Popular in Course
Popular in Mathematics (M)
This 11 page Class Notes was uploaded by Miss Hillary Grady on Thursday October 29, 2015. The Class Notes belongs to MATH 490 at College of William and Mary taught by Junping Shi in Fall. Since its upload, it has received 14 views. For similar materials see /class/231141/math-490-college-of-william-and-mary in Mathematics (M) at College of William and Mary.
Reviews for Seminar
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
Date Created: 10/29/15
PHYSICAL REVIEW E 69 061917 2004 Pattern formation and nonlocal logistic growth Nadav M Shnerb Department of Physics BarIlan University Ramat Gan 52900 Israel Received 8 August 200339 revised manuscript received 12 November 200339 published 22 June 2004 Logistic growth process with nonlocal interactions is considered in one dimension Spontaneous breakdown of translational invariance is shown to take place at some parameter region and the bifurcation regime is identi ed for short and longrange interactions Domain walls between regions of different order parameter are expressed as soliton solutions of the reduced dynamics for nearestneighbor interactions The analytic results are con rmed by numerical simulations DOI 101103PhysRevE69061917 I INTRODUCTION Logistic growth is one of the basic models in population dynamics First introduced by Verhulst for saturated prolif eration at a single site it has been extended to include spatial dynamics by Fisher 1 and by Kolmogoroff et al 2 In its onedimensional continuum version one consider the con centration of a reactant cxt with time evolution that is given by the rate equation W DV2cxt acxt rbc2xt 1 where D is the diffusion constant a is the growth rate and b is the saturation coef cient set by the carrying capacity of the e ium The Fisher process is a generic description of the invasion of a stable phase into an unstable region It is applicable to wide range of phenomena ranging from genetics the origi nal context of Fisher work proliferation of a favored muta tion or gene to population dynamics chemical reactions in unstirred reactors hydrodynamic instabilities invasion of normal states by superconducting front spinodal decomposi tion and many other branches of natural sciences A compre hensive survey may be found in recent review article by van Saarloos 3 The Fisher process ends up with a Lmiform saturated phase in contrast with other nonlinear and reactive systems which yields spatial structures with no underlying inhomo geneity These patterns are usually related to an instability of the horn ogenous solution most commonly of Turing or Hopf types 4 Spontaneous symmetry breaking of that type mani fests itself in vegetation patterns where competition of ora for common resource water induces an indirect interaction and may lead to a Turinglike spatial segregation 5 The basic motivation of this work mes from recent study of nonTuring mechanism for pattern formation in the vegetationwater system which yields ordered or glassy structures 6 Basically it is easy to realize that competition for common resource induced some indirect repulsion among agents which may lead to spatial segregation As an example consider the vegetation case there is a constant ow of water into the system rain and the water dynamics evaporation percolation diffusion is much faster than the dynamics of the ora Now let us assume the existence of a 15393755200469606191762250 69 0619171 PACS numbers 8717Aa 0545Yv 8717Ee 8240Np large amount of ora say a tree at certain spatial point One may expect the water density to adjust almost instantly to the tree and to equilibrate in some water pro le that is lower around its location The immediate neighborhood of the tree though is less favorable for a second tree to ourish instead one may expect the next to grow up some typical distance away reducing the water level between them even more This seems to be a plausible and generic mechanism for seg regation induced by resource competition These arguments may be relevant to the dynamics of almost any Lmstirred reactive system interesting example is the process of evolu tionary speciation where new species may survive only far enough in the genome space where the spatial structure is given say by Hamming distance from its ancestor in order to nd a nonoverlapping biological niche SLu39prisingly it turns out that the partial differential equa tions that describe this process here presented in a nondi mensionalized form where w stands for water density 1 for ora and R is the rain 15w Vlb 7M1 wb Wxt DV2wRrwrwb 2 yield only a linearly stable homogenous solution In order to get patterns one should add a crossdiffusion effect slowing down of the water diffusion in the presence of ora that leads to Turinglike instability as in Ref 5 but this is a different mechanism and one may wonder about the validity of the basic intuitive argument presented above Recent work 6 suggests a hint for the answer It seems that a continuum and local description of a reactive system fails to capture the competition induced segregation dis cussed above The continuum process is trying to smear the reactant pro le and instead of getting spatially segre gated structure of large biomass Lmits trees it favors ho mogenous pro le of grass covering all the area In Ref 6 a biomass Lmit was allowed for long time survival only if it exceeds some predetermined threshold and simulation of the system reveals an immediate appearance of spontaneous seg regation