Anomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in 2-D

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1 Anomalous Scaling of Hopf Bifurcation Thresholds for the Stability of Localized Spot Patterns for Reaction-Diffusion Systems in 2-D J. C. Tzou, M. J. Ward, J. C. Wei July 5, 27 Abstract For three specific singularly perturbed two-component reaction diffusion systems in a bounded 2-D domain admitting localized multi-spot patterns, we provide a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes. The two key bifurcation parameters in each of the RD systems are the reaction-time parameter τ and the inhibitor diffusivity D. In the limit of large diffusivity D = D /ν, with D = O(), ν /logǫ and ǫ 2 denoting the activator diffusivity, a leading-order-in-ν analysis shows that the linear stability of multi-spot patterns is determined by the spectrum of a class of nonlocal eigenvalue problems (NLEPs). The specific form for these NLEPs depends on whether τ = O() or τ. For D < D c, where D c > is some critical threshold, we show from a new parameterization of the NLEP that no Hopf bifurcations leading to temporal oscillations in the spot amplitudes can occur for any O() value of the reaction-time parameter τ. This resolves a long-standing open problem in NLEP theory (see [33] [J. Wei, M. Winter Mathematical aspects of pattern formation in biological systems, Applied Mathematical Science Series, Vol. 89, Springer, (24)]). Instead, by deriving a new modified NLEP appropriate to the regime τ, we show for the range D < D c that a Hopf bifurcation will occur at some τ = τ H, where τ H has the anomalous scaling law τ H ν ε τc, for some τ c satisfying < τ c < 2. The anomalous exponent τ c is calculated from the modified NLEP for each of the three RD systems. Introduction In the large diffusivity ratio limit, two-component reaction-diffusion (RD) systems can exhibit a surprisingly wide variety of 2-D spatially localized structures including spot, stripe, and labyrinthian patterns, depending on the specific form of the nonlinear reaction kinetics (cf. [2], [5], [6]). Localized spot patterns are those for which one of the two solution components is highly localized at certain discrete points in the 2-D domain. In the absence of any O() time-scale instability of the spot profile, the spatial locations of the spots will typically evolve slowly towards some steady-state spot configuration. For a few specific RD systems in the singularly perturbed limit of a large diffusivity ratio, localized spot patterns have been shown to exhibit various types of instabilities and dynamical behaviors such as spot self-replication, temporal oscillations in the spot amplitudes, spot annihilation due to over-crowding, and slow spot drift (cf. [27], [], [28] [3], [3], [], [3], [7], [9], [8], [9], [23]). In this paper we provide a detailed study of the onset of temporal oscillations in the spot amplitudes for three prototypical singularly perturbed two-component RD systems in bounded 2-D domains, resolving a key open problem arising in the original linear stability analyses of [27], [28] [3], and [3]. In the simpler -D setting, the conditions for the onset of oscillations for localized spike solutions due to a Hopf bifurcation have been established for many RD models (see [5], [6], [25], [2], [2], [4], [2] and some of the references therein). More recently, [22] has developed a weakly nonlinear theory to unfold the Hopf bifurcation and determine whether -D spike amplitude oscillations are subcritical or supercritical. In contrast to this now well-studied -D problem, the corresponding problem for 2-D spot amplitude Dept. of Mathematics, University of British Columbia, Vancouver, BC, Canada tzou.justin@gmail.com Dept. of Mathematics, University of British Columbia, Vancouver, BC, Canada (corresponding author ward@math.ubc.ca) Dept. of Mathematics, University of British Columbia, Vancouver, BC, Canada jcwei@math.ubc.ca

2 oscillations is less well understand. By using a theoretical framework based on a nonlocal eigenvalue problem (NLEP) we will discuss and then address a key open question regarding the onset of spot amplitude oscillations for multi-spot patterns in 2-D domains. Our analysis will be applied to three prototypical systems posed in a bounded 2-D domain Ω. The first model is the standard Gierer-Meinhardt (GM) system (see [27]) v t = ε 2 v v +v 2 /u, τu t = D u u+ε 2 v 2, x Ω. (.) The second system is the non-dimensional Gray-Scott (GS) model, given in the scaling regime of [3] by v t = ε 2 v v +Auv 2, τu t = D u+ u ε 2 uv 2, x Ω. (.2) Lastly, in the scaling regime of [3], our third system is the non-dimensional Schnakenberg model given by v t = ε 2 v v +Auv 2, τu t = D u+ ε 2 uv 2, x Ω. (.3) In each of these RD systems we impose the no-flux conditions n v = and n u = on Ω. In (.2) and (.3) we refer to A = O() as the feed-rate parameter. In the limit ε, each of these three RD systems admits localized N-spot quasi-equilibrium solutions where v concentrates on a set of discrete points x,...,x N in Ω. For these solutions, we will analyze the onset of spot amplitude temporal oscillations due to a Hopf bifurcation as the reaction-time parameter τ exceeds a critical value. Our analysis will focus on the regime, considered originally in [27], [28] [3], and [3], where D = O(ν ) with ν /logε. It is on this range of D where, to leading order in ν and for τ = O(), the linear stability properties of a quasi-equilibrium N-spot pattern are characterized by the spectrum of a radially symmetric NLEP of the form L Ψ χ(τλ)w 2 wψρdρ w 2 ρdρ = λψ; Ψ () =, Ψ as ρ, (.4a) where χ(z) is called the multiplier of the nonlocal term. Here Ψ = Ψ(ρ) and the local operator L is defined by L Ψ ρ Ψ Ψ+2wΨ, (.4b) where ρ ρρ +ρ ρ. In this NLEP, w(ρ) is the unique radially symmetric ground-state solution satisfying ρ w w +w 2 =, < ρ < ; w() >, w () = ; w as ρ. (.4c) The eigenvalues λ of the NLEP (.4) in Re(λ) > correspond to instabilities in the amplitudes of the spots in the sense that, for some function Φ j (ρ) (related to Ψ(ρ)), the perturbation to the quasi-equilibrium solution v e has the form v = v e + N [ c j Φ j ǫ x x j ] e λt. (.5) j= For each of these three RD systems, in the asymptotic regime where τ = O() and D = D /ν, with D = O(), there are exactly two possible choices for the multiplier χ(z) corresponding to either synchronous or asynchronous perturbations in the spot amplitudes characterized by either (c,...c N ) = (,...,) T or N j= c j =, respectively ([27], [28] [3], and [3]). For the N distinct asynchronous modes, an analysis of the resulting NLEP for each of these three RD systems ([27], [28] [3], [3]) has proved that there is a model-specific critical value D c of D for which the multi-spot pattern is linearly stable to asynchronous perturbations if and only if D < D c. For D > D c, this linear instability of the spot amplitudes is referred to as a competition instability since, owing to N j= c j =, it preserves the average spot amplitude. From full numerical computations of the RD systems this linear instability mechanism has been found to trigger a nonlinear process through which one or more spots are annihilated in finite time (cf. [3], [7], [9]). For the synchronous mode, where c j = for j =,...,N, for each of the three RD systems the multiplier χ(z) of the NLEP is a bilinear function when D = D /ν and τ = O(). From an analysis of the NLEP (.4) with this bilinear multiplier, the synchronous mode is known to undergo a Hopf bifurcation at some value τ = τ H whenever D > D c ([27], 2

3 [28] [3], and [3]). As a result, on the range D > D c where the asynchronous modes are unstable for any τ >, an additional instability due to a Hopf bifurcation associated with the synchronous mode will occur when τ exceeds some threshold τ H. This threshold depends on D but is independent of the spatial configuration of spots. A key long-standing open problem in NLEP theory for 2-D spot patterns has been to determine whether there is a Hopf bifurcation for the synchronous mode on the range D < D c where the asynchronous modes are linearly stable (cf. [33]). For D < D c, and for each of the three RD systems above, the rigorous NLEP results in [27], [28] [3] and [3] for the synchronous mode have established that Re(λ) < when either < τ < τ 2 or τ > τ 3 for some positive constants τ 2 and τ 3. Therefore, no Hopf bifurcations occur for the synchronous mode when τ is either sufficiently small or large. Since it is unknown whether τ 3 = τ 2 or τ 3 > τ 2, a central open problem (cf. [33]) in NLEP theory for 2-D spot patterns is to determine whether a Hopf bifurcation can occur for some intermediate range of τ when D < D c. For each of the three RD systems we will use a combination of analytical and numerical methods, based on the NLEP (.4), to show that there is no O() Hopf bifurcation threshold value of τ for the synchronous mode when D < D c. By rederiving and then analyzing a new class of NLEPs appropriate to the regime D = D /ν, but with τ, we will show analytically that when D < D c there is a Hopf bifurcation threshold τ H of τ with the anomalous scaling τ H ν ε τc, for some τ c satisfying < τ c < 2. The key implication of our main result is that when D < D c /ν and for any τ = O(), quasi-equilibrium multi-spot patterns are linearly stable on O() time-scales and will therefore exhibit slow spot dynamics (cf. [3], [], []) towards a true steady-state configuration. A detailed outline of this paper is as follows: In 2 we develop a new rigorous winding number approach to analyze the NLEP (.4) when the multiplier χ(z) is a bilinear function. As discussed above, such a bilinear multiplier is relevant for analyzing the linear stability of multi-spot patterns for our three RD systems when D = D /ν and τ = O(). We derive some new specific conditions in terms of the coefficients in the bilinear multiplier that determine the number of unstable eigenvalues in Re(λ) > of the NLEP (.4). These results are then applied to our three RD systems, but are not sufficiently refined for resolving whether there is a Hopf bifurcation for the synchronous mode when D < D c. To address this issue for each of our three RD systems, in 2 we formulate and numerically implement a new parameterization of any Hopf bifurcation threshold τ H of τ for the NLEP (.4) with a bilinear multiplier. For D > D c our parameterization shows that there is a unique Hopf threshold τ = τ H on D > D c, with corresponding Hopf eigenvalue λ = iλ IH and λ IH >. As D D + c we will show that τ H + and λ IH +, and derive a new analytical scaling law for this limiting behavior. Most importantly, however, is that our new parameterization reveals that there is no O() Hopf bifurcation threshold value of τ for the synchronous mode when D < D c. Thus, we conclude from this leading-order-in-ν NLEP theory that N-spot quasi-equilibrium patterns are linearly stable for any τ = O() whenever D < D c /ν. Motivated by our observation that τ H + as D D c +, in 3 we analyze a new class of NLEPs, not considered previously in [27], [28] [3], and [3], that are relevant to the asymptotic regime D = O(ν ), τ, with τλ/d. The asymptotic derivation of these new modified NLEPs for each of the three models is given in the three appendices. By analyzing these modified NLEPs for each of the three RD systems, in we will show that when D < D c there is a Hopf bifurcation threshold τ H with the anomalous scaling τ H ν ε τc for some τ c in < τ c < 2. An asymptotic expansion for this anomalous exponent τ c and for the corresponding Hopf eigenvalue will be derived for each of three RD systems. This anomalous exponent is found to be independent of the spatial configuration of spots. Our asymptotic result for the anomalous Hopf bifurcation threshold is not uniformly valid as D Dc. For the GM (.) and Schnakenberg (.3) systems, in 4 we numerically compute the Hopf bifurcation threshold from the modified NLEP for the simple spatial configuration of a ring pattern of spots inside the unit disk. For the Schnakenberg model (.3), in 5 we extend the leading-order-in-ν NLEP theory to derive a globally coupled eigenvalue problem (GCEP) that characterizes the linear stability of multi-spot patterns, while accounting for all terms in powers of ν. Although the study of this GCEP is intractable analytically, we numerically compute from it the Hopf bifurcation threshold for a ring pattern of spots in the unit disk. The resulting asymptotic prediction of the Hopf threshold for spot amplitude oscillations is then favorably compared with full numerical results computed from the Schnakenberg model (.3) using FlexPDE6 [7]. Finally, in 6, we conclude with a brief discussion. 3

