Kondo model of pattern formation Model Kondo tworzenia się wzorów

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1 Uiwersytet Wrocławski Wydział Mateatyki i Iforatyki Istytut Mateatyczy specjalość: ateatyka teoretycza Szyo Cyga Kodo odel of patter foratio Model Kodo tworzeia się wzorów Praca agisterska apisaa pod kierukie prof. dr hab. Grzegorza Karcha Wrocław,

2 ... (iioa i azwisko) (aktualy adres do korespodecji)... (adres e-ail)... (wydział)... (kieruek studiów)... (uer PESEL) (pozio i fora studiów) (uer albuu) Załączik Nr do Zasad OŚWIADCZENIE O PRAWACH AUTORSKICH I DANYCH OSOBOWYCH Ja iżej podpisay/a.. studet/ka Wydziału. kieruek.. oświadcza, że przedkładaa praca dyploowa a teat:.. jest ojego autorstwa i ie arusza autorskich praw w rozuieiu ustawy z dia lutego r. o prawie autorski i prawach pokrewych (tekst jedolity: Dz. U. z r. Nr, poz., z póź. z.) oraz dóbr osobistych chroioych prawe; ie zawiera daych i iforacji uzyskaych w sposób iedozwoloy; ie była wcześiej przediote iej urzędowej procedury związaej z adaie dyplou uczeli wyższej lub tytułu zawodowego; treść pracy dyploowej przedstawioej do obroy, zawarta a przekazay ośiku elektroiczy, jest idetycza z jej wersją drukowaą. Oświadcza, iż zostałe/a poiforoway/a o prawie dostępu do treści oich daych osobowych oraz ich poprawiaia. Udostępieie przez ie daych osobowych a charakter dobrowoly. Wyraża zgodę, a: udostępieie ojej pracy dla celów aukowych i dydaktyczych; przetwarzaie oich daych osobowych w yśl ustawy z dia sierpia r. o ochroie daych osobowych (tekst jedolity: Dz. U. z r., poz., z póź. z.); uieszczeie ojej pracy w bazie daych Uczeli i jej przechowywaie przez okres stosowy do potrzeb Uczeli; wykorzystaie ojej pracy jako eleetu koparatywej bazy daych Uczeli; udostępieie ojej pracy iy podioto cele prowadzeia kotroli atyplagiatowej prac dyploowych i iych tekstów, które zostaą opracowae w przyszłości; porówywaie tekstu ojej pracy z tekstai iych prac zajdujących się w bazie porówawczej systeu atyplagiatowego i zasobach Iteretu. Wrocław,. (rrrr - dd) (czytely podpis autora pracy)

3 Cotets Itroductio. Priary odel Equivalet odel Liearized odel. Stateet of the proble Laplace operator properties Existece of statioary solutios Stability of statioary solutios Copact operators Noliear odel. Existece of statioary solutio Noliear fuctioal Existece by the bifurcatio theore Stability Nuerical Siulatios. Nuerical schee Siulatio results Liear odel Noliear odel Coclusio

4 Chapter Itroductio The reactio-diffusio equatios ca be widely used to describe the behaviour of ay biological pheoea like forig patters i a orgais. Japaese biologist Shigeru Kodo preseted i his works, [], [], [] a process of forig patter o a certai species of guppy fish. He observed that two dyes that are preset i a fish orgais ifluece each other. That results i forig a diversified patters o a fish surface. Shigeru Kodo proposed a explaatio of this pheoea usig a iproved reactio-diffusio syste. He claied that the chage rate of a dye depeds o a dye desity ad locatio. It follows i soe areas dye chage rate depeds positively o dye volue ad i soe areas the depedece is egative. The positive depedece is called activatio, ad the egative depedece is called ihibitio.. Priary odel Matheatical odel proposed by Shigeru Kodo was a differetial equatio which describes the cocetratio of a specific substace o a fish ski. Cosider bouded ad coected subset of plae Ω R. Let u deote the cocetratio of a substace. The followig odel was used u t = (S a)u (.) where a > is a costat cell destructio rate ad S correspods to the cell sythesis. Cell sythesis is a process of sophisticated cell iteractios, depedet o stiulatio operator. Shigeru Kodo i his work claied that stiulatio operator is a covolutio with a radial kerel Sti(x, y) = u(x ξ, y η)kerel( ξ + η )dξdη. (.) The kerel used i siulatios is desiged to have positive ad egative parts as well. It was observed that without ihibitio or activatio part o diversified patters ca be achieved ad the solutio is trivial. I sae cases uwated, high level of cells desity ca be obtaied. To eliiate this proble the saturatio fuctio was itroduced, to esure that the desity of a substace is tepered. The saturatio fuctio was give with the followig forula, Sti <, S = Sti, < Sti M, M, S > M. (.)

