A TIME-DELAY IN THE ACTIVATOR KINETICS ENHANCES THE STABILITY OF A SPIKE SOLUTION TO THE GIERER-MEINHARDT MODEL

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1 A TIME-DELAY IN THE ACTIVATOR KINETICS ENHANCES THE STABILITY OF A SPIKE SOLUTION TO THE GIERER-MEINHARDT MODEL NABIL T. FADAI, MICHAEL J. WARD, AND JUNCHENG WEI Abstract. We study the spectrum of a ne class of nonlocal eigenvalue problems (NLEPs) that characterize the linear stability properties of localized spike solutions to the singularly perturbed to-component Gierer-Meinhardt (GM) reaction-diffusion (RD) system ith a fixed time-delay T in only the nonlinear autocatalytic activator kinetics. Our analysis of this model is motivated by the computational study of Seirin Lee et al. [Bull. Math. Bio., 7(8), ()] on the effect of gene expression time delays on spatial patterning for both the GM and some related RD models. For various limiting forms of the GM model, e sho from a numerical study of the associated NLEP, together ith an analytical scaling la analysis valid for large delay T, that a time-delay in only the activator kinetics is stabilizing in the sense that there is a ider region of parameter space here the spike solution is linearly stable than hen there is no time delay. This enhanced stability behavior ith a delayed activator kinetics is in marked contrast to the de-stabilizing effect on spike solutions of having a time-delay in both the activator and inhibitor kinetics. Numerical results computed from the RD system ith delayed activator kinetics are used to validate the theory for the -D case.. Introduction. For activator-inhibitor to-component reaction-diffusion (RD) systems, it is a ell-knon result, originating from Turing [], that a small perturbation of a spatially uniform steady-state solution can become unstable hen the diffusivity ratio is large enough. This initial instability then leads to the generation of large-amplitude stable spatial patterns. Although this mechanism for the development of spatially inhomogeneous patterns is ell-understood, and has been applied to a broad range of specific RD systems (cf. [7]) and modeling scenarios on various spatial scales, hat is less ell-understood is the effect on pattern development of any time-delays in the reaction kinetics. Although there are no general results for the linear stability of spatially uniform steady-states under the effect of a time-delay (cf. []), hich have been applied to the Gierer-Meinhardt (GM) model ith saturated and time-delayed reaction kinetics in [], the effect of time-delays in the reaction kinetics on localized RD patterns is not nearly as ell understood. Time-delays in the reaction kinetics for modeling RD patterns on a cellular spatial scale are thought to be an important biological mechanism, as there typically exists a time-delay beteen the initiation of protein signal transduction and the time at hich genes are ultimately produced (cf. [7]). In an effort to understand the effect of such gene expression timedelays on pattern formation, various ne to-component activator-inhibitor RD models ith a fixed time delay in the reaction kinetics ere developed based on various hypothethical sub-cellular gene expression processes in [7], [5], [6], [7] (see also the survey [8]). In these studies, pattern formation aspects of the time-delayed RD models ere examined through a Turing-type linear stability around a homogeneous steady-state and from large-scale numerical computations of the PDE system on both fixed and sloly groing spatial domains. For some of these activator-inhibitor models ith time-delayed kinetics, related to the classical GM model and its variants (cf. [8]), it as shon that the delay induces temporal oscillations in the spatial patterning, and that these oscillations can become very large and uncontrolled as the time-delay increases, thereby suggesting a global breakdon of a robust stable patterning mechanism. In [6], this de-stabilizing effect of a time-delay in the reaction kinetics as analyzed in detail for a general class of GM models ith a time-delay in either the inhibitor kinetics or in both the activator and inhibitor kinetics. This study of [6], and the numerical simulations in [5], leads to the question of hether a time-delay in the reaction-kinetics can ever be a stabilizing effect on pattern formation. The main ne goal of this paper is to sho that a time-delay in only the autocatalytic term of the activator kinetics actually leads to an increase in the parameter range here localized spatial patterns are linearly stable. This stabilizing effect of a time-delay in the activator kinetics ill be analyzed in detail for Oxford Center for Industrial and Applied Mathematics (OCIAM), Oxford University, Oxford, UK Dept. of Mathematics, UBC, Vancouver, Canada. (corresponding author ard@math.ubc.ca) Dept. of Mathematics, UBC, Vancouver, Canada.

2 the GM model ith prototypical exponents of the nonlinearities. The GM model [8] for the prototypical exponent set in the semi-strong interaction limit is a singularly perturbed to-component RD system ith a large diffusivity ratio. For the case here only the nonlinear activator kinetics has a finite time delay T, the activator and inhibitor concentrations v and u, respectively, in a bounded domain Ω R N are solutions to the non-dimensional system v t = ε v v +v T/u, x Ω; τu t = D u u+ε N v T, x Ω; n u = n v =, x Ω. (.) Here < ε, v T v(x,t T), and the reaction-time constant τ and inhibitor diffusivity D are both O() positive parameters. In the limit ε, it is ell-knon that the steady-state GM model has localized spike solutions hereby v concentrates at certain points in Ω. We ill study the linear stability of these spike solutions for three different variants of this GM model ith delayed activator kinetics: a one-spike solution for the -D shado problem corresponding to the large inhibitor diffusivity limit D, a one-spike solution in -D on the infinite line, and an M-spot pattern ith M in a bounded -D domain in the eak-coupling regime D = O( logε). The study of these three variants of the GM model ill sho in a rather broad sense the stabilizing effect of delayed activator kinetics on the linear stability of localized spike solutions. In the traditional scenario here there is no activator delay, results characterizing the existence of a Hopf bifurcation threshold value of τ have been derived for the -D shado problem in [] and [], for the -D infinite-line problem in [5] and [3], and for the -D multi-spot problem in [6]. These previous results are all based on linearizing the RD system around a localized spike (-D) or spot (-D) steady-state solution and then studying the spectrum of a nonlocal eigenvalue problem (NLEP) that arises from the linearized stability problem. For the to -D problems, these previous results ith undelayed kinetics sho that a one spike-solution is linearly stable only hen < τ < τ H, and that a Hopf bifurcation occurs as τ crosses above some threshold τ H. Qualitatively similar results ere found for the -D problem in [6], although the analysis and results in [6] ere more intricate oing to the existence of to distinct modes of instability for the amplitudes of the spots, representing either in-phase (synchronous) or anti-phase (asynchronous) instabilities of the amplitudes of the spots. Overall, the qualitative mechanism for an oscillatory instability is that as τ increases, the inhibitor field can only respond relatively sloly to any local increase in the activator concentration due to the autocatalytic term. Such a slo response by the inhibitor field leads to an instability of the spike. The goal of this paper is to sho for each of these three variants of the GM model that the effect of a time-delay in the activator kinetics is stabilizing in the sense that there is a larger parameter range of τ, as compared ith the corresponding undelayed problems, here the steady-state spike (-D) or spot (-D) patterns are linearly stable. In particular, for the -D shado and infinite-line problems e ill sho, using a combined analytical and numerical study of the associated NLEP, that the Hopf bifurcation threshold for τ is a monotone increasing function of the delay T. A simple scaling la for this Hopf bifurcation threshold for the case of large delay T is derived analytically for the to -D problems. Similar results shoing that an activator time-delay has a stabilizing effect on -D multi-spot patterns are obtained from a combined analytical and numerical study of the to specific NLEPs that are associated ith either asynchronous or synchronous instabilities of the amplitudes of the spots. The mathematical challenge of this study is that the NLEP under the effect of activator delay is difficult to analyze oing to the effect of the time-delay in both the local part of the operator as ell as in the multiple of the nonlocal term. In the -D case, this NLEP has the form L µ Φ χ(τλ)µ Φdy dy = λφ, < y < ; Φ H (R), µ e λt, (.)