4 2 A Class of NLEPs with a Bilinear Multiplier For the parameter range where D = O(ν ) with ν /logε and for τ = O(), the linear stability on an O() time-scale of N-spot quasi-equilibrium solutions for either (.), (.2), or (.3) can be reduced to the study of an NLEP of the form (.4a), where the multiplier χ(τλ) of the nonlocal term is a bilinear function of λ of the form χ(τλ) = c+dτλ e+fτλ, (2.) where c, d, e, and f are real positive constants and τ. It is then readily shown that the discrete eigenvalues of (.4) are the roots λ of g(λ) =, defined by g(λ) = C(λ) F(λ), (2.2) where C(λ) χ(τλ) = e+fτλ c+dτλ, F(λ) ] w[(l λ) w 2 ρdρ. (2.3) w 2 ρdρ To calculate the number M of discrete eigenvalues of (.4) in Re(λ) >, we use the argument principle of complex analysis to determine the number of zeroes of g(λ) in Re(λ) >. We determine the winding number of g over the counterclockwise contour composed of the imaginary axis ir c Imλ ir c and the semi-circle Γ Rc, given by λ = R c > for arg λ π/2, and then let R c +. Since c > and d > in (2.3), C(λ) is analytic in Re(λ) >. In contrast, F(λ) is analytic in Re(λ) > except at the simple pole λ = ν >, where ν is the unique positive eigenvalue of L (see Theorem 2.2 of [4]). For R c, we have on Γ Rc that C(λ) is bounded and F(λ) = O(/ λ ). As such, there is no change in the argument of g over Γ Rc as R c. Then, by using g(λ) = g(λ), and assuming that there are no zeros of g(λ) on the imaginary axis, we get M = + π [argg] Γ I, (2.4) where [argg] ΓI denotes the change in the argument of g along the semi-infinite imaginary axis Γ I = iλ I with λ I <, traversed in the downwards direction. To calculate [argg] ΓI, where λ = iλ I with < λ I <, we decompose g(iλ I ) into real and imaginary parts, as g(iλ I ) = g R (λ I )+ig I (λ I ), to get where g R (λ I ) = C R (λ I ) F R (λ I ), g I (λ I ) = C I (λ I ) F I (λ I ), (2.5) C R (λ I ) = ec+dfτ2 λ 2 I c 2 +d 2 τ 2 λ 2, F R (λ I ) I C I (λ I ) = τλ I(fc de) c 2 +d 2 τ 2 λ 2 I [ ρwl L 2 +λ 2 ] w I 2 dρ, (2.6a) ρw 2 dρ ρw [ L 2 +λ 2 ] w I 2 dρ, F I (λ I ) λ I. (2.6b) ρw 2 dρ As established rigorously in [24], F R and F I have the following qualitative properties: F R (λ I ) κ c λ 2 I +, as λ I + ; F R(λ I ) < for λ I > ; F R ( ) =, F I (λ I ) λ I /2, as λ I + ; F I (λ I ) > for λ I > ; F I ( ) =, (2.7a) (2.7b) where κ c > is given by κ c ρ(w+ρw /2) 2 dρ ρw 2 dρ.436. (2.7c) The numerical value for κ c was obtained by computing the ground-state w(ρ) in (.4c) numerically. With these basic properties we now establish a few key results for the discrete spectrum of (.4). 4

5 Lemma 2. Let τ > and suppose that C R () = e/c < and e/c > C R ( ) = f/d. Then, M =, and so any discrete eigenvalue of (.4) satisfies Re(λ) <. In addition, if τ = and e/c <, then Re(λ) <. Proof: Let τ >. We have g() = e/c < and g(i ) = f/d >, and since fc de < we have upon using (2.7) and (2.6) that Im(g(iλ I )) = g I (λ I ) < for any λ I >. This proves that [argg] ΓI = π. Therefore, M = from (2.4), and so Re(λ) < for any eigenvalue of (.4). If τ =, we have g() = e/c < and g(i ) = e/c >, and g I (λ I ) = F I (λ I ) < for all λ I >. Thus, [argg] ΓI = π, and M = from (2.4). Hence, Re(λ) < for any eigenvalue of (.4). We now establish a condition for which there is a unique positive real eigenvalue for the NLEP (.4). Lemma 2.2 Suppose that e/c >. If τ =, then M =. If τ >, and f/d > then M = for all τ >. Proof: If e/c > and τ =, then from (2.7) and (2.6), we get g() = e/c > and g(i ) = e/c >. Since g R (λ I) >, owing to the fact that F R < from (2.7), we conclude that g R(λ I ) > for λ I >. Therefore, [argg] ΓI = and so from (2.4) we get M =. If τ >, C R () = e/c > and C R ( ) = f/d >, then since C R (λ I ) is either monotonically increasing or decreasing, we have C R (λ I ) > for all λ I. Since F R () = and F R (λ I ) is monotone decreasing on λ I >, it follows that g R (λ I ) > on λ I. As a result, [argg] ΓI =, and so from (2.4) we get M =. Lemma 2.3 Let τ > and suppose that C R () = e/c < and e/c < f/d = C R ( ). Then, either M = or M = 2. If < τ, then M =. In contrast, if τ, then M = 2 when f/d > and M = when f/d <. Proof: Since, from (2.6), C R (λ I ) is monotone increasing, while from (2.7), F R (λ I ) is monotone decreasing, then g R (λ I ) = has a unique root λ I >. We have g() = e/c < and g(i ) = f/d >. If g I (λ I) <, then [argg] ΓI = π, and M = from (2.4), while if g I (λ I) >, then [argg] ΓI = π, and M = 2. If < τ, we obtain that λ I = O(), but C I (λ I) = O(τ) from (2.6). Therefore, from (2.7) we have g I (λ I) = F I (λ I)+O(τ) <, which yields M =. Next, suppose that τ. If f/d >, then the unique root λ I > to g R (λ I ) = satisfies λ I = O(τ ). Since fc de >, we have C I (λ I) = O() > and F I (λ I) = O(τ ) from (2.6) and (2.7). This yields that g I (λ I) >, and consequently M = 2. Alternatively, if τ and f/d <, then λ I = O() and C I (λ I) = O(τ ). This yields that g I (λ I) <, so that M =. Lemma 2.3 establishes that Re(λ) < for τ whenever c/e > and d/f >. The next result establishes a precise bound on τ for which Re(λ) < holds in this parameter regime. Lemma 2.4 Suppose that e/c < and f/d <. Then, any discrete eigenvalue of (.4) satisfies Re(λ) < when τ > τ B 2(cf ed)/[3(d f) 2 ]. Proof: From equation (5.62) of [32], we have that if λ R = Re(λ), then ( ρw 3 dρ ρw 2 dρ) χ(τλ) 2 +Re [ λ(χ(τλ) ) ]. (2.8) Since ρw 3 dρ/ ρw 2 dρ = 3/2 (see (B.) and (B.5) of [24]), and upon defining z τλ, (2.8) becomes 3 χ(z) 2 + Re[ z(χ(z) )]. (2.9) 2 τ Then, upon using χ(z) = [(c e)+(d f)z]/(e+fz), we calculate from (2.9) that if z R then 3 c e+(d f)z 2 + Re[ z(c e+(d f)z)(e+f z)], 2 τ 3 c e+(d f)z τ Re[ f(d f) z z 2 +e(d f) z 2 +(c e)f z 2 +e(c e) z ]. 5

