Numerical Simulation of Reaction-Diffusion Systems of Turing Pattern Formation

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1 Inernaional Jornal of Modern Nonlinear Theory and Applicaion Pblished Online Deceber 15 in SciRes. hp:// hp://dx.doi.org/1.436/ina Nerical Silaion of Reacion-Diffsion Syses of Tring Paern Foraion Gendai G Hongxiao Peng School of Maheaics and Physics Norh China Elecrical Power Universiy Baoding China Received 7 Ocober 15; acceped 9 Noveber 15; pblished 1 Noveber 15 Copyrigh 15 by ahors and Scienific Research Pblishing Inc. This work is licensed nder he Creaive Coons Aribion Inernaional License (CC BY). hp://creaivecoons.org/licenses/by/4./ Absrac Differenial ehod and hooopy analysis ehod are sed for solving he wo-diensional reacion-diffsion odel. And he srcre of he solions is analyzed. Finally he hooopy series solions are silaed wih he aheaical sofware Malab so he Tring paerns will be prodced. Overall analysis and experienal silaion of he odel show ha he differen paraeers lead o differen Tring paern srcres. As ie goes on he srcre of Tring paerns changes and he final solions end o saionary sae. Keywords Differenial Mehod Hooopy Analysis Mehod Reacion-Diffsion Model Tring Paerns 1. Inrodcion In ie or space paerns have nonnifor acroscopic srcre wih reglariy. Fro he herodynaic poin of view he nare of he paern foraion can be divided ino wo caegories. One is presening in he herodynaic eqilibri condiions sch as he crysal srcre of inorganic cheisry he self-organic paern foraion of organic polyers and so on. The oher is he saion far fro he herodynaic eqilibri condiions sch as he ripples of he sea he srface paerns of he anial he srip clods in he sky and so on. Reacion-diffsion syse is one of he fndaenal eqaions which describe he oion of he nare. I no only has a wide pracical backgrond b also is sed in any fields for exaple predaor-prey odel spread of infecios diseases igraion of poplaion and spread of fores fires. Is aheaical odel is a special kind of parabolic parial differenial eqaions. As for reacion-diffsion syses he copling of nonlinear dynaical and linear diffsion leads o sponaneosly prodcing a variey of ordered or disordered paern of he syse. This is he paern dynaics of he reacion-diffsion syses [1]. According o he paern dynaics he ordered paerns can be divided ino wo caegories: saionary sae (sch as Tring paerns) and raveling How o cie his paper: G G.D. and Peng H.X. (15) Nerical Silaion of Reacion-Diffsion Syses of Tring Paern Foraion. Inernaional Jornal of Modern Nonlinear Theory and Applicaion hp://dx.doi.org/1.436/ina