5 Let K be a covolutio kerel ad f be a saturatio fuctio. The Kodo odel ca be rewritte as u t = au + f(k u). (.) Shigeru Kodo observed that the odel coefficiets, destructio rate a, kerel shape, ihibitio, activatio rate ad saturatio rate plays the iportat role i the shape of the obtaied patter. First of all, specific requireets for coefficiets eeds to be claied to esure that the solutio will be o-trivial. Secodly, the utual relatios betwee coefficiets deterie the properties of the obtaied patter. I soe cases the patter ca be syetric, atisyetric or coected. Geerally, the patters are regular i soe sese. We will explai this pheoea i this work later.. Equivalet odel The Kodo odel preseted i (.) is difficult to aalyse, sice it is a covolutio differetial equatio. I this work we will propose a alterative sei reactio - diffusio odel, to approxiate the covolutio odel. We will show eve the siplified odel, is sufficiet to prove the existece ad stability of o trivial statioary solutios. We itroduce the secod substace v which will be used to substitute covolutio kerel with Laplace equatio. Let Ω R be bouded copact ad coected. Let u, v(x, t) Ω R + R be a fuctios describig the substaces desity. Cosider the syste u t = au + f(v), =cu + (d + )v, x Ω, v =, u(x, ) =u (x), x Ω, where a >, c, d R, f C (R). We applied Neua boudary coditio because i Kodo odel substace does ot diffuse outside of fish orgais. We claied that the substace v does ot follow the law of reactio. Observe that our syste correspods to the specific covolutio differetial equatio with kerel K equal to shifted Gree kerel. The followig forula holds (.) v(x) = c Ω K(x, y)u(y)dy. (.) The value of d deteries the core syste properties sice if d is equal to the eigevalue of Laplace operator the the secod equatio ca ot have solutio for every u. This paper is orgaized as follows. I chapter we will cosider liearised odel, with liear fuctio f. We will prove the existece ad stability of statioary solutios uder specific coditios for odel coefficiets. We will exted our arguet for Laplasia substituted with ay operator B such that B is copact. I chapter we will prove the existece of o costat statioary solutios for o liear odel usig Rabiowitz Bifurcatio theore. The stability of solutios will be prove oly i diesio, sice restrictio for supreu or are required. I chapter we will preset soe uerical siulatio to explai the process of forig patters fro rado iitial coditios. I chapter we will draw a coclusio.

6 Chapter Liearized odel. Stateet of the proble Let Ω R be a ope, bouded, coected doai with C boudary. Cosider the liearised syste with the Neua boudary coditio u = au + bv x Ω, t = cu + dv + v v = u (x) = u(x, ). x Ω, where a > ad b, c, d R. It is coveiet to represet the syste (.) i atrix for ( u t ) = ( a b c d + ) (u v ) x Ω, v = u(x, ) = u (x). Let us defie a weak solutio of proble (.). x Ω, Defiitio... A pair of fuctios u C ([, ), W, (Ω)), v C ([, ), W, (Ω)) is a weak solutio of proble (.) if for every φ W, (Ω) the followig equality hold true Ω u t φ = Ω auφ + Ω bvφ, = Ω cuφ + Ω dvφ + Ω v φ. We will prove that if coefficiets a, b, c, d fulfils soe specific assuptios the there exists a weak solutio of (.). Moreover, we will prove its stability.. Laplace operator properties To prove the existece ad stability of solutios we eed to recall soe basic facts about eigevalues of Laplacia. (.) (.) (.)

7 Theore.. (Spectral theore for Laplace operator with Neua boudary coditios). Let Ω R be a bouded coected doai with C boudary. The proble has coutably ay eigevalues which satisfy u = λu x Ω, u = x Ω. (.) > λ λ λ ad orthooral basis of eigefuctios {e j } j= of W, (Ω) such that Proof. The proof ca be foud i [, page ]. e j = λ j e j. Existece of statioary solutios Theore... Assue that a >, b, c, d. There exists a o zero weak statioary solutio of proble (.) if ad oly if ad bc a = λ k for soe k N Moreover each such statioary solutio is of the for k ṽ = C l e jl ad ũ = b aṽ, l= where e jl are eigefuctios correspodig to the eigevalue λ k, k deotes the ultiplicity of λ k eigevalue ad C l are arbitrary costats. Proof. A statioary solutio of (.) fulfils ( ) = ( a b c d + ) (ũ ṽ ) x Ω, ṽ = x Ω. Evaluatig ũ fro the first equatio ad applyig it to the secod oe yields to the proble ad + bc ṽ = ṽ a x Ω, ṽ = x Ω. It follows fro Theore.. that syste (.) has a solutio if ad oly if ad + bc = λ k a for soe k. There exists k eigefuctios correspodig to λ k ad k idices fulfillig ad + bc = λ jl = λ k for each l {,,, k } a (.) (.)

