On the numerical simulation of population dynamics with density-dependent migrations and the Allee effects

Transcription

1 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 On he nmerical simlaion of poplaion dynamics wih densiy-dependen migraions and he Allee effecs HN Sweilam MM Khader and FR Al-Bar Deparmen of Mahemaics and PhysicsQaar Universiy PO Box 7 Doha- Qaar n_sweilam@yahoocom Deparmen of Mahemaics Facly of Science Benha Universiy Benha-Egyp Deparmen of Mahemaics Facly of Science Umm Al-Qra Universiy KSA Absrac In his paper he variaional ieraion mehod (VIM) and he Adomian decomposiion mehod (ADM) are presened for he nmerical simlaion of he poplaion dynamics model wih densiy-dependen migraions and he Allee effecs The convergence of ADM is proved for he model problem The resls obained by hese mehods are compared o he exac solion I is fond ha hese mehods are always converges o he righ solions wih high accracy Frhermore VIM needs relaive less compaional work han ADM Inrodcion Recenly mch aenion has been devoed o varios nmerical mehods which do no reqire discreizaion of space-ime variables or linearizaion of he nonlinear differenial eqaions among which he variaional ieraion mehod (see [] [6] [9]-[] []-[] and he reference cied herein) and he Adomian decomposiion mehod (see [] [] [5] [7] [8] [4] and he reference cied herein) are widely sed for his prpose Many ahors poined o ha he variaional ieraion mehod has meris over oher mehods and can overcome he difficlies arising in calclaion of Adomian s polynomials in Adomian decomposiion mehod (see [6] [7] [9] and he references herein) The aim of his paper is o develop VIM and ADM o simlae he solions of he model of poplaion dynamics wih densiy-dependen migraions and he Allee effec [4] [8] This model can be described by he ransien non-linear advecion-diffsion-reacion eqaion of he form: U U [ Θ(U)U D ] F(U)U T X X X Ω T > () The nknown field U U(X T) is he poplaion densiy in Ω R and T U changes in space and ime de o he non-linear velociy field Θ Θ(U) he diffsion D and he inrinsic growh rae F(U) which incldes all local processes (sch as birh deah and predaion/harvesing) To whom any correspondence shold be addressed c 8 IOP Pblishing Ld

2 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 The model () specifies ha he spaial disribion is affeced by wo physical processes he advecion and he isoropic diffsion of Fickian ype [4] [8] Here we also consider a biological mechanism on he advecion process in order o inclde he case when he species prposely migraes in some pariclar direcion de o some chemical commnicaion These assmpions yield he following non-linear velociy field Θ Θ and he diffsion coefficien D are consans yielding Θ(U) Θ Θ U () In his speed of migraion model () Θ is he densiy-independen migraion velociy which is known or migh come from a hydrodynamic solver The model () also assmes he exisence of a densiy-dependen migraion ha varies linearly wih he poplaion densiy where Θ depends on he species axis We assme here for simpliciy ha he flid is incompressible ( div( Θ ) ) and U U U ( Θ Θ U) D F(U)U T X X Consider he growh dynamics wih Allee effecs given by F(U)U ~ α U(U K )(K () U) (4) Where K is he carrying capaciy and K is he measre of he Allee effecs When K is consan i is convenien o se he dimensionless variable U/K so ha (4) is re-wrien as: f() α ( β)( where β K /K represens he srengh of he Allee effecs The srong and he weak Allee effecs occr when < β < and - < β < respecively The parameer α α (β ) is a normalizaion consan which is defined by a maximm growh rae leading o a family of models The qaliaive resls regarding he Allee effecs and asympoic raes of spread are independen from he choice of he normalizaion consan Wih his assmpion and sing αk Tα K and x X eqaion D () can be wrien in he following dimensionless form: (θ θ ) - β ( β) - x xx (6) Θ where we sed he addiional dimensionless parameers Θ θ and θ K α D Hence he α D poplaion densiies have been re-scaled so ha [] in [T final ] Travelling wave solions are considered so ha he (6) is solved in an nbonded domain wih he following condiions a infiniy: For (he species is a is carrying capaciy); for x (he species is absen) some iniial condiion Under hese bondary condiions one can find in [4] and he references sied herein he asympoic sabiliy analysis of he ravelling wave for he scaled diffsion reacion eqaion ) (5)