2 wave (sch as spiral paern). As for he Tring paerns of reacion-diffsion syses in 195 Tring [] for he firs ie showed ha he hoogeneos syse will be sable wih lile disrbance for he absence of proliferaion while i becoes nsable wih he spaial disrbance for oining he seady-sae diffsion. This is he Tring insabiliy of reacion-diffsion eqaions. The classical ehod o sdy he Tring paerns of reacion-diffsion is he analysis of linear sabiliy ehod [3]-[8]. Firsly he nonlinear reacion-diffsion odel is rned ino linear odel. Then perrbaion heory is ilized o sdy is solion which can be seen as he lile perrbaion of eqilibri sae (i.e. nifor and sable sae). And we can obain he condiions of generaing Tring paern hrogh sdying he linear eqaions wih perrbaion by he analysis of sabiliy ehod. Finally hrogh he nerical silaion he reacion-diffsion odel can obain differen Tring paern srcres. This ehod shows he relaionship beween he paraeers of generaing Tring paern. B for he srong nonlinear probles his ehod ay no be applicable. The range of paraeers of he Tring paern can be obained by he analysis of linear sabiliy ehod. Based on he paraeers which lii o he range his aricle solves he reacion-diffsion odel by he se of he cobinaion of differenial ehod and hooopy analysis ehod. Then he changes of he echanis of Tring paerns can be nder conrol by he silaion of experienal daa and analysis of ilizing his ehod. Finally he sdy of he concree case shows he feasibiliy and effeciveness of his ehod.. Hooopy Analysis Solion of Reacion-Diffsion Model The general aheaical represenaion of wo-diensional reacion-diffsion odel is as follows = f( xy µ ) + D v = g( xy µ ) + Dv v (1) v = = ( xy ) Γ n ( xy ) n ( xy ) ( xy ) = µ 1( xy ) vxy ( ) = µ ( xy ) Their definiion are as follows: and v are differen reacan concenraions vecor; µ is a conrol paraeer; f and g represen he nonlinear dynaics fncion of he syse; D and D v describe he diffsion coefficien; is Laplace operaor; Γ= [ αβ ] [ αβ ] describes he regional space of he research for he odel. Discree he reacion-diffsion odel in discree nodes x = α + ih y = α + h ( h ( β α) M i 1 M) = =. The discreizaion of he odel akes he following for And i i+ 1 + i 1 + i i 1 4i = fi + D h vi vi+ 1 + vi 1 + vi vi 1 4vi = gi + Dv h ( xi y ) = µ 1( xi y) v( xi y ) = µ ( xi y) ( ) ( ) ( ) ( ) i ( i µ ) i ( i µ ) i i i i i i i = x y v = v v x y (3) f = f x y g = g x y (4) According o he hogh of hooopy analysis ehod we ake he vale d dv L = + λ Lv = + λv (5) d d () 16

3 i i+ 1 + i 1 + i i 1 4i N= fi D h (6) vi vi+ 1 + vi 1 + vi vi 1 4vi Nv= gi Dv h and ipose he iniial gess solions i and v i according o he iniial and bondary condiions. Therefore i and v i are called he zero-order approxiae solion of proble (). Inrodce he ebedded variable q [ 1]. Axiliary adsing paraeers ee he condiions and v while axiliary conrol fncions ee he condiions H i ( ) and Hv i ( ). For any i ( = 1 M) zero-order deforaion eqaion is as follows: ( 1 q) L( ϕi ( q ) i ( ) ) = qh ( ) N ( ( )) i ϕi q ( 1 q) Lv( φi ( q ) vi ( ) ) = vqhv ( ) N( ( )) i v φi q (8) ϕ i ( q) = µ 1 ( xi y) φ i ( q) = µ ( xi y) (7) The proble (8) shows ha wo cases. One is q = ha ϕi ( ) i ( ) φi ( ) vi ( ) q = 1 ha ϕi ( 1 ) = i ( ) φi ( 1) = vi ( ). The solions of zero-order eqaions ϕ i ( q ) and φ i ( q ) are coninos fro iniial gess solions ( ) and v ( ) o he solions (i.e. ( ) and v ( ) proble () while ebedded variable q is coninos changing fro o 1. q φ q express he for of Taylor expansion as follows ϕ ( ) and ( ) where i i i ( ) ϕ i i ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i q = i + i 1 q+ + i q + φ 1 ϕ =! q i q = vi + vi 1 q+ + vi q + i q= v ( ) i 1 φ =! q i q= = = ; he oher is i i = 1 M; = 1 i ) of The wo sides of zero-order Eqaions (8) are solved -order derivaive abo q and divided! a he sae ie. Take q = can obain he -order deforaion eqaion as follows where The solions of proble (1) are ( ) ( ) ( ) χ ( ) = ( ) ( ) L i i 1 H R i i 1 L v vi ( ) χvi 1 ( ) = vhv ( ) R ( ) i vi 1 (1) i ( ) = vi ( ) = ( ( )) R ( ( )) 1 =! 1 i 1 1 R v 1 =! 1 N N i 1 1 ( ϕi. ( q )) q ( φi. ( q )) q 1 = 1 χ = > 1 q= q= ( ) ( ) λ λτ ( ) = ( ) + ( ) ( ) i χ e e H τ R τ dτ i i 1 i 1 λ λτ ( ) = ( ) + ( ) ( ) i v χ v e e H τ R v τ dτ i i 1 v v i 1 (9) (11) 17