8 It follows that the solutio is of the for ṽ = k l= C l e j l for arbitrary costat C l. Fially, which copletes the proof. ũ = b aṽ,. Stability of statioary solutios We are ow ready to forulate ad prove a stability theore for liear odel (.). We will prove stability for wider assuptios for coefficiet a, b, c, d that arises fro the Kodo odel. Theore... Assue that a >, b, c, d. Let (ũ, ṽ) be a o zero statioary solutio of proble (.), correspodig to eigevalue λ k. The solutio (ũ, ṽ) is stable if ad oly if ad + bc + aλ j d + λ j for each j N, where {λ j } j= are eigevalues recalled i Theore... Proof. Fro Theore.. the statioary solutio is give by (ũ, ṽ) = ( b a k l= k Cl e jl, Cl e jl ). (.) Cosider the proble with disturbed iitial coditio (u, v ) = (ũ + u, ũ + v ). A solutio of proble (.) ca be expressed usig orthooral properties of eigefuctios l= u(t, x) = ũ + a j (t)e j (x), j k v(t, x) = ṽ + b j (t)e j (x). j k with suitably chose, a j ad b j. Substitutig those fuctios ito equatio (.) yields to the fact that it is eough to aalyse the stability of the trivial solutio (, ). Hece, (.) ( j a j (t)e j(x) ) = ( a b c d + ) (ũ + j a j (t)e j (x) ṽ + j b j (t)e j (x) ) = ( a c b d + ) ( j a j (t)e j (x) j b j (t)e j (x) ). (.) Sice {e j } j= is a orthooral basis, the solutio of syste (.) has the followig property for each j ( a j (t) ) = ( a b ) ( a j(t) c d + λ j b j (t) ). (.) Thus, the stability of solutios to proble (.) is equivalet to the stability of solutio of syste (.) for each j N. Substitutig b j fro secod equatio ito the first equatio lead us to a j(t) = aa j (t) cd d + λ j a j = a j ad + bc + aλ j d + λ j = µ j a j. (.) The solutio of equatio (.) is stable if ad oly if µ j is o egative for each j N. By assuptio, the ubers µ j are well defied ad o egative for each j. Moreover if j µ j <, because for each j, µ j ad li j µ j = a. It follows that a j are doiated decreasig fuctios ad the statioary solutio is stable.

9 It ca be observed that coditios stated i Theore.. deterie the rage of variability for coefficiet d. Collary... Deote by λ k, λ k+ eigevalues adjacet to λ k ad differet fro λ k. If the solutio (ũ, ṽ) of proble (.) correspodig to eigevalue λ k, k > is stable the d ( λ k, λ k+ ) { λ k }. If the statioary solutios correspods to the first eigevalue λ the d (, λ + ) { λ }. Proof. If d = λ j for soe j, the fro equatio (.) we iediately obtai that the coefficiets (a j (t), b j (t)) = (, ) for ay t, which cotradicts (a j (), b j ()) = (< u, e j >, < v, e j >) Moreover, oiators of µ j fors a icreasig diverget sequece ad fro theore.., the uber µ k =. Thus stability is esured if deoiator of µ j is positive for j k ad egative for j k+. Sice, λ j is ootoic ad decreasig, the thesis holds if λ k < d < λ k+. We will ow prove the followig lea which will be useful i the subsequet part of this work. Lea... If the assuptios of Theore.. hold, the every stable solutio of proble (.) with iitial coditio u fulfils ad for soe costats C, µ >. u(t) C µt u for all t > (.) Proof. Let us express u, v i the orthooral basis {e j }. We obtai u = a j e j j u = a j j v = b j e j, j v = b j. j (.) The explicit solutio of the liear proble for each a j follows fro ordiary differetial equatio theory ad it is give by forula a j (t) = a j e µjt (.) Sice e j is orthooral basis followig iequality holds for each t u(t) = a j(t) = a je µjt a je ifj µj = u e µt (.) where µ = if j N µ j. The uber if j N µ j exists because {µ j } j N is boude fro below. Moreover

10 . Copact operators The reasoig preseted i this chapter ca be exteded for wilder scope of operators geeralizig the Laplace operator. Let Ω R be a ope, bouded, coected, doai with C boudary. Let K W, (Ω) W, (Ω) be a liear operator, such that K is copact, bouded ad self-adjoit. Cosider the geeralized liearised syste u = au + bv x Ω, t = cu + dv + K(v) u (x) = u(x, ). (.) We will prove the existece ad stability of statioary solutios usig the sae techiques as i the previous sectio. We eed to recall the Hilbert - Schidt Theore. Theore.. (Hilbert - Schidt spectral theore). Let X be Hilbert space. Let H X X be a copact, bouded, self-adjoit operator. The there exist a fiite or ifiite sequece of o zero eigevalues {η j } j. Moreover if the sequece is ifiite the li j η j =. Furtherore there exist a orthooral basis {ϕ i } i= of X, such that Proof. Proof ca be foud i [, page ]. Hϕ j = η j ϕ j. Reark... Operator K is copact. Fro Theore.. there exists a sequece of eigevalues λ j = η j of K satisfyig li j λ j =. The sequece {λ j } ca be diverget to ± or ca have o liit. It is coveiet to idex the eigevalues with itegers ad order the i a icreasig sequece. Observe that, because there is o liit poit i set of eigevalues we have λ k λ k λ < < λ λ k λ k+. Theore... Assue that a >, b, c, d. There exists a o zero weak statioary solutio of proble (.) if ad oly if ad bc a = λ k for soe k. Moreover each such statioary solutio is of the for k ṽ = C l e kl ad ũ = b aṽ, l= where e kl are eigefuctios correspodig to the eigevalue λ k, k deotes the λ k ultiplicity ad C l are arbitrary costats. Theore... Assue that a >, b, c, d. Let (ũ, ṽ) be a statioary solutio of proble., correspodig to eigevalue λ k, k Z. The solutio (ũ, ṽ) is stable if ad oly if ad + bc + aλ j d + λ j for each j A Z where {λ j } j A Z are eigevalues recalled i Collary...

