Research Article Dynamics of a Stage Structure Pest Control Model with Impulsive Effects at Different Fixed Time

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1 Discree Dynamics in Naure and Sociey Volume 29, Aricle ID , 15 pages doi:1.1155/29/ Research Aricle Dynamics of a Sage Srucure Pes Conrol Model wih Impulsive Effecs a Differen Fixed Time Bing Liu, 1 Ying Duan, 2 and Yinghui Gao 3 1 Deparmen of Mahemaics, Anshan Normal Universiy, Anshan, Liaoning 1147, China 2 Deparmen of Mahemaics, Liaoning Normal Universiy, Dalian, Liaoning 11629, China 3 School of Mahemaics and Sysem Sciences & LMIB, Beihang Universiy, Beijing, 183, China Correspondence should be addressed o Bing Liu, liubing529@126.com Received 29 Sepember 29; Acceped 16 November 29 Recommended by Leonid Berezansky Many exising pes conrol models, which conrol pess by releasing naural enemies, neglec he effec ha naural enemies may ge killed. From his poin of view, we formulae a pes conrol model wih sage srucure for he pes wih consan mauraion ime delay hrough-sage ime delay and periodic releasing naural enemies and naural enemies killed a differen fixed ime and perform a sysemaic mahemaical and ecological sudy. By using he comparison heorem and analysis mehod, we obain he condiions for he global araciviy of he pes-eradicaion periodic soluion and permanence of he sysem. We also presen a pes managemen sraegy in which he pes populaion is kep under he economic hreshold level ETL when he pes populaion is uniformly permanen. We show ha mauraion ime delay, impulsive releasing, and killing naural enemies can bring grea effecs on he dynamics of he sysem. Numerical simulaions confirm our heoreical resuls. Copyrigh q 29 Bing Liu e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. 1. Inroducion The warfare beween man and pess has been on for housands of years, and pes oubreaks ofen cause serious ecological and economic problems. Wih he developmen of sociey, human beings have come up wih numerous mehods o conrol pess, for insance, biological, culural, physical, mechanical, and chemical ools in a way ha minimize economics. Among hose mehods, releasing naural enemies is an effecive and basic one, and i canno pollue he environmen. Morever, i is relaively simple o implemen. The pes conrol models by releasing naural enemies have been sudied by many researchers 1 3. All of hem have invariably assumed ha he releasing enemies can all have an effec on he models, whereas i is ofen he case ha he naural enemies may ge killed. For example, releasing frogs may preven locuss. Bu in many places in China, especially in

2 2 Discree Dynamics in Naure and Sociey some souhern areas, i is cusomary o ea frogs, and in markes he price of a frog is 15 yuan; so he number of he released frogs is reduced in a period as a resul of heir geing killed by people. According o he above biological background, we formulae a pes conrol model wih periodic impulsive releasing naural enemies and naural enemies killed a differen fixed ime. We assume ha he pes has a sage srucure wih consan mauraion ime delay hrough-sage ime delay. Many sage srucure models wih ime delay were exensively sudied see 4 9. In pracice, from he principle of ecosysem balance, we need only o conrol he pes populaion under he economic hreshold level ETL and no o eradicae naural enemy oally and hopes pes populaion and naural enemy populaion can coexis when he pess do no bring abou immense economic losses. Then he quesions ha arise here are he following: how do he impulse period and ime delay affec he exincion of he pes and permanence of he sysem? How many naural enemies should we release o conrol pess? How o keep he pess under ETL? For hese purposes, in Secion 2 we sugges a delay impulsive differenial equaion o model he process of impulsive releasing naural enemies and naural enemies killed a differen fixed ime and inroduce lemmas which will be used in his paper. Impulsive differenial equaions are found in almos every domain of applied science and have been sudied in many invesigaions 1 17, and i can describe populaion dynamic models, since many life phenomena and human exploiaion are almos impulsive in he naural world, and impulsive delay differenial equaions are almos analyzed in heory see Time delay and impulse are inroduced ino pes conrol models wih sage srucure, which grealy enrich biologic background, bu he sysem becomes nonauonomous and quie complicaed, which causes grea difficulies for us o sudy he model. In Secions 3 and 4, he condiions for he global araciviy of he pes-eradicaion periodic soluion and permanence of he sysem are obained. We give a brief conclusion of our resuls in he las secion. Numerical simulaions are presened o illusrae our heoreical resuls. 2. Model Formulaion and Auxiliary Lemmas A model of single pes populaion growh incorporaing sage srucure as a reasonable generalizaion of he logisic model was derived as follows in 21 : dx j αx dx j αe dτ x τ, dx αe dτ x τ βx 2, 2.1 where x j and x represen he immaure and maure pes populaion densiies, respecively. This model assumes an average age o mauriy which appears as a consan ime delay reflecing a delayed birh of immaure and a reduced survival of immaure o heir mauriy. α> is he coefficien of birh rae of he maure populaion in his environmen; τ, called mauraion ime delay, is he ime o mauriy; d> is he coefficien of deah rae of he immaure populaion; β> is he maure pes deah and overcrowding rae; he erm αe dτ x τ represens he immaure pess which are born a he ime τ i.e., αx τ and survive a he ime wih he immaure pes deah rae d and herefore, represens he ransformaion of immaures o maures.