6 Therefore, if z R, then 3 2 (c e+(d f)z R) 2 + τ z Rf(d f) z 2 + τ (c e)fz2 R + τ e(d f)z2 R + e τ (c e)z R +z 2 I [ 3 2 (d f)2 ] (cf ed). (2.) τ If c/e >, z R, and d/f >, then the first term is positive, and all remaining terms in (2.) except the last one are non-negative. Hence, we have a contradiction to z R if 3(d f) 2 /2 (cf ed)/τ. Thus, if τ > τ B 2(cf ed)/[3(d f) 2 ] then Re(λ) <. 2. The Gierer-Meinhardt Model In [27] the linear stability on an O() time-scale of an N-spot quasi-equilibrium pattern for the GM model (.) was analyzed in the parameter regime D = O(ν ), with ν /logε, and where τ > satisfies τ = O() as ε. To leading order in ν, it was proved in [27] that the linear stability of an N-spot pattern on an O() time-scale is determined by the spectrum of (.4) where χ has either of the two possible forms χ a 2 +µ, χ s = 2 +µ ( ) +µ+τλ +τλ, where µ 2πND Ω. (2.) Here D Dν and Ω is the area of Ω. The multipliers χ a and χ s correspond to either asynchronous or synchronous perturbations of the spot amplitudes, respectively (see [27]). For the asynchronous mode, with multiplier χ a, we identify from (2.) that e/c = (+µ)/2 and τ =. Since e/c < iff µ <, we conclude from the second statement of Lemma 2. that Re(λ) < when < µ <. Alternatively, when µ >, the first statement of Lemma 2.2 proves that there is a unique real unstable eigenvalue in Re(λ) >. Therefore, µ = is the stability threshold for the asynchronous modes. Next, consider the synchronous mode, with multiplier χ s given in (2.). From (2.) we identify that e/c = /2 and f/d = (+µ)/2. Since e/c < and e/c < f/d for all µ >, we conclude from the first statement of Lemma 2.3 that Re(λ) < for < τ. Since f/d > iff µ >, we conclude from the second statement of Lemma 2.3 that if µ > there are exactly two discrete eigenvalues of (.4) in Re(λ) > when τ. Since the multiplier χ s in the NLEP depends on the product τλ, eigenvalues cannot enter Re(λ) > through the origin λ = as τ is increased. Therefore, by continuity in τ, there must be a Hopf bifurcation (HB) value of τ and complex conjugate eigenvalue pair λ = ±iλ IH with λ IH >, possibly non-unique, for the synchronous mode when µ >. Alternatively if < µ <, for which f/d <, we conclude from Lemma 2.3 that Re(λ) < when τ. More precisely, Lemma 2.4 yields the bound that Re(λ) < when τ > τ B = 4µ)/[3( µ) 2 ]. A key open question, unresolved from the analysis above or in [27], is whether there is a HB value of τ for the synchronous mode when < µ <. Our rigorous results above, and obtained in [27], establish that Re(λ) < on < µ < whenever τ is either sufficiently small or sufficiently large. The open question of [27] for the synchronous mode is to establish whether or not Re(λ) < for all τ > when < µ <. We examine this open question numerically, by developing a new parameterization of any HB value τ = τ H (µ), as obtained by setting g(iλ I ) = in (2.5) with (2.) for χ s, which yields ( ) (+µ) +iτλi = F R (λ I )+if I (λ I ). (2.2) 2 +µ+iτλ I We let µ = µ(λ I ) and τ H (λ I ) for λ I >, and after separating (2.2) into real and imaginary parts, we obtain the parameterization ( ) 4 F 2 + F R µ =, τ H = 2 F 2 F R, (2.3) 2F R λ I F I where F 2 = F 2 R +F2 I. Here we have labelled F R F R (λ I ) and F I F I (λ I ). 6

7 τh 5 τh µ λih.5 λih µ µ (a) τ H versus µ µ (b) λ IH versus µ Figure : Plot of the HB threshold τ H (left panel) and imaginary eigenvalue λ IH (right panel) versus µ for the synchronous mode of instability for the GM model, as obtained from (2.3). There is no HB threshold for µ <, and τ H + while λ IH + as µ +. The inserts validate the asymptotic results of (2.4) (dashed curves) for τ H and λ IH as µ +. By varying λ I on < λ I <, and numerically computing F R and F I from (2.6), we obtain the Hopf bifurcation curve in the µ > and τ > parameter plane as shown in Fig. (a). The corresponding eigenvalue, denoted by λ IH, is plotted versus µ in Fig. (b). From Fig. (a,b), we observe that there is a unique HB threshold τ H for the synchronous mode only on the range µ >, and that τ H + and λ IH + as µ +. Since our computations show that there is no HB threshold value of τ for the range < µ <, we conclude that the synchronous mode is linearly stable when µ < for any τ >, which resolves the open problem of [27]. Since the asynchronous mode is also linearly stable when < µ <, as we showed above, we conclude that, for any τ > independent of ε, an N-spot quasi-equilibrium pattern is linearly stable on the parameter regime D = O(ν ) whenever D < Ω /(2πN), where D = D /ν. The asymptotic behavior as µ + and in the shadow-limit µ of the HB threshold in Fig. is readily established. For µ +, we have that λ IH λ IH, where λ IH.59 is the unique root of F R (λ I ) = /2. The uniqueness of this root follows from the monotonicity property of F R in (2.7). This yields from (2.3) that τ H τ H 2F I (λ IH )/λ IH.563 as µ. Now for µ +, for which λ IH, we use the local behavior as λ I of F R and F I given in (2.7b) to readily derive from (2.3) that µ +λ 2 I( 2κ c )+ and τ H 2/λ 2 I. In terms of κ c.436, as defined in (2.7c), this yields the limiting behavior τ H 2( 2κ c) µ 2.2 The Schnakenberg Model, λ IH µ 2κ c, as µ +. (2.4) In [3] the linear stability of an N-spot quasi-equilibrium pattern for the Schnakenberg model (.3) was analyzed for the parameter range D = O(ν ), with ν /logε, where τ = O() as ε. To leading order in ν, the analysis in [3] proved that the linear stability of an N-spot pattern on an O() time-scale is determined by the spectrum of (.4) where there are two choices for the multiplier χ, corresponding to either asynchronous, χ a, or synchronous, χ s, perturbations of the spot amplitudes. These multipliers are given by χ a 2 +α, χ s = 2(µ+τλ) µ+(α+)τλ, where µ 2πND, α 4π2 bd N 2 Ω Ω 2 A 2. (2.5) Here D = D /ν, Ω is the area of Ω, and b ρw 2 dρ 4.935, where w(ρ) is the ground-state of (.4c). For the asynchronous mode, with multiplier χ a, (2.) yields that e/c = (+α)/2 and τ =. From Lemma 2. and Lemma 2.2 we conclude that Re(λ) < iff α < and that there is a unique unstable real eigenvalue of (.4) when α >. By using (2.5) for α, we conclude that an N-spot quasi-equilibrium pattern is linearly stable to asynchronous 7

8 perturbations in the spot amplitudes iff A > A c 2πN Ω bd, b ρw 2 dρ (2.6) τh 5 τh α λih.5 λih α α (a) τ H versus α 2 3 α (b) λ IH versus α Figure 2: Plot of the HB threshold τ H (left panel) and imaginary eigenvalue λ IH (right panel) versus α for the synchronous mode of instability for the Schnakenberg model when µ = 2, as obtained from (2.8). There is no HB threshold for α <, and τ H + while λ IH + as α +. The asymptotic results given in (2.9) (dashed curves) for τ H and λ IH as α + shown in the two inserts agree well with the numerical results from (2.8). Next, we consider the synchronous mode with multiplier χ s. From (2.5) and (2.) we identify that e/c = /2, f/d = (+α)/2, and so e/c < f/d for all α >. From Lemma 2.3 we conclude that Re(λ) < for < τ. In addition, for α > there are exactly two discrete eigenvalues of (.4) in Re(λ) > when τ, and so by continuity in τ there must be a HB value of τ, possibly non-unique, for the synchronous mode when α >. In contrast, when < α <, Lemma 2.3 yields that Re(λ) < when τ. This qualitative behavior of a HB threshold for the Schnakenberg model is identical to that for the GM model considered in (2.2). The key open question, unresolved from the analysis above or in [3], is whether there is a HB threshold for τ for the synchronous mode when < α <. To address this question numerically, we seek a parameterization of any HB value τ = τ H (α) by setting g(iλ I ) = in (2.5) with (2.5) for χ s. This involves taking the real and imaginary parts of (α+)τλ I +µ 2µ+2iτλ I = F R (λ I )+if I (λ I ). (2.7) With this characterization, and analogous to (2.3), we readily obtain the following parametric description of the HB threshold in the form α = α(λ I ) and τ H (λ I ) for λ I > : ( ) 4 F 2 + F R α =, τ H = µ(2f R ). (2.8) 2F R 2λ I F I By using the local behavior for F R and F I as λ I given in (2.7b), we calculate from (2.8) that τ H µ( 2κ c) α, λ IH α 2κ c, as α +. (2.9) By using the parameterization (2.8), in Fig. 2 we plot the HB threshold τ H and corresponding eigenvalue λ IH versus α for a fixed value µ = 2. These results show that there is a unique HB threshold for the synchronous mode only on the range α >, and that the limiting asymptotic behavior in (2.9) as α + agrees well with the numerical results from (2.8). There is no HB threshold value of τ for the range < α < for the synchronous mode, and so we conclude for this mode that Re(λ) < when < α < for any τ >. Owing to the fact that the asynchronous mode is also linearly stable when < µ <, we conclude that, for any τ > with τ = O(), an N spot quasi-equilibrium pattern is linearly stable on an O() time-scale on the parameter regime D = O(ν ) when A > A c, where A c is defined in (2.6). 8