and stable patterns Similar situation appears presumably in the process of bacterial colony growth where the food supply is limited As noted by BenJacob et al 7 spatial segregation and branch 2004 The American Physical Society NADAV M SHNERB ing are induced by the competition of bacteria for diffusive food A communicating walkers model is used by these au thors to simulate the branching on a petridish where con tinuum equation dictates the food dynamics while the indi vidual bacteria are discrete objects The discreteness of bacteria adds some weak threshold to the system and induces segregation Note however that the model admits weak de pendence of the diffusion on the local bacterial density at the boundaries of the colony In this work I consider the onespecies analogy of the competition problem namely a logistic growth with nonlo cal interactions where the carrying capacity at a site is re duced due to the presence of life in another site Nonlocal competition has been recently considered by Sayama el al 8 and by Fuentes el al 9 Both groups uncover the pos sibility of spontaneous symmetry breaking and patterns de pending on the strength and the smoothness of the weight function that controls the nonlocality It seems that nonlocal interactions are not simply an effective model obtained by integrating out the fast degrees of freedom39 rather it incor porates some nonlinear effects like the threshold and allow for linear instability that manifests the intuitive competition induced segregation argument Sayama el al 8 deal with a twodimensional model of population dynamics with no diffusion term Both the local growth term and the carrying capacity at a site depend not in the same way on the population of neighboring sites in a crowded neighborhood the growth term becomes larger due to offspring migration while the carrying capacity decreases as a result of longrange competition The conditions for an instability of the horn ogenous solution have been found ana lytically and demonstrated numerically for a stepwise weight function taking as the effective neighborhood the average density inside a prescribed radius around the site It was also pointed out that a Gaussian weight function yields no instability In the numerical work of Ref 9 a onedimensional re alization of diffusing reactants has been considered equiva lent to Fisher equation with nonlocal interactions Again it was shown that a stepwise weight function may lead to in stability while Gaussian weights lead to stable homogenous solution39 the authors proceed to consider intermediate weight functions that interpolate between Gaussian and a step func tlon As in any case of spontaneous symmetry breaking the system falls locally into one of the minima of the order parameter and typically domains are formed These domain walls determine the low lying excitation spectrum of the sys tem as their movement is smooth if the broken symmetry is continuous the resulting Goldstone modes may destroy the longrange order at nite temperature and the same is true for the domain walls if discrete symmetry is broken Al though we are dealing with an out of equilibrium system one may guess that the response to small noise is determined by these domain walls The goals of this work are twofold in the following Sec I will try to give more comprehensive discussion of the in stability condition with and without diffusion and its depen dence on the weight functions it turns out that it depends on the minimal value of its Fourier transform The third section PHYSICAL REVIEW E 69 061917 2004 is devoted to the appearance of topological defects in the segregated phase Finally in Sec IV some discussion and possible implications are presented II lNSTABILITY CONDITIONS The model is a onedimensional realization of longrange competition system on a lattice with lattice spacing lo and the continuum limit is trivially attained at ZOHO In the generic case of diffusion and nonlocality the time evolution of the reactant density at the nth site is given W 3 92 260 Ema 6710 1160 bear 760 Mata a an 3 where f is the diffusion constant and 111 are the corre sponding reaction rates The de nition of dimensionless quantities 7 at c bEa 4 h D lm the new diffusion constant is D WZFO where W E wD a is the width of the Fisher front provides the dim en sionless equation 00 30 n Dl zcn Cn1 Cnrll cnlt17 cn E Vrlcrm P1 3 7 c gt 5 that may be expressed in Fourier space with Ak EEK cne mo as 14k akAk E BkiquAkim 6 q where oak E l 2Dl cosklo 7 BF 1 22 ycosltrklo 8 P1 As on is positive semide nite A0 is always macroscopic Any mode is suppressed by A0 and for small y one expects only the zero mode to survive 10 If on the other hand y increased above some threshold bifurcation may occur with the activization of some other k modes and the homog enous solution becomes unstable This is the situation where patterns appear and translational symmetry breaks To get a basic insight into the problem let us consider the case with no diffusion D0 04151 Eq 5 becomes 0619172 PATTERN FORMATION AND NONLOCAL LOGISTIC GROWTH PHYSICAL REVIEW E 69 061917 2004 TABLE I The function Bk and the instability conditions for various types of nonlocal interactions The results for the Gaussian case are ti in the continuum approxima on Type 3 Bk Instability condition Sinhlln71l E tial 3 N 39 t bil39 Xp m W coshomwvaicoswo ms a y 7 771 We2 c 6 Quadratic r Zl 12y1 K7 wide2 7 71k7710 ka s 1 rep A k 1 12 81 7 0 DP Sin 2 argep c 3 2 kl 2 39202 gt NE 77 