11 Collary... If the solutio (ũ, ṽ) of proble (.) correspodig to eigevalue λ k, is stable ad λ k is either axiu or iiu eigevalue the d ( λ k, λ k+ ) { λ k }. If the statioary solutios correspods to the axiu or iiu eigevalue the we obtai d (, λ k+ ) { λ k }, if λ k is iial eigevalue d ( λ k, ) { λ k }, if λ k is axial eigevalue where λ k, λ k+ deotes eigevalues adjacet to λ k Lea... If the assuptios of Theore.. hold, the every stable solutio of proble (.) with iitial coditio u fulfils ad for soe costats C, µ >. u(t) C µt u for all t > (.) The proofs of Theore.., Theore.., Lea.. ad Collary.. are copletely aalogous to the proofs of Theore.., Theore.., Lea.. ad Collary.., hece we skip it.

12 Chapter Noliear odel Let Ω R be a ope, bouded, coected set with C boudary. Cosider the oliear odel u t = au + f(v) x Ω, = cu + dv + v, v = u = u(, x). x Ω, For our purpose we will clai that f() =, f C (R), aely f, f, f are cotiuous ad bouded. We will also clai that a >, c, d.. Existece of statioary solutio I this sectio we will prove the existece of o costat statioary solutios of proble (.), which satisfies = au + f(v) x Ω, = cu + dv + v, (.) v = x Ω. Proble (.) ca be reduced ito oe equatio by assigig the fuctio u = f(v) fro first a equatio ad applyig it ito the secod oe. This lead us to the followig proble Deote g(v) = c f(v). We obtai the followig syste a (.) = c f(v) + dv + v x Ω, a (.) v = x Ω. = g(v) + dv + v x Ω, v = x Ω. (.) Notice that if c a is positive the g has the sae properties as f. Moreover if d = λ k for soe λ k the the operator (d + ) is irreversible, sae like i liear odel. We will use this fact later. We will ow costruct a oliear fuctioal correspodig to proble (.).

13 .. Noliear fuctioal Let G (u) = g(u). Let defie us a fuctioal J W, (Ω) R. We will ow prove the followig properties of the fuctioal J. Lea... Fuctioal J W, (Ω) R is cotiuous. Proof. Let We will show that J(u ) J(u ). J(u) = u + G(u) (.) u, u W, (Ω) ad u u W,. J(u ) J(u ) = u u + G(u ) G(u ) (.) The covergece of the first itegral is deteried by the cotiuity of the or. u u u W, u W, u u W, (.) Fuctio G fulfils global Lipschitz coditio G(u u ) < C u u. It follows that G(u ) G(u ) C ( (G(u ) G(u )) ) C u u W, (.) We obtaied that J(u ) J(u ). Lea... There exists the Gateaux derivative of fuctioal J W, (Ω) R. Moreover DJ W, (Ω) L(W, (Ω), R) is cotiuous ad J C (W, (Ω), R). Proof. Let v W, (Ω). The Gateaux derivative of J o the vector v is give by the forula d dt J(u + tv) t= = ( u + t v) v t= + g(u + tv)v t= = u v + g(u)v. (.) We eed to show, that if u u W,, the DJ(u ) DJ(u ) teds to i L(W, (Ω), R). Recall that the or of a fuctioal l L(W, (Ω), R) is give by It follows that (DJ(u ) DJ(u )) L(W, (Ω),R) = l L(W, (Ω),R) = sup v W, (Ω) v = sup x W, (Ω) x = l(x) (u u ) v + ( ( (u u )) ) + ( (g(u ) g(u )) ) sup v W, (Ω) v = sup v W, (Ω) v = sup v W, (Ω) v = ( ( v) ) (.) (g(u ) g(u ))v ( ( v) ) u u W, + ( (g(u ) g(u )) ). (.)

