Bifurcation analysis of reaction diffusion Schnakenberg model

Transcription

1 J Math Chem (13) 51:1 19 DOI 1.17/s x ORIGINAL PAPER Bifurcation analysis of reaction diffusion Schnakenberg model Ping Liu Junping Shi Yuwen Wang Xiuhong Feng Received: 1 January 13 / Accepted: 1 May 13 / Published online: 1 June 13 Springer Science+Business Media New York 13 Abstract Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction diffusion system modeling Schnakenberg chemical reaction. The existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns. Keywords Schnakenberg model Steady state solution Hopf bifurcation Steady state bifurcation Pattern formation Mathematics Subject Classification () 5F7 5C 5C15 3C3 35B 35B3 1 Introduction Schnakenberg model [13] is a basic differential equation model to describe an autocatalytic chemical reaction with possible oscillatory behavior. The trimolecular reactions between two chemical products X, Y and two chemical source A, B areinthe following form: Ping Liu: Partially supported by NSFC Grant and NCET Grant 151-NCET-. Junping Shi: Partially supported by NSF Grant DMS-1. Yuwen Wang: Partially supported by NSFC Grant and Natural Science Foundation of Heilongjiang Province of China A11. P. Liu (B) Y. Wang X. Feng Y.Y. Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang 155, People s Republic of China liuping5@gmail.com J. Shi Department of Mathematics, College of William and Mary, Williamsburg, VA , USA 13

2 J Math Chem (13) 51:1 19 A X, B Y, X + Y 3X. (1.1) Using the law of mass action, one can obtain a system of two reaction diffusion equations for the concentrations u(x, t) and v(x, t) of the chemical products X and Y describing the reactions in (1.1). The non-dimensional form of the equations is u t u = a u + u v, v t d v = b u v. In the above equations, d are the diffusion coefficients of the chemicals X, Y, and a, b are the concentrations of A and B.ItisalsoassumedthatAand B are in abundance so a and b remain as constants. Because of its algebraic simplicity, Schnakenberg model has been used by many people as an exemplifying reaction diffusion system to study spatiotemporal pattern formation [,11,1]. In this paper we consider the reaction diffusion Schnakenberg model in a bounded domain : u t u = a u + u v, x, t >, v t d v = b u v, x, t >, u(x, ) = u (x), v(x, ) = v (x), x, (1.) u n = v =, x, t >. n Here R N,(N 1) is a bounded connected domain with a smooth boundary ; u(x, t) and v(x, t) are the concentrations of the chemical products X and Y at time t and location x ; a no-flux boundary condition is assumed so that the chemical reactions are in a closed environment; a, b,, d are positive constants and the initial concentrations u,v C( ) are non-negative functions. For simplicity, in this paper we only consider the case N = 1 and = (, lπ) for some l >. In this paper, we analyze the stability of the unique positive steady state solution (u,v ). When (u,v ) loses its stability, we study the associated Hopf bifurcations and steady state bifurcations which give rise to temporal oscillations or non-constant stationary patterns. For simplicity of calculation, we use two new parameters: b + a = β, b a = α. (1.3) We fix the value of β>, and vary the value of α in ( β,β). Our main results can be summarized as follows (a more detailed version is in Sect. ): 1. When β is large, then the unique positive steady state (u,v ) is locally asymptotically stable for any α ( β,β). Hence pattern formation is not likely.. If <β<1and d / is small, then (u,v ) is locally asymptotically stable for α ( β,β 3 ), and it loses the stability at α = β 3 through a Hopf bifurcation. In this case, a time-periodic pattern is likely to emerge for α (β 3,β). 3. If <β<1and d / is large, then there exists an α S j (,β 3 ) such that (u,v ) is locally asymptotically stable for α ( β,α S j ), and it loses the stability at α = α S j through a steady state bifurcation. In this case, a stationary spatially non-homogenous pattern is likely to appear for α (α S j,β). 13