9 2.3 The Gray-Scott Model In [3] the linear stability of an N-spot quasi-equilibrium pattern for the Gray-Scott (GS) model (.2) was analyzed for the parameter range D = O(ν ) and A = O(), with ν /logε, where τ = O() as ε. A key feature of the GS model in this regime is that quasi-equilibrium multi-spot patterns exhibit a saddle-node bifurcation structure in which the feed-parameter A is related to the common spot amplitude, characterized by the parameter S, through the nonlinear algebraic equation (see Appendix B) A µ /2 = (+µ)s + b, µ 2πND, A A S Ω Ω 2πN. (2.2) Here D = D /ν and b ρw 2 dρ 4.935, where w(ρ) is the ground-state satisfying (.4c). For µ = 2, the bifurcation diagram of S versus A is shown in Fig. 3. The saddle-node bifurcation point where da /ds = occurs at A = A f 2[b(+µ)/µ] /2 when S = S f [b/(+µ)] /2. At each point on the bifurcation curve (2.2), the linear stability of the N-spot quasi-equilibrium pattern on an O() time-scale is characterized by the spectrum of the NLEP (.4) where the multiplier χ has, once again, two distinct forms corresponding to either asynchronous or synchronous perturbations of the spot amplitudes (see Appendix B). These multipliers are χ a 2S2 b+s 2, χ s = 2S 2 (+µ+τλ) b(+τλ)+s 2. (2.2) (+µ+τλ) 4 S 3 S c 2 S f A Figure 3: The bifurcation diagram of the common spot amplitude, measured by S, versus the feed-rate parameter A from (2.2) for the GS model with µ = 2. The saddle-node point is at S = S f = [b/(+µ)] /2 and A = A f 2[b(+µ)/µ] /2, where b On the upper (solid) branch, where S > S c b, the asynchronous mode is linearly stable, and the synchronous is linearly stable for all τ >. On the middle (dashed) portion, where S f < S < S c, the asynchronous mode is unstable, and the synchronous mode undergoes a Hopf bifurcation at some τ = τ H(S ) >. On the lower (dashed-dotted) branch, where S < S f, the asynchronous mode is unstable, and the synchronous mode is unstable for any τ. Fortheasynchronous mode,withmultiplierχ a,wecompare(2.2)and(2.)toidentifythatthate/c = (b+s 2 )/[2S 2 ] and τ =. We calculate e/c > iff S < S c b. From Lemma 2. and the first statement of Lemma 2.2 we conclude that Re(λ) < when S > S c, and that there is a unique unstable real eigenvalue of (.4) whenever S < S c. As shown in Fig. 3 it is only along the solid-shaded portion of the bifurcation curve that the N-spot pattern is linearly stable to asynchronous perturbations. We refer to S c b, and the corresponding feed-rate value A = A c (2+µ) b/µ, as the competition-stability threshold. For the synchronous mode with multiplier χ s, we compare (2.2) and (2.) to identify e/c and f/d as e c = 2 + b 2S 2(+µ), f d = 2 + b 2S 2. (2.22) Since µ >, it follows that e/c < f/d. We observe that e/c < iff S > S f, whereas f/d < iff S > S c. On the lower branch (dashed-curve) of the bifurcation curve in Fig. 3 where S < S f we have < e/c < f/d. From the second 9

10 statement of Lemma 2.2 we conclude that the NLEP (.4) has a unique positive real eigenvalue for any τ. Therefore, this lower solution branch is unconditionally unstable to synchronous perturbations of the spot amplitudes. Next, along the middle branch (dashed-dotted curve) of the bifurcation curve of Fig. 3 where S f < S < S c, we have e/c < while f/d >. We conclude from Lemma 2.3 that Re(λ) < when τ and that there are exactly two discrete eigenvalues of (.4) in Re(λ) > when τ. Therefore, along this middle branch, where S f < S < S c, there must be a HB value of τ, possibly non-unique, for the synchronous mode. A parameterization for this Hopf bifurcation curve is given below in (2.24). Finally, along the upper branch (solid curve) of the bifurcation curve of Fig. 3 where S > S c, in which the asynchronous mode is linearly stable, we have e/c < f/d <. We conclude from Lemma 2.3 that Re(λ) < when < τ and when τ, with no stability information for intermediate values of τ. The primary open question, unresolved this analysis and that in [3], is to determine whether there is a HB threshold for τ for the synchronous mode along the upper solution branch when S > S c. For this range of S, we recall that the asynchronous mode is linearly stable. τh τh S λih λih S S f S S c (a) τ H versus S S f S S c (b) λ IH versus S Figure 4: Plot of the HB threshold τ H (left panel) and imaginary eigenvalue λ IH (right panel) versus S for the synchronous mode of instability for the GS model on the interval S f < S < S c when µ = 2, as obtained from (2.24). There is no HB threshold for S > S c, and τ H + while λ IH + as S S c. The asymptotic results given in (2.25b) (dashed curves) for τh and λih as S S c shown in the two inserts agree well with the numerical results from (2.24). To study this question, we numerically seek a parameterization of any HB threshold in the form τ = τ H (λ I ) and S = S (λ I ), which implicitly yields τ = τ H (S ). By setting g(iλ I ) = in (2.5), we obtain from (2.2) with χ s the complex-valued equation S(+µ+iτλ 2 I )+b(+iτλ I ) 2S 2(+µ+iτλ = F R (λ I )+if I (λ I ), (2.23) I) which yields +µ+ b S 2 ( ) b = 2(+µ)F R 2τλ I F I, τλ I S 2 + = 2F I (+µ)+2τλ I F R. By eliminating τλ I, we obtain a quadratic equation for b/s, 2 which has two possible roots. After some algebraic simplification we obtain that τ H = [ ] µ(2f R ) µ 4λ I F 2 (2F R ) 2 6F 2I (+µ), (2.24a) I S = [ ( b (2F R ) + µ ) ± ] /2 µ (2F R ) 2 6FI 2(+µ). (2.24b) By sweeping in λ I and using (2.24), in Fig. 4 we plot the HB threshold τ H and corresponding eigenvalue λ IH versus S for a fixed value µ = 2. Our results show that a unique HB threshold occurs for the synchronous mode only on the middle (dashed) branch of the bifurcation curve in Fig. 3 for which S f < S < S c. There is no HB value of τ for the synchronous mode on the upper branch where S > S c. As a remark, in our sweep in λ I using (2.24), both choices of

11 the sign in (2.24) were needed to cover the full range S f < S < S c, with the discriminant of the square root vanishing at the intermediate value λ I.4962 when µ = 2. Finally, we use (2.24) to determine the limiting behavior as S S + f and as S Sc of the HB threshold τ H, as observed in Fig. 4(a). To calculate the limiting behavior of τ H as S S + f, we substitute the local behavior for F R and F I as λ I from (2.7b) into (2.24a) and choose the minus sign. Then, by using the asymptotic estimate x x 2 h h/(2x) for h and x >, with x = O(), we obtain [ τ H = lim λ I 2λ 2 µ ( ] [ ] 2κ c λ 2 ) I µ 2( 2κ c λ 2 2 I) 4λ 2 4λ 2 I (+µ) = lim I(+µ) I λ I 2λ 2 I 2µ ( 2κ c λ 2 ). I This yields that τ H (+µ) µ, as S S + f b +µ. (2.25a) For µ = 2, this yields τ H.5 as S S + f, and is consistent with the results in Fig. 4(a). Next, we consider the limiting behavior for τ H and λ IH as S Sc. We use the local behavior for F R and F I as λ I in (2.24b) with the plus sign and in (2.24a) with the minus sign. This leads to b/s 2 +λ 2 ( I 2κc +µ ) and τ H µ/λ 2 I. This yields the limiting behavior b S 2 λ IH S 2( 2κ c +µ ), τ H S2 [+µ( 2κ c )] b S 2, as S Sc. (2.25b) In the inserts of Fig. 4, this limiting behavior in (2.25b) is favorably compared with numerical results computed from the parameterization (2.24). 3 Anomalous Scaling of the Hopf Bifurcation Threshold The central result of 2 for the GM, Schnakenberg, and GS models is that, for the range D = D /ν and τ = O(), there is no synchronous oscillatory instability of the spot amplitudes for an N-spot quasi-equilibrium pattern in the parameter range where the asynchronous mode is linearly stable. However, the key limitation on the NLEP analysis in 2 is that the bilinear multiplier χ in (2.) is valid only under the assumption that D = O(ν ) and τ = O(). Since the HB threshold τ H was found in 2 to become unbounded at the edge of the instability threshold for the asynchronous mode, this suggests that we need to re-derive the appropriate form of the NLEP and its multiplier for the case where τ can be asymptotically large as ε. By studying this modified NLEP problem we will show that there is a HB theshold τ H, for which τ H as ε, in the parameter range where the asynchronous mode is linearly stable. 3. The Gierer-Meinhardt Model As shown in Appendix C, the modified NLEP problem for Ψ = Ψ(ρ), valid for D = D /ν and arbitrary τ, is L Ψ σ j (λ)w 2 wψρdρ w 2 ρdρ = λψ, Ψ as ρ, (3.a) where the local operator L is defined in (.4b), and where w(ρ) is the ground-state solution satisfying (.4c). In (3.a), there are N choices of the multiplier σ j (λ), given by σ j (λ) = 2 +µ (+2πνκ λ,j), j =,...,N ; µ 2πND. (3.b) Ω Here D = D /ν, and κ λ,j are the eigenvalues of the λ-dependent Green s matrix G λ given by G λ v j = κ λ,j v j, j =,...,N. (3.2)