14 Sice g fulfils global Lipschitz coditio g(u ) g(u ) C u u. Thus we obtai ad the assuptio holds. ( (g(u ) g(u )) ) C u u W, (.) Collary... There exists the Fréchet derivative of J ad its equal to Gateaux derivative Proof. This is a well-kow result ad the proof ca be foud i [, page ] Lea... There exists the secod Gateaux derivative of fuctioal J W, (Ω) R. Moreover D J W, (Ω) L(W, (Ω), L(L(W, (Ω), R)) is cotiuous ad J C (W, (Ω), R). Proof. Secod derivative is a operator D J W, (Ω) L(W, (Ω), L(W, (Ω), R)). Recall that the or of a fuctioal l L(Ω, L(Ω, R)) is give by l L(Ω,L(Ω,R)) = sup x Ω l(x) L(Ω,R) = sup x Ω sup l(x)(y), (.) where l(x)(y) deotes fuctioal l(x) applied o y. Let w W,. Secod derivative is a fuctioal that ca be evaluated as follows (D J(u)w) v = ( d dt DJ(u + tw) t= ) v y Ω = d dt (u + tw) t= v + d dt g(u + tw) t= (.) = w v + g (u)vw. To show cotiuity of the secod derivative we eed to show that if u u W, the D J(u ) D J(u ) L(W, (Ω),L(W, (Ω),R)). D J(u ) D J(u ) = sup w W, (Ω) w = sup v W, (Ω) v = w v g (u )vw sup w W, (Ω) w = sup v W, (Ω) v = w v + g (u )vw (g (u ) g (u )) vw. (.) By the assuptio the fuctio g fulfils the Lipschitz coditio, hece fro the Schwartz iequality it follows that (f (u ) f (u )) vw C u u W, ( (vw) ) C u u W, v L w L. To prove the covergece we eed to show that v L w L is bouded. Let us recall Sobolev ebeddig Theore... Sice the diesio is less the, we obtai that W, (Ω) L (Ω) with a cotiuous ebeddig ad cosequetly v L w L C v W, w W, Collary... There exists the secod Fréchet derivative of J ad its equal to Gateaux derivative Proof. This is a well-kow result ad the proof ca be foud i [, page ].

15 .. Existece by the bifurcatio theore Let X be a real Hilbert space, Ω X be a eighbourhood of. Let L Ω X be a liear cotiuous operator ad let H C(Ω, X). Set H(u) = o( u ) as u. Cosider equatio Obviously, there exists a trivial solutio (λ, ) R X for each λ. Lu + H(u) = λu. (.) Defiitio... A poit (µ, ) R X is called a bifurcatio poit for equatio (.) if every eighbourhood of (µ, ) cotais otrivial solutio of (.). Lea... If (µ, ) is a bifurcatio poit the µ belogs to the spectru of operator L. Proof. Let (µ, ) be a bifurcatio poit for equatio (.). Cosider the faily of balls B ((µ, ), ) R X. For each there exists (µ, u ) B satisfyig Lu + H(u ) = µ u (.) Obviously, we have µ µ. Divide both sides of this equatio by the or of u ad cosider the weak solutio u u v + H(u ) u v = µ u u v By assuptio for H, if we obtai H(u) u v. Put w = bouded, hece it is weakly copact. Let w k w. It follows L(w)v = µwv u u. Sequece {w } = is The equality holds for each v W,, hece µ belogs to the spectru of operator L. Theore.. (Rabiowitz Bifurcaio Theore). Let X be a real Hilbert space, Ω a eighbourhood of i X ad h C (Ω, R) with h (u) = Lu + H(u), L beig liear ad H(u) = o( u ) at u =. If µ is a isolated eigevalue of L of fiite ultiplicity, the (µ, ) is a bifurcatio poit for (.). Moreover, at least oe of the followig alteratives occurs:. (µ, ) is ad isolated solutio of (.) i {µ} X. There is oe-sided eighbourhood, Λ of µ such that for all λ Λ {µ}, equatio (.) posses at least two distict otrivial solutios.. There is a eighbourhood I of µ such that for all λ I {µ}, equatio (.) posses at least oe o trivial solutio. Proof. Proof ca be foud i [, page ]. We are ow ready to prove the existece of statioary otrivial solutios for proble (.). First, we eed to trasfor the equatio. ito the for of Lu + H(u) = λ k u (.)

16 for soe eigevalue of Laplacia λ k recalled i Theore... Addig λ k yields to The operators L, H are give by λ k v = c a f(v) + (d + λ k)v + v x Ω, (.) v = x Ω. L(v) = v ad H(v) = c a f(v) + (d + λ k)v. Let g = c a f ad G = g. Let us defie the oliear fuctioal I W, (Ω) R I(v) = v + G(v) + (d + λ k )v. (.) Theore... Cosider syste (.). If f C (R) fulfils the global Lipschitz coditio f() = ad f () = a(d+λ k), the there exist a sequece {d c } = R, which coverges to d ad a sequece of ocostat fuctios {u } =, {v } = W, (Ω), such that (ũ, ṽ ) is a statioary solutio of (u) t = au + f(v) x Ω, for each N. Proof. Let I(v) be as i (.) the = cu + d v + v, v = DI(v) = Lv + H(v). x Ω. Observe that fuctioal I is a odificatio of fuctioal J defied i (.) I(v) = J(v) + (d + λ k )v = J(v) + R(v) (.) Fro Lea.. we obtai that J C (W, (Ω), R). Applyig Lea.. for the fuctio f R R, f(v) = v gives R C (W, (Ω), R). To prove that H(u) = o( v ) W, (Ω), we eed to check that if v W, (Ω), the H(v) W,. (.) v W, Sice f() = we obtai c a f(v) + (d + λ k)v W, v W, = c a f(v) c a f() + (d + λ k)v W,. (.) v W, The right had side teds to the ius Fréchet derivative of f if v W,. It follows fro the assuptio that the oiator teds to. Now we ca apply Theore.. to obtai that (λ k, ) is a bifurcatio poit for syste (.). It follows that there exists a sequece of {λ k } = coverget to λ k ad a sequece of ocostat fuctios {v } = W, (Ω) such that λ kv = c a f(v ) + (d + λ k )v + v x Ω, v = x Ω. (.)