3 J Math Chem (13) 51: In Sect., we do a Hopf bifurcation analysis with parameter α, and in Sect. 3, we analyze the steady state solution bifurcations. Some numerical simulations and a summary of change of dynamics are given in Sect.. Our analysis follows the approach given in Yi et al. [] for diffusive predator-prey systems, and similar approach has also been used in [3,,9,1,19 1] for other reaction diffusion models. Note that both patterns described above were first considered by Turing [15] in his seminal work years ago. There have been some previous work on reaction diffusion Schnakenberg model. It has been shown in [5,17,1] that for small values of the system (1.) exhibits various stationary multi-spot and spike patterns. Our work explains such multi-spike solutions from the view of bifurcation. Very recently the steady state solutions of (1.)(ormore general form) have also been considered in [,], and the Hopf bifurcations of (1.) has been investigated in [19]. Our results here use a simpler parametrization (1.3) which simplifies the expression of many calculations. Our results not only sharpen many previous results, but also is the only one to provide a panoramic view of the dynamics of (1.). We also mentioned that it is known that the limit cycle of the corresponding ODE system of (1.) is unique (see []). In the paper we use (u α,v α ) to denote the unique constant steady state solution of (1.). We denote by N the set of all the positive integers, and N = N {}. Hopf bifurcations In this section, we consider the Hopf bifurcations for the Schnakenberg model. In particular we derive an explicit algorithm for direction of Hopf bifurcation and stability of the bifurcating periodic solutions for the reaction diffusion system of Schnakenberg model with Neumann boundary condition. While our calculations can be carried over to higher spatial domains, we restrict ourselves to the case of one-dimensional spatial domain (, lπ), for which the structure of the eigenvalues is clear. The Schnakenberg model in the spatial domain = (, lπ) is in form u t u xx = a u + u v, x (, lπ), t >, v t d v xx = b u v, x (, lπ), t >, u(x, ) = u (x), v(x, ) = v (x), x (, lπ), u x (, t) = u x (lπ, t) = v x (, t) = v x (lπ, t) =, t >. (.1) First we consider the corresponding kinetic system of (.1). u = a u + u v, t >, v = b u v, t >, u() = u >, v() = v >. (.) An equilibrium point (u,v)of (.) satisfies { a u + u v =, b u v =. 13

4 J Math Chem (13) 51:1 19 By a simple calculation, of (.). Let ( a + b, ) b (a + b) is the unique positive equilibrium point a + b = β, b a = α. Then a = β α, b = β + α, and a >, b > is equivalent to β > and β>α> β. We shall rewrite the equilibrium point by (β, β + α β ) (u α,v α ).In the following we choose a fixed parameter β>, and use α as the main bifurcation parameter. The Jacobian matrix of system (.) evaluated at (β, β + α β ) is α β L (α) = β 1 α. (.3) β β The characteristic equation of L (α) is μ T (α)μ + D(α) =, (.) where T (α) = β + α β, D(α) = β. The eigenvalues μ(α) of L (α) are given by μ(α) = T (α) ± T (α) D(α). A Hopf bifurcation value α satisfies the following condition: T (α) =, D(α) >, and T (α) =. Clearly we always have D(α) >, and T (α) = implies α = β 3. Finally T (α) = 1/β for any β>. Then α = β 3 is the only Hopf bifurcation point for (.). Since we assume that β>α, hence when <β<1, there exists a unique Hopf bifurcation point α = β 3. When β 1 and β>α, the unique equilibrium point of (.) is always locally asymptotically stable. Next we consider Hopf bifurcations from the constant steady state (u α,v α ) of the reaction diffusion system (here we use equivalent parameters α and β) u t = u xx + β α u + u v, x (, lπ), t >, v t = d v xx + β + α u v, x (, lπ), t >, (.5) u x (x, t) = v x (x, t) =, x =, lπ, t >, u(x, ) = u (x), v(x, ) = v (x), x (, lπ). 13