3 Discree Dynamics in Naure and Sociey 3 The basic model ha we consider in his paper is he one based on he ideas ha he naural enemies may be killed when hey are released o conrol pes populaion; hen we formulae he following wo-sage pes conrol model wih sage srucure for pes and pulse releasing of naural enemies killed a differen fixed ime: dx j αx dx j αe dτ x τ, dx αe dτ x τ d 1 x 2 βx y, dy λβx y d 2 y x j x j, x x, y ( 1 p ) y, n k 1 T, / nt, / n k 1 T, 2.2 x j x j, x x, y y μ, nt, where x j, x, andy represen he densiy of he immaure pes, maure pes, and naural enemy populaion a ime, respecively; n Z,andZ {1, 2,...}; d 2 > ishe naural enemy deah s rae; μ> is he naural enemy releasing amoun a every impulsive period nt; p 1 is he killed rae of naural enemies a every impulsive period n k 1 T. The meanings of oher parameers are he same as hose of Model 2.1. The firs equaion in sysem 2.2 may be rewrien as follows: x j τ αe d s x s ds, 2.3 and x j τ αe ds x s ds; haisx j can be linear expression by x, andx j does no appear in he second and hird equaions in sysem 2.2, herefore in he res of his paper, we will sudy he subsysem of 2.2 as follows: dx αe dτ x τ d 1 x 2 βx y, dy λβx y d 2 y, x x, y ( 1 p ) y, n k 1 T, / nt, / n k 1 T, 2.4 x x, y y μ, nt.

4 4 Discree Dynamics in Naure and Sociey The iniial condiions for 2.4 are ( ϕ1 s,ϕ 2 s ) ) C C( τ,,r 2, ϕ i >, i 1, From he biological poin of view, we only consider sysem 2.4 in he biological meaning region: D {x,y x,y }. Lemma 2.1 see 22. Consider he following equaion: dx ax τ bx cx 2, 2.6 where a, b, c, τ are all posiive consans, x >, for τ ; one has he following. 1 If a<b,henlim x. 2 If a>b,henlim x a b /c. Lemma 2.2. For each soluion of sysem 2.2 wih being large enough, one has x j,x,y L, where L λα d 3 2 /4λd 3 d 1 μe d 3T / e d 3T 1 and d 3 min{d 2,d}. Proof. Le V λx j λx y, d 3 min{d 2,d}. If / nt and / n k 1 T, we have dv λ α d x λd 1 x 2 d 3 V λ α d 3 2 4d 1 d 3 V, 2.7 If n k 1 T, V n k 1 T V n k 1 T ;if nt, V nt V nt μ. We have V V e d3 λα d 3 2 e d3 s ds μe d 3 nt 4λd 3 d 1 <nt< λ α d 3 2 μed3t L, 4d 3 d 1 e d 3T 1 as. 2.8 So V is uniformly ulimaely bounded. Hence, by he definiion of V, we obain ha each posiive soluion of sysem 2.2 is uniformly ulimaely bounded. The proof is complee. Lemma 2.3 see 23. Consider he following sysem: dy d 2 y, / nt, / n k 1 T, y ( 1 p ) y, n k 1 T, 2.9 y y μ, nt,