7 0 Gauss1an 31 If 77 lo 2 mel exp 4 unw No 1nstabihty 039quot 0n 1 r 0n r E 7rcnr cw 9 r and division by cH yields for the steady state the linear equation Q gy where Q is a circular matrix Q is the vector ofcn s andy7llll The sum ofthe elements of row of 7Q is the same so the homogenous state scalar multiplication of y should be an eigenvector On the other hand if Q is nonsingular it must admit a full set of mutually orthogonal eigenvectors Only the constant eigenvector of Q has nonvanishing projection on y so the only positive de nite nondiverging steady state 650 cngt0Vn solution is the hamagenaus one 051 3 As implied by Eq 6 the homogenous steady state is unstable iff for some k Bklt0 In that case bifurcation oc curs and the new steady state is a combination of the zero mode and the k mode with equal weights A0AklBO 3k The function Bkke07rlo is discrete for nite sys tems and becomes continuous at the thermodynamic limit If Bk never crosses zero there is no bifurcation and the homog enous solution 01130 is stable The results for few types of interaction ranges with the critical value fl where the instability occurs and k the rst excited mode are surn marized in Table It is interesting to note that these expressions may be gen eralized to yield a full period doubling type instability cas cade The mth instability involves 2 quot modes and the steady state is 12 Bk where the sum runs over all the active modes The condition for the ml bifurcation activation of another 2de modes is the existence of a wave number q such that 2k quklt0 with the sum runs again over all 2 quot active k s There are however some obstacles for the impli cation of these solutions above the rst bifurcation Degen eracies in Bk eg for y 874 both klo7r4 and kl0 377 4 are minima and solitons between different stable phases described below may blur the native state In this paper though Bk is used only for the rst patternforming instability criteria and the details of the emerged structure are presented just for nearestneighbor interaction Once diffusion is added to the system its features changes but not so much The horn ogenous state is still char acterized by cH l BO and the rst pattern formation instabil ity appears when some k mode satis es Bk lt ZBODU cosklo 10 Above this instability the amplitudes of the modes are not equa 04k 041430 3k 30 A A 11 30 k waka and there are no zeroes of c This result ts perfectly with the numerical data presented in Fig 1 Again the mth insta bility involves the activation of 2 quot modes although the sta bility analysis is more complicated The question of pattern instability is thus translated to the determination of the minimal value of the Fourier coef cient of the weight function or the weight series y If the minimal value is smaller than some prescribed number zero if there is no diffusion instability takes place and patterns emerge Unfortunately I am not familiar with a general theo rem that sets bounds on the minimal value of the Fourier coef cient of a function based on its smoothness or other analytic properties so any case should be considered sepa rately with the generic examples given in Table I III DOMAIN WALL STRUCTURE Above the pattern formation threshold generic initial con ditions fail to yield perfect lattice as different domains reach saturation with different phases These domains are connected by solitonlike solutions of the time independent equation in the following sense any stable solution Cxj 0 should satisfy in the continuum limit 0619173 NADAV M SHNERB 1r I Q Q 08 bxo 3 x 06 Q5 3 I I 1 l9 km s Q 02 Q 0 I I A 002 004 006 o s 01 D FIG 1 Maximal amplitude of cquot differences maxc 7mincn circles for a sample of 1024 sites periodic boundary conditions and the predicted difference according to Eq 11 for nearestneighbor interaction with y08 dashed line The agree ment is up to the numerical error dzc x D dxg 006 Cx x ycydy 12 thus it looks like a trajectory of mechanical particle with mass D in a nonlocal potential with x as the time A domain wall is a nite size structure so it must connect xed points of this ctitious dynamics ie a domain wall corre sponds to heteroclinic orbit In this section we consider these solitons and look for their shape and size at different condi tions In order to simplify the discussion only the nearest neighbor case is considered both with and without diffusion With no diffusion and mm competition Eq 9 takes the form 90 67 n Cn1 Cn 7Cnl Cnil The uniform solution in this case is c1 12 7 and the nonuniform solution is either cn1 for odd n and cn0 for even or vice versa Stability analysis shows that the uniform solution becomes unstable at ya 1 2 and the zeroone phase is stable above this value One may expect though to see a jump from homogenous to patterned zeroone phase at 70 However if the initial conditions are taken from ran dom distribution there is a chance for a domain wall be tween two regions as indicated by the numerical results pre sented in Fig 2 Clearly such a soliton should be a solution of the map 1 cn Cnl Cnil of course 01010101 odd 0 s and 101010101 even 0 s are already solutions of this equation We are looking for the solution that connect these two xed points Such a tra jectory begins in say 010101010 state but then after the zero it gives not 1 but XI The dynamics now continue in a PHYSICAL REVIEW E 69 061917 2004 1 505 O 0 39 2 g J 1 o 140 150 160 170 FIG 2 A typical soliton of length L20 an outcome of for ward Euler integration of Eq 13 