DYNAMICS AND SPATIOTEMPORAL PATTERN FORMATIONS OF A HOMOGENEOUS REACTION-DIFFUSION THOMAS MODEL

. In this paper, we are mainly considered with a kind of homogeneous diﬀusive Thomas model arising from biochemical reaction. Firstly, we use the invariant rectangle technique to prove the global existence and uniqueness of the positive solutions of the parabolic system, and then use the max- imum principle to show the existence of attraction region which attracts all the solutions of the system regardless of the initial values. Secondly, we con- sider the long time behaviors of the solutions of the system; Thirdly, we derive precise parameter ranges where the system does not have non-constant steady states by using use some useful inequalities and a priori estimates; Finally, we prove the existence of Turing patterns by using the steady state bifurcation theory.

1. Introduction. In this paper, we consider a reaction-diffusion biochemical Thomas model which was first proposed by Thomas in [9] to explain the observed oscillatory behavior in substrate-inhibition chemical reaction involving the substrates oxygen and uric acid which react in the presence of the enzyme uricase.
The dimensionless form of the empirical rage equations for the uric acid (denoted by u) and the oxygen (denoted by v) can be written in the following form: x ∈ Ω, t > 0, where Ω is an open bounded domain in R n , n ≥ 1, with smooth boundary ∂Ω; u = u(x, t) and v = v(x, t) stand for the concentrations of the uric acid and the oxygen at the position x and the time t respectively; d 1 and d 2 are diffusion coefficients of u and v respectively. u 0 , v 0 ∈ C 2 (Ω) ∩ C 0 (Ω) and the Neumann boundary conditions indicate that there are no flux of the chemical substances of u and v on the boundary. Here the parameters a, α, b, ρ and k are all positive. Basically, the acid u and the oxygen v are assumed to be supplied at a constant rates a and αb, degrade linearly proportional to their concentrations and both are used up in the reaction at a rate ρuv/(1+u+ku 2 ), which exhibits substrate inhibition. The parameter k is a measure of the severity of the inhibition.
The Thomas model (1) has been studied extensively by several authors, but most of the research focuses either on the corresponding ODE system. Thomas [9] considered the boundedness of the solutions of ODE system of system (1) by proving the existence of invariant rectangles. To the best of our knowledge, in the existing literatures, there are few works investigating the dynamics of the reaction diffusion equations (1). It is then our purpose to consider the dynamics and pattern formations of this particular biochemical model. The Thomas model (1) is similar to but slight different from the Seelig model ( [7,10]). In a recent paper due to Yi et al [10], the authors considered the dynamics of the Seelig model. They not only proved the global existence and boundless of the in-time solutions, but also show the existence of attraction regions which attract all the solutions of the system regardless of the initial values. Furthermore, the considered the existence and non-existence of Turing patterns for the Seelig model. Motivated by the observations of [10], we are concerned with whether the Thomas model has the same dynamics as that of the Seelig model. This paper is organized as follows. In Section 2, we study the boundedness and uniqueness of global-in-time solutions of the parabolic system (1). In particular, we show that an invariant rectangle exists which attracts all the solutions of the parabolic system (1) regardless of the initial values. Then, we consider the long time behaviors of the solutions of system (1), and derive precise conditions so that the solutions of R-D (1) converge exponentially either to its unique constant steady state solution, or to its stable spatially homogeneous orbitally periodic solutions. In Section 3, we derive conditions so that system (1) does not have non-constant positive steady states, including Turing patterns. In Section 4, we use global bifurcation theory to prove the existence of Turing patterns.
2. Attraction region and large time behaviors of the solutions. For convenience of our discussions, we copy (1) here: x ∈ Ω, t > 0, To begin with, we derive precise conditions so that system (2) has a unique positive constant equilibrium solution (u * , v * ).
Proof. By Substituting (6) into (5), we have Define We then have Let ∆ H be the discriminant of the function H(u), and u H be the symmetry axis of the function H(u). Then Clearly, F(a) = ρab > 0, F(0) = −a < 0. Then, there must exist at least one zero point u * in (0, a). Under (3), we have H(u) ≥ 0 for all u ∈ (0, a), which implies that the zero point u * is unique in (0, a). On the other hand, under (4), we have ∆ H > 0, u H < 0, H(0) ≥ 0, then H(u) > 0, u ∈ (0, a). Again, we can conclude that that the zero point u * is unique in (0, a).
Finally, by a b < α, we can obtain the positivity of v * via the positivity of u * .