12 The entries of this Green s matrix are obtained from the reduced-wave Green s function G λ (x;x i ), defined by G λ θλg 2 λ = δ(x x i ), x Ω; n G λ =, x Ω, (3.3a) G λ (x;x i ) = 2π log x x i +R λ,i +o(), as x x i, (3.3b) where R λ,i is the regular part of G λ (x;x i ) at x = x i. Here θ 2 λ ν(+τλ)/d. In (3.2), and in terms of the spatial configuration x,...,x N of the localized spots, the entries of the symmetric N N Green s matrix G λ are (G λ ) ij = G λ (x j ;x i ), i j; (G λ ) jj = R λ,j. (3.4) We first show how to recover the NLEP for the GM model given by (.4a), where the two multipliers are given in (2.). For τ = O(), we readily calculate from (3.3) that for ν G λ = D ν(+τλ) Ω +G D +O(ν), R λ = ν(+τλ) Ω +R +O(ν), (3.5) where G is the Neumann Green s function with regular part R, satisfying (A.4). Thus, for ν, the Green s matrix G λ is D N G λ = ν(+τλ) Ω E +G +O(ν), E = N eet, (3.6) where e = (,...,) T. Now since Ee = e and Eq j = for j = 2,...,N, where q T j e =, the eigenvalues κ λ,j of G λ are κ λ, D N/[ν(+τλ) Ω ] and κ λ,j = O() for j = 2,...,N. In this way, we get 2πνκ λ, µ +τλ ; 2πνκ λ,j = O(ν), for j = 2,...,N. (3.7) Upon substituting (3.7) into (3.b) we recover the multipliers for the synchronous and asynchronous modes given in (2.) for the case where D = D /ν and τ = O(). In this parameter regime, the N modes v j = q j for j = 2,...,N, correspond to asynchronous perturbations in the spot amplitudes, which conserve the sum of the spot amplitudes via q T j e =, while v = e is the mode for which this perturbation is synchronous. Our main focus here is to consider the new limit where τ so that ( + τλ)ν/d in (3.3). In this limit, G λ (x;x i ) decays rapidly away from x i so that, except within a thin boundary layer near Ω, we have G λ (x;x i ) 2π K (θ λ x x i ), θ λ (+τλ)ν D. (3.8) Here we must specify the principal branch of the square root so that G λ (x;x i ) decays exponentially away from x i. By using the well-known behavior K (z) logz + log2 γ + O(z 2 logz), where γ is Euler s constant, we calculate that the regular part of (3.8) is independent of x i, and as x x i G λ (x;x i ) 2π log x x i +R λ ; R λ 2π ( 2 log(ν(+τλ))+log (2 D ) γ ). (3.9) When θ λ, the off-diagonal entries of the Green s matrix are exponentially small, and satisfy ( ) (G λ ) i,j = O x i x j /2 e θ λ x i x j, for x i x j = O(), (3.) since Re(θ λ ) >. Therefore, when θ λ, we have that G λ is, asymptotically, a multiple of the identity, given by G λ R λ I. As a result, the eigenvalues of G λ in (3.2) are, asymptotically, the common value κ λ,j R λ for j =,...,N. Weremarkthatinthisregimewhere θ λ, thetemporaloscillationsofthespotamplitudescannolongerbeclassifiedas being either synchronous or asynchronous. In fact, since G λ is, asymptotically, a multiple of the identity, the perturbations of the spot amplitudes, characterized by the vectors v j for j =,...,N, now span all of R N. 2

13 A key observation is that in order to make νκ λ,j = O() in the multiplier (3.b) of the NLEP (3.a) we need to allow for τ and introduce a re-scaling of this parameter in terms of ε. This is done by defining a new parameter τ c > via τ ε τc ν, τ c >. (3.) When τ c >, we have τ, with the scaling being anomalous in ε. However, from the self-consistency condition of (A.) of Appendix C we require that ε 2 τλ/d. With (3.) and D = D /ν, this requirement is satisfied provided that τ c is not too large, i.e. that ε 2 τc λ D. (3.2) By combining (3.) with (3.9), we calculate that 2πνκ λ,j = τ c 2 +νk ε, K ε )+log (2 ) 2 log(λ+νετc D γ, (3.3) so that the N multipliers in (3.b) collapse to a common multiplier when θ λ, given by σ j (λ) = σ gm (λ) 2 ( τ ) c +µ 2 +νk ε. (3.4) The discrete eigenvalues of the modified NLEP (3.a) with multiplier (3.4) are the roots λ of g gm (λ) =, where g gm (λ) (+µ) 2 ] ( τ ) w [(L c 2 +νk λ) w 2 ρdρ ε F(λ), F(λ). (3.5) w 2 ρdρ We now determine a HB threshold value of τ c for the modified NLEP (3.a) with multiplier σ gm in (3.4), by examining whether g gm (λ) = in (3.5) can have any purely complex conjugate roots. To motivate the analysis below we first neglect νk ε in (3.5) and readily conclude from Lemmas 2. and 2.2 that Re(λ) < iff σ gm >, which holds when τ c < µ. When τ c = µ, we have λ =. We observe that τ c µ > on the parameter range < µ < where the asynchronous mode in 2. was found to be linearly stable and where the synchronous mode had no HB threshold when τ = O(). With this motivation, we fix µ in < µ < and use (3.5) to determine a HB threshold where τ c is near the critical value µ and where λ = iλ I satisfies < λ I. To do so, we first separate (3.5) into real and imaginary parts to get τ c ( µ) = (τ c 2)[ Re(F(iλ I ))]+2νRe[K ε F(iλ I )], (3.6a) (τ c 2)λ I = 4νIm[K ε F(iλ I )], (3.6b) where K ε is defined in (3.3). In (3.6) we then use (2.7) to express the local behavior of F(iλ I ) as F(iλ I ) + iλ I 2 κ cλ 2 I +, for λ I, (3.7) where κ c is defined in (2.7c). By using (3.7), we will show for ν that (3.6) has a root with τ c ( µ) νlogν+ and λ I = O(ν). For < µ <, so that τ c >, and with λ I = O(ν), we estimate from (3.3) that K ε = K +O(νε τc ), where K = iπ 4 2 logλ I +β k, β k log (2 ) D γ. (3.8) Therefore, in (3.3) and in (3.6) we can replace K ε with K when τ c >. By asymptotically calculating the root of (3.6) with (3.7) and (3.8) when ν, we obtain the following main asymptotic result characterizing the HB threshold for a multi-spot pattern for the GM model (.) in the regime where D = D /ν and µ = 2πND / Ω < : 3

14 Proposition 3. Consider an N-spot quasi-equilibrium pattern for the GM model (.) in the regime where D = D /ν with ν. Define µ 2πND / Ω and suppose that < µ <. Then, then the modified NLEP (3.) has a Hopf bifurcation, corresponding to temporal oscillations in the spot amplitudes, when τ = τ H and λ = ±iλ I, where λ I. For ν, the asymptotic expansion of this HB threshold and Hopf frequency is given by τ H ν ε τc, τ c = ( µ) νlogν +τ c ν +τ c2 ν 2 + ; λ iν ( λ I +ν 2 λ I2 + ), (3.9) where the coefficients in this expansion are defined by τ c = 2β k logλ I, τ c2 = π 2 4(+µ) ( 4κ c), λ I = Here β k and κ c are defined in (3.8) and (2.7c), respectively. π (+µ), λ I2 = π 3 ( ) (+µ) 3 4 2κ c. (3.2) τc τh µ λih µ µ (a) τ H versus µ (b) λ IH versus µ Figure 5: Left panel: HB threshold τ H (solid curve) versus µ for a one-spot solution to the GM model (.) centered at the origin of the unit disk with ν =., as computed from (3.23). The dot-dashed curve is the HB threshold for the NLEP bilinear multiplier, obtained from (2.3). In the insert, τ c νlog(τ Hν), as computed from (3.23), is seen to compare well with the asymptotic result in (3.9) on µ <, except near µ =. Right panel: λ IH versus µ from (3.23) (solid curve), from the NLEP theory with bilinear multiplier from (2.3) (dot-dashed curve for µ > ), and from the asymptotic result in (3.9) (dot-dashed curve for µ < ). Proof: We substitute (3.7) and (3.8) into (3.6), and use Re(z z 2 ) = Re(z )Re(z 2 ) Im(z )Im(z 2 ) together with Im(z z 2 ) = Re(z )Im(z 2 )+Re(z 2 )Im(z ), to get τ c ( µ) (τ c 2)κ c λ 2 I +2νRe(K )( κ c λ 2 I) Im(K )νλ I, (3.2a) [ ] λi (τ c 2)λ I 4ν 2 Re(K )+Im(K )( κ c λ 2 I), (3.2b) where we calculate from (3.8) that Im(K ) = π/4 and Re(K ) = β k 2 logλ I. Then, since λ I, from the first and second equations in (3.2) we get τ c µ and (τ c 2)λ I 4νIm(K ), respectively. This second expression yields λ I νπ/(+µ). To systematically determine the higher order correction terms we write λ I = ν λ I, so that (3.2) becomes τ c ( µ) = νlogν νlog λ I +2νβ k +(τ c 2)κ c ν 2 λ2 I + π 4 ν2 λi +O(ν 3 logν), (τ c 2) λ I = π + λ [ ] I νlogν νlog λ I +2νβ k +πκ c λ2 Iν 2 +. (3.22a) (3.22b) Now define τ c µ and λ I π/(+µ). From (3.22a) we get τ c = τ c νlogν +O(ν). Next, we expand τ c as in (3.9) and λ I as λ I = λ I +νλ I +ν 2 λ I2 +, and substitute into (3.22). By comparing the O(ν) terms in (3.22a) we get that τ c = 2β k logλ I. From the O(ν) terms in (3.22b), we conclude that (τ c 2)λ I = λ I (2β k logλ I τ c ). 4