3 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 The exisence of wave frons (x ) g() xx U(x c) was derived relying on he properies of g Here g() -β ( β) - and has a leas wo disinc zeros g and g ; if here exiss a srong Allee effec here sill is anoher zero beween g and g a which he percapia growh rae is posiive For more deails on his model see [4] and [8] Implemenaion of VIM In his secion VIM will apply o he following nonlinear parial differenial eqaion of he form: (θ θ ) x xx (7) β ( β) (8) sbjec o he iniial condiion (x) f(x) Firs we consrc he correcion fncional: n n λ( τ )[ (θ θ û ) û û βû ( β)û û ] dτ nτ n nx nxx n n n where λ is a general Lagrange mliplier û n û nx û nxx denoe resriced variaions ie (9) δ û n δ û nx δ û nxx Making he above correcion fncional saionary we obain he following saionary condiion: λ( τ ) λ( τ ) τ The Lagrange mliplier herefore can be defined in he following form: λ( τ ) Sbsiing from () ino (9) resls he following ieraion formla: n n [ (θ θ ) β ( β) ] dτ nτ n nx nxx n n n Now if we sar wih he following iniial approximaion () () β exp(λ exp(λ (x) exp(λ exp(λ () where x ϕ i ; λ β/ and λ / and ϕ ϕ are arbirary consans i i Using he recrrence relaion () we obain he firs componens of he solion in he case ( θ θ ) in he following form: (x ) (x)

4 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 (x ) (x ) (β - β exp(λ exp(λ [ [-β exp(λ )) )( β λ ) ( exp(λ exp(λ λ ) exp(λ λ (- β)(β - β (λ λ ) ) exp(λ )( β λ ) - exp(λ λ ( β)( β λ λ λ ( ( β) ( β)λ λ ( β) λ λ ) ] ] exp(λ β (β λ λ ) λ ) and so on The res of componens of he ieraive formla () were obained in he same manner sing he Mahemaica package The exac solion of he eqaion (8) [in he case ( θ θ )] β exp(λ exp(λ nder he iniial condiion () is given by: (x ) where exp(λ exp(λ x - η ϕ i ; η ( β) - λ ; λ β/ and λ / and are i i i i i ϕ ϕ arbirary consans Here we se β ϕ and ϕ The error behavior for differen ime vales are shown in figres -4 where he nmerical resls are obained by sing wo erms only from he ieraive formla () I is eviden ha he overall errors can be made smaller by adding new erms from he ieraion formla Implemenaion of ADM In his secion he ADM will apply o (8) and () so we rewrie (8) in he following form: L θ β N() xx x () where L is linear operaor N() θ ( β) - is nonlinear operaor x By aken he inverse operaor L () () d of () hen he solion of () can be wrien in he form (x ) (x) L [ - θ - β xx x N() ] (4) The ADM assmes ha he nknown solion (x ) can be expressed by an infinie series of he form: (x ) n (x ) n (5) and he nonlinear operaor erm N() can be decomposed by an infinie series of polynomials given by: N() A n n (6) 4