17 for each N. Applyig d = d + λ k λ k ad u = c (d v + b ) yields to the result. Collary... If fuctio f is a liear fuctio equal to f(v) = bv the the coditio i.. is equal to the coditio i theore.... Stability Let (ũ, ṽ) be a statioary ocostat solutio of proble (.). Cosider the disturbace of statioary solutio u = ũ + ϕ, (.) v = ṽ + ψ. Applyig this ito the proble (.) yields to ϕ t = a(ϕ + ũ) + f(ṽ + ψ) = cϕ + dψ + ψ x Ω, ψ = Sice, aũ = f(ṽ), the by Taylor expasio we obtai x Ω. (.) ϕ t = aϕ + f (ṽ)ψ + R(ũ)ϕ = cϕ + dψ + ψ x Ω, ψ = x Ω. Propositio.. (Liearised Stablity). The styste (.) is liearly stable if the syste (.) is stable. ϕ t = aϕ + f (ṽ)ψ = cϕ + dψ + ψ x Ω, ψ = x Ω. (.) We will prove the stability of solutios i case Ω R. This assuptios is required to esure, that if the solutio is i W, (Ω) the the L or is fiite. This follows fro Sobolev ebeddig theore for p = ad =. Theore.. (Sobolev ebeddig theore). Let Ω R be a ope set with boudary of class C (Ω). Let p <. The the followig ebeddigs hold W,p (Ω) L p, where p = p N, if p < W,p (Ω) L q, where q [p, ), if p = W,p (Ω) L, if p > Proof. Proof of the theore ca be foud i [], p.

18 Ufortuately siilar relatio does ot hold i case of Ω R, so we are ot able to prove stability i two diesioal case. To achieve required estiate, we eed to ivolve soe regularity theory. We are ow ready to forulate ad prove that solutios achieved by bifurcatio theore are stable uder specific coditios. Theore... Let Ω = R. Let the syste (ũ, ṽ) be the statioary ocostat solutios of (.) obtaied through Rabiowitz Bifurcatio Theore. The solutio is liearly stable if ad + f ()c + aλ j d + λ j for each j N, where {λ j } j= are eigevalues recalled i Theore... Proof. Let (ϕ, ψ) be the solutio of syste ϕ t = aϕ + f (ṽ)ψ = cϕ + dψ + ψ x Ω, ψ = x Ω. (.) ad (ϕ, ψ) be the solutio of syste ϕ t = aϕ + f ()ψ = cϕ + dψ + ψ x Ω, ψ = x Ω. (.) The solutio of (ϕ, ψ) is give by the explicit forula i theore... Defie θ = ϕ ϕ, ω = ψ ψ. We subtract the first equatio i. fro the first equatio i. ad the secod equatio i. fro the secod equatio i. θ t = aθ + f ()ω + (f (ṽ) f ())ψ, = cθ + dω + ω. (.) Evaluatig ω fro the secod equatio ad applyig to the first gives θ t = aθ f ()c(d + ) θ + (f (ṽ) f ()) (ω + ψ), (.) Multiplyig by θ ad itegratig both sides over Ω gives θ t θ = a θ f ()c ((d + ) θ) θ + (f (ṽ) f ()) (ω + ψ)θ (.) Ω Ω Ω Ω The first itegral is equal to the tie derivative of L or θ t θ = d Ω dt θ L.

19 To show the stability of solutios we eed to prove that the itegrals o the right had side are bouded. Fro lea.., we have a Ω θ f ()c Ω ((d + ) θ) θ Ω ( a f ()c(d + ) ) θ θ µ θ L (.) for all µ. Sice f () a(d+λ k), coefficiet µ <. Next, for (.) ad Cauch - Schwartz c iequality we have Ω (f (ṽ) f ()) ωθ Ω (f (ṽ) f ()) ( c(d + ) ) θ θ f (ṽ f ()) L ( c(d + ) ) θ L θ L Sice, ṽ L is sall ad f is Lipschitz fuctio the f (ṽ f ()) L < ε. Operator c(d+ ) is bouded i L, thus we obtai Last itegral ca be estiated as follows Ω (f (ṽ) f ()) ωθ εc θ L. Ω (f (ṽ) f ()) ψθ f (ṽ f ()) L ψ L θ L Ce µt θ L Fially, we obtai d dt θ L µ θ L + εc θ L + Ce µt θ L = θ L ( µ + εc + Ce µt ). (.) For sufficietly sall ε, the expressio ( µ+cε) is egative. It follows that for large t, ( µ + εc + Ce µt ) is egative ad less the δ <. Therefore θ L is bouded by C e δt for sufficietly large t. t Sice θ L >, the θ L.