5 Discree Dynamics in Naure and Sociey 5 where d 2,p,μ,kare all posiive consans; hen, sysem 2.9 has a globally asympoic sable posiive periodic soluion: μe d 2 n 1 T 1 ( 1 p ), n 1 T < n k 1 T, e d 2T y μ ( 1 p ) e d 2 n 1 T 1 ( 1 p ), n k 1 T < nt. e d 2T 2.1 Lemma 2.4 see 24. Le V : R R 3 R and V V. Assume ha D V, x g, V, x, / nt, V, x ψ n V, x, nt, 2.11 where g : R R R is coninuous in nt, n 1 T R and for ν R,n Z, lim,y nt,ν g, y g nt,ν exis, and ψ n : R R is nondecreasing. Le r be he maximal soluion of he scalar impulsive differenial equaion: du g, u, u ψ n u, u u, / nt, nt 2.12 exiing on,. ThenV,x u implies ha V, x r,, wherex is any soluion of Global Araciviy of he Pes-Eradicaion Periodic Soluion Define ζ p e d 2T αe dτ /βμ 1 p e d 2T. Theorem 3.1. If ζ 1 globally aracive. < 1 holds, he populaion-exincion periodic soluion,y of 2.4 is Proof. From he second equaion in sysem 2.4, we have dy / d 2 y ; hen consider he following sysem: dz 1 d 2 z 1, / nt, / n k 1 T, z 1 ( 1 p ) z 1, n k 1 T, 3.1 z 1 z 1 μ, nt.

6 6 Discree Dynamics in Naure and Sociey According o Lemma 2.3, he globally asympoic sable posiive periodic soluion of sysem 3.1 is μe d 2 n 1 T 1 ( 1 p ), n 1 T < n k 1 T, e d 2T z 1 μ ( 1 p ) e d 2 n 1 T 1 ( 1 p ), n k 1 T < nt. e d 2T 3.2 According o Lemma 2.4, for any ɛ 1 >, here exis a n 1 such ha y >z 1 ɛ 1 > μ( 1 p ) e d 2T 1 ( 1 p ) e ɛ d 1 : η, nt < n 1 T, n > n T From he firs equaion of sysem 2.4, we can ge dx <αe dτ x τ d 1 x 2 βx η. 3.4 Then consider he following comparison equaion: dz αe dτ z τ d 1 z 2 βz η. 3.5 Since ζ 1 < 1, according o Lemma 2.1, we obain lim z ; hen by he comparison heorem in differenial equaions, we ge lim x. Wihou loss of generaliy, we assume ha <x <ɛ,. 3.6 Consider he following sysem: dy ( ) λβɛ d 2 Y, Y ( 1 p ) Y, / nt, / n k 1 T, n k 1 T, 3.7 Y Y μ, nt. The globally asympoic sable posiive periodic soluion of sysem 3.7 is μe λβɛ d 2 n 1 T 1 ( 1 p ), n 1 T < n k 1 T, e λβɛ d 2 T Y μ ( 1 p ) e λβɛ d 2 n 1 T 1 ( 1 p ), n k 1 T < nt. e λβɛ d 2 T 3.8

7 Discree Dynamics in Naure and Sociey 7 By 3.3 and Lemma 2.4, for any ɛ 2 >, here exis a T, when >T : z 1 ɛ 2 <y <Y ɛ Le ɛ, we can ge y ɛ 2 <y <y ɛ for being large enough, which implies lim y y. This complees he proof. 4. Permanence Define ζ p e d 2T αe dτ /βμe d 2T. Definiion 4.1. Sysem 2.4 is said o be permanen if here are consans l, L > and a finie ime T such ha for every posiive soluion x j,x,y R 3 wih iniial condiions 2.5 saisfies l x j L, l x L, l y L for all T.HereT may depend on he iniial condiion 2.5. Theorem 4.2. If ζ 2 > 1 holds, here exiss a posiive consan q such ha each posiive soluion x,y of 2.4 saisfies x q wih being large enough. Proof. The firs equaion of sysem 2.4 may be rewrien as follows: dx [ ] αe dτ d 1 x βy x αe dτ d x s ds. τ 4.1 Define V x αe dτ τx s ds. 4.2 Calculaing he derivaive of V along he soluion o 2.4 gives dv [ ] αe dτ d 1 x βy x. 4.3 Since ζ 2 > 1, we can choose m 1.Leɛ> be small enough such ha d 2 λβm 1 > and αe dτ d 1 m 1 βσ 1 >, 4.4 where σ μe d 2 λβm 1 T / 1 1 p e d 2 λβm 1 T ɛ. For any posiive consan >, we claim ha he inequaliy x <m 1 canno hold for all. Oherwise, here is a posiive