4 λ Choose he axiliary fncion H ( ) H ( ) e e λ = = hen i vi ( ) ( ) λ ( ) = ( ) + e ( ) χ R τ dτ i i 1 i 1 λ ( ) = ( ) + e ( ) v χ v R v τ dτ i i 1 v i 1 In order o ake Taylor series are convergen a q = 1 we can choose he vale of hv. Obain he solions of proble () as follows ( ) ϕ ( 1) ( ) ( ) ( ) ( ) φ ( 1) ( ) ( ) ( ) = = i i i i 1 i v = = v + v + + v + i i i i 1 i In sary he -order hooopy approxiae solions of proble (1) are 3. Solving of Exaple and Analysis ( i ) i ( ) + i 1 ( ) + + i ( ) ( i ) i ( ) + i 1 ( ) + + i ( ) x y v x y v v v 3.1. Hooopy Analysis of Brsselaor Reacion Diffsion Model The ypical Brsselaor reacion diffsion odel is as follows = a ( b+ 1) + v+ D v = b v + Dv v v = = ( xy ) Γ n ( xy ) n ( xy ) where ab are conrol paraeers for he syse; D and (1) (13) (14) (15) D v represen he coefficien of diffsion and l x l y l ; h describes he space sep (i.e. he sep of x and y are he sae lengh h) and * * nifor seady-sae solion o he reacion-diffsion syse is ( ) choose he vale 1 λ = and H ( ) e H ( ) i vi M =. The h v a b =. By division 1 a h = M = = = e. Cobined he significance of reacion-diffsion od- el and he seady-sae solion of he hoogeneos perrbaion analysis he gess of iniial solion can be as follows ( ) = a + aδ cos k hi cos k h i 1 b b vi ( ) = + δ cos k1hi cos kh a a where δ is a perrbaion paraeer and k 1 and k represen he wave nber of x-direcion and y-direcion respecively. Base on he ehod above hree order approxiae solion for he proble of wo diensional reacion diffsion odel is as follows where c n and ( ) ( ) 3 1 x y = c e i i w n w= n= ( ) ( ) 3 1 v x y v = d e i i w n w= n= d n are he fncions which depend on he paraeers ab k1 k n n δ. (16) (17) 18

5 Brsselaor odel is a classical dissipaive srcre odel which has been sdied by any researchers. The condiions for Tring bifrcaion of Brsselaor odel is as follows 3.. The Effecive Range of ( 1 v ) b < + a D D ( 1 1) Dv > Da + a Paraeers have he abiliy o reglae and conrol he convergen region of series solion and he speed of convergence of he series solion. We can obain an appropriae hrogh he effecive range abo crve v of physical qaniy Φ ( ) = and Ψ ( ) =. The Figres 1-4 show he effecive x= y= = x= y= = ranges of in differen condiions. The effecive ranges in Table 1 show ha we will ge differen effecive ranges nder differen condiion of differen paraeers. (18) axiliary paraeer Figre 1. The effecive area abo of when a = 3 b = axiliary paraeer Figre. The effecive area abo of v when a = 3 b =

6 axiliary paraeer Figre 3. The effecive area abo of when a = 4 b = axiliary paraeer Figre 4. The effecive area abo of v when a = 4 b = 1. Table 1. The effecive area abo nder differen paraeers. Paraeers The effecive area abo of The effecive area abo of v a = 3 b = 9.5 [ 3 1] [.7.5] a = 4 b = 1 [.5 1] [.5.5] 3.3. Silaion of Tring Paerns abo Reacion-Diffsion Paraeer have he abiliy o reglae and conrol he convergen region of series solions and he speed of convergence of he series solions. The analysis of he basis fncion in he hooopy analysis ehod shows ha when h and h v are liied vale he hooopy series solions are convergen. Choose he sep h = 1 and oher paraeers l = a = 3 b = 9.5 D = 3 Dv = 1 k1 =.1 k =.1 δ =.1. In his case as ie goes by he changes of Tring paerns srcres show as Figres 5-1.