3 for some χ(τλ) that is analytic in Re(λ). In (.) the delayed local operator is defined by L µ Φ Φ Φ+µΦ, here = 3 sech (y/). We ill provide a ne detailed analytical study of spectra of the delayed local eigenvalue problem L µ φ = λφ ith φ H (R). By reducing this spectral problem to the study of hypergeometric functions, similar to that in [4], e derive transcendental equations characterizing all complex or real-valued spectra of L µ. After deriving a fe key properties of the NLEP (.), and then shoing ho the nonlocal term eliminates unstable spectra of L µ, e use a simple numerical method together ith an analytical scaling la to determine the boundary τ H = τ H (T) in the τ versus T parameter space here (.) undergoes a Hopf bifurcation for both the -D shado problem and -D infinite-line problem. Similar results for a related NLEP are derived in a -D context. A key qualitative conclusion from our analysis is that a time-delay in only the activator kinetics has a stabilizing influence on the stability of localized spike solutions. This result is in marked contrast to the results found in [6] here a time-delay as assumed in both the activator and inhibitor kinetics, or only in the inhibitor kinetics. In the former case, the associated NLEP is (.) here χ(τλ)µ is replaced by χ(τλ)µ, hereas in the latter case (.) holds, but ith L µ Φ replaced ith its undelayed counterpart LΦ Φ Φ + Φ. With either of these to seemingly slight modifications of the NLEP (.) induced by delayed inhibitor kinetics, it as shon in [6] that the Hopf bifurcation threshold τ H is monotone decreasing in the delay T, and that, moreover, there is a finite non-zero critical value T of T for hich a one-spike steady-state solution in -D is linearly unstable for all τ. In this sense, [6] shoed that a steady-state spike solution can become highly unstable hen inhibitor delay effects are included. The outline of this paper is as follos. In, e describe the three variants of the GM model ith activator time-delay and, for each variant, give the specific NLEP hose spectrum characterizes the linear stability of spike solutions. In 3, e derive a fe explicit and rigorous results for spectral properties of the delayed local operator and for a general class of NLEPs in -D and in -D that are associated ith delayed activator kinetics. In 4, e use the NLEP to calculate the Hopf bifurcation threshold for both the -D shado and infinite-line problems, and e derive a scaling la for these boundaries in the large delay limit. Hopf bifurcation thresholds associated ith the -D NLEP, corresponding to the -D multi-spot problem, are studied in 5 for both synchronous and asynchronous instabilities of the spot amplitudes. A brief discussion in 6 concludes the paper.. Formulation of the NLEP Problems. We first study the linear stability of a one-spike steady-state solution to to variants of the prototypical GM model [8] in one-space dimension hen there is a time-delay in only the activator kinetics. Our first variant is the infinite-line problem here, ithout loss of generality, e take the inhibitor diffusivity as D =. This problem is formulated as v t = ε v xx v +v T/u, τu t = u xx u+ǫ v T, < x <, t >, (.) here v T v(x,t T). For ε it as shon in [3] that a one-spike steady-state solution v e, u e to (.), hich is centered at x =, is given by v e (x) U ( ε x ) (, u e (x) U e x, U = dy), (.) here (y) = 3 sech (y/) is the homoclinic solution on < y < to + = ith () >, () =, and as y. In addition, e ill study the so-called limiting shado problem for v(x,t) and u(t) (cf. []) on the interval x, hich corresponds to taking the large inhibitor diffusivity limit D in (.). This shado problem is v t = ε v xx v +v T/u, < x <, v x (±,t) = ; 3 τu t = u+ ε v T dx. (.3)

4 For ε, a one-spike steady-state solution v e, u e for (.3) is given by (cf. []) ( v e u e (x/ε), u e dy). (.4) To study the linear stability of the steady-state solution for each of these to models e linearize either (.) or (.3) about the steady-state by introducing v = v e +e λt Φ(x/ε), and u = u e +e λt η. After a short calculation, similar to that done in [6], e obtain that Φ(y) and λ are eigenpairs of the nonlocal eigenvalue problem (NLEP) L µ Φ χ(τλ)µ Φdy dy = λφ, < y < ; Φ as y, µ e λt, (.5a) here the delayed local operator, L µ, is defined by L µ Φ Φ Φ + µφ. In (.5a), the multiplier χ of the nonlocal term for either the infinite-line or shado problems is χ(τλ) +τλ, (shado problem); χ(τλ), (infinite-line problem). +τλ In (.5b) e must specify the principal branch of +τλ (cf. [3]). (.5b) The NLEP (.5) characterizes the linear stability of a one-spike solution on an O() time-scale to perturbations in the amplitude of the spike (cf. [3], [6]). It is readily shon that any unstable eigenvalue of (.5), satisfying Re(λ) >, must be a root of g(λ) =, here g(λ) µχ(τλ) F µ(λ); F µ (λ) [ (L µ λ) ] dy dy, µ e λt. (.6).. The -D NLEP Problem. Next, e formulate the linear stability problem for an M-spot pattern, ith M, for the GM model (.) ith delayed activator kinetics in a bounded -D domain. A localized spot pattern is one for hich the steady-state solution for v concentrates at a discrete set of points x j Ω, for j =,...,M, as ǫ. For the -D GM model ith no delayed reaction kinetics, in [6] the linear stability properties of such patterns ere analyzed in the eak coupling regime D = O(ν ) here ν = /lnǫ. By setting D = D /ν in (.) ith N =, e can readily extend the analysis of [6] to sho that the linear stability of an M-spot solution ith delayed activator kinetics, and ith τ = O(), is characterized by the spectrum of the NLEP L µ Φ χ(τλ)µ ρφdρ ρ dρ = λφ, < ρ < ; Φ () =, Φ as ρ, (.7a) here L µ Φ ρ Φ Φ + µφ, µ e λt, and ρ Φ ρρ Φ + ρ ρ Φ: Here (ρ) is the positive radially symmetric ground-state solution to ρ + = ith () >, () =, and as ρ. In (.7a), χ(τλ) can assume either of the to forms (cf. [6]) χ(τλ), (asynchronous mode); χ(τλ) +β +β ( ) +β +τλ, (synchronous mode), (.7b) +τλ here β πmd / Ω and Ω is the area of Ω. These to choices for χ correspond to either asynchronous (anti-phase) or synchronous (in-phase) instabilities of the amplitudes of the spots (cf. [6]). Both such modes of instability are possible hen M. Although there are M possible anti-phase modes of instability of the spot amplitudes, from the leadingorder asymptotic theory of [6] leading to (.7b) ith χ = /(+β), these modes have a common stability threshold. With either form for χ, the discrete eigenvalues of the NLEP (.7) are roots g(λ) =, here g(λ) µχ(τλ) F ρ [ (L µ λ) ] dρ µ(λ); F µ (λ), µ e λt. (.8) ρ dρ 4