5 J Math Chem (13) 51: Again (.5) has a unique positive constant steady state (u α,v α ) = (β, α + β β ). We choose the fixed parameters β,l properly, and use α as the main bifurcation parameter. Define X ={(u,v) H [(, lπ)] H [(, lπ)] u () = v () = u (lπ) = v (lπ) = }. The linearized operator of system (.5) at(β, α + β β ) is L(α) = x + α β 1 α β It is well-known that eigenvalue problems β d x. β ψ = μψ, x (, lπ), ψ () = ψ (lπ) =, has eigenvalues μ n = n /l (n =, 1,,...), with corresponding eigenfunctions ψ n (x) = cos n l x.let ( ) φ = ϕ n= ( an b n ) cos n l x (.) be an eigenfunction for L(α) with eigenvalue μ(α), that is, L(α)(φ, ϕ) T = μ(α)(φ, ϕ) T. From [], there exists n N such that L n (α)(a n, b n ) T = μ(α)(a n, b n ) T, where L n is define by n L n (α) = l + α β 1 α β The characteristic equation of L n (α) is where β d n. (.7) l β μ T n (α)μ + D n (α) =, n =, 1,,..., (.) T n (α) := α β β n l ( + d ), (.9) n ( D n (α) := d l + β αd ) n β l + β. (.1) 13

6 J Math Chem (13) 51:1 19 and the eigenvalues μ(α) of L n (α) are given by μ(α) = T n(α) ± Tn (α) D n(α), n =, 1,,... We identify the Hopf bifurcation value α satisfying the following condition. (H 1 ) There exists n N, such that T n (α) =, D n (α) >, T j (α) =, D j (α) =, for any j = n. Let the unique pair of complex eigenvalues near the imaginary axis be γ(α)± iω(α), then the following transversality condition holds: For j N, define α H j = β 3 + β j then T j (α H j ) = and T i (α H j ) = fori = j. Define l n = n γ (α) =. (.11) l ( + d ), (.1) + d 1 β, n N. (.13) Then for l n < l l n+1, we have exactly n + 1 possible Hopf bifurcation points α = α H j ( j n) defined by (.1), and these points satisfy that α H (=β3 )<α H 1 < <αh n <β. Next we verify whether D i (α H j ) = for all i N, and in particular, D j (α H j )>. We claim that if then D i (α H j )>for any i N and α H j from (.1), 13 ( d 1) <β<1, (.1) (,β). Indeed since α H j /β < 1, then ( D i (α H j ) = pi d + p i β α H ) j d + β β > p i d + p i ( β d ) + β, (.15)

7 J Math Chem (13) 51: where p i = i l.ifβ d /, then β d and D i (α H j )>from(.15). If then from (.15), ( 1) d <β< d, D i (α H j )>pi d + p i ( β d ) + β β d ( 1d β 1 d = d ( 1 β ( + 1) d )( β ( 1) d ) >. d ) Finally let the eigenvalues close to the pure imaginary one near at α = α H j iω(α). Then be γ(α)± γ (α H j ) = T j (α H j ) = 1 β >. Now by using the Hopf bifurcation theorem in [], we have Theorem.1 Let l n as defined in (.13), and assume that l n < l l n+1 for some n N. Suppose that, d and β satisfy (.1). Then for (.5), there exist n +1 Hopf bifurcation points α H j ( j n) defined by (.1), satisfying β 3 = α H <αh 1 <αh < <αh n <β. At each α = α H j, the system (.5) undergoes a Hopf bifurcation, and the bifurcating periodic orbits near (α, u, v) = α H j (,β, α H ) j + β β can be parameterized as (α(s), u(s), v(s)), so that α(s) C in the form of α(s) = α H j + o(s), s (,δ)for some small δ>, and ( u(s)(x, t) = β + s a n e πit/t (s) + a n e πit/t (s)) cos n l x + o(s), v(s)(x, t) = α H j + β β + s (b n e πit/t (s) + b n e πit/t (s)) cos n (.1) l x + o(s), where (a n, b n ) is the corresponding eigenvector, and T (s) = π/ D j (α H j ) + o(s) (D j is defined in (.1)). Moreover 1. The bifurcating periodic orbits from α = α H = β 3 are spatially homogeneous, which coincide with the periodic orbits of the corresponding ODE system;. The bifurcating periodic orbits from α = α H j are spatially nonhomogeneous. 13