20 Chapter Nuerical Siulatios. Nuerical schee It this chapter we will preset uerical siulatios obtaied for the aalysed odel. Siulatios were obtaied usig followig uerical schee. Cosider syste of equatios u t = au + f(v) x Ω, = cu + dv + v, v = x Ω, u = u(, x). (.) Let Ω R be copact, ad coected doai. For our purpose we will clai that Ω is a square doai Ω = [, π] [, π]. We will clai that costats a, c, d R {}, ad a >, b >. The first equatio ca be approxiated usig differece quotiet with appropriate h. The first copoet i the differece quotiet u is obtaied fro the syste defiitio. Thus we obtai u + u = au + f(v ), h u + = ( au + f(v )) h + u. (.) The fuctio v ca be evaluated fro secod equatio cu = dv + v, (.) suppleeted with the Neua boudary coditios. To evaluate the value of v we will express the fuctio u i the basis of eigefuctios {ϕ j } j= of Laplacia with Neua boudary coditios. Let u (t, x) = a j (t)ϕ j (x) j= a j =< u, ϕ j >= Ω u ϕ j (x) (.) It follows that v = a j (t) c ϕ j (x). (.) d + λ j j=

21 To esure uerical efficiecy we will approxiate v with fiite su of order N N v = a j (t) c ϕ j (x). (.) d + λ j The eigefuctios {ϕ j } j= for quadratic doai are give by explicit forula j= {ϕ j } j= = {cos ( x ) cos (y ),, N} =,=. (.) To facilitate further aalysis we will chage the idexig of eigefuctios such that ϕ, (x, y) = cos ( x ) cos (y ) It follows iediately that the eigevalue λ, correspodig to ϕ, is equal to λ, = +. Notice that λ, = λ,. The dot product over quadratic doai is equal to the double itegral < u, ϕ, >= π π u (x, y)cos ( x ) cos (y ) dxdy I our uerical siulatios itegratig over doai is approxiated by fiite sus over hoogeeous doai partitio.. Siulatio results Paraeters that were used i uerical siulatios are the coproise betwee required coputatio accuracy ad accepted calculatio tie. Paraeters used i siulatio are followig Grid size = x, Step size h =., Nuber of eigefuctios N =... Liear odel Results preseted i this sectio were obtaied for liear odel, with fuctio f(v) = bv. Siulatio results are show of Fig., Fig., Fig. ad Fig.. The patter of solutio u is preseted i the first row. The coefficiets of eigefuctios ϕ, are show i the secod row. Each colu shows the solutio i tie t =,,,. Fuctio u is chose radoly with two diesioal uifor distributio. Syste coefficiets are fixed ad a =, b =., c =. Coefficiet d is chose to fulfil aλ, + ad + bc = for soe λ,. It ca be see that all eigefuctios which do ot correspods to eigevalue λ, coverges to zero. Notice that for d =. ad d =. the statioary solutio is syetric. Those values correspods to the sigle eigevalues λ, ad λ.,. respectively. It iplies that there exists oly oe eigefuctio for each eigevalue which is syetric, so the solutio is syetric as well. If d =. or d =. the there exists two eigefuctios for each eigevalue which do ot coverge to. Those eigefuctios are ϕ,., ϕ., ad ϕ.,., ϕ.,. for d =. ad d =. respectively. The coefficiets for each eigevalue are deteried by the iitial coditio which is rado. It follows that the statioary solutio is ot syetric.

22 t = t = t = t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Figure.: Patters ad eigefuctios coefficiets for t =,,,. Paraeter value: d =. t = t = t = t = Eigefuctio coefficiets t =... Eigefuctio coefficiets t =. Eigefuctio coefficiets t = -. Eigefuctio coefficiets t = Figure.: Patters ad eigefuctios coefficiets for t =,,,. Paraeter value: d =.

23 t = t = t = t = Eigefuctio coefficiets t =. Eigefuctio coefficiets t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Figure.: Patters ad eigefuctios coefficiets for t =,,,. Paraeter value: d =. t = t = t = t = Eigefuctio coefficiets t =.. Eigefuctio coefficiets t = Eigefuctio coefficiets t = -. Eigefuctio coefficiets t = Figure.: Patters ad eigefuctios coefficiets for t =,,,. Paraeter value: d =.