Next, we prove that (2) has a unique solution (u(x, t), v(x, t)) defined for all t > 0 and x ∈ Ω, and is bounded by some positive constants depending on a, α, b, ρ, k, and the maximum and minimum of the initial conditions, u 0 (x) and v 0 (x). Proposition 1. Suppose that a, α, b, ρ, k are all positive, a b < α holds. If additionally, either (3) or (4) holds so that (u * , v * ) is the unique positive constant equilibrium solution. Then, for any d 1 , d 2 > 0, the parabolic system (2) has a unique solution (u(x, t), v(x, t)), defined for all x ∈ Ω and t > 0. Moreover, there exist two positive constants C 1 and C 2 , depending on a, α, b, ρ, K, u 0 (x) and v 0 (x), such that Proof. By [2], we can prove the existence and uniqueness of local-in-time solutions to the parabolic system (2). We now use the techniques in [5,3,10] to prove the global existence and the boundedness of the solutions. We consider two cases: in the following way: Clearly, u 0 (x) and v 0 (x) are closed by the rectangle R. We now prove that the vector field points inward on the boundary of R. In fact, On the bottom side of On the right side of R, On the left side of R, is the invariant rectangle for the vector field. Choosing C 1 = min{U 1 , V 1 } and C 2 = max{U 2 , V 2 }, we obtain the desired results.

Case 2. Suppose that max
x∈Ω v 0 (x) > a/ρ holds. In this case, the aforementioned R is not the invariant rectangle anymore, since the last inequality in (16) fails. Since the inequalities in (13), (14) and (15) still hold, we can conduct the same analysis as in [10] to show that one can construct a new invariant rectangle as we did in Case 1. This leads to another suitable positive constants C 1 and C 2 . So far, we have proved the global existence and boundedness of the solutions.
We then show that system (2) has an attraction region defined by in the phase plane which actually attracts all solutions of this system, regardless of the initial values u 0 and v 0 .
be the unique solution of system (2). Then, for any x ∈ Ω, we have Proof. 1). Firstly, we prove that lim inf t→∞ v > αb α + ρ . By Proposition 1, there exists a sufficiently small τ > 0 such that for all x ∈ Ω and t > 0, ρuv Let v τ be the unique solution of the following ODE: Defining , and by (2) and (19), we have By the maximum principle for parabolic equations, it follows 2). We now prove that lim t→∞ sup v < b. By Proposition 1, there exists a sufficiently small 0 < δ < b such that for all x ∈ Ω and t > 0, δ < ρuv/(1 + u + ku 2 ) holds.
Let v δ = v δ (t) be the unique solution of the following ODE: Letting , and by (2) and (21), we have By the maximum principle for parabolic equations, we have p 2 (x, t) < 0, or euqiv., v(x, t) < v δ (t), for all x ∈ Ω and t ≥ 0. Since lim 3). We then prove that lim t→∞ sup u < a. By Proposition 1, there exists a sufficiently small 0 < ζ < a such that for all x ∈ Ω and t > 0, ζ < ρuv 1 + u + ku 2 holds.
Let u ζ = u ζ (t) be the unique solution of the following ODE: Defining , and by (2) and (23), we have By the maximum principle of parabolic equations, we have w 1 (x, t) < 0, or equiv., u(x, t) < u ζ (t), for all x ∈ Ω and t ≥ 0. Since lim t→∞ u ζ (t) = a − ζ, we have lim t→∞ sup u < a.

4). Finally, we prove that lim
we can find a finite number t 0 , depending on u 0 and v 0 , such that for any t ≥ t 0 and all By Proposition 1 and (26), there exists a sufficiently small χ > 0, such that for all x ∈ Ω and t ≥ t 0 , one has ρuv This can be done by choosing χ > 0 sufficiently small, since when χ = 0, (27) holds automatically. Let u χ be the unique solution of the following ODE: Letting , and by (2) and (28), we have Then by the maximum principle for parabolic equations, we have In what follows, we consider the long time behaviors of the solutions of the system (2). Following [1], we define σ := dλ 1 − Q, where λ 1 is the principal eigenvalue of −∆ on Ω subject to homogeneous Neumann boundary conditions, d := min{d 1 , d 2 }, and Q := sup where where Obviously, for u, v > 0, the following inequalities hold For any (u, v) ∈ A, defined precisely in (17), we have Thus, The following results show that, under certain conditions, the solutions of system (1) either converge exponentially to the unique positive constant steady states or to the spatially homogeneous periodic solutions.  , t), v(x, t)) of system (2) converges exponentially to (u * , v * ); while if (45) holds, then every solution (u(x, t), v(x, t)) of system (2) converges exponentially to the spatially homogeneous periodic solutions.
Proof. By Theorem 2.2, it follows that, there exists T > 0, such that for any t > T , the solution (u(x, t), v(x, t)) ∈ A for all x ∈ Ω. Without loss of generality, we can assume that T = 0.
Then by [1,3], there exist constants N i > 0, i = 1, 2, 3, such that, for any solution (u(t, x), v(t, x)) of system (2) where u, v are the average of u and v over Ω respectively satisfying Moreover, the ω−limit set of (38) is the subset of the ω−limit set of the following We now consider the dynamics of the ODE system of the original system (2), which was given by The linearized operator of system (40) evaluated at (u * , v * ) is given by where Then, the characteristic equation of (41) is given by Suppose that a, α, b, ρ, k are all positive, a b < α holds. If additionally, either (3) or (4) is satisfied so that (u * , v * ) is the unique positive constant equilibrium of (2). If holds, then (u * , v * ) is locally asymptotically stable in system (40). However, if holds, then (u * , v * ) is unstable in system (40), and the system (40) has a locally orbitally stable periodic orbit, denoted by (p(t), q(t)). If (44) holds, then (u * , v * ) is locally asymptotically stable; While if (45) holds, then (u * , v * ) is unstable. By Theorem 2.2, the solutions is bounded. Thus, from Poincare-Bendixson theorem, we conclude the existence of a locally orbitally stable periodic orbit, denoted by (p(t), q(t)).