8 J Math Chem (13) 51:1 19 Next we follow the methods in [] to calculate the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits bifurcating from α = α H.Wehave the following result. Theorem. For equation (.5), when <β<1, the Hopf bifurcation at α H = β 3 is supercritical. That is, for a small ε>and α (α H,αH + ε), there is a small amplitude spatially homogenous periodic orbit, and this periodic orbit is locally asymptotically stable orbits. Proof Here we follow the notations and calculations in []. When α = α H = β3, Eq. (.) has a pair of purely imaginary eigenvalues μ =±iβ. For the Jacobian matrix ( L (α) = an eigenvector q of eigenvalue iβ satisfying β β 1 β β L q = iβq ), (.17) can be chosen as q := (a, b ) T = ( β,β i) T. Define the inner product in X C by U 1, U = lπ (u 1 u + v 1 v )dx, where U i = (u i,v i ) T X C, i = 1,. We choose an associated eigenvector q for the eigenvalue μ = iβ satisfying L q = iβq, q, q =1, q, q =. ( 1 iβ Then q := (a, b )T = βlπ, i ) T. lπ Let f (u,v)= a u +u v and g(u,v)= b u v, by calculation, at (β, β + ) β3 β, we have g uu = f uu, g uv = f uv, g uuv = f uuv, f vv = f vvv = f uvv = f uuu = g uuu = g vv = g uvv = g vvv =, f uu = 1 + (.1) β, f uv = β, f uuv =. β By direct calculation, it follows that c = f uu a + f uva b + f vv b = β 3β3 + β i, d = g uu a + g uva b + g vv b = c. e = f uu a + f uv (a b + a b ) + f vv b = β 3β 3, 13

9 J Math Chem (13) 51: f = g uu a + g uv (a b + a b ) + g vv b = e. g = f uuu a a + f uuv ( a b +a b )+ f uvv ( b a + b a )a = β 3 β i, h = g uuu a a + g uuv ( a b + a b ) + g uvv ( b a + b a ) = g. Denote Q qq = ( c d ), Q q q = ( ) ( ) e g, C f qq q =. (.19) h Then q, Q qq = q, Q q q = q, C qq q = q, Q qq = q, Q q q = lπ lπ lπ lπ lπ q, Q qq = q, Q qq, q, C qq q = q, Q q q. ( 1 + iβ βlπ c + i ) lπ d dx = c β, ( 1 + iβ βlπ e + i ) lπ f dx = e β, ( 1 + iβ βlπ g + i ) lπ h dx = g β, ( βc + (β + i)d ) dx = c β, ( βe + (β + i) f ) dx = e β, Hence H = (c, d ) T + c β (a, b ) T + c β (a, b ) T = c (1, 1) T + c ( 1, 1) T =, H 11 = (e, f ) T + e β (a, b ) T + e β (a, b ) T = e (1, 1) T + e ( 1, 1) T =, which implies that ω = ω 11 =, then q, Q ω11 q = q, Q ω q =. (.) 13

10 1 J Math Chem (13) 51:1 19 v u v u Fig. 1 Phase portraits and periodic orbits for the ODE system corresponding to (.5) whenβ =.9. The horizontal axis is u,andthevertical axis is v. Left: α =.7, a small amplitude limit cycle; Right: α =., a large amplitude limit cycle. Here the Hopf bifurcation point α H =.79 Thus { i Re(c 1 (α)) = Re q, Q qq q, Q q q + 1 } ω q, C qq q = 1 3β 3β = 1, Moreover Re(c 1 (α)) <, since T (α) = β + α β, hence T (α) = 1 >. Therefore β when α>α H = β3, the equilibrium point of (.) is unstable, and the system must have a periodic orbit by the Poincaré Bendixson theorem. From the result in [], the periodic orbit must be unique. From calculation above, the Hopf bifurcation at α = α H is supercritical; and when α (α H,αH + ε), the bifurcating periodic orbit is locally asymptotically stable. The periodic orbits of system (.) for some parameters are shown in Fig. 1, and a bifurcation diagram of the periodic orbits is shown in Fig.. 3 Steady state bifurcation In this section, we consider the steady state bifurcations of the system (.1). We consider the system of two coupled ordinary differential equations: u + β α u + u v =, x (, lπ), d v + β + α u (3.1) v =, x (, lπ), u () = u (lπ) = v () = v (lπ) =, which is a special case of more general steady state equation for a higher dimensional domain: 13