15 By using our expression for τ c we get that λ I =. Then, by using log λ I = log λ I +O(ν 2 ), we obtain from the O(ν 2 ) terms in (3.22) that τ c2 and λ I2 satisfy (τ c 2)λ I2 +τ c2 λ I = πκ c λ 2 I, τ c2 = (τ c 2)κ c λ 2 I + π 4 λ I. This determines τ c2 and λ I2 as written in (3.2). This completes the derivation of (3.9). We now make a few remarks regarding this main result. We first observe that since < µ <, our scaling threshold for τ c and λ I satisfies the required consistency condition (3.2) for any < µ <. Secondly, since λ I, the temporal oscillations of the spot amplitudes for the Hopf bifurcation are of low frequency. Next, we observe that in the regime where τ the HB threshold for the spot amplitudes is, asymptotically, independent of the spatial configuration of the spots and depends only on the number of spots. This qualitative feature that the HB threshold is independent of the spot locations was also found to hold for the conventional NLEP analysis in 2. for the regime where D = D /ν and τ = O(). Finally, our analysis of the anomalous HB threshold in the regime where < µ < is not uniformly valid in the limit µ where τ c +. When µ, we will show in 4 that the spatial configuration of the spots is important for determining the HB threshold. In Fig. 5 we illustrate our main result for the HB threshold and eigenvalue for the special case of a one-spot solution centered at the origin of the unit disk with ν =.. From the modified NLEP (3.), and by calculating the reduced-wave Green s function (3.3) analytically, the HB threshold τ H and λ IH is the root of Re(κ λ, )+ 2πν ( (+µ)f R 2(F 2 R +F2 I ) ) =, Im(κ λ, )+ (+µ)f I 4πν (FR 2 =, (3.23a) +F2 I ) where κ λ R λ, is given explicitly in terms of modified Bessel functions as R λ, = [ 2 (2 2π log(ν(+τλ))+log ) ] D γ K (θ λ ) 2π I (θ λ), θ λ ν(+iτλ IH ) D, (3.23b) where γ is Euler s constant. We remark that the term in (3.23b) involving modified Bessel functions represents the effect of the finite domain. In Fig. 5 we compare the HB threshold and eigenvalue as computed from (3.23), from the NLEP theory with bilinear multiplier in (2.3) on µ >, and from the new asymptotic scaling law in (3.9) valid on µ <. We observe from this figure that this new scaling law accurately determines the HB threshold and eigenvalue on < µ <, except near µ =. 3.2 The Schnakenberg Model An analysis similar to that in 3.2 can be done for the Schnakenberg model (.3). As shown in Appendix A, the modified NLEP problem, valid for D = D /ν and for arbitrary τ, is (3.a) where the N choices for the multiplier of the NLEP are σ j (λ) = 2(+2πνκ λ,j) +α+2πνκ λ,j, j =,...,N ; α 4π2 bd N 2 Ω 2 A 2. (3.24) Here Ω is the area of Ω, and b ρw 2 dρ 4.935, where w(ρ) is the ground-state satisfying (.4c). In (3.24), the κ λ,j for j =,...,N are the eigenvalues of the Green s matrix in (3.4) where the entries of this Green s matrix are now defined in terms of the Green s function G λ (x;x i ) satisfying G λ ντλ D G λ = δ(x x i ), x Ω; n G λ =, x Ω, (3.25a) G λ (x;x i ) = 2π log x x i +R λ,i +o(), as x x i. (3.25b) We recall from the conventional NLEP analysis in 2.2 for the regime where D = D /ν and τ = O() that, for any fixed µ > and α <, the asynchronous mode is linearly stable while the synchronous mode is also linearly stable for any 5

16 τ >. We showed that the synchronous mode does undergo a Hopf bifurcation when α >, with the HB threshold value of τ tending to infinity and the corresponding HB frequency tending to zero as α +. By analyzing the modified NLEP (3.) with multiplier (3.24), we will show that there is a HB threshold τ H in the regime where α <, with τ H + as ε. We use the same approach as in 3.. We assume that τλν/d so that G λ (x;x i ) 2π K (ω λ x x i ), where ω λ is the principal branch of ω λ τλν/d. When x i x j = O() for i j, we get G λ R λ I, where R λ is given in (3.9) upon replacing (+τλ) with τλ. Then, upon introducing the scaling law τ = ν ε τc, we obtain that 2πνκ λ,j = τ c 2 +νk, K 2 logλ+β k, β k log (2 ) D γ, (3.26) where γ is Euler s constant. Therefore, when τ, the N multipliers in (3.24) collapse to a common multiplier σ j (λ) = σ sc (λ) 2( τ c/2+νk ) (+α τ c /2+νK ). (3.27) The discrete eigenvalues of the modified NLEP (3.a) with multiplier (3.27) are the roots of g sc (λ) =, where g sc (λ) (+α τ c/2+νk ) 2( τ c /2+νK ) F(λ), F(λ) ] w[(l λ) w 2 ρdρ. (3.28) w 2 ρdρ We now examine whether (3.28) can have any purely complex conjugate roots. If we neglect νk in (3.28), we conclude from Lemmas 2. and 2.2 that Re(λ) < iff σ c >, which holds when τ c < 2( α). When τ c = 2( α), we have λ = is a root of (3.28). Notice that τ c = 2( α) > on the parameter range < α < where the asynchronous mode in 2.2 was found to be linearly stable and where the synchronous mode had no HB threshold when τ = O(). For a fixed α in < α < the asymptotic analysis to characterize the HB threshold and frequency when τ c is near the critical value 2( α) parallels that in 3.. We let λ = iλ I and upon setting g sc (iλ I ) = in (3.28), we obtain that +α τ c 2 (2 τ c)f(iλ I ) = νk [2F(iλ I ) ]. (3.29) We then substitute the local behavior (3.7) for F(iλ I ) into (3.29) and take real and imaginary parts of the resulting expression. This yields that α + τ c 2 +(2 τ c)κ c λ 2 I νre(k ) ( 2κ c λ 2 ) I νλi Im(K ), (3.3a) ( τc ) 2 λ I νim(k ) ( 2κ c λ 2 I) +νλi Re(K ), (3.3b) where Im(K ) = π/4 and Re(K ) = β k 2 logλ I. Since λ I, the leading-order terms in (3.3) are τ c τ c 2( α) and (τ c /2 )λ I νπ/4, which yields λ I πν/(4α). Since λ I = O(ν) and τ c = 2( α) satisfies < τ c < 2 on < α <, the required consistency condition in (3.2) is always satisfied. By introducing an asymptotic expansion to systematically calculate the correction terms to this leading-order HB threshold and frequency, as in the proof of Proposition 3., we obtain the following main result: Proposition 3.2 Consider an N-spot quasi-equilibrium pattern for the Schnakenberg model (.3) in the regime where D = D /ν with ν. Suppose that the feed-rate parameter A satisfies A > A c 2πN Ω bd, where b ρw 2 dρ 4.935, and w(ρ) is the ground-state satisfying (.4c). Then, the modified NLEP (3.a) with multiplier (3.24) has a Hopf bifurcation, corresponding to temporal oscillations in the spot amplitudes, when τ = τ H and λ = ±iλ I, where τ H ν ε τc, τ c = 2( α) νlogν +τ c ν +τ c2 ν 2 + ; λ iν ( λ I +ν 2 λ I2 + ). (3.3) Here the coefficients are given explicitly by τ c = 2β k logλ I, τ c2 = π2 8α ( 2κ c), λ I = π 4α, λ I2 = π3 6α 3 Here α, β k, and κ c are defined in (3.24), (3.26), and (2.7c), respectively. ( ) 4 κ c. (3.32) 6

17 τc.5.5 τh 5.5 α λih α (a) τ H versus α α (b) λ IH versus α Figure 6: Left panel: HB threshold τ H (solid curve) versus α for a one-spot solution to the Schnakenberg model (.3) centered at the origin of the unit disk with ν =.25 and µ = 2, as computed from (3.33). The dot-dashed curve is the HB threshold for the NLEP with a bilinear multiplier (2.8). In the insert, τ c νlog(τ Hν) from (3.23) agrees well with the asymptotic result in (3.3) on α <, except near α =. Right panel: λ IH versus α from (3.33) (solid curve), from the NLEP with a bilinear multiplier (2.8) (dot-dashed curve for α > ), and from the asymptotic result (3.3) (dot-dashed curve for α < ). In Fig. 6 we plot the HB threshold and eigenvalue for a one-spot solution centered at the origin of the unit disk with ν =.25 and µ = 2, for which D =. In place of (3.23a), the HB threshold τ H and λ IH for the modified NLEP (3.a) with multiplier (3.24) is the root of Re(κ λ, )+ 2πν ( ) α( 2F R ) + ( 2F R ) 2 +4FI 2 =, Im(κ λ, )+ πν αf [ I ( 2F R ) 2 +4FI 2 ] =, (3.33) where κ λ R λ, is given in (3.23b), where θ λ is now defined by θ λ iντλ IH /D. In Fig. 6 we compare the HB threshold and eigenvalue, as computed from (3.33), with that from the conventional NLEP theory (2.8), valid on α >, and with that from the new asymptotic scaling law in (3.3), valid on α <. 3.3 The Gray-Scott Model A similar analysis can be done for the GS model (.2). As shown in Appendix B, the modified NLEP problem, valid for D = D /ν and for arbitrary τ, is (3.a) where the N choices for the multiplier of the NLEP are σ j (λ) = 2(+2πνκ λ,j) +b/s 2 +2πνκ, j =,...,N. (3.34) λ,j Here Ω is the area of Ω, b ρw 2 dρ where w(ρ) is the ground-state satisfying (.4c), while S parameterizes the steady-state bifurcation diagram in terms of the feed-rate parameter as in (2.2) (see Fig. 3). In (3.34), the κ λ,j for j =,...,N are the eigenvalues of the Green s matrix G λ in (3.4) where the matrix entries are defined in terms of the reduced-wave Green s function of (3.3). In our conventional NLEP analysis of 2.3 for the regime D = D /ν and τ = O() we showed that on the upper branch of the bifurcation diagram in Fig. 3, where S > S c b, the asynchronous mode is linearly stable and the synchronous mode is also linearly stable for any τ >. We showed that the synchronous mode does undergo a Hopf bifurcation when S f < S < S c (see Fig. 3), with the HB threshold value of τ tending to infinity and the corresponding HB frequency tending to zero as S S c. We now use the modified NLEP (3.) with multiplier (3.34) to show that there is a HB threshold τ H along the upper branch of Fig. 3, where S > S c b, for which τ H + as ε. We proceed as in 3. and 3.2. When τ O(ν ), so that (+τλ)ν/d, G λ (x;x i ) is approximated by the free-space Green s function in (3.8) and the Green s matrix G λ satisfies G λ R λ I, where R λ is given as in (3.9). We then 7