5 3. Some General Results for the Delayed Operator. For either N = or N =, in this section e derive some properties for the general NLEP for Φ = Φ(y), given by L µ Φ χ(τλ)µ Φ = λφ, Φ H (R N ); L µ Φ = Φ Φ+µΦ, µ e λt, (3.) here χ(τλ) is given in (.5b) hen N = and in (.7b) hen N =. For simplicity of notation, for N = in (3.) e have Φ = Φ(y), = (y) = 3 sech (y/) and the integration in (3.) is over the real line. For N =, e have Φ = Φ(ρ) and = (ρ), ith ρ = y, and the integration in (3.) is over R. With this notation, it readily follos that the eigenvalues λ of (3.) are the roots of g(λ) =, here g(λ) χ(τλ)µ F µ(λ), F µ (λ) [ (Lµ λ) ]. (3.) To analyze (3.), e must first consider some properties of the delayed local eigenvalue problem, defined by L µ Φ = λφ, Φ H (R N ). (3.3) This is done in detail for N = in 3. by using an analysis based on hypergeometric functions. Some partial results for the case N = are given in Delayed Local Operator: One-Dimensional Case. When N =, e can derive transcendental equations for all of the eigenvalues (complex or real) of (3.3) by folloing the approach used in [5] for the case here µ is a fixed complex constant. For the convenience of the reader, the derivation of these equations is given in Appendix A. As shon in (A.) of Appendix A (see also page 7 of [5]), e obtain that any eigenvalue λ of (3.3) hen N = must be a root of one of the transcendental equations K l (λ) =, for l =,,,..., defined by K l (λ) 4 +λ+ +48µ+l =, l =,,,... ; µ e λt. (3.4) We first observe that the translation mode λ =, Φ =, must be an eigenpair for all T. This eigenvalue corresponds to setting l = in (3.4). Next, e characterize any non-zero real-valued eigenvalue satisfying λ > that exists for all T. It is easy to see that such eigenvalues can only occur hen l = or l = in (3.4). In particular, for the case l =, e calculate that K () <, K (λ) + as λ +, and K (λ) is monotone increasing on λ >. Consequently, (3.4) ith l = has a unique root λ = λ (T) > for any T. We readily find from (3.4) that λ () = 5/4, λ log()/t for T, and λ (T) is monotone decreasing in T. In the left panel of Fig. 3. e plot λ (T) versus T, as computed numerically from (3.4). The only other real non-zero eigenvalue that exists for any T is obtained from (3.4) ith l =. Since K ( ) <, K () = >, and K (λ) is monotone increasing in λ, K (λ) = has a unique root λ (T) satisfying < λ (T) < for any T. We readily find that λ 3/4 as T and λ log(3/5)/t for T. Next, e observe from (3.4) for l 3 that a discrete real eigenvalue emerges from the continuous spectrum λ hen the delay T exceeds a critical threshold T l edge >. By setting K l( ) = and solving for T, e identify that ( l Tedge l ) +l log, l = 3,4,..., (3.5) so that T l+ edge > Tl edge. Curiously, e find T3 edge =, so that a discrete real eigenvalue emerges as soon as the delay is turned on. Since for each l 3 e have K l () >, K l (λ) is monotone increasing on < λ <, and K l ( ) < henever 5

6 T > Tedge l, it follos that there is a unique root λ l(t) to (3.4) in < λ < hen T > Tedge l. A simple calculation using (3.4) shos that, for each l 3, this eigenvalue has the folloing limiting asymptotics: ( ) λ l (T) c l /T, c l log, as T. (3.6) (l+3)(l+) This expression shos that all of these discrete eigenvalues that bifurcate from the continuous spectrum at critical values of the delay eventually accumulate on the stable side of the origin λ = as T. λ(t) Im(λ) λ λ λ 3 λ 4 λ T 4... Re(λ) Fig. 3.: Left panel: The positive real eigenvalue λ (T) of the delayed local problem L µφ = λφ hen N = as obtained by solving (3.4) numerically ith l =. Right panel: all the paths of complex-valued spectra of L µ in Re(λ) for TH T TH, 5 as computed from (3.4) for l =. Here T j H for j is the j-th value of T here Lµ has a pure imaginary eigenvalue λ = iλi ith λ I.5 (see (3.)). The path ith imaginary eigenvalue hen T = T j H is labeled by λj. For even larger values of T, these paths all tend to the origin λ =, but in the half-space Re(λ) >. For each path, e also plot its continuation into Re(λ) < for smaller delays. In summary, ith regards to real-valued eigenvalues of L µ hen N =, the spectral problem (3.3) has exactly three real eigenvalues that exist for any T. They are λ = λ (T) >, λ =, and λ = λ (T), here < λ (T) <. In addition, real eigenvalues bifurcate from the edge λ = of the continuous spectrum at the critical values T l edge for l = 3,4,... of the delay, as given explicitly in (3.5). For each l 3, this additional real eigenvalue remains in < λ < for all T Tedge l, and it tends to the origin as T ith the asymptotic rate given in (3.6). Next, e consider complex-valued roots of (3.4). We ill only characterize those complex-valued branches of roots of (3.4) as T is varied that can exist in the unstable right-half plane Re(λ). To do so, e first focus on determining any values of the delay T for hich (3.3) has a pure imaginary eigenvalue λ = iλ I ith λ I >. As shon in Appendix A, (3.4) can only have a pure imaginary root λ = iλ I for the case l =. As derived in Appendix A, such a pure imaginary eigenvalue occurs at the discrete values TH n, given by T = T n H (θ +πn) λ I, n =,,3,.... (3.7) Here λ I, hich is independent of n, is given explicitly by λ I = ( + 8 Re ) ( ) +48µ Im +48µ, µ e i(θ+πn) = cosθ isinθ, (3.8) in terms of the unique root θ in π/3 θ of the function H(θ), defined by H(θ) (4cosθ 7) ( cos θ 8cosθ + ) sin θ. (3.9) The uniqueness of this root follos from the fact that H() >, H( π/3) <, and H(θ) is monotonic on π/3 < θ <. 6