11 J Math Chem (13) 51: Max(V) H alpha Fig. Bifurcation of periodic orbits for the ODE system corresponding to (.5) when β =.9. The horizontal axis is α, and the vertical axis is max v(t) for the limit cycle (u(t), v(t)). Here the Hopf bifurcation point α H =.79 u = β α u + u v, d v = β + α u v, u n = v n =, x, x, x. (3.) The following a priori estimate can be easily established via maximum principle, and we omit the proof (see [,]). Lemma 3.1 Suppose that β> and β>α> β. Then any positive solution (u,v) of (3.) satisfies β α u(x) β + d (β + α), x, (3.3) (β α) and α + β [ β + d ] (β + α) v(x) (β α) (β + α), x. (3.) (β α) Recall D n (α) and T n (α) defined in (.9) and (.1). Now we identify steady state bifurcation values α of steady state system (3.1), which satisfy the following condition: (H ) there exists n N such that D n (α) =, T n (α) =, D j (α) =, T j (α) =, for any j = n; (3.5) 13

12 1 J Math Chem (13) 51:1 19 and d dα D n(α) =. (3.) Apparently, D (α) = β = forα >, hence we only consider n N. Inthe following we fix an arbitrary β>, to determine α values satisfying condition (H ), we notice that D n (α) = is equivalent to ( D(α, p) = d p + β αd ) p + β =, (3.7) β where p = n. Solving α from (3.7), we obtain that l α(p) = βp + β3 d p + d β 3. (3.) We also solve p from the equation (3.7), and we have p = p ± (α) = αd β 3 ± ( β 3 αd ) d β. (3.9) d β Since we require that α<β, we define l n,+ := n p+ (β), n l n, :=, n = 1,,..., (3.1) p (β) where p ± are defined in (3.9). For a fixed n N, if l n,+ < l < l n,, then there exists a unique αn S := α(n /l ) such that D n (αn S) =. These points αs n are potential steady state bifurcation points. The function α(p) and the functions p ± (α) satisfy the following properties. Lemma 3. Define p = β d1 d, ( α = β + β d d ). (3.11) Then the function α(p) : (, ) R + defined in (3.) has a unique critical point at p, which is the global minimum of α(p) in (, ), and lim α(p) = lim α(p) = p + p. Consequently for α α := α(p ), p ± (α) are well defined as in (3.9); p + (α) is strictly increasing and p (α) is strictly decreasing, and p + (α ) = p (α ) = p. Proof Differentiating D(α(p), p) = twice and letting α (p) =, we obtain that 13 d dp D(α(p), p) = pd + β α (p)p d β + α(p)d β =,