18 introduce τ c > via the scaling law (3.) and obtain that 2πνκ λ,j = τ c 2 +νk ε, K ε )+β 2 log(λ+νετc k, β k log (2 ) D γ, (3.35) where γ is Euler s constant. Upon substituting (3.35) into (3.34) we obtain that the N multipliers collapse to a common multiplier given by 2( τ c /2+νK ε ) σ j (λ) = σ gs (λ) (+b/s 2 τ c/2+νk ε ). (3.36) The discrete eigenvalues of the modified NLEP (3.a) with multiplier (3.36) are the roots of g gs (λ) =, where g gs (λ) ( +b/s 2 τ c /2+νK ε ) 2( τ c /2+νK ε ) F(λ), F(λ) ] w[(l λ) w 2 ρdρ. (3.37) w 2 ρdρ To determine whether (3.37) can have complex conjugate imaginary roots for some critical value of τ c, we simply observe that (3.37) is identical in form to that in (3.28) for the Schnakenberg model if we replace α and K in (3.28) with b/s 2 and K ε. Therefore, the leading-order HB threshold occurs for the GS model occurs when τ τ c 2 ( b/s 2 ). Since τ c > when S > S c b, we can use K ε = K +O(νε τc ) (see (3.8)) to approximate K ε in (3.37) by K with an asymptotically negligible error. The asymptotic analysis to calculate a refined HB threshold and frequency is then exactly the same as for the Schnakenberg model and is summarized as follows. Proposition 3.3 Consider an N-spot quasi-equilibrium pattern for the GS model (.3) in the regime where D = D /ν, with ν, and along the upper branch of the bifurcation diagram in Fig. 3 where S > b. Then, the modified NLEP (3.a) with multiplier (3.34) has a Hopf bifurcation, corresponding to temporal oscillations in the spot amplitudes with the scaling law τ H (, τ ν ε τc c = 2 b ) S 2 νlogν +τ c ν +τ c2 ν 2 + ; λ iν ( λ I +ν 2 λ I2 + ). (3.38) Here b ρw 2 dρ 4.935, where w(ρ) is the ground-state solution (.4c), while the coefficients in (3.38) are given explicitly by τ c = 2β k logλ I, τ c2 = π2 S 2 8b ( 2κ c), λ I = πs2 4b, λ I2 = π3 6 Here β k and κ c are defined in (3.35) and (2.7c), respectively. ( ) S 2 3 ( ) b 4 κ c. (3.39) The HB threshold and eigenvalue for a one-spot solution centered at the origin for the GS model (.2) is plotted in Fig. 7 when ν =.25 and µ = 2, for which D =. For the GS model, this threshold is the root of (3.33) where θ λ ν(+iτλ IH )/D, and α = b/s 2, where S > S f b/(+µ) parametrizes the upper branch of the bifurcation diagram of Fig. 3. In Fig. 6 we compare these HB threshold values with that from the conventional NLEP theory (2.24), valid on S f < S < S c b, and with that from the asymptotic scaling law in (3.38), valid for S > S c. 4 Refined HB Thresholds for Multi-Spot Patterns The leading-order conventional NLEP analysis of 2 for the parameter regime D = D /ν and τ = O() showed that the HB threshold depends only on the number of spots, and is independent of their spatial arrangement in the domain. This qualitative property of the HB threshold also occurs in the limit τ for the modified NLEP analysis of 3, which leads to the new scaling laws in Propositions In this section, we numerically study the modified NLEPs of 3 for the parameter range where the decay rate θ λ in the reduced-wave Green s function (3.3) is O(). It is in this transition regime where the HB threshold depends on the spot locations, and where there can be N distinct HB values corresponding to the distinct eigenvalues of the Green s matrix (3.4). 8

19 τh 5 τc S λih S f S S c (a) τ H versus S S f S c S (b) λ IH versus S Figure 7: Left panel: HB threshold τ H (solid curve) versus S on the range S > S f = b/(+µ) (see Fig. 3) for a one-spot solution to the Gray-Scott model (.2) centered at the origin of the unit disk with ν =.25 and µ = 2, as computed from (3.33) in which α = b/s 2. The dot-dashed curve is the HB threshold for the conventional NLEP (2.24). In the insert, τ c νlog(τ Hν) from (3.33) is in close agreement with the asymptotic result in (3.38) on S > S c b. Right panel: λ IH versus S from (3.33) (solid curve), from the NLEP with a bilinear multiplier (2.24) (dot-dashed curve for S f < S < S c), and from the asymptotic result (3.38) (dot-dashed curve for S > S c). We will illustrate our theory for a quasi-equilibrium pattern where N spots are equally-spaced on a ring of radius r, with < r <, which is concentric within the unit disk, for which the spot centers are x j = r (cos ( ) ( )) T 2πj 2πj, sin, j =,...,N. (4.) N N For r we will take the steady-state ring radius determined in [] from the leading-order-in-ν analysis for the D = D /ν regime. In the unit disk. G λ (x;ξ) in (3.3) and its regular part are (see Appendix A. of [3]) G λ (x;ξ) = 2π K (θ λ x ξ ) 2π n= σ n cos(n(ψ ψ )) K n(θ λ ) I n(θ λ ) I n(θ λ r)i n (θ λ r ), where σ, σ n 2 for n, while x r(cosψ,sinψ) T, and ξ r (cosψ,sinψ ) T. The regular part is R λ = [ log (2 D ) γ ] 2π 2 log(ν(+τλ)) 2π n= σ n K n(θ λ ) I n(θ λ ) [I n(θ λ r )] 2, (4.2a) (4.2b) For a ring pattern, the symmetric Green s matrix G λ in (3.4) also has a cyclic structure. This Green s matrix is obtained by a cyclic permutation of its first row a λ (a λ,,...,a λ,n ) T, which is defined term-wise by a λ, R λ ; a λ,j = G λ,j G λ (x j ;x ), j = 2,...,N. (4.3) The matrix spectrum G λ v j = κ λ,j v j is readily calculated as in 6 of [8]. The synchronous eigenpair of G λ is κ λ, = N a λ,n, v = (,...,) T, (4.4a) n= while the other eigenvalues, corresponding to the asynchronous modes for which v T j v = for j = 2,...,N, are κ λ,j = N n= ( 2π(j )n cos N 9 ) a λ,n+, j = 2,...,N. (4.4b)

20 Since κ λ,j = κ λ,n+2 j for j = 2,..., N/2, there are N/2 pairs of degenerate eigenvalues for G λ. Here the ceiling function x is defined as the smallest integer not less than x. When N is even, there is an eigenvalue of multiplicity one given by κ λ, N 2 + = N n= ( )n a n+. The other eigenvectors for j = 2,..., N/2 are v j = v N+2 j = (,cos (,sin ( 2π(j ) N ( 2π(j ) N ),...,cos ),...,sin ( )) T 2π(j )(N ), N ( )) T 2π(j )(N ). Finally, when N is even, there is an additional eigenvector given by vn 2 + = (,,..., )T. By using the explicit formulae for G λ and its regular part from (4.2), the eigenvalues κ λ,j of the Green s matrix G λ are then easily computed from (4.4a) and (4.4b). N (4.4c) τh τc µ λih µ µ (a) τ H versus µ (b) λ IH versus µ Figure 8: HB threshold τ H (left panel) and eigenvalue λ IH (right panel) versus µ for a two-spot ring pattern in the unit disk for the GM model (.) when r =.452 and ν =.. The solid curves in the left and right panel are for the synchronous mode j = mode, while the dotted curves are for the asynchronous j = 2 mode, both computed from (4.5). The dot-dashed curves for µ > are from the conventional NLEP theory (2.3), while the dot-dashed curves for µ < are from the asymptotic scaling law (3.9). In the insert in the left panel we plot τ c νlog(τ Hν). The HB thresholds for the synchronous and asynchronous modes are clearly distinct on.8 < µ <, with the synchronous mode yielding the smaller of the two values for τ H. The HB eigenvalue for the asynchronous mode vanishes just beyond µ =. To determine the HB thresholds for the GM model (.) from the modified NLEP (3.), we must numerically determine the roots of Re(κ λ,j ) = ( (+µ)f ) R 2πν 2(FR 2 +F2 I ), Im(κ λ,j ) = (+µ)f I 4πν (FR 2, j =,...,N, (4.5) +F2 I ) where F R (λ I ) and F I (λ I ) are defined in (2.6). For a two-spot ring pattern with r =.452 and ν =., in Fig. 8 we plot the HB threshold values τ H and λ IH versus µ for j = (synchronous, solid curves) and for j = 2 (asynchronous, dotted curves), as computed from (4.5). Results from the conventional NLEP theory for µ > (2.3) and from the asymptotic scaling law for µ < (3.9) are indicated by the dot-dashed curves. We observe that on the range µ <, the HB thresholds for τ H and λ IH for the synchronous and asynchronous modes essentially coincide only up to around µ =.8. On the range.8 < µ <, the HB thresholds for the two modes are distinct, with the synchronous mode yielding the smaller τ H. From the right panel of Fig. 8, the HB eigenvalue for the asynchronous mode is positive for µ <, but vanishes at some critical value of µ that is slightly above µ =. This is consistent with conventional NLEP theory, in that it was shown in 2. for the regime τ = O() and D = D /ν that the asynchronous mode does not undergo a Hopf bifurcation when µ >, but is unstable due to a positive real eigenvalue. This mechanism whereby the purely imaginary pair of complex conjugate eigenvalues associated with the asynchronous mode collides at the origin λ = of the spectral plane at some critical value µ c of µ, and produces a positive real eigenvalue when µ > µ c, is similar to that found in the study of the stabiliy of GM spike patterns in -D (see Fig. 8 of [25]). 2