on 1024 lattice points with peri odic boundary conditions and random initial conditions at 3 0505 just above the bifurcation There is a perfect agreement with the theoretical prediction Eq 21 up to the accuracy of the numerics here four to ve signi cant digits different trajectory but x1 should be selected such that after L steps of the map for domain wall of size L the 101010 solution is rendered In a matrix form the condition that determined xf x1 for a given L is 1 L 0 1 xf xi 7 07 0 M 15 1 1 1 xi 0 10 1 00 where we assume symmetry of the soliton so L must be even In other words the condition that determine xf is L xi 0 0 16 1 Q 1 O Q 1 O 1 O O O 1 1 1 1 O O Diagonalization of M is given by the matrix S 1M5D 17 where 0 0 D 0 e i9 0 18 0 e and 6arctan4y21arccos127 The eigenvalue problem 16 may be written in terms of the diagonal matrix 19 r23 L0 COS 7 7 and using S12ST3rlzew and S122Sf2k3r23eh7 it implies that 0619174 PATTERN FORMATION AND NONLOCAL LOGISTIC GROWTH 0397 I I 0393 I I J I I I I39 0 200 400 600 800 1000 X FIG 3 Solitons for nn interaction with y0505 and D 0001 circles D00012 heavy line and D000124 line Dc000124378 1 l 1 T 12 2 1 L1 7 20 where Tnx is the nth Chebyshev polynomial of the rst kind After determining x the same method may be used to derive a general expression for all the elements of the size L soliton m where lmL 1 WW 2y1 27 1 81270 TWO271 21 and it ts perfectly the numerical experiment presented in Fig 2 see captions The above analysis gives the shape of a soliton for any prescribed length L but simulation indicates that only one soliton size L is selected for any set of parameters and its length diverges as 1 approaches its critical value Looking carefully at the solutions 21 one realizes that all other pos sible solitons admit values for some of the xm s that are either negative or larger than 1 so these options are unphysical negative or unstable to small perturbations The actual Ly may be forecasted by a rough argument based on a continuum approach De ning the local deviation from the onezero solution cn1 and cni1 0 lt 039 on 1 5m 22 Cnil 57211 and plugging it into Eq 14 gives 5 5nl nil n 7 Subtracting of 25 from both sides and taking the continuum limit ie assuming that the changes in 5 from site to site are small compared to lo here taken to be unity one gets PHYSICAL REVIEW E 69 061917 2004 300l 250 so 200 1so 2 a a 100 50 I I I 40 60 BO 100 1N8 FIG 4 Soliton size L as a function of 1 6 y yo for a nearestneighbor interaction without diffusion The circles are the results of a numerical simulation and the line is 7T In the inset the characteristic length of the soliton tail 6 is plotted against 1 77 for y0505 for the solitons shown in Fig 3 The circles come from the numerics and the line is the best linear t that give a slope of 0198 to be compared with 1 0177 0 I I 46 12e V2600 6x 24 with EE 7 yc goes to zero at the transition so 12ez 1 The solution of Eq 24 that satis es the boundary condi tions 600 together with 6L1 is sin2 ex sin2 18L 39 This expression fails to converge smoothly to the back groun ordered 010101 phase at x0 it has a nite slope and is also asymmetric Put it another way there are no non trivial heteroclinic orbits for a parabolic potential On the other hand close to the transition where the size of the do main wall is large it seems that it should t a solution of the continuum approximation The only way out is to pick a soliton size L such that the continuum equation admit no solution at all ie 5x 2 5 L 27 26 Such a choice forces us back to the discrete equation 14 and its solution 21 This argument turns out to give the right length of the stable soliton in the large Le gt 0 limit as shown in Fig 4 Let us consider now the domain walls for the nite diffu sion case As seen in Fig 3 there are also solitons for the nite diffusion case but now they admit tails that asymptoti cally looks like 5 expx E and E diverges at the transi tion eg at Dc2y 142y1 for nearestneighbor in teraction De ning a vectorial order parameter according 0619175 PHYSICAL REVIEW E 71 066201 2005 Bifurcations in reaction diffusion systems in chaotic ows Shakti N Menon and Georg A Gottwald School of Mathematics amp Statistics University of Sydney New South Wales 2006 Australia Received 14 January 200539 published 2 June 2005 We study the behavior of reacting tracers in a chaotic ow In particular we look at an autocatalytic reaction and at a bistable system which are subjected to stirring by a chaotic ow The impact of the chaotic advection is described by a onedimensional phenomenological model We use a nonperturbative technique to describe the behavior near a saddle node bifurcation We also nd an approximation of the solution far away from the bifurcation point The results are con rmed by numerical simulations and show good agreement DOI 101103PhysRevE7l066201 I INTRODUCTION In recent years interest has risen in the dynamics of re acting tracers in a complex ow environment Apart from the purely theoretical challenge this is due to the environmental and industrial applications Examples are ubiquitous in na ture and industry and include mixing of reactants within continuously fed or batch reactors 12 the development of plankton blooms and ochu rence of plankton patchiness 376 and increased depletion of ozone caused by chlorine laments 7 In typical chaotic ows uid parcels are deformed Cha otic advection gives rise to regions of stretching and folding causing uid parcels to form lamental