7 From simple numerical computations based on this explicit characterization, e obtain that θ.9946 and λ I.5. The first fe critical values of the delay are T H.585, T H 5.584, T 3 H 8.498, T 4 H.488, T 5 H (3.) With these explicit values for hich (3.4) has a pure imaginary eigenvalue hen l =, e then readily use Neton s method on (3.4) ith l = and numerically path-follo branches of spectra of L µ for T > TH n. As shon in the right panel of Fig. 3. for n = 4, these branches lie in the unstable right-half plane Re(λ) > and accumulate near λ = as T. By path-folloing these same branches on the range < T < TH n these branches of spectra are in Re(λ) <, as expected, and ere created from a singular limit process T + and λ ith Re(λ) <. This singular behavior as T + is a ell-knon feature of quasipolymomials in the eigenvalue parameter that arise as the characteristic equation for traditional ODE delay equations. Finally, e briefly discuss a general method to determine all the eigenvalues of L µ ith N = that exist near the origin λ = in the limit T. This approach is based on the folloing result: Lemma 3.. Consider the auxiliary spectral problem for Ψ(y) and ξ on < y < given by Ψ Ψ+ξΨ =, Ψ H (R). (3.) There is a countably infinite number of eigenvalues ξ l for l =,,,..., ith ξ l < ξ l+ for l, given explicitly by ξ =, ξ = ; ξ l = (l+)(l+3) 6 The first to eigenfunctions are Ψ = and Ψ =., l =,3,.... (3.) Proof: We simply set λ = and µ = ξ/ in (3.4) to obtain 5+l = +4ξ. Upon solving for ξ, e obtain (3.). The first to eigenfunctions follo trivially from using + = and ( ) + =. We no illustrate the use of this lemma by seeking all eigenvalues of L µ near λ = hen T. We let λ c/t for T and from (3.3) obtain that Φ Φ+ [ e c +O(T ) ] Φ = O(T )Φ. To leading order e put Φ = Ψ + O(T ) so that Ψ is an eigenfunction of (3.) ith eigenvalue ξ e c. These eigenvalues are given in (3.). If e set ξ =, e have = e c, so that c = log() + nπi for n =,±,±,... This gives λ [log()+nπi]/t for T. Setting n =, e obtain the asymptotics of the positive real eigenvalue λ (T) log()/t, derived earlier. In addition, the choices n = ±,±,... correspond to the limiting behavior as T of the paths of complex-valued eigenvalues in Reλ > (see the right panel of Fig. 3.). In contrast, if e use ξ l in (3.) ith l, e obtain e c = ξ l / for l. This yields the large T behavior for the negative real eigenvalues λ l (T) for l, as given in (3.6). 3.. Delayed Local Operator: To-Dimensional Case. When N =, the explicit expression (3.4) no longer holds, and so only e must proceed indirectly and through numerical computations. For positive real eigenvalues e can still claim the folloing: Lemma 3.. For each T, there exists a unique eigenvalue real positive eigenvalue λ = λ (T) to (3.3). Proof: For each µ (,), it is it is easy to see that there exists a unique principal eigenvalue, called λ(µ), to (3.3), ith positive principal eigenfunction. In fact, it admits the variational characterization λ(µ) = inf φ H (R ),φ ( φ +φ ) µφ φ 7. (3.3)

8 We only consider the range µ > /, since if µ < /, i.e. µ <, it holds from Lemma 5. (part ) of [5] ( φ +φ ) φ, (3.4) here the equality holds iff φ = C. Hence for µ /, λ(µ) and if µ = /, then λ(/) = is the principal eigenvalue ith eigenfunction. For µ > /, e have λ(µ) >, as can be verified by using the trial function φ = in (3.3), hich yields λ(µ) ( + ) µ 3 = ( µ) 3, (3.5) This implies λ(µ) > hen µ > /. Next, for µ = e recall that L admits only one positive eigenvalue ν, and that the second eigenvalue is zero. By a min-max variational characterization of eigenvalues based on (3.3), it follos that the eigenvalues of L µ are monotone decreasing in µ. Thus for µ <, the second eigenvalue L µ, is strictly less than the second eigenvalue of L, hich is. Therefore, the only positive eigenvalue is the first eigenvalue and hence it is principal, hich must be simple by definition. By uniqueness, λ(µ) is a smooth function of µ. Then, recalling µ = e λt, the problem (3.3) can be reritten as the transcendental equation λ(µ) = logµ T. (3.6) Note that λ(/) = hile λ() = ν, and λ(µ) is an increasing of µ. On the other hand, the right-hand side of (3.6) is a decreasing function of µ. Hence there exists a unique µ satisfying (3.6). This implies that there exists a unique positive real eigenvalue λ = λ to (3.3). We no relabel this unique positive eigenvalue of (3.3) as λ (T). Next, for N =, e determine approximations to λ (T) that are valid hen either T or T. We first determine an approximation valid for small delay. When T =, e kno that L Ψ = νψ ith Ψ as ρ has a unique positive eigenvalue ν ith eigenfunction Ψ > (cf. [6]). To determine an approximation for T e perform a standard perturbation analysis of a simple eigenvalue. We rite λ = ν +Tν + and φ = Ψ+Tφ +. Upon substituting these expansions into L µ φ = λ φ, and equating the O(T) terms, e obtain that (L ν )φ ρ φ φ +φ ν φ = ν Ψ +ν Ψ. Since (L ν )Ψ =, and L is self-adjoint, the solvability condition for this problem is that the right-hand side of this expression is orthogonal to Ψ. This determines ν. To determine an approximation for large delay e use the scaling la ansatz λ λ c /T for T, hich yields that ρ φ φ+ [ e λc + ]φ = O(T ). Therefore, φ = +O(T ), and upon using ρ =, e obtain that this equation holds hen ( e λc ) =. This yields that λ c = log. In this ay, e conclude that λ (T) has the folloing limiting asymptotics for small and large delay: ρψ λ (T) ν ν T dρ ρψ dρ +, as T ; λ (T) log/t +, as T. (3.7) In the left panel of Fig. 3. e sho numerical results of a Neton iteration scheme applied to L µ Φ = λφ to compute λ (T) for any T > hen N =. The limiting asyptotics (3.7) are found to compare favorably ith these results. Next, e study complex eigenvalues of (3.3). We first observe that if Φ, λ is an eigenpair of (3.3) then so is Φ and λ. We first seek a necessary condition for hich λ = iλ I ith λ I > is an eigenvalue of (3.3). We rite Φ = Φ R +iφ I and µ = e iλit = µ R +iµ I in (3.3), and after separating into real and imaginary parts e get Φ R Φ R +µ R Φ R µ I Φ I = λ I Φ I, Φ I Φ I +µ R Φ I +µ I Φ R = λ I Φ R. (3.8) 8