8 8 Discree Dynamics in Naure and Sociey consan such ha x <m 1 for all. From he second and he fourh equaions in sysem 2.4, we have dy ( ) λβm 1 d 2 y, y ( 1 p ) y, / nt, / n k 1 T, n k 1 T, 4.5 y y μ, nt. According o Lemma 2.4, here exiss T 1 τ such ha for T 1, μe λβm 1 d 2 T y < 1 ( 1 p ) e λβm 1 d 2 T ɛ : σ. 4.6 From 4.3 and 4.6 we have dv > [ αe dτ d 1 m 1 βσ ]x ( d 1 m 1 βσ )( ) αe dτ d 1 m 1 βσ 1 x, T Le m 2 min T1,T 1 τ x, we show ha x m 2 for all >T 1. Oherwise, here exiss a nonnegaive consan T 2, such ha x m 2, for T 1,T 1 τ T 2, x T 1 τ T 2 m 2 and x T 1 τ T 2 <. Thus, from he firs equaion of sysem 2.4 and 4.5, we easily see ha dx T 1 τ T 2 αe dτ x T 1 T 2 d 1 x 2 T 1 τ T 2 βx T 1 τ T 2 y [ αe dτ d 1 m 1 βσ ]m 2 > ( d 1 m 1 βσ )( ) αe dτ d 1 m 1 βσ 1 m 2 >. 4.8 This is a conradicion. So we obain ha x m 2, for all >T 1. From 4.7, we have dv > ( d 1 m 1 βσ )( ) αe dτ d 1 m 1 βσ 1 m 2, 4.9 which implies V as. This is a conradicion o V L 1 ατe dτ. Therefore, for any posiive consan, he inequaliy x <m 1 canno hold for all. If x m 1 holds rue for all large enough, hen our aim is obained. Ohervise, x is oscillaory abou m 1.Le q min { m1 2,m 1e d 1 β Lτ }. 4.1

9 Discree Dynamics in Naure and Sociey 9 In he following, we will show ha x q for being large enough. There exis wo posiive consans, ω such ha ( ) ( ) x x ω m 1, x <m 1 for << ω Since x is coninuous and bounded and is no effeced by impulses, we conclude ha x is uniformly coninuous. Then here exiss a consan T 3, such ha x >m 1 /2 for all T 3. If ω T 3, our aim is obained. If T 3 <ω τ, from he firs equaion of 2.4 we have ha dx ( d 1 β ) Lx for < ω Then we have x m 1 e d 1 β Lτ for < ω τ I is clear ha x q for < ω. If ω>τ, by he firs equaion of 2.4, hen we have ha x q for < τ. Thus, proceeding exacly as he proof for above claim, we can obain he inequaliy x <m 1 canno hold for all > τ, so he same argumens can be coninued for for τ < ω, and we can ge x q for τ< ω. Since he inerval, ω is arbirarily chosen we only need o be large, wegehax q for being large enough. In view of our argumens above, he choice of q is independen of he posiive soluion of 2.4 which saisfies ha x q for being sufficienly large. This complees he proof. Theorem 4.3. If ζ 2 > 1, hen sysem 2.4 is permanen. Proof. Suppose ha x,y is any soluion of sysem 2.4 wih iniial condiion 2.5. Le q μ 1 p e d 2T /1 1 p e d 2T,by 3.3 we know y q, and according o Theorem 4.2, here exis posiive consans q, such ha x q.se D { x,y q x L, q y L } Then, D is a bounded compac region which has posiive disance from coordinae axes. By Theorem 4.2, one obains ha every soluion o sysem 2.4 wih he iniial condiion 2.5 evenually eners and remains in he region D. The proof is compleed. The immaure pes is hardly any harmful for he crop; so we jus consider he effec of he maure pes. From he sandpoin of ecological balance and saving resource, we only need o mainain he maure pes populaion under he economic hreshold level ETL and no o eradicae he pes oally; hen we have he following heorem.

