in R + such ha every soluion of sysem (1) wih posiive iniial values will evenually ener and remain in region D. 3 The soluion of sysem (1) denoed by X() = (S(), I(), Z()): R + R + is coninuously differeniable on (nt, (n + 1)T] R 3 +, n Z + = {1,2,3,, } and he limi X(nT + ) = lim X() exiss and is finie for nt + n Z +. The global exisence and uniqueness of soluion of sysem (1) is guaraneed by he smoohness properies similar as in [5]. Below we sae some lemmas whose proofs are obvious. Lemma 3.1 Suppose ha X() is a soluion of (1) wih X(0 + ) 0 for all > 0. Furher, if X(0 + ) > 0 hen X() > 0 for all > , IJAERA - All Righs Reserved 426

4 Lemma 3.2 [5] Le V: R + R 3 + R and V V 0. Assume ha D + V(,x) g(, V(, x)), nt, V(, X( + )) ψ n V(, X()), = nt, where g: R + R + R is coninous in (nt, (n + 1)T] R + and for each v R 3 +,n Z + lim g(, y) = (,y) (nt +,v) g(nt+, v) exiss and is finie. Le ψ n : R + R + is non-decreasing and R() be he maximal soluion of he scalar impulsive differenial equaion u () = g(,u), nt, u( + ) = ψ n (u()), = nt, u(0 + ) = u 0, defined on [0, ). Then V(0 +,x 0 ) u 0 implies ha V(, x()) R(), 0, where x() is any soluion of sysem (1). IV. BOUNDEDNESS In his secion, we prove he boundedness of he sysem (1). Lemma 4.1 [5] Le he funcion m PC [R +,R] and m() be lef-coninuous a k, k = 1,2,3, saisfy he inequaliies m () p()m() + q(), 0, k, m( k ) d k m( k ) + b k, = k, k = 1,2,3, (2) where p, q PC [R +,R] and d k 0, b k are consans, hen m() m( 0 ) d k exp ( p(s)ds) + 0 < k < ( d j exp ( p(s)ds) 0 0 < k < 0 < j < ) b k + 0 s< k < d k exp ( p(σ)dσ q(s) ds), 0 s 0. (3) If all he direcions of inequaliies in (2) are reversed, he inequaliy (3) also holds rue for he reversed inequaliy. Theorem 4.2 There exis a posiive consan L such ha S() L, I() L,Z() L, for each soluion (S(), I(), Z()) of sysem (1) wih posiive iniial values, where is large enough. Proof. Define a funcion V such ha V() = S() + I() + Z(). Then for nt, D + V() + dv() = S () + I () + Z () + d(s() + I() + Z()) Le d = min{d 1, d 2 }. We obain D + V() + dv() (1 + d)s() S 2 () M 0, where M 0 = ( 1+d) 2. 4 When = nt, V( + ) = V() + u + v. Using Lemma 4.1, we ge V() = V(0)e d + M 0 e d( s) ds + 0 0<kT< (u + v)e d( kt) M 0 d + (u+v)e dt,as. e dt 1 Consequenly, by he definiion of V() we obain ha each soluion of (1) wih posiive iniial values is uniformaly ulimaely bounded. This complees he proof , IJAERA - All Righs Reserved 427