structures Tracers are advected with these laments which leads to an in creased surface area of the tracers In the case of reacting tracers this has strong implications on the reaction kinetics and gives rise to phenomena that are not observed in a non stirred ow For example differential uid ow can generate a nonTuring mechanism for pattern formation 8 chaotic ow can determine synchronization in oscillatory media 910 or cause clustering ll Chaotic stirring also implies a dependence of mixing results on the initial condition 12 In principle these phenomena can be studied directly in a two or threedimensional 2D or 3D reactionadvection diffusion system with huge computational effort An analyti cal treatment of the full system is prohibited by the compli cated nature of the underlying equations which involve multiplescale processes Simpli ed models are needed to capture essential features of the in uence of the stirring on the reaction kinetics Such a model was rst introduced in 45 They replaced the twodimensional problem of react ing tracers by a onedimensional one of the form a a 32 3 101 MEC Dxa xzcx1cxgtkxgt 1 for n reacting tracers cl with diffusion coef cients D1 reac tion rates kl and stirring rate A single reacting tracer with fcc and fccl 0 was studied in 45 but the idea has been taken up by several authors and was applied to more interacting tracers in bistable and excitable media in Corresponding author Electronic address gottwaldmathsusydeduau 15393755200571606620172300 0662011 PACS numbers 8240Bj 8240Ck 0545a 4752j several physical chemical and biological contexts 13718 The phenomenological model I can be justi ed by the fol lowing considerations The chaotic advection causes la ments to be stretched in one direction and compressed in another In the stretched direction the concentration is ho mogenized and gradients along the laments can be ne glected This motivates a onedimensional reduction for the concentration in the direction transverse to the lament sub ject to the effect of stirring and compression The parameter can be thought of as the Lagrangian mean strain in the contracting direction and is given by the absolute value of the negative Lyapunov exponent For a different approach to this problem see 19 In 13718 it was numerically shown that the behavior of the onedimensional lament model I qualitatively de scribes the behavior of the corresponding full 2D reaction advectiondiffusion system In particular a saddle node bi furcation was observed The saddle node can be phenomenologically understood as the competition of stir ring and reaction If the stirring is too strong ie it occurs on a faster time scale than the reaction the laments become thin and in the case of a closed ow soon cover the whole uid container or in the case of an open ow leave the uid container Consequently perturbations are either carried out of the container or laments are too thin to cause spread of reaction In some cases an asymptotic theory could be developed for slow stirring rates far away from the saddle node 14 However the bifurcation point and the pulse be havior close to the saddle node have not been previously described to our knowledge We use a nonperturbative non asymptotic technique developed for excitable media in 20 to describe the behavior near the saddle node We consider a bistable and an autocatalytic system and determine the criti cal bifurcation parameter and the pulse shape close to the bifurcation point as a function of the equation parameters Moreover we apply the same technique to describe the form of the solution far away from the bifurcation point going beyond the asymptotic analysis of 14 In the next section we present the two models under con sideration In Sec III we review the perturbation technique developed in 20 and in Sec IV we show results of our perturbation technique for the models presented in Sec II close to the saddle node bifurcation and compare with nu merical results In Sec V we nd an approximate solution for the front solutions far away from the bifurcation point 2005 The American Physical Society S N MENON AND G A GOTTWALD II THE MODELS We use two different onecomponent models to illustrate our method We study the same models used in 14 Therein also the behavior of the full 2D reactionadvectiondiffusion systems for closed and open chaotic ows was investigated We follow their notation and rescale Eq 1 by introducing nondimensional variables I I and x Dx to obtain omitting the prim es 9 a 7 32 arc xaxc xz where the Damkohler number Dak measures the ratio of the time scales of uid motion and reaction Small Damkohler numbers correspond to fast stirring andor slow reaction For large Damkohler numbers the system behaves asymptotically like an unstirred system For the reaction term fc we use the Fisher KolomogorovPetrovskyPiscounoff type 2122 0 Da fc 2 a a 7 32 ato xaxc axzcDa cl c 3 This equation has two equilibrium points an unstable xed point 00 and a stable one 01 It describes the propagation of an unstable phase into a stable phase The reaction term arises naturally for autocatalytic reactions ABgt2B and was rst introduced in the context of population dynamics in 21 and in the context of combustion in 22 Equation 3 has recently been used as a caricature to model plankton blooms 4 As a second model we introduce a generic bistable model a a 32 a Icixgcg x2cDa cozccil 4 where 0 lt ozlt 1 This system has two stable xed points 00 and 01 which are separated by an unstable xed point at 004 