9 .5 λ(t) T Fig. 3.: Left panel: the positive real eigenvalue λ (T) of L µφ = λφ hen N = (solid curve), as computed numerically from a BVP solver. The asymptotic results (3.7) for small and large delay are the dashed curves. Upon multiplying the first equation by Φ I and the second by Φ R, e subtract the resulting expressions and use Green s identity to obtain µ I (Φ I +Φ R) = λi (Φ I +Φ R). (3.9) Since λ I >, this shos that µ I = sin(λ I T) <. Next, e multiply the first equation in (3.8) by Φ R and the second by Φ I, and add the resulting equations. After using (Φ R Φ R ) = Φ R +Φ R Φ R. This readily yields µ R (Φ I +Φ R) = ( ΦR + Φ I +Φ R +Φ I), (3.) so that µ R = cos(λ I T) > at any HB point. In summary, e have shon that at any HB point ith λ I > e must have cos(λ I T) > and sin(λ I T) <. The next result establishes the existence of a HB point for L µ hen N =. Lemma 3.3. The delayed local eigenvalue problem (3.3) in -D has a HB point. Proof: Let (ρ) be the ground-state solution to ρ + = and define θ λ I T. We no sho that there is a value θ of θ for hich ρ Φ Φ+e iθ Φ = iλ I Φ, Φ H (R ), (3.) has a solution λ I >. In fact, e consider the folloing eigenvalue problem ρ Φ Φ+e iθ Φ = λφ, Φ H (R ), (3.) here e vary θ ( π 3,). When θ =,e iθ =, it is knon that (3.) has a unique positive eigenvalue. On the other hand, hen θ = π/3, so that e iθ = +i 3, e claim that all eigenvalues of (3.) must lie on the left-hand side of the complex λ plane. To see this, e multiply (3.) by the conjugate of Φ and integrate to obtain the identity ( Φ + Φ cos(θ) Φ ) = λ R Φ, (3.3) and sin(θ) ( Φ ) = λ I 9 Φ. (3.4)

10 Upon setting cos(θ) = /, e observe from (3.3) and (3.4) that λ R, so that λ R. If λ R =, e must have Φ = c for some constant c. Substituting into the equation, e see that this is impossible. By the continuity argument (see [3]), as θ varies from to π 3, the eigenvalue must cross the imaginary axis at some θ ( π 3,). From (3.3) and the fact that sin(θ) in this open interval, e have that λ. Therefore the crossing point must be a Hopf bifurcation point. From (3.4) e see that λ I >. Since θ ( π 3,), cos(θ ) > and sin(θ ) <, the HB values of T are T n H (θ +πn)/λ I for n =,,3,... From simple numerical computations of a matrix eigenvalue problem obtained from a finite difference approximation of (3.), e calculate that θ.933 and λ I.69. The first fe critical values of the delay are then T H.989, T H 4.34, T 3 H 6.659, T 4 H 8.995, T 5 H.33. (3.5) Paths in the complex plane for T > T j H are then similar to those shon in the right panel of Fig. 3. for the case N = A Monotonicity Property for F µ (λ). Next, e sho for either N = or N = that F µ (λ), as defined in (3.), is monotone increasing in λ > real ith λ λ (T), here λ (T) > is the unique real positive eigenvalue of L µ Φ. We first compute that L µ = (µ ), (L µ λ) ( ) = Hence (3.) becomes µ (L µ λ) (L µ ) = µ + λ µ (L µ λ) (). (3.6) g(λ) = χ(τλ)µ F µ(λ), F µ (λ) µ + λ (Lµ λ) P(λ), here P(λ). (3.7) µ Therefore, upon setting g(λ) =, e conclude that λ must satisfy χ(τλ)µ = F µ(λ) = µ + λ P(λ), (3.8) µ here P(λ) is defined in (3.7). We no claim that P(λ), and hence F µ (λ) is monotone increasing. Lemma 3.4. For λ > real and λ λ (T), e have P (λ) >. Proof: We define ψ (L µ λ), so that ψ solves Upon differentiating this equation ith respect to λ e obtain ψ ψ +e λt ψ = λψ +. (3.9) ψ ψ +e λt ψ λψ = Te λt ψ +ψ. (3.3) Hence, upon using (3.7) for P(λ), and integrating by parts, e obtain that P (λ) = ψ = (Lµ λ) (Te λt ψ +ψ) = Te λt (Lµ λ) (ψ) = Te λt ((Lµ λ) )(ψ) + + (Lµ λ) ψ (Lµ λ) [ ψ = Te λt + (Lµ λ) ψ = Te λt + (Lµ λ) ] >,,,

11 hich proves the claim. As a remark, by using L = +ρ / for N = and L P(λ) > P() = = +y / for N =, e can calculate that L = 4 N. 4 As a numerical confirmation of this key monotonicity result for F µ, for both N = and N = in Fig. 3.3 e plot F µ (λ), defined in (3.), on the interval < λ < λ (T) of the real λ-axis for three values of the delay T. This confirms that F µ (λ) is monotone increasing on this interval ith F µ (λ) + as λ λ (T) from belo. Using L =, it also follos that F µ () = for all T. We remark that this monotonicity property is qualitatively similar to that for the undelayed operator L (see [] and [3]). 8 8 Fµ(λ) 6 4 Fµ(λ) T = λ.5 T = λ Fig. 3.3: Plot of the numerically computed function F µ(λ), defined in (3.), hen N = (left panel) and hen N = (right panel) on the positive real axis for < λ < λ (T) for three values of the delay T as indicated in the figure. We confirm that F µ(λ) is monotone increasing on < λ < λ (T), ith F µ(λ) + as λ λ (T) from belo. This monotonicity result yields to key qualitative results for the roots of (3.8), hich follo immediately from the facts that F µ () =, F µ (λ) is monotone increasing on < λ < λ (T) and F µ (λ) as λ λ (T) from belo. Theorem 3.. Consider the NLEP (3.), for hich the eigenvalues are roots of (3.8). (I) Suppose that χ() <. Then, the NLEP has a positive real eigenvalue on < λ < λ (T) for any T and τ. (II) Fix T, and suppose χ() >, here χ(τλ) is given in (.5b) and (.7b). Then, there are at least to roots to (3.8), and hence at least to positive real eigenvalues for the NLEP (3.), hen τ Continuous dependence on T. We rite (3.) in the folloing form: L λ Φ := L µ Φ χ(τλ)µ Φ = λφ. (3.3) WeclaimthatbranchesofeigenvaluesofthisNLEParecontinuousinT ons {λ Re(λ) >, χ(τλ) is analytic}. We ill only ork in the class of radially symmetric functions. We first look at Fredholm properties. Since the map Φ µφ χ(τλ) is relatively compact as a map of H to L, e see that the operator L µ λ is Fredholm of index zero. Next e note that, if λ is an eigenvalue of L λ Φ = λφ then its geometric multiplicity on L r, here L r is the operator L µ restricted to the radial class, is unless λ = λ. To see this, note that if e have to linearly independent eigenfunctions, a possible combination ill make φ =, and hence e obtain that λ satisfies the local eigenvalue problem (3.3), hich is impossible. The analyticity of our operator in λ on the set S, hich can be seen from the definitions, the Fredholm property and Theorem 3.6 of Gokhberg and Krein (cf. [9]) implies that all eigenvalues of (3.3) are isolated. Finally, e sho that the algebraic multiplicity is also one. This follos from Dancer s argument (see page 48 of [3]).