5 V. STABILITY In his secion, we sudy he sabiliy of pes eradicaion periodic soluion of sysem (1) using Floque heory of impulsive differenial equaions. The condiion for global aracion of pes eradicaion period is also esablished. Lemma 4.3. Sysem u () = w u(), nt, u() = μ, = nt. (4) has a posiive soluion u (), for every soluion u() of his sysem wih posiive value u(0 + ), u() u () 0 as, wher u () = μe w ( nt) 1 e wt u (0 + ) = μ 1 e wt. Proof. The proof is obvious, in fac, since he soluion of (4) is u() = (u(0 + μ ) 1 e wt) e wt + u (). nt < (n + 1)T. When S() 0 for all 0, we ge he subsysem of sysem (1) I () = d 1 I(), nt, Z () = d 2 Z(), I() = u, = nt, n Z + = {1,2,3,, } (5) Z() = v. In his sysem, we can see here is no relaion beween I() and Z(). Thus, we can solve hem independenly. By Lemma 3.6, we ge he following resul. Theorem 4.3 Sysem (5) has a unique posiive periodic soluion I () = ue d 1 ( nt) 1 e d 1T, Z () = ve d 2 ( nt), for nt < (n + 1)T. 1 e d 2T Where I (0 + u ) =, 1 e d 1T Z (0 + ) = v. 1 e d 2 T In addiion for every soluion of he sysem (5) wih iniial values I(0 + ) > 0, Z(0 + ) > 0, i follows ha I() I (),Z() Z (), as. Thus, he complee expression for he suscepible pes-eradicaion soluion of sysem (1) is obained as (0, I (),Z ()). Theorem 5.1 Le (S(),I(), Z()) be any soluion of (1) (i) The rivial soluion (0,0,0) is unsable. (ii) If T < ( αv d2 + βu d1 ), hen he suscepible pes eradicaion periodic soluion (0, I (), Z ()) is locally asympoically sable. Proof: (i) To prove he local sabiliy of rivial soluion (0,0,0), we use small-ampliude perurbaion mehod. Le S() = p(), I() = q(), Z() = r(), where p(), q(), r() are small perurbaions. Then sysem (1) can be linearized by using Taylor expansions afer neglecing higher-order erms as: , IJAERA - All Righs Reserved 428

6 p () = p(), q () = d 1 I (), nt r () = d 2 Z (), p(nt + ) = p(nt), (6) q(nt + ) = q(nt), = nt, n Z + = {1,2,3,, } r(nt + ) = r(nt), Le () be he fundamenal soluion marix of (6). Then () saisfy d () = ( 0 d d 1 0 ) (), 0 0 d 2 (0) = I 3 is he ideniy marix. Hence he fundamenal soluion marix is e T 0 0 () = ( 0 e d 1 T 0 ) 0 0 e d 2 T Also, he fourh, fifh and sixh equaions in (6) read as p(nt + ) p(nt) ( q(nt + )) = ( 0 1 0) ( q(nt) ) r(nt + ) r(nt) M = ( 0 1 0) (T). Therefore he eigen values of M are λ 1 = e d 2 T < 1,λ 2 = e d 1 T < 1, λ 3 = e T > 1. Since λ 3 > 1, rivial soluion (0,0,0) is unsable (ii) Now for local sabiliy of periodic soluion (0, I (), Z ()), he small-ampliude perurbaion implies S() = p(), I() = q() + I (), Z() = r() + Z (), where p(), q(), r() are small perurbaions. Then sysem (1) can be linearized by using aylor expansions and afer neglecing higher-order erms we ge p () = p() α p()z () βp()i (), q () = βp()i () d 1 q() d 1 I (), nt r () = α p()z () d 2 r() d 2 Z (), p(nt + ) = p(nt), (7) q(nt + ) = q(nt), = nt, n Z + = {1,2,3,, } r(nt + ) = r(nt), Le () be he fundamenal soluion marix of (7). Then () saisfy 1 α Z () βi () 0 0 d () = ( βi () d d 1 0 ) (), α Z () 0 d 2 Where (0) = I 3 is he ideniy marix. Hence he fundamenal soluion marix is , IJAERA - All Righs Reserved 429