21 τh 3 2 τc µ λih µ µ (a) τ H versus µ (b) λ IH versus µ Figure 9: HB thresholds for the GM model (.) for a five-spot ring pattern with r =.6252 and ν =.. For N = 5, there is a HB for the synchronous mode j = (solid curves), and for two distinct asynchronous modes j = 2,3 (dotted curves), as obtained numerically from (4.5). The dot-dashed curves for µ > are from the conventional NLEP theory (2.3), while the dot-dashed curves for µ < are from the asymptotic scaling law (3.9), where we plot τ c νlog(τ Hν). The HB threshold value of τ is smallest for the synchronous mode. For each of the two asynchronous modes, the HB thresholds exist on µ > only up to some critical value near µ =, where the Hopf eigenvalue vanishes. A very similar scenario occurs for larger N. In particular, in Fig. 9 we plot the HB threshold values, as computed from (4.5), for a five-spot pattern with ν =. and ring radius r = For N = 5, we have κ λ,i = κ λ,7 i for i = 2,3, and so there are exactly two distinct asynchronous modes. From the insert in the left panel of Fig. 9, and from the right panel of Fig. 9, we conclude that the HB values of τ H and λ IH for the synchronous and asynchronous modes essentially coincide on µ <.65, where they are well-approximated by the scaling law (3.9). In addition, the conventional NLEP theory (2.3) provides a decent approximation to the HB threshold for the synchronous mode when µ >. Finally, the HB eigenvalue for each of the two asynchronous modes vanishes at some critical value of µ near µ =. A qualitatively similar behavior of the HB threshold occurs for a ring-pattern of spots for the GS model (.2) and the Schnakenberg model (.3). For illustration, we will only consider the latter model here. For the Schnakenberg model, any HB value for the modified NLEP (3.a) with multiplier (3.24) is a root of Re(κ λ,j ) = 2πν ( ) α( 2F R ) + ( 2F R ) 2 +4FI 2, Im(κ λ,j ) = πν αf [ I ( 2F R ) 2 +4FI 2 ], j =,...,N. (4.6) For the unit disk, µ = 2πND / Ω = 2ND and α is related to the feed-rate parameter A in (.3) (see (3.24)) by α = 4bD N 2 A 2, where b ρw 2 dρ In Fig. we plot the HB threshold values versus α, as computed from (4.6), for a two-spot pattern with ν =.25, ring radius r =.452, and with µ = 4, so that D =. We observe that the HB values of τ H and λ IH for the synchronous and asynchronous modes essentially coincide on α <.85, where they are well-approximated by the new scaling law (3.3). Moreover, the conventional NLEP theory, given by the parameterization (2.8), approximates well the HB threshold for the synchronous mode when α >. We observe that the HB eigenvalue for the asynchronous mode vanishes at some critical value of α slightly above α =. Finally, in Fig. we plot the HB threshold values versus α for a five-spot ring pattern with ν =.25, ring radius r =.6252, and with µ =, so that we maintain D =. The results are qualitatively identical to that discussed in the caption of Fig. 9 for a five-spot pattern for the GM model. 5 HB Thresholds for Moderately Small ν: Numerical Validation In this section we show how to accurately determine the HB threshold for a ring pattern of spots for the Schnakenberg model (.3) when ν = /logε is only moderately small. In contrast to NLEP theory, which is a leading-order-in-ν 2

22 5.5 2 τh τc α λih α (a) τ H versus α α (b) λ IH versus α Figure : HB threshold τ H (left panel) and eigenvalue λ IH (right panel) versus α for a two-spot ring pattern in the unit disk for the Schnakenberg model (.3) when r =.452, ν =.25, and µ = 4, so that D =. The solid and dotted curves, computed from (4.6), are for the synchronous (j = ) and asynchronous (j = 2) modes, respectively. The dot-dashed curves for α > and for α < are from the conventional NLEP theory (2.8) and the new scaling law (3.3), respectively. From the left panel, the HB eigenvalue for the asynchronous mode vanishes just beyond α =. τh τc α λih α (a) τ H versus α α (b) λ IH versus α Figure : HB threshold τ H (left panel) and eigenvalue λ IH (right panel) versus α for a five-spot ring pattern in the unit disk for the Schnakenberg model (.3) when r =.6252, ν =.25, and µ =, so that D =. The solid and dotted curves, computed from (4.6), are for the synchronous (j = ) and asynchronous (j = 2,3) modes, respectively. The dot-dashed curves for α > and for α < are from the conventional NLEP theory (2.8) and the new scaling law (3.3), respectively. From the left panel, the HB eigenvalue for each of the two asynchronous modes vanishes near α =. 22

23 theory, the resulting problem formulated below for the HB threshold effectively incorporates all terms in powers of ν. The formulation of this problem relies on some results in Appendix A. For a quasi-equilibrium ring pattern of N spots, and with D = D /ν, the common spot source strength is S j = S c = A ν/(2πn D ) for j =,...,N. From (A.), the common core problem near each spot is ρ V c V c +U c V 2 c =, V c() =, V c as ρ, (5.a) ρ U c = U c V 2 c, U c() =, U c S c logρ+d(s c ), as ρ. (5.b) Upon linearizing around this particular quasi-equilibrium solution, we use (A.6) and (A.) to obtain the following spectral problem (I(+νB c )+2πνG λ )C =, (5.2) where C (C,...,C N ) T and B c = B c (λ,s c ) is obtained from the BVP system ρ Φ c Φ c +2U c V c Φ c +V 2 cn c = λφ c, Φ c() =, Φ c as ρ, (5.3a) ρ N c V 2 cn c = 2U c V c Φ c, N c() =, N c logρ+b c. as ρ, (5.3b) In (5.2), G λ is the eigenvalue-dependent Green s matrix defined in (3.4) in terms of the Green s function of (3.25). For this ring of patterns, the matrix spectrum of G λ is as given in (4.3) (4.4c). In this way, we obtain from (5.2) that the discrete eigenvalues λ of the governing the linear stability of a ring pattern of spots are the union of the roots of κ λ,j = 2πν (+νb c), j =...,N, (5.4) where κ λ,j is given explicitly in terms of the first row of the Green s matrix by (4.4b). In (5.4), B c = B c (λ,s c ) is to be computed numerically from (5.3) in terms of the numerical solution to the core problem (5.). To determine the HB threshold for the j = synchronous mode, and for the remaining asynchronous modes with j >, we set λ = iλ I in (5.4) and use Newton s method on the complex-valued equation (5.4) to compute the HB values τ H and λ IH for each j =,...,N j = j = 2 j = 3 numerical.2.8 τh 2 5 λih α.2 j = j = 2 j = α (a) τ H versus α (b) λ IH versus α Figure 2: HB threshold τ H (left panel) and eigenvalue λ IH (right panel) versus α for a four-spot ring pattern in the unit disk for the Schnakenberg model (.3) when r =.5986, ε =., and D = D /ν where D = and ν = /logε. The solid curves were computed from the full asymptotic theory (5.4), which is accurate to all orders in ν. The j = mode is the synchronous mode, while the j = 2 and j = 3 modes correspond to asynchronous oscillations. The j = 4 curve is identical to that for j = 2. Left panel: As τ increases, the synchronous mode is the first to go unstable. The j = 3 curve ends at α.95. For α >.95, there is a real positive eigenvalue for any value of τ, corresponding to a competition instability. The open circles correspond to thresholds obtained from determining the HB threshold from full numerical solutions of the Schnakenberg PDE (.3). Right panel: the HB eigenvalue λ IH. As α approaches the competition threshold, the eigenvalue for the j = 3 mode approaches the origin along the imaginary axis. 23

24 To illustrate the theory, we let N = 4 and take r =.5986, which is the steady-state ring radius when N = 4 (cf. []). For ε =. and D = D /ν with D =, in Fig. 2 we plot the HB values, computed from (5.4), as a function of α where α 4bD N 2 A 2 with b From Fig. 2(a) we observe that the HB value of τ for the synchronous mode j = is the smallest, and so it is this mode which sets the HB threshold. The two distinct asynchronous modes have larger HB values for τ and, as expected, the thresholds for each of these modes exist only when α is less than some critical value near α =. From Fig. 2(b), we observe that the corresponding Hopf frequency vanishes at these critical values of α. Finally, we compare our asymptotic theory with results from full PDE numerical simulations of the Schnakenberg model (.3) computed using the adaptive finite element solver FlexPDE6 [7] for the specific four-spot pattern of Fig. 2 with A = For this value of A, we have α.684, and Fig. 2(a) for the j = mode yields the asymptotic prediction τ H 53. For the parameter value τ = 58, which slightly exceeds the HB prediction τ H 53, in Fig. 3 we show that a random initial perturbation of the spot amplitudes will eventually lead to synchronized spot amplitude oscillations as time increases. Since the range of these oscillations is small, the full numerical results in Fig. 4 over a much longer time interval than in Fig. 3(b) suggest that the Hopf bifurcation for this parameter set is supercritical. spot amplitude (a) four-spot ring pattern at steady-state spot spot 2 spot 3 spot t (b) spot amplitudes versus t Figure 3: Full numerical computations in the unit disk for the Schnakenberg model (.3) using FlexPDE6 [7] showing synchronous oscillations in the spot amplitudes for a four-spot ring pattern with ring radius r =.5986 and with parameters ε =., A = 22.8, τ = 58, and D = D /ν where D = and ν = /logε. This corresponds to α =.684 in Fig. 2. Left panel: when r =.5986, the four-spot ring pattern is at a steady-state configuration. Right panel: Growing temporal oscillations of the amplitudes of the four spots when τ = 58 is set slightly above the HB threshold τ H 53, as predicted by the asymptotic theory (5.4). The HB thresholds for the asynchronous modes are τ H2 32 and τ H The steady-state was perturbed by a small random perturbation. However, the oscillations quickly synchronize as predicted by the asymptotic theory, with the amplitudes of the four spots exhibiting in-phase oscillations of increasing amplitude. The angular frequency of the oscillations is approximately.3, close to the predicted value of λ IH =.334 from (5.4). 6 Discussion We have provided a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes for multi-spot patterns associated with three singularly perturbed two-component RD systems in a bounded 2-D domain. In these systems, the two bifurcation parameters are the reaction-time parameter τ and the inhibitor diffusivity D. In the limit of large diffusivity D = D /ν, with D = O(), the linear stability properties of multi-spot patterns on O() time-scales is determined by the spectra of an NLEP. For the conventional regime where τ = O(), considered originally in [27], [28] [3], and [3], from a new parameterization of the NLEP we have shown that synchronous temporal oscillations in the spot amplitudes due to a Hopf bifurcation can only occur on the range D > D c, where D c > is the competition stability threshold. For D < D c no spot amplitude oscillations occur for any τ = O(). To analyze whether a Hopf bifurcation can occur when τ and D < D c we derive a new modified NLEP appropriate to this regime by considering a new distinguished limit of the reduced-wave Green s function. From an analysis of this modified NLEP we have shown for the range D < D c that a Hopf bifurcation will occur at some threshold τ = τ H, where τ H ν ε τc, for some τ c satisfying < τ c < 2. The exponent τ c in this anomalous scaling law was calculated for 24