12 In conclusion, e have shon that for each fixed T, the eigenvalues of (3.3) are isolated ith geometric and algebraic multiplicity one. Applying the classical theory of Kato [3], e conclude the eigenvalues vary continuously in T..5 TH 5 5 λih χ χ Fig. 3.4: HB threshold for (3.3) versus the constant multiplier χ of the nonlocal term in (3.3). Left panel: The minimum value T H of T versus χ here a HB occurs. Right panel: The HB frequency λ IH versus χ. A HB occurs only on χ < ith λ IH + and T H + as χ. For χ > the NLEP (3.3) does not have any HB as T is increased. Toillustratetheuseofthisresult, eletn = andstudynumericallyhotheunstablecomplexconjugateeigenvalues for L µ are pushed into the stable left half-plane Re(λ) < for the folloing NLEP ith a constant multiplier χ > : L µ Φ χ µ Φ = λφ, Φ H (R ); L µ Φ = Φ Φ+µΦ, µ e λt. (3.3) As a starting point, e let χ = and choose the minimum value TH of the delay for hich L µ has a pure imaginary eigenvalue λ IH, and e then numerically track this HB point as χ is increased. With this homotopy, the HB threshold value of T and frequency as a function of χ are shon in the left and right panels of Fig From this figure, e observe that a HB occurs only hen χ <, and that λ IH + and T H + as χ. For χ > the NLEP (3.3) does not undergo any HB as T is increased. Since hen T = e have Re(λ), ith equality holding iff Φ = (cf. [4]), and that unstable eigenvalues can only enter Re(λ) > through a HB, e conclude, by continuity in T, that Re(λ) for any T > henever χ >. We remark that an extension of this result to N = is used in 5. to characterize HB points for the NLEP (.7) ith the constant multiplier χ = /( + β), hich applies to asynchronous instabilities of multi-spot patterns in -D. 4. The -D Problems: NLEP Computations and a Scaling La. In this section, e consider the NLEP (.5) for the -D case. From this spectral problem, e numerically determine the threshold conditions for a Hopf Bifurcation (HB) for both the infinite-line (.) and the shado (.3) problems. We also derive a scaling la for the HB thresholds in the limit of large delay T. We first note that hen τ = in (.5), e have χ(τλ) =, and so by the result in 3.4 (see Fig. 3.4), e have Re(λ) for all T. Fixing T, e have from Theorem 3. that there are at least to positive real eigenvalues hen τ is large enough. By continuity in τ, there must be a HB at some τ = τ H > depending on T. To determine the threshold conditions for such a HB, e set Re(g(iλ IH )) = and Im(g(iλ IH )) = in (.6), here λ IH >. This yields a nonlinear system for the HB values τ H and λ IH at a particular value of the delay T. By using Neton s method on this system, together ith a numerical computation of (L µ iλ IH ), in the left panel of Fig. 4. e plot the numerically computed HB boundary τ H versus T for the shado problem (.3). Our numerical results sho that the spike solution is linearly stable for τ < τ H. The corresponding HB frequency λ IH is plotted versus T in the middle panel of Fig. 4.. In the right panel of Fig. 4. e sho that τ H /T and λ IH T both tend to finite non-zero limiting

13 values as T. As shon in Fig. 4., e obtain qualitatively similar results for a HB for the infinite-line problem (.). Our key conclusion that τ H is monotone increasing in T shos that a time-delay in only the activator kinetics has a stabilizing effect on a spike solution for (.3) and (.). τh λih λ IHT τ H T T 5 5 T 5 5 T Fig. 4.: HB threshold τ H (left panel) and frequency λ IH (solid curve in middle panel) versus T, as computed from (.6) for the shado problem (.3). For τ < τ H (shaded region), the spike solution is linearly stable. The dashed curve in the middle panel is the large-delay asymptotic result for λ IH given in (4.4). Right panel: plot of τ H/T (monotone decreasing solid curve) and λ IHT (monotone increasing solid curve), as computed from (.6). The asymptotes (dashed lines) are the theoretically predicted limiting values lim T τ H/T 3/π and lim T λ IHT π/3, as obtained from (4.4) λ IHT τ H T τh 4 λih T 5 5 T 5 5 T Fig. 4.: HB values for the infinite-line problem (.), ith the same caption as in Fig. 4.. In the middle panel, the large-delay asymptotic result (dashed curve) is λ IH c /T ith c.78. In the right panel the theoretically predicted horizontal asymptotes are lim T τ H/T.99 and lim T λ IHT.78, as obtained from (4.8). The numerical results shon in Fig. 4. and Fig. 4. sho that the HB threshold and frequency satisfy τ H and λ IH as T, respectively, ith τ H /T and λ IH T both tending to finite non-zero limiting values as T. We no characterize this large delay limiting behavior analytically. Our explicit results, described belo, are shon by the dashed lines in the middle and right panels of Fig. 4. and Fig. 4.. For T, e pose τ H τ T and λ ic /T for some c > and τ > to be found. With this scaling la, (.5a) reduces to leading-order to Φ Φ+ [ e + ]Φ [ ic χ e + ] ic Φdy [ ] dy = ic T + Φ, (4.) here χ χ(ic τ ). Since + =, (4.) yields that Φ +O(T ), provided that c and τ are roots to e ic = χ. (4.) 3