7 T [1 α Z () βi ()]d e () = ( e d 1 T 0 ) 0 e d 2 T Also, he fourh, fifh and sixh equaions in (4) read as p(nt + ) p(nt) ( q(nt + )) = ( 0 1 0) ( q(nt) ) r(nt + ) r(nt) Hence, if all eigen values of M=( 0 1 0) (T) have absolue values less han 1, hen he periodic soluion (0, I (),Z ()) is locally asympoically sable. The eigen values of M are λ 3 = e T 0 [1 α Z () βi ()]d. λ 1 = e d 2 T < 1,λ 2 = e d 1 T < 1, I follows ha λ 3 < 1 if and only if T < ( αv + βu d2 d1 ) holds. Thus he Floque heory of impulsive differenial equaions, in his siuaion, implies ha he suscepible pes-eradicaion periodic soluion (0, I (),Z ()) is locally asympoically sable. The proof is complee. Theorem 5.2 If T < ( αv + βu ), hen he periodic soluion (0, d2 d1 I (),Z ()) is globally asymoically sable for he sysem (1). Proof: By given condiion and Theorem 5.1, i is easy o know ha (0, I (), Z ()) is locally asympoically sable. Therefore, we only need o prove is global aracion. Since T < ( αv + βu ), we can choose a ϵ d2 d1 1small enough such ha Besides, we have T 0 [1 α (Z () ϵ 1 ) β(i () ϵ 1 )]d = σ < 0. I () = βs()i() 1 + ys() d 1 I() d 1 I(). From Lemmas 3.2 and 4.3, here exiss a n 1 such ha for I() I () ϵ 1, for n 1 T, Similarly, here exiss a n 2 (n 2 > n 1 ) such ha Z() Z () ϵ 1, for n 2 T. Thus, for n 2 T, we have S αs() Z() () = S()(1 S()) 1 + xs() βs()i() 1 + ys() αs() Z() S() 1 + xs() βs()i() 1 + ys() S() (1 α(z () ϵ 1 ) β(i () ϵ 1 ) ). 1 +xs() 1+yS() From he above inequaliy, we ge α(z [1 () ϵ1 ) β(i () ϵ1 ) ]d 1+xS() S() S(n 2 T)e n2t 1+yS() S(n 2 T)e kσ. Where ((n 2 + k)t,(n 2 + k + 1)T],k Z +. Since σ < 0, we can easily see ha S() 0 as k +. Thus for arbirary posiive consan ϵ 2 small enough, here exis n 3 (n 3 > n 2 ) such ha , IJAERA - All Righs Reserved 430

8 S() < ϵ 2 for all n 3 T. From which we ge I () = βs()i() d 1 I() (βϵ 2 d 1 ) I(), From Lemmas 3.2 and 4.3, here exiss a n 4 (n 4 > n 3 ) such ha I() I 2 () + ϵ 1, for n 4 T, where I 2 () = ue (d1 ϵ2 )( kt) 1 e (d1 ϵ2 ) T, for (kt,(k + 1)T],k Z +. By similarly argumen, here exiss a n 5 (n 5 > n 4 ) such ha Z() Z 2 () + ϵ 1, for n 5 T, where Z 2 () = ve (d2 βϵ2 )( kt) 1 e (d2 βϵ2 ) T, for (kt,(k + 1)T],k Z +. Noe ha ϵ 1, ϵ 2 are posiive consans small enough and I 2 () I (),Z 2 () Z () as. Therefore, he periodic soluion (0, I (), Z ()) is globally asymoically sable. VI. REFERENCES [1] DD Bainov and PS Simeonov (1993). Impulsive differenial equaions: periodic soluion and applicaions. London: Longman. [2] P Debach and D Rosen (1991). Biological conrol by naural enemies. 2nd ed. Cambridge: Cambridge Universiy Press. [3] B Goh (1980). Managemen and analysis of biological populaions. Amserdam, Oxxford, New york: Elsevier Scienific Publishing Company. [4] Y Ding, S Gao, Y Liu and Lan (2010). A pes managemen epidemic model wih ime delay and sage srucure, Appl. Mah. 1 pp [5] V Lakshmikanham, DD Bainov and PS Simeonov (1989). Theory of impulsive differenial equaions. Singapore: world Scienific. [6] R Shi and L Chen (2010). An impulsive predaor-prey model wih disese in he prey for inegraed pes managemen, Commun Nonlinear Sci Numer Simula, pp [7] S Zhang and L Chen (2005). A Holling II funcional response food chain model wih impulsive perurbaions, Chaos Solions and Fracals 24, pp [8] HJ Freedman (1976). Graphical sabiliy, enrichmen and pes conrol by a naural enemy, Mah Biosci, 31 pp [9] ML Luff (1983). The poenial of predaors for pes conrol, Agri Ecos Environ, 10 pp [10] JC Van Leneren (2000). Measures of success in biological conrol of anhropoids by augmenaion of naural enemies, In: Wraen S, Gurr G, ediors, Measures of success in biological conrol, Dordrech; Lluwer Academic Publishers, pp [11] H Zhang, J Jiao, and L Chen (2007). Pes managemen hrough con-inuous and impulsive conrol sraegies. Biosysems 90, pp , IJAERA - All Righs Reserved 431