24 .. Noliear odel Results preseted i this sectio were obtaied for oliear odel (.), with saturatio fuctio f R R equal to, x <, f(x) = x, x x, (.) x, x > x. Coefficiets a is the sae as i liear odel. Modificatio of a coefficiet is ot required to achieve all possible patters. Variatio of a deteries the covergece rate of eigevalue coefficiets, however required patters ca be achieved by appropriate odificatios of reaiig paraeters. Paraeters c was set to. Larger values of c esures that ore eigefuctios becoes ustable, which results i ore coplicated ad diversified patters. The value of paraeter x is set to. It was observed that the values of lower ad upper saturatio liits does ot chage solutio patter, but results oly i rescalig ad shiftig of the obtaied solutio. Various patters ca be achieved by odifyig d coefficiets, sice it deteries the rage of ustable eigefuctios. Siulatio results are preseted i Fig.., Fig., Fig.. Notice that the stabilizatio tie of the solutio is oticeably shorter tha i liear case. This pheoea occurs, sice i liear case all but oe eigefuctios were decreasig, ad the reaiig oe stayed costat. I oliear case soe eigefuctios are ustable, ad this epowers the relative differece betwee stable ad ustable eigefuctios. Observe that patters o Fig.. ad Fig.. are regular i soe sese. Patters obtaied at level, are siilar to patters obtaied at level. Patter i the Fig.. is ot regular. Patter at level is sigificatly differet the patter at level. The crucial differece betwee those patters is that i case ad, there exist a few eigevalues which are sigificatly larger the the rest of eigefuctios. t = t = t = t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Figure.: Patters ad eigefuctios coefficiets for t =,,,. Paraeter value: d =.

25 t = t = t = t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = Figure.: Patters ad eigefuctios coefficiets for t =,,,. Paraeter value: d =. t = t = t = t = Eigefuctio coefficiets t = Eigefuctio coefficiets t = - - Eigefuctio coefficiets t = - - Eigefuctio coefficiets t = Figure.: Patters ad eigefuctios coefficiets for t =,,,. Paraeter value: d =.

26 Chapter Coclusio I this work we have attepted to explai the process of forig patter obtaied by Shigeru Kodo i uerical siulatios. We proposed a alterative odel sei reactio-diffusio odel which is sigificatly sipler to aalyse tha covolutio differetial syste proposed by Shigeru Kodo. Our odel correspods to the specific kerel i the Kodo odel, with kerel K equal to the Gree kerel. Moreover, by substitutig the Laplace operator with other copact operator we ca exted our result for wider class of covolutio kerels. Usig base Laplace operator properties we have prove that o certai coditios, o costat stable statioary solutios ay exist. The shape of the obtaied patter strogly depeds o the coefficiets value ad iitial coditios. I the liear case we have prove that the stable statioary solutio is a liear cobiatio of eigefuctios. The eigevalue correspodig to those eigefuctios fulfils specified coditio for odel coefficiets. The value of eigefuctios are deteried by the iitial coditio. Notice that we did ot use the fact that the diesio is equal to. All of those theores holds for ay R. Results fro this sectio are perfectly cofired by uerical siulatios. Eigefuctios that do ot correspod to the distiguished eigevalue are vaishig expoetially. I the o liear case we have prove the existece usig Rabiowitz Bifurcatio Theore. We have obtaied, that uder specific assuptios for derivative of f there ay exist a set of coefficiets d such that bifurcatio occurs ad o costat statioary solutio exists. However, proposed theore is isufficiet to decide whether there exists statioary solutio for a give d. It follows fro the properties of ivolved theore. We obtai oly the discotiuous brach of bifurcatio poits. To obtai the coplete theore for ay d we would eed to ivolve the bifurcatio theore with cotiuous brach of bifurcatio poits. Notice that, i Kodo odel saturatio fuctio does ot follow the C assuptio. To tackle with this proble, oe ca approxiate polylie with appropriate sooth fuctio. Nuerical siulatios for o liear odel deostrates sigificatly differet echais of forig statioary solutios. Obtaied statioary solutios are ot sall. It ca be observed that for specific valued of f, c, d there exists a set of eigefuctios, that are ustable. As tie passes, those eigefuctios are growig expoetially, util first eigefuctio reaches saturatio level. The the solutio cocetrates ear the values of u ax ad u i ad subsequetly it stabilizes. Those patters atch the patters obtaied by Shigeru Kodo. It shows, that there exists differet echais of patter foratio that eed to be ivestigated. To prove the stability of o costat solutios, we required the L estiates for fuctio u. Ufortuately, we are ot able to esure appropriate estiates i diesio larger the. To subit proper proof for = we would eed to ivolve relevat theores fro regularity theory.

27 Bibliography [] A reactio-diffusio wave o the ski of the arie agelfish Poacathus, S. Kodo, R. Asai, Nature vol, August, [] S. Kodo, M. Iwashita, M. Yaaguchi, How aials get their ski patters: fish piget patter as a live Turig wave, It. J. Dev. Biol. : - () [] S. Kodo, A updated kerel-based Turig odel for studyig the echaiss of biological patter foratio, Joural of Theoretical Biology () [] P. H. Rabiowitz, A bifurcatio for Potetial Operators, Joural of fuctioal aalysis, - () [] P. Drábek, J. Milota, Methods of Noliear Aalysis. Applicatios to Differetial Equatios, ISBN ----, Birkhäuser Verlag AG [] M. Reed, B. Sio, Methods of oder atheatical physics: Fuctioal Aalysis, ISBN ---, Califoria [] H. Brezis, Fuctioal aalysis, Sobolev Spaces ad Partial Differetial Equatios, ISBN ---- [] J. Jost, Partial Differetial Equatios ISBN --- [] L. C. Evas, Partial Differetial Equatios, ISBN ---- [] A. Abrosetti, G. Prodi, A Prier of Noliear Aalysis, ISBN ---