13 J Math Chem (13) 51: d dp D(α(p), p) = d α (p)p d β α (p) d β =, If α (p) =, then α (p)p d β d =, which is α (p) = β p >. Therefore for any critical point p of α(p), wemusthaveα (p) >, and thus the critical point must be unique and it is a local minimum point. It is easy to see that lim α(p) = lim α(p) =, hence the unique critical p + p point p is the global minimum point. Since (3.9) is also obtained by (3.7), then ={(α(p), p) : < p < } and the curve (α, α ± (p)) are identical. Then the properties of α(p) determine the monotonicity and limiting behavior of p ± (α). From Lemma 3., ifforsomei, j with i < j, wehaveα(p i ) = α(p j ), then p (αi S) = p +(α S j ) must hold. In this case, for α = αs i = α S j, is not a simple eigenvalue of L(α), and we shall not consider bifurcations at such points. We notice that, from the properties of p ± (α) in Lemma 3., the multiplicity of as an eigenvalue of L(α) is at most. On the other hand, it is also possible that some αi S = α H j (a Hopf bifurcation point). So the dimension of center manifold of the steady state (u α,v α ) can be from 1 to. Following an argument in [], we can prove that there are only countably many l >, in fact only finitely many l (, M) for any given M >, such that α = αi S = α H j or αi S = α S j for these l and i, j N. We define the set of these l to be L E ={l > : α(i /l )=α(j /l ) or α(i /l )= α(j /l ), α [α, ), i, j N}, (3.1) where α(p) is the function defined in (3.), and α(p) = β 3 + β( + d )p, (3.13) following (.1). Then the points in L E can be arranged as a sequence whose only d limit point is. Finally we show that dα D j(α S j ) =. By direct calculation, we have d dα D j(α S j ) = p d j β <, p j = j l. Now we state the main result about the steady state solution bifurcation of (3.1). Theorem 3.3 Assume that <β<( d 1), (3.1) 13

14 1 J Math Chem (13) 51:1 19 and let n N. Suppose that l ( l n,+, l n, ) \ L E, where l n,± is defined in (3.1) and L E is a countable subset of R + defined in (3.1). Then αn S = α(n l ) satisfies α <αn S <β, and α = αs n is a bifurcation point for (3.1). Moreover, 1. There exists a smooth curve Ɣ n C of positive solutions of (3.1) bifurcating from (α, u,v) = (αn S, u αn S,v αn S ), with Ɣ n contained in a global branch C n of positive solutions of (3.1).. Near (α, u,v) = (αn S, u αn S,v αn S ), Ɣ n = {(α n (s), u n (s), v n (s)) : s ( ε, ε)}, where u n (s) = β +sa n cos(nx/l)+sψ 1,n (s), v n (s) = β + αs n β +sb n cos(nx/l)+ sψ,n (s), fors ( ε, ε) for some C smooth functions α n,ψ 1,n,ψ,n such that α n () = αn S, ψ 1,n() = ψ,n () = ; Here a n, b n satisfy L(α S n )[(a n, b n ) T cos(nx/l)] =(, ) T. (3.15) 3. Either C n contains another (α, u,v)= (αm S, u αm S,v αm S ) for m = n, or the projection proj α C n of C n on the α-axis satisfies for some δ>. ( β + δ, β) proj α C n (αn S,β), (3.1) Proof If β satisfies (3.1), then α <α S n <β. Since we have verified conditions (H ) in previous paragraphs, then a bifurcation of steady state solutions occurs at α = α S n. Note that we exclude L E (as defined in (3.1)), so α = α S j is always a bifurcation from a simple eigenvalue point. From the global bifurcation theorem in Shi and Wang [1], Ɣ n is contained in a global branch C n of solutions. Hence the results stated here are all proved except (i) C n consists of positive solutions only; and (ii) part 3. Firstly we prove that any solution on C n is positive if α ( β,β). This is true for solutions on Ɣ n as β> and α S n +β>. Suppose that there is a solution on C n which is not positive. Then by the continuity of C n, there exists an (α, u,v ) C n such that α ( β,β),u (x), v (x) forx, and there exists x, such that u (x ) = orv (x ) =. Suppose that u (x ) =, then u reaches its minimum at x.ifx, then we get a contradiction with u (x ) = (β α)/ >. If x, then near x = x,wehave u (x )> and x is the local minimum, then n u (x )< which contradicts with the zero Neumann boundary condition. Thus u (x ) = is impossible. Similarly we can prove that v (x ) = is impossible. This shows that any solution of (3.1)onC n is positive as long as α ( β,β). Secondly we notice that when α = β, the only solution of (3.1) is(u,v) = ((β α)/, ). And as we have shown here the only bifurcation points for steady state solutions are α = α S n. Thus proj αc n must have a lower bound which is strictly larger than β. From Lemma 3.1, all positive solutions of (3.1) are uniformly bounded for α [ β,β δ). Hence C n cannot be unbounded for α [ β,β δ). From the global bifurcation theorem in [1], either C n contains another (α, u,v)= (α S m, u α S m,v α S m ) for m = n, orc n is unbounded, or C n intersects the boundary of ( β,β) X X, 13