it is well known that in the unstirred case an initial perturbation which is larger than 04 over a nite range will spread over the whole domain if 0 lt alt05 If 05 lt ozlt 1 an initial perturbation will decay to the stable state 00 For the nonstirred case both systems are well known and well described in textbooks such as 23726 The stirred cases were investigated numerically in 145 In the stirred case stationary fronts exist for large enough values of the Damkohler number for both models The existence of sta tionary fronts in systems 3 and 4 is due to a balance of the xdependent stirring and the counterpropagating fronts An initial suf ciently large perturbation seeded at x0 spreads as a front driven by its reaction kinetics and diffusion until it reaches the location x where its velocity equals the ambient spatially dependent velocity of the chaotic stirring xquot It has been observed for both models in 14 that there is a critical Damkohler number such that no stationary pulses exist for DaltDa ie when the time scale of the chaotic advection 7f 1 becomes too fast with respect to the time scale of the reaction tl k For large Damkohler numbers an asymptotic expression for the scaling of the total concen tration was developed in 14 However these techniques PHYSICAL REVIEW E 71 066201 2005 cannot describe the behavior close to the bifurcation point It is this saddle node bifurcation which we are mainly con cerned with in this work III NONPERTURBATIVE METHOD A method was developed in 20 to study critical wave propagation of single pulses and pulse trains in excitable media in one and two dimensions It was based on the ob servation that close to the bifurcation point the pulse shape is approximately a bellshaped function Numerical simulations show that this is the case for both systems 3 and 4 close to the bifurcation point at Dag Atest function approximation that optimizes the two free parameters of a bellshaped func tion ie its amplitude and its width allows us to nd the actual bifurcation point Dag and determine the pulse shape for closetocritical pulses at Damkohler numbers near Dag We note that the framework of asymptotic techniques such as inner and outer expansions where the solution is separated into a steep narrow front and a at plateau are bound to fail close to the bifurcation point as the pulse is clearly bell shaped and such a separation is not possible anymore We shall make explicit use of the shape of the pulse close to the critical point and parametrize the pulse appropriately as is done in the method of collective coordinates in the studies of solitary waves 26 We choose 0 of the general form 006 foC77 with 77wx 5 where C77 is a symmetric bellshaped function a Gauss ian for example of unit width and height and f0 is the amplitude of the pulse Numerical simulations reveal that close to the saddle node the solution is asymptotically given by a Gaussian However our result does not depend on the speci c choice of the test function and the numerical values differ only marginally when sech functions are used We re strict the solutions to a subspace of a bellshaped function C77 which is parametrized by the amplitude f0 and the inverse pulse width w These parameters are determined by minimizing the error made by the restriction to the subspace de ned by 5 This is achieved by projecting Eq 3 or Eq 4 onto the tangent space of the restricted subspace which is spanned by 303f0C and 303w 77C w This assures that the error made by restricting the solution space to the test functions is minimized We set the integral of the product of Eq 3 or Eq 4 with the basis functions of the tangent space over the entire 77 domain to zero This will lead to algebraic equations for the amplitude f0 and the inverse pulse width w and also yield the critical Damkohler number Dag Moreover for the solution at large Damkohler numbers far way from the bifurcation point where the solution takes the shape of a well de ned front we may use a superposition of tanh functions for the test function C77 Here the free parameters are the inverse width of the interface and the total width of the front We can apply the same technique to de termine these two free parameters This will be done in Sec V 0662012 BIFURCATIONS IN REACTIONDIFFUSION SYSTEMS 08 02 93 20 10 H 10 20 30 FIG 1 The steady solutions of the autocatalytic reaction for logarithmically spaced values of Da between DaDac and Da100 IV BEHAVIOR CLOSE TO THE SADDLE NODE BIFURCATION In this section we apply the technique described in the previous section to describe the behavior near the saddle node bifurcation where the solution is well approximated by a bellshaped function with two free parameters namely the amplitude f0 and the inverse pulse width w This is a purely numerical observation and has no further analytical justi ca tion A Autocatalytic system We rst investigate the autocatalytic system 3 As has been observed numerically in 14 steady solutions to the onedimensional problem can be obtained for values of DagtDac As we approach the bifurcation point the ampli tudes of the solutions to the autocatalytic reaction decrease to zero see Fig 1 Conversely with increasing Damkohler number the pulse Width increases and the maximal amplitude saturates around Cx 1 Here the solution is a regular front solution with a well de ned plateau and a narrow steep front We are interested in steadystate solutions and set ac at0 in Eq 3 We obtain the ordinary differential equa tion 13 C1 fC 0 6 