7 Figre 5. The paerns in space of () when = Figre 6. The paerns in space of v() when = Figre 7. The paerns in space of () when = Figre 8. The paerns in space of v() when =

8 Figre 9. The paerns in space of () when = Figre 1. The paerns in space of v() when = Figre 11. The paerns in space of () when = Figre 1. The paerns in space of v() when = 1.

9 The paraeers abd Dv δ are consan and k1 = 5 k = 5. In his case as ie goes by he changes of Tring paerns srcres show as Figres 13-. As ie goes on he final paern srcre will end o a seady sae. Tring paerns srcre of hooopy series solions are sensiive o he selecion of iniial gess solion and are affeced by he wave nber. In sary in he range of paraeer abo Tring paerns he syse will appear sriped paerns poin paerns and he coexisence of sriped and poin paern wih he ie going on Figre 13. The paerns in space of () when = Figre 14. The paerns in space of v() when = Figre 15. The paerns in space of () when =.5. 3

10 Figre 16. The paerns in space of v() when = Figre 17. The paerns in space of () when = Figre 18. The paerns in space of v() when = Figre 19. The paerns in space of () when = 1. 4

11 Figre. The paerns in space of v() when = Conclsions In his paper a new ehod based on differenial ehod and hooopy analysis ehod is sed o solve he ypical wo-diensional reacion diffsion odel. Differen shapes of Tring paern can be obained hrogh Malab aheaical sofware on experienal daa silaion of he srcre of he solion. And i is proved ha he proposed ehod which o solving nonlinear reacion diffsion probles is feasible and effecive. The new ehod no only redces he diension abo differenial in space b also ges he analyical expression wih physical paraeers hrogh he hooopy in ie. I will faciliae he analysis of he inflence of paraeers variaion on Tring paern srcre. The ehod which is differenial in space and hooopy in ie doain has enoros poenial for solving definie solion of nonlinear parial differenial and i has grea prooional vale. The innovaion poin of his aricle is aking se of he new ehod o solve he wo-diensional reacion diffsion odel. The new ehod is cobining he differenial ehod and hooopy analysis ehod. Fnding This work is sppored by he Naional Sciences Fondaion of People s Repblic of China nder Gran References [1] Oyang Q. (1) Nonlinear Science and Paern Dynaics. Peking Universiy Press Beiing. [] Oyang Q. () Reacion Diffsion Syse Paern Dynaics. Shanghai Science and Technology Edcaion Press Shanghai. [3] Li P.P. (9) A Raio Dependen Predaor-Prey Model of Spaial Paern Foraion Research. Maheaics in Pracice and Theory [4] Wang Y. Cao J.D. Sn G.-Q. and Li J. (14) Effec of Tie Delay on Paern Dynaics in a Spaial Epideic Model. Physica A: Saisical Mechanics and Is Applicaions hp://dx.doi.org/1.116/.physa [5] Parshad R.D. Kari N. Kasiov A.R. and Abderrahane H.A. (13) Tring Paerns and Long-Tie Behavior in a Three-Species Food-Chain Model. Maheaical Biosciences hp://dx.doi.org/1.116/.bs [6] Li S.H. and G Y.X. (1) Copled Reacion-Diffsion Syse in he Sperlaice Paern. Hebei Universiy (Naral Science Ediion) [7] D Y.-K. and X R. (14) Paern Foraion in Two Classes of SIR Epideic Models wih Spaial Diffsion. Chinese Jornal of Engineering Maheaics [8] Li X.Z. Bai Z.G. Li Y. Zhao K. and He Y.F. (13) Doble Nonlinear Copling Reacion-Diffsion Syses in Coplex Tring Paerns. Chinese Jornal of Physics