15 J Math Chem (13) 51: D(α,p)= p T(α,p)= D(α,p)= p 3 1 T(α,p)= β 3 α α α * α β 3 * Fig. 3 Graph of T (α, p) = andd(α, p) =. (Left) β =., =., d =.1 andl = 1; (Right) β =.3, =.1, d = 1andl = 3. The horizontal lines are p = n /l where X is a function space. But either one in the latter two alternatives implies that proj α C n (αn S,β). This completes the proof. Remark The continuum C n in Theorem 3.3 is possibly unbounded as α β as the upper bounds in Lemma 3.1 tend to infinity as α β. But it is also possible that C n can be extended to α = β, which corresponds to the case a = forthe original parameters. Since a is not physically reasonable, we do not discuss the case that a.. If d, then all the bifurcating steady state solutions on Ɣ n near α = αn S are unstable since αn S >α >β 3 /d β 3 = α H, which is the primary Hopf bifurcation point. Numerical simulations and discussion We have identified two critical parameter values for the system (1.): α = β 3, which is the smallest Hopf bifurcation point, and α = α (defined as in (3.11)). The constant steady state (u α,v α ) is locally asymptotically stable when α<min{β 3,α }. When β 3 <α, then (u α,v α ) loses the stability at α = β 3 through a Hopf bifurcation; when α <β 3, a steady state bifurcation is likely to happen for some α (α,β).the left panel of Fig. 3 shows the graph of T (α, p) = and D(α, p) = with a set of parameters so that β 3 <α, while the right panel shows the one for the case α <β 3. For the latter case, one can choose the length parameter l so that there are steady state bifurcation points in (α,β). Indeed for the parameters given in the right panel of Fig. 3, if we choose l = 3, then we have α5 S =.13 <αs =.19 <αs =. <αs 7 =.1 <α S =.5 <αh =.7 <αs 3 = αs 9 =.33. (.1) We use several numerical simulations to illustrate and complement our analytical results. The simulations are generated by using the Matlab pdepe solver which solves initial-boundary value problems for parabolic-elliptic PDEs in 1-D. For the parameters 13

16 1 J Math Chem (13) 51:1 19 u1(x,t) u(x,t) Time t Distance x Time t Distance x 1 Fig. Numerical simulation of the system (1.). (Left) u(x, t); (Right) v(x, t). Here α =.5,β =., =., d =.1, l = 3, t 3, and the initial values u (x) =. +.1cos(x); v (x) = cos(x). The solution converges to a spatially homogenous periodic orbit u1(x,t) u(x,t) Time t Distance x Time t Distance x Fig. 5 Numerical simulation of the system (1.). (Left) u(x, t); (Right) v(x, t). Here α =.,β =.3, =.1, d = 1, l = 3, t 1, and the initial values u (x) = sin(x); v (x) = The solution converges to a spatially non-homogenous steady state solution 1 given in the left panel of Fig. 3 with l = 3, a simulation with α =.5 >α H =.51 is shown in Fig., which is dominated by the ODE limit cycle dynamics. On the other hand, for the parameters given in the right panel of Fig. 3 with l = 3, several steady state bifurcations occur at α-values smaller than α H (see (.1)). Figures 5, and 7 show three different solution trajectories with different initial conditions. While each shows a stationary spatial pattern, the one in Fig. 5 appears to have spatial period 3π/, which corresponds to n = (mode cos(x/3)); the one in Fig. appears to have spatial period π, which corresponds to n = (mode cos(x)); and finally the one in Fig. 7 is not spatially periodic but with peaks of different height. For the simulations in Figs. and 7, if the integration time is chosen longer, then the observed patterns can switch to a different pattern. This suggests the spatial patterns observed in earlier time may be unstable steady states. Note that such asymmetric peak solutions have also been observed in [17]for(1.). From (.1), the parameter α =. in Figs. 5, and 7 is between α S =.19 and αs =.. Our analytical results given here and the numerical simulations guided by the analytical results show a rough picture of the dynamics in terms of system parameters. 13