6772 a 0 a where 77wx As described in Sec III we need to project Eq 6 onto the tangent space of the restricted solution submani fold We require ltw2C7777 72C77Da 0 w2 ltw2C nCDa C1 foCgtlnCgt 0 8 where the angular brackets indicate integration over the whole 7 domain Using CWCgt ltC 7gt2lt7CWC7gt and nCnC C H a1 we can simplify the set of equa tions to get an expression for the amplitude of the form PHYSICAL REVIEW E 71 066201 2005 04 035 03 025 a X 08 06 04 02 0 05 35 b FIG 2 Comparison of numerical simulations of our analytical results continuous lines with the autocatalytic model Eq a Pulse Cx at Da135 b Pulse amplitude f0 versus Damkohler number Da The continuous line is our analytical result for the stable branch of the saddle node bifurcation ltCZgt4D31 2mm 2 Li C3 5Da fo Choosing a Gaussian test function C exp 772 this reduces further to f 1g Dal 9 0 5 Da This immediately yields the critical Damkohler number Dac 1 which is veri ed by numerical simulation of the full autocatalytic system 3 see Fig 2b Using the result 9 for the amplitude f0 we can calculate the inverse pulse width w from either 7 or 8 We obtain w 7 12Da 10 For values of Dagt 35 Eq 10 yields purely imaginary val ues indicating that our method breaks down and that at these 0662013 S N IVIENON AND G A GOTTWALD 10 FIG 3 The steady solutions of the bistable reaction with a 02 for logarithmically spaced values of Da between DaDac and Da1000 Damkohler numbers the solution cannot be approximated by a bellshaped function anymore We note that the solution saturates to become a frontlike solution see Fig 1 at Dam 10 However the solution loses its bellshaped charac ter before that saturation point Figure 2 shows a comparison of our analytical results 9 and 10 with numerical simulations of Eq 3 The analyti cal results for the amplitude t progressively better as we approach the saddle node corresponding to the fact that the solution is well approximated by a bellshaped function the closer it is to the saddle node B Bistable system We can apply the same methodology used in Sec IV A to the bistable system 4 The steady solutions of the bistable system have the same behavior as those in the autocatalytic system 3 Close to the bifurcation point at Dac the solution takes the form of a bellshaped function see Fig 3 whereas the solution approaches a front solution for higher values of the Damkohler number as is evidenced in Fig 3 As in Sec IV A we look at stationary front solutions in the study of 4 and consider d2C dC WZW7IEID3CCYf0CUOC10 11 Integrating the product of Eq 11 with C and with 7 6C 67 over the 7 domain leads to expressions for the amplitude f0 and the inverse width w We obtain a quadratic equation for the amplitude Af3BfoC0 12 where as before the coef cients can be obtained explicitly for a speci c choice of test function Choosing a Gaussian test function we have PHYSICAL REVIEW E 71 066201 2005 3 51 14 B a C x21Daa 3J5 Da 39 This yields two solutions for the amplitude f0 one corre sponding to a stable branch and one corresponding to an unstable branch These two branches collide at the critical Damkohler number and disappear via a saddle node bifurca tion An expression for the critical Damkohler number for any given value of a can be obtained from Eq 12 with the condition BZ4AC0We nd that 1 th 25 W1 2 q w 2 a q 81 This poses an upper bound for a 1 2q x1 4q a max which is approximately amaX04744 Hence the chaotic stining changes the Maxwell point which in the nonstirred case is at a05 As in Sec IVA the inverse width can be calculated as well In Fig 4 we show a comparison of our analytical results 12 and 13 with numerical simulations of Eq 11 In Fig 4 we see that the correspondence of our analytical results with the numerical simulation of the full system 11 is much better for the unstable branch than for the stable branch As a matter of fact the unstable solutions obtained by integrating 11 by means of a shooting method stay close to a bell shaped function even far away from the bifurcation point at DaDac 13 Dac V BEHAVIOR FAR AWAY FROM THE BIFURCATION In this section we apply the technique described in Sec III to describe the behavior far away from the saddle node bifurcation For large Damkohler numbers the solution is not bell shaped anymore but instead becomes a front solution with a well de ned plateau see Figs 1 and 3 Numerical simulations show that the solution in this regime is well ap proximated by a test function of the following form Cx12tanhwx 12 tanhwx v 14 Again we have two free parameters namely the total width V and the inverse interface width w This in principle pro vides two conditions by projecting onto the tangent space of the restricted solution space spanned by aC 39w and aCau In the literature of lamellar onedimensional model equa tions one encounters the following phenomenological argu ment for the location of the front We recall that a stationary front is given through a balance of the front velocity v with the velocity of the chaotic stirring x The front has a zero velocity when U x which implies v V If we now approxi mate the front velocity v by its unstirred value we can cal culate v as a function of the Damkohler number Our nonperturbative technique is able to deduce this phe nomenological formula for the front width v for the bistable cases There is strong agreement between our theory and the 0662014 BIFURCATIONS IN REACTIONDIFFUSION SYSTEMS 09 I I I 08 07 06 3 05 6 04 03 02 017 08 06 Ste 2 047 08 1O 12 14 16 18 20 