5 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 he componens n (x ) will be deermined recrrenly and An are he Adomian s polynomials of defined by: n d n i A n [ N( s ) ] n n! n i s ds i Sbsiing from (5) (6) in (4) we can obaain he sbseqen componens: A A (x ) ( β) (x) (-6-6 n - θ - (x ) - θ x ( β) L ( ( - θ - β ) nxx nx n One can se he general form of formla (7) for A n as follows: A ( β) - x - -θ x 4 ) θ ( x L (A ) n x n ) ) x A ( ( β) ( ) - 6θ ( 6 x x x For nmerical comparisons prpose based on he ADM we consrced he solion (x ) as: (7 ) (8) )) x n- Φ n (x ) (x ) n lim Φn (x ) m (x ) n where m (9) To obain he componens of he solion we sar by sbsiing he iniial condiion () in (8): (x) ( exp(λ exp(λ exp(λ ( β) (x ) (x) λ ( β)(β β (λ λ ) ) exp( λ (β - λ ) exp(λ ( β λ ) λ ( β) ( β λ λ λ ) exp(λ λ ( ( β) λ λ ( β) λ ) ]] and so on he oher erms can be obained in he case ( θ θ ) λ λ exp(λ ) [ [ β exp(λ )( β λ ) exp(λ β (β β λ ) Convergence Analysis of he ADM In his secion we will prove he convergence of ADM applied o eqaion (8) Le s define he Hilber space H L ((α(β) [T]) as a se of all applicaions : ( α β ) [T] R wih (xs)ds dτ < ( α β ) [T] 5

6 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 Consider (8) wih he noaion L() hen we can wrie (8) in he following operaor form: L() xx θ x θ x β ( β) () Theorem: ( Sfficien condiions of convergence ) The ADM applied o he nonlinear eqaion () is converges owards a pariclar solion if he following wo hypoheses are saisfied: (H) : ( L() - L(v) - v ) m - v m > v H ; (H) : C(K) > K > sch ha v H wih K v K we have ( L()-L(v)w ) C(K) -v w w H Proof : To verify ( H) for he operaor L() we have L() L(v) (-v) θ (-v) θ ( v ) β(-v) ( β)( v ) ( v ) x x x Then we claim: ( L() Since L(v) and x x v) ( x ( - v) β( - v - v) v) θ ( ( β)( x ( - v) - v) v - v) θ ( ( x v - v) ( v ) - v) are differenial operaors in H hen here exis consans δ and δ : ( ( v) v ) δ v v δ v x () ( ( v) v ) δ v - v δ v x () () and ( ( v ) v) δ - v - v x his according o Schwarz ineqaliy Now by sing he mean vale heorem and he above relaion we ge: ( ( v ) v) δ - v - v δ η - v x where < η < v and v K Therefore ( ( v ) v) δ K v (4) x Also we have ( v v) v v v (5) ( v v) v v η v K v (6) 6

7 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 ( v v) v v η v K v (7) sbsiing from ()-(7) ino () we ge (L() L(v) v) (δ θ δ 5θ δ K β ( β)k K ) - v where m δ θ δ 5θ δ K β ( β)k To verify (H) for he operaor L() we have K Hence we verified (H) m - v ( L() L(v) w) ( (-v) w) θ ( (-v) w) θ ( ( v ) w) x x x β(-v w) ( β)( v w) ( v w) herefore (L() L(v) w) ( θ 5θ K β ( β)k K ) - v w Hence we verified (H) where C( K) θ 5 θ K β ( β)k K C(K) - v w (8) 4 Special Cases of he Model Case I: No Migraion: θ θ and (6) redces o: - β ( β) xx - β exp(λ exp(λ The exac solion of (9) is (x ) where x - η ϕ exp(λ exp(λ i i i i ; η ( β) - λ ; λ β/ and λ / and are arbirary consans i i ϕ ϕ (9) Case II: Densiy-Independen Migraion: In he case ha he speed of he species migraion does no depend on he poplaion densiy eg when drifing wih he wind he dynamics of he poplaion are described by he following eqaion: θ - β ( β) x xx - where θ is he speed of advecion Considering raveling wave coordinaes (x ) (z ) where z x - θ so ha û(z ) from () we obain û û - β û ( β) xx û - û () () 7