17 J Math Chem (13) 51: u1(x,t) u(x,t) Time t 1 Distance x Time t Distance x 1 Fig. Numerical simulation of the system (1.). (Left) u(x, t); (Right) v(x, t). Here α =.,β =.3, =.1, d = 1, l = 3, t, and the initial values u (x) = sin(3x); v (x) = cos(x). The solution shows a spatially periodic pattern with wave length π u1(x,t) u(x,t) Time t 1 Distance x Time t 1 Distance x 1 Fig. 7 Numerical simulation of the system (1.). (Left) u(x, t); (Right) v(x, t). Here α =.,β =.3, =.1, d = 1, l = 3, t, and the initial values u (x) =.3+.1 sin(5x/3); v (x) = cos(5x/3). The solution shows a spatially asymmetric pattern There are several parameter regimes where the dynamical behavior of (1.) are drastically different. 1. When β is large, then for any α ( β,β),, d >, the solution of (1.) appears to converge to the constant steady state. Indeed our analytical results show that when <β<min{β 3,α },(α is defined in (3.11)), that is equivalent to { β>max 1,( 1) d }, (.) then (u α,v α ) is locally asymptotically stable for any α ( β,β). The convergence to the constant steady state can be easily seen by simulation, but the analytical proof of global stability for PDE remains open. We also conjecture that (u α,v α ) is globally asymptotically stable for any α ( β,] regardless the value of β (which corresponds to the case b < a in terms of original parameters.) 13

18 1 J Math Chem (13) 51:1 19. When <β 3 <β<α is satisfied, that is ( d 1) <β<1, (.3) then there are a sequence of Hopf bifurcation points β 3 = α H <α1 H < < αn S <βwhere periodic orbits of (1.) bifurcate out from the constant steady state (u α,v α ) (see Theorem.1). In particular, (u α,v α ) loses the local stability to a spatially homogenous periodic orbit at α = β 3. On the other hand, since β<α, there is no steady state bifurcation for any α ( β,β). Hence the parameter regime given by (.3) is dominated by time-periodic patterns but probably not stationary spatially nonhomogeneous patterns. The number n of spatially nonhomogeneous Hopf bifurcation points depends on the length parameter l. 3. When <α <β<β 3 is satisfied, that is 1 <β<( d 1), (.) then there exist a number of steady state solution bifurcation points α S j for m j n and m, n N where spatially nonhomogeneous steady state solutions bifurcate from the constant steady state (u α,v α ) (see Theorem 3.3). Again the number n m of spatially nonhomogeneous Hopf bifurcation points depends on the length parameter l. On the other hand, since β < β 3, there is no Hopf bifurcation for any α ( β,β). Hence the parameter regime given by (.) is dominated by stationary spatially nonhomogeneous patterns but probably not time-periodic patterns.. Finally if { <β<min 1,( 1) d }, (.5) then < max{α,β 3 } <β. In this case both the results in Theorem.1 and the ones in Theorem 3.3 are applicable. Hence possibly both Hopf bifurcations and steady state bifurcations occur for α (min{α,β 3 },β), and these bifurcation points form an intertwining sequence of bifurcation points. The pattern formation in the parameter regime (.5) is perhaps the most complicated one. We also comment that near the codimension-two Hopf Turing bifurcation point, where the two instabilities happen at the same parameters, these instabilities can compete with each other which results in more complicated spatiotemporal patterns. The Hopf Turing point in this model is the intersection point of curves T (α, p) = and D(α, p) = infig. 3. Some previous studies in Hopf Turing bifurcations can be found in [1,7,1]. Acknowledgments We thank two anonymous reviewers for their helpful comments and suggestions. 13

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