b Da FIG 4 Comparison of numerical simulations of our analytical results continuous lines with the bistable model Eq 11 solved by a shooting method a Pulse Cx at Da9 b Pulse amplitude f0 versus Damkohler number Da The continuous and dashed lines show the stable and the unstable branches respectively of the saddle node bifurcation according to our analytical result 12 phenomenological formulae up to 01 Therefore for simplicity in the following sections we use the phenomeno logical argument to close the equations for the two free vari ables w and V A Autocatalytic system For the autocatalytic system 3 the front velocity for the unstirred case is given by v 2 Da provided that the initial condition is of a form such as 14 2324 Hence the phe nomenological argument yields v2D a 15 Figure 5a shows that the phenomenological argument in deed is a good approximation Equation 15 can now be used to close one of the two conditions of the projection method Without loss of gener PHYSICAL REVIEW E 71 066201 2005 0 200 400 600 800 1000 61 Da O 200 400 600 800 b Da 1000 FIG 5 Solution behavior for large Damkohler numbers Da far away from the saddle node bifurcation Numerical simulations of the full autocatalytic system 3 are depicted by stars the analytical results are depicted by continuous lines a Total width V as a function of the Damkohler number The continuous line shows the phenomenological formula 15 b Inverse interface width w as a function of the Damkohler number The continuous line shows our analytical result ality we choose the projection onto aC 39w The resulting equation is wzCW77C77Da C1 C7C7gt0 16 Here we choose 14 as a test function and express V by 15 The resulting equation for w is transcendental and we need to evaluate it numerically In Fig 5b a comparison of our result with the numerical simulation of Eq 3 is shown B Bistable system For the bistable system the front velocity for the unstirred case is given by v2Da 1 2 a 2324 Hence our phe nomenological argument now yields 0662015 S N lVIENON AND G A GOTTWALD 0 200 400 600 800 1000 60 Da 0 200 400 600 800 b Da 1000 FIG 6 Solution behavior for large Damkohler numbers Da far away from the saddle node bifurcation Numerical simulations of the full bistable system 4 are depicted by stars the analytical results are depicted by continuous lines a Total width V as a function of the Damkohler number The continuous line shows the phenomenological formula 17 b Inverse interface width w as a function of the Damkohler number The continuous line shows our analytical result 18 V 2Da a 17 Figure 6a shows again good agreement of the phenomeno PHYSICAL REVIEW E 71 066201 2005 logical argument with the actual dynamics of the full system Again Eq 17 can be used to calculate the inverse inter face width w from the condition that the projection of Eq 4 onto aC 39w vanishes This condition is given by wzCW nC Da Ca CC 17C7 0 18 where as above we use 14 as a test function and express V by 17 As for the autocatalytic system the inverse width w can only be given by numerically evaluating 18 In Fig 6b a comparison of our result with the numerical simula tion of Eq 4 is shown VI SUMMARY AND DISCUSSION We studied the solution behavior near the saddle node bifurcation which occurs in onedimensional simpli ed mod els of reactiondif 1sion equations subjected to chaotic ad vection The interplay of reaction dynamics with the chaotic stining leads to stationary fronts in the onedimensional model equation corresponding to laments with a well de ned width in the 111 twodimensional system Depending on the Damkohler number which measures the ratio of the time scales of the chaotic uid motion and the reaction ki netics the system undergoes a saddle bifurcation when the uid motion is much faster than the reaction kinetics We applied a technique originally developed for excitable media 20 to study this saddle node bifurcation We deter mined the critical Damkohler number and described the so lution close to the bifurcation point with good agreement with numerical simulations of the full partial differential equations By choosing a frontshaped test function we were able to apply the technique originally developed to study behavior close to the saddle node bifurcation to describe fully devel oped fronts far away from the bifurcation point The two conditions given by the variational technique for the two free parameters of such a stationary front ie its inverse inter face width w and its total width V accurately reproduced the numerical results Moreover we were able to reproduce a widely used phenomenological argument relating the front width to the front velocity of the unstirred case A compari son with numerical simulations justi ed our approach ACKNOWLEDGEMENTS GAG gratefully acknowledges support by the Australian Research Council DP0452147 SNM was supported by a University of Sydney Postgraduate award We thank Stephen Cox for valuable discussions 1 I R Epstein Nature London 374 321 1995 2 M A Allen J Brindley J Merkin and M J Piling Phys Rev E 54 2140 1996 3 E R Abraham Nature London 391 577 1998 4 A P Martin J Plankton Res 22 597 2000 5 P McLeod A P Martin and K J Richards Ecol Modell 158 111 2002 6 E HernandezGarcia and C Lopez Ecol Complexity 1 193 2004 7 S Edouard B Legras F Lefevre and R Eymard Nature London 384 444 1996 8 A Rovinsky and M Menzinger Phys Rev Lett 69 1193 1992 9 C Zhou J Kurths Z Neufeld and I Z Kiss Phys Rev Lett 0662016
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'