10 1 Discree Dynamics in Naure and Sociey Theorem 4.4. Under he condiion of Theorem 4.3, if ( d1 E αe dτ)( 1 ( 1 p ) e ) d 2T βμ ( 1 p ) > 1, 4.15 e d 2T hen he maure pes populaion is evenually under he economic hreshold level E. Proof. From 3.3 we know for any ɛ 1 > ha here exis a n 1 such ha y > μ( 1 p ) e d 2T 1 ( 1 p ) e ɛ d 1 : η, nt < n 1 T, n > n T From he firs equaion of sysem 2.4, we can ge dx <αe dτ x τ d 1 x 2 βx η Then consider he following comparison equaion: dz αe dτ z τ d 1 z 2 βz η If ζ 2 > 1 holds, we can ge αe dτ >βη;hen according o Lemma 2.1,weobain lim z αe dτ βη d 1 According o Lemma 2.4, we have x αe dτ βη /d 1 compleed. < E, as. The proof is 5. Discussion In his paper, we discuss a pes conrol model wih sage srucure for he pes wih consan mauraion ime delay hrough-sage ime delay and periodic releasing naural enemies and naural enemies killed a differen fixed ime. From Theorems 3.1, 4.2, and 4.3, we can observe ha he exincion and permanence of he populaion are very much dependen on T, τ, μ, p.if μ is oo large, from he condiion of Theorem 3.1 we know ha he populaion will be exinc. Alhough in heory pess can be eliminaed compleely, in fac i is hard o implemen. In he naural world, he immaure pes does no have any effec on he crop; so we only consider he maure pes in his paper. Even hough pes is exinc, he food chain beween pes and naural enemy is broken, ha would be anoher disaser. Therefore we only need o conrol he maure pes populaion under ETL and no o eradicae naural enemy oally and hope ha pes populaion and naural enemy populaion can coexis when he pess do no bring abou immense economic losses.

11 Discree Dynamics in Naure and Sociey 11 x xj.3.1 a b.6 y c Figure 1: Time series of sysem 2.4 wih parameers d.6; d 1 ; α.8; τ.8; d 2.5; β.8; λ.9; p.3; μ ; T. xj a x y b c Figure 2: Time series of sysem 2.4 wih parameers d.6; d 1 ; α.8; τ.8; d 2.5; β.8; λ.9; p.3; μ.5; T.

12 12 Discree Dynamics in Naure and Sociey xj a x.1 b.6 y c Figure 3: Time series of sysem 2.4 wih parameers d.6; d 1 ; α.8; τ 2; d 2.5; β.8; λ.9; p.3; μ ; T. xj a x y b c Figure 4: Time series of sysem 2.4 wih parameers d.6; d 1 ; α.8; τ.8; d 2.5; β.8; λ.9; p.1; μ ; T.

13 Discree Dynamics in Naure and Sociey 13 x xj a b y c Figure 5: Time series of sysem 2.4 wih parameers d.6; d 1 ; α.8; τ.8; d 2.5; β.8; λ.9; p.3; μ ; T 5. To verify he heoreical resuls obained in his paper, in he following we will give some numerical simulaions and ake d.6; d 1 ; α.8; τ.8; d 2.5; β.8; λ.9; p.3; μ ; T see Figure 1 ; here we can compue ζ > 1; from Theorem 4.3 we know ha sysem 2.4 is permanen. If we increase he naural enemy inpu amoun o μ.5 ζ < 1 or increase he mauraion ime delay τ 2 ζ < 1, oher parameers are he same wih hose in Figure 1, and he pes will be exinc see Figure 2 and Figure 3. If we decrease he killed rae of naural enemies o p.1, hen ζ < 1, he pes also will be exinc see Figure 4. I shows ha if he mauraion ime delay is oo long, or he naural enemy releasing amoun is oo large, or fewer naural enemies are killed, he permanence of he sysem disappears and he pes populaion dies ou. This implies ha pulse releasing naural enemies, he mauraion ime delay and naural enemies killed bring grea effecs on he dynamics behaviors of he model. Suppose ha he economic hreshold level, E, is.3, hen Figure 5 shows ha if we choose an appropriae pulse releasing naural enemy period wih T, we can conrol he pes populaion under he ETL and no eradicae naural enemy oally. This gives us some reasonable suggesions for pes managemen. Acknowledgmens This work is suppored by he Naional Naural Science Foundaion of China and and he Science and Research Projec Foundaion of Liaoning Province Educaion Deparmen.

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16 Advances in Operaions Research Advances in Decision Sciences Journal of Applied Mahemaics Algebra Journal of Probabiliy and Saisics The Scienific World Journal Inernaional Journal of Differenial Equaions Submi your manuscrips a Inernaional Journal of Advances in Combinaorics Mahemaical Physics Journal of Complex Analysis Inernaional Journal of Mahemaics and Mahemaical Sciences Mahemaical Problems in Engineering Journal of Mahemaics Discree Mahemaics Journal of Discree Dynamics in Naure and Sociey Journal of Funcion Spaces Absrac and Applied Analysis Inernaional Journal of Journal of Sochasic Analysis Opimizaion