8 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 Eqaion () coincides wih (9) and hs he exac solion of (9) gives also an exac solion of () wih he obvios change x z Case III: Densiy-Dependen Migraion: In his secion we consider he case when he densiy-independen advecion cased by environmenal facors is absen and migraion akes place de o biological mechanisms which are assmed o be densiy-dependen Then θ and from (6) we arrive a he following eqaion: The exac solion of () is given by: θ - β ( β) x xx (x ) - β exp( ω ψ ) exp( ω ψ ) exp( ω ψ ) exp( ω ψ ) where ψ x - q ε q ( β ) ν ( θ ν ) ω ; i ; ω β / ν i i i i i ν 5( θ θ 8) and are arbirary consans ϕ ϕ ω / ν sch ha Case IV: General Case: In a general case migraions can ake place de o boh densiy-dependen and densiy-independen facors The dynamics of a given poplaion are hen described by fll (6) where now θ and θ The exac solion in his case exac solion: β exp [ ω ( x ( q θ ) ε )] exp [ ω ( x ( q θ ) ε )] (x ) exp [ ω ( x ( q θ ) ε )] exp [ ω ( x ( q θ ) ε )] where he noaions are he same as in () The figres -4 simlae he error beween he exac solion and he boh mehods approximae solion of he above for cases respecively Error Error () Figre : (Case I) The error a x Figre : (Case II) The error a x 65 and θ θ and θ θ 8

9 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 Error Error Figre : (Case III) The error a x 5 Figre 4: (Case IV) The error a x 5 and θ θ and θ θ 5 Conclsion In his paper VIM and ADM are applied o solve he model of poplaion dynamics wih densiydependen migraions and he Allee effecs he mehods need mch less compaional work compared wih radiional mehods We achieved a very good approximaion wih he acal solion of he model by sing wo erms of he ieraion scheme derived above in he ADM and VIM I is eviden ha he overall resls come very close o he exac solion even sing only few erms of he ieraion formla Errors can be made smaller by aking new erms of he ieraion formlas I is fond ha hese mehods are always converges o he righ solions wih high accracy We fond ha he variaional ieraion mehod can overcome he difficlies arising in calclaion of Adomian s polynomials in Adomian decomposiion mehod Frhermore VIM needs relaive less compaional work han ADM References [] Abbasbandy S and Darvishi T M 5 Applied Mahemaics and Compaion 6 65 [] Abdo A M and Soliman A A 5 Physica D [] Adomian G 989 Nonlinear Sochasic Sysems and Applicaions o Physics ( Klwer Academic Pblishers-Dordrech) [4] Almeida C R Delphim A S and Cosa S I M 6 Ecological Modelling 9 6 [5] Bhaacharyya K R and Bera K R 4 Applied Mahemaics Leers 7 7 [6] Biazar J Ghazvini H 7 In J Nonlinear Sci 8 [7] Bl H Ergü M Asil V and Bokor H R 4 Applied Mahemaics and Compaion 5 7 [8] Gellal S Grimal P and Cherral Y 997 Compers Mah Appl 5 [9] He J H 999 In J Non-Linear Mech [] He J H Applied Mahemaics and Compaion 4 5 [] He J H and X-Hong W 6 Chaos Solions and Fracals 9 8 [] He J H 6 In J Modern Physics 4 [] He J H 6 Perrbaion mehods: Basic and Beyond (Elsevier Amserdam) [4] Kaya D and El-Sayed M S Physics Leers A 8 [5] Kaya D and Inan E I 5 Applied Mahemaics and compaion 6 5 [6] Lesnic D Compers and Mahemaics wih Applicaions 44 [7] Lesnic D 5 Commnicaions in Nonlinear Science and Nmerical Simlaion 58 [8] Perovskii S and Bai-Lian Li Mahemaical Biosciences [9] Sofyane A and Bolmalf M 5 Applied Mahemaics and Compaion

10 7 Inernaional Symposim on Nonlinear Dynamics (7 ISND) IOP Pblishing Jornal of Physics: Conference Series 96 (8) 8 doi:88/ /96//8 [] Sweilam H N and Al-Bar R F 7 Compers and mahemaics wih Applicaions [] Sweilam H N Khader M M and Al-Bar R F 7 Physica Leers A 7 6 [] Sweilam H N 7 J Comp Appl Mah 7 64 [] Sweilam H N and Khader M M 7 Chaos Solions and Fracals 45 [4] Vadas P and Olek S In J Hea and Mass Transfer 4 75 [5] Wazwaz A M 997 Appl MahComp 8 65