14 For the shado problem, here from (.5b) e have χ = /(+τλ), the complex-valued equation (4.) becomes e ic = ic τ +ic τ. (4.3) By setting the modulus of the right-hand side of (4.3) to unity, and defining ξ c τ, e get ξ = +ξ, so that c τ = / 3. Then, (4.3) yields e ic = e iπ/3, hich has c = π/3 as its minimal root. In this ay, the scaling la for a HB for the shado problem (.3) is τ H 3 π T ; λ = iλ IH, λ IH π, as T. (4.4) 3T Theasymptoticsforλ IH, givenbythedashedcurveinthemiddlepaneloffig.4., agreeelliththenumericalresults. In addition, the theoretically predicted horizontal asymptotes lim T τ H /T = τ.55 and lim T λ IH T = c.47, shon in the right panel of Fig. 4., agree ell ith the numerically computed results from (.6). A similar analysis can be done for the infinite-line problem here, from (.5b), χ = / +τλ. Then, (4.) becomes e ic = [ +iτ c ] +iτ c. (4.5) By setting the modulus of the right-hand side of (4.5) to unity, e get that z = z, here z +iξ and ξ c τ. Upon separating z into real and imaginary parts, as z = z R +iz I, e get α+ α 3(zR +zi) = 8z R 4, here z R =, z I =, α +ξ, (4.6) hich can be combined to obtain 3α = 4 (+α) 4. This yields that α = 4(+ )/9, and consequently 95+3 ξ = c τ = lim λ IHτ H = (4.7) T 9 With c τ knon, e take the imaginary part of (4.5) to obtain that sin(c ) = z I /(zr +z I ). From the expressions for z R and z I in (4.6), e obtain c and τ in terms of α = 4(+ )/9 as ( ) 95+3 c = sin α.786, τ = (4.8) α 9c With these values of c and τ, the scaling la for a HB for the infinite-line problem (.) is τ H τ T and λ = iλ IH ith λ IH c /T as T. The asymptotics λ IH c /T is shon by the dashed curve in the middle panel of Fig. 4.. The theoretically-predicted limiting values lim T τ H /T = τ.99 and lim T λ IH T = c.78, given by the horizontal asymptotes in the right panel of Fig. 4., agree ell ith the numerically computed results from (.6). 4.. Numerical Validation of the Theory. In order to readily compare our theoretical results for the HB threshold ith full numerical results from the delayed RD system e need to extend our theory to the case of a finite -D domain x L so as to be more readily able to compute solutions to the full PDE system. As such, e consider a one-spike solution centered at x = for the finite-domain problem v t = ε v xx v +v T/u, τu t = u xx u+ǫ v T, x L, t >, (4.9) here v x = u x = at x = ±L. By first constructing a one-spike solution and then analyzing the linear stability problem, a simple calculation, similar to that in [6] and [3], shos that the linear stability problem reduces to determining the roots of (.6), here χ(τλ) is no defined in terms of the principal branch of +τλ by ( ) tanh(l) χ(τλ) = +τλ tanh ( L +τλ ). (4.) 4

15 L =. L = L = L = 8 τh T Fig. 4.3: Plot of the HB threshold τ H versus T for the -D finite-domain problem (4.9) on x L here L =. (dashed curve), L = (dashed-dottedcurve), L = (solidcurve), andl = (heavysolidcurve). Thethresholdascomputednumerically from (.6) ith χ(τλ) as given in (4.). The one-spike solution is linear stable hen τ < τ H. The threshold for L = closely approximates that for the infinite-line problem, hich as given in the left panel of Fig. 4.. We then set λ = iλ IH and numerically compute the roots of g(iλ IH ) = using a Neton iteration scheme, here g(λ) is defined in (.6) in terms of χ as given in (4.). The resulting HB curves τ H versus T for four values of L are shon in Fig As L increases e see there is a ider range of τ at a given delay T here a one-spike solution is linearly stable Activator v(,t).5.55 Activator v(,t).5.55 Activator v(,t) t t t Fig. 4.4: Plot of the amplitude v(,t) of the spike versus t for τ = 5.3 (left panel), τ = 5.6 (middle panel), and τ = (right panel), as computed numerically by discretizing (4.9) ith 5 spatial meshpoints on [,] ith ǫ =.5 and T =. The theoretical HB prediction is τ H (see Fig. 4.3 ith L = and T = ). The numerics shos a sloly decaying (groing) oscillation hen τ = 5.3 (τ = 5.6), respectively. A large oscillation leading to a collapse of the spike occurs hen τ is ell-above the HB threshold (right panel). As a partial verification of the results of Fig. 4.3 for the HB stability threshold, e took ǫ =.5, L =, and T =, and discretized (4.9) ith 5 spatial meshpoints using a method of lines approach. We then used the dde3 solver of MATLAB to solve the system of delay (ordinary) differential equations (DDEs) for a value of τ slightly belo and then slightly above the theoretically predicted HB threshold of τ H The numerical results shon in the left and middle panels of Fig. 4.4 confirm our prediction of the HB threshold. In the right panel of Fig. 4.4 e sho that the spike amplitude first oscillates and then collapses for a value of τ that is considerably above the HB threshold. 5. The -D NLEP Problem: Computations and a Scaling La. In this section e study the NLEP (.7) characterizing the linear stability of an M-spot pattern, ith M, for the GM model (.) ith delayed activator kinetics in a bounded -D domain. For both the synchronous and asynchronous modes of instability, e ill sho that a 5

16 time-delay in the activator kinetics leads to a ider parameter range here spot patterns are linearly stable. 5.. The Synchronous Mode. We first consider the NLEP (.7) for the synchronous mode by determining a parameterization of the HB curve in the τ versus β plane for a fixed delay T. We substitute χ for the synchronous mode, as given in (.7b), into (.8) and set λ = iλ I, ith λ I, to obtain that ( ) (+β) e iλit +iτλi = F µ (iλ I ). (5.) +β +iτλ I We solve this equation for β to get β = + iτλ I F µ (iλ I ) e iλit (+iτλ I ) F µ (iλ I ), (5.) and, upon taking the imaginary part of this expression, e conclude that ( ) iτλ I F µ (iλ I ) Im e iλit =. (5.3) (+iτλ I ) F µ (iλ I ) We then separate F µ (iλ I ) into real and imaginary parts as F µ (iλ I ) = F Rµ (λ I )+if Iµ (λ I ). (5.4) Upon substituting (5.4) into (5.3), e solve for τ = τ H (λ I ) to obtain the parameterization here F Rµ F Rµ (λ I ), F Iµ F Iµ (λ I ), and F µ Similarly, if e solve (5.) for iτλ I, e get τ H = F µ F Rµ cos(λ I T) F Iµ sin(λ I T) λ I (F Iµ cos(λ I T) F Rµ sin(λ I T)) F Rµ +F Iµ. ( F µ (iλ I ) e iλit ) iτλ I = (+β) e iλit (+β) F µ (iλ I ), (5.5). (5.6) By setting the real part of the right-hand side of (5.6) to zero, and upon solving the resulting equation for β, e obtain the parameterization β = β H (λ I ) here β H = 4 F µ + 4F Rµ cos(λ I T) 4F Iµ sin(λ I T) F Rµ cos(λ I T)+F Iµ sin(λ I T). (5.7) The expressions (5.5) and (5.7) parametrize the HB threshold in the τ versus β plane in terms of λ I >, at a fixed value of the delay T. We remark that if e replace F µ ith F, corresponding to setting T = in (5.5) and (5.7), e recover the parameterization given in equation (4.6) of [6] (see also Fig. 4. of [6]) for the -D GM model ith no activator or inhibitor delays. To implement (5.5) and (5.7) numerically, e rite F µ (iλ I ) in terms of the complex-valued ψ (L µ iλ I ) as F µ (iλ I ) = ρψdρ ρ dρ, here ρψ ψ +(cos(λ I T) isin(λ I T))ψ iλ I ψ =, (5.8) here (ρ) > is the ground-state solution satisfying ρ + =. We solve for the ground-state numerically and then determine ψ = ψ R +iψ I from a BVP solver applied to ψ. In this ay, e can readily compute F Rµ (λ I ) and F Iµ (λ I ) from (5.8), hich is needed in our expressions (5.5) and (5.7) for the HB threshold. 6

17 τh 5 5 T = T = T = T = 5 T = Asymptotics (T ) λih T = T = T = T = 5 T = Asymptotics (T ) β 3 4 β Fig. 5.: HB thresholds for the synchronous mode computed from (5.5) and (5.7). Left panel: The HB threshold τ H versus β for T = (heavy solid curve). The spot pattern is linearly stable for all β < and for τ < τ H hen β >. From bottom to top, the various dashed curves are HB thresholds for T =, T =, T = 5 and T =. The thin solid curve is the asymptotic scaling la (5.) for τ H hen T =. The effect of activator delay leads to a ider parameter range for stability. Right panel: plot of the HB frequency λ IH versus β on β > ith the same labeling as in the left panel, except that no as λ IH decreases as T increases. The thin solid curve is the asymptotic scaling la (5.) for λ IH hen T =. With this numerical approach, in the left panel of Fig. 5. e plot the HB threshold in the τ versus β plane for T = and four nonzero values of the delay. The corresponding HB frequency, λ IH, is plotted versus β in the right panel of Fig. 5.. From this figure, e observe that a HB exists only on the range β >, and that τ H + as β +. We also observe that at a fixed β >, τ H increases as T increases, so that the effect of delayed activator kinetics is to increase the parameter range for the linear stability of the multi-spot pattern. To determine the limiting behavior of τ H for T at a fixed β >, e pose the scaling la τ H τ T, λ ic /T here c > and τ > are to be found. Upon substituting this into (3.) e get ρ Φ Φ+ [ e + ]Φ [ [ ] ic χ e + ] ic Φρdρ ρdρ = ic T + Φ, (5.9) hereχ = χ(ic τ )isgivenin(.7b). SettingΦ = +O(T ), andusing ρ + =, eget +( χ )e ic =. This yields that e ic = +β ( ) +β +iτ c +iτ c Upon taking the modulus of (5.), and solving for τ c, e get τ c = = β ( ) iτ c. (5.) +β +iτ c β, here f f +β, (5.) provided that f >, hich implies that β >. Then, by taking the imaginary part of (5.) e get ( ) c = sin fτ c +τ. c In this ay, e obtain for β πmd / Ω >, that as T +, there is a HB for the synchronous mode ith the scaling la ( ) τ H τ T, τ = f sin ( ), f β f /f β +, (5.a) λ = iλ IH, λ IH c /T, c = sin ( ) f /f. (5.b) 7

18 We observe that for a fixed T ith T, e have τ H + and λ IH as β +. A plot of the asymptotic results for λ IH and τ H versus β on β > hen T = are shon in Fig. 5. to compare very ell ith full numerical results computed from the parameterization (5.5) and (5.7). 5.. The Asynchronous Mode. WenoconsidertheNLEP(.7) for theasynchronous modehereχ = /(+β). We recall from Theorem 3. that there is a positive eigenvalue for any T hen χ <. Consequently, e conclude that the multi-spot pattern is unstable for any T hen β πmd / Ω >. Next, e calculate the HB threshold in the T versus β plane, obtained by solving (.8), hich e rite as a coupled system for T = T H and λ IH, satisfing (+β) cos(λ I T) = F Rµ (λ I ), (+β) sin(λ I T) = F Iµ (λ I ), (5.3) here F µ (iλ I ) = F Rµ (λ I ) + if Iµ (λ I ). This system is solved as β is varied by using Neton iterations. As a starting point, e let β and choose the minimum value T H of the delay for hich L µ has a pure imaginary eigenvalue iλ IH. We then numerically path-follo this HB point as β is decreased. With this numerical approach, the HB threshold value of T H and frequency λ IH as a function of β are shon in the left and right panels of Fig. 5.. From this figure, e observe that a HB occurs only hen β >, and that λ IH + and T H + as β +. For β < the NLEP (.7) for the asynchronous mode does not have a HB for any T. Although there is a HB value of T hen β >, the entire region β > is linearly unstable for any T, as the NLEP alays has a positive real eigenvalue. In contrast, hen β <, e conclude that Re(λ) for any T..5 TH 5 5 λih β β Fig. 5.: HB threshold for the asynchronous mode computed from (5.3). Left panel: The minimum value T H of T versus β here a HB occurs. Right panel: The HB frequency λ IH versus β. We observe that a HB occurs only on β > ith λ IH + and T H + as β +. For β > the NLEP (.7) for the asynchronous mode also has a positive real eigenvalue for any T. When β <, e predict that the mutli-spot pattern is linearly stable for any T. Finally, e compare our HB threshold for the asynchronous mode ith only activator delay ith corresponding thresholds for the case of both activator and inhibitor delay, and ith only inhibitor delay. In the former case vt /u in (.) is replaced ith v T /u T, hereas hen there is only inhibitor delay v T /u and ε N v T in (.) are replaced ith v /u T and ε N v. When there is both activator and inhibitor delay, χµ in (.7) is replaced ith χµ, and a HB of the NLEP must be a root λ I = λ IH and T = T H of the coupled system (+β) cos(λ I T) = F Rµ (λ I ), (+β) sin(λ I T) = F Iµ (λ I ), (5.4) In contrast, hen there is only inhibitor delay, L µ in (.7) is replaced by its undelayed counterpart L Φ ρ Φ Φ+Φ, 8