So far, under (46) and 0 < a−α ≤ 1, by Dulac criteria, system (40) does not have closed orbits in the first quadrant. By Theorem 2.2, it follows that the solution is bounded. Thus, by Poincare-Bendixson Theorem, we know that (u * , v * ) is globally asymptotically stable in ODEs.
3. Non-existence of Turing patterns: Some estimates. This part, we discuss the non-existence of the non-constant positive steady state solutions of the system: x ∈ ∂Ω. holds. If additionally, either (3) or (4) is satisfied, Let (u(x), v(x)) be any given positive steady state solution of system (1). Then, for any x ∈ Ω, the following conclusions hold: For a steady state solution pair (u(x), v(x)) of system (50), we define (52) Multiplying the first equation of (50) by −1, and adding the second equation of (50), we can obtain that Integrating (53) over Ω, we obtain Define and We are now stating the following useful estimates on the steady state solutions: is the solution pair of (50), and let φ(x), ψ(x) be defined in (52). Then, we have where λ 1 is the principle eigenvalue of −∆ on Ω subject to the homogeneous Neumann boundary conditions.
For the later discussions, we define , where u are defined precisely in (52).
From Lemma 3.1, it follows that, any positive solutions (u, v) of system (50) satisfies (u, v) ∈ A, where A is defined in (17). Define We are now in the position to state the following theorem regarding the nonexistence of non-constant positive solutions of the system (50): Proof. We first prove that if (d 1 , d 2 ) ∈ Σ 1 ∪ Σ 2 , then system (50) does not have non-constant positive solutions.
Multiplying the second equation of (50) by ψ and integrating over Ω, we have A direct calculation shows that On the other hand, we have (87) By (87) and (86), we have Then we discuss in two cases: By (88) and Lemma 3.2, we have For any (d 1 , d 2 ) ∈ Σ 1 , we have For any (d 1 , d 2 ) ∈ Σ 2 , we have Remark 3. In particular, when we have Thus, we have ∇ψ ≡ 0. By Lemma 3.3, we have ∇φ ≡ 0. Then system (50) does not have non-constant positive solutions.
Multiplying the second equation of (50) by φ and integrating over Ω, we have A direct calculation shows that Then we discuss in two cases: By (96) and Lemma 3.2, we have For any (d 1 , d 2 ) ∈ Σ 3 , we have (98) Thus, we have ∇φ ≡ 0. By Lemma 3.3, we have ∇ψ ≡ 0. Then system (50) does not have non-constant positive solutions.
Case 2. d 2 − αd 1 < 0. By (96) and Lemma 3.2, we have For any (d 1 , d 2 ) ∈ Σ 4 , we have For any (d 1 , d 2 ) ∈ Σ 4 , we have ∇φ ≡ 0. By Lemma 3.3, we have ∇ψ ≡ 0. Thus, system (50) does not have non-constant positive solutions. 4. Existence of Turing patterns: Global steady state bifurcations. In this section, we use the global steady state bifurcation theory to consider the existence of positive non-constant of steady state system (50), in particular the existence of Turing patterned solutions.
Let h 1 and h 2 be defined precisely in (42). Then, if h 1 < − 1 ρ holds, system (2) is a substrate-inhibition system. That is., the Jacobian matrix of the corresponding ODEs evaluated at (u * , v * ) takes in the form of And if a, α, b, ρ, k are all positive, a b < α holds. If additionally, either (3) or (4) is satisfied, and is satisfied, then (u * , v * ) is positive and stable in the corresponding ODEs system. Thus, in the rest of the paper, we always assume that a b < α holds. If additionally, either (3) or (4) is satisfied, and are satisfied. The linearized operator of system (50) evaluated at (u * , v * ) is given by (choosing d 1 as the bifurcation parameter) Let λ i and ξ i (x), i ∈ N 0 , be the eigenvalues and the corresponding eigenfunctions of −∆ in Ω subject to Neumann boundary conditions. Then, by [5,11], the eigenvalues of L(d 1 ) are given by those of the following operator L i (d 1 ): whose characteristic equation is where T i (d 1 ) : = −(d 1 + d 2 )λ i − (1 + ρh 1 + α + ρh 2 ), D i (d 1 ) : = d 1 d 2 λ 2 i + (d 2 + ρh 1 d 2 + αd 1 + ρh 2 d 1 )λ i + α + ρh 1 α + ρh 2 . (106) According to [8,11], if there exist i ∈ N 0 and d * 1 > 0, such that and the derivative d dd 1 D i (d * 1 ) = 0, then a global steady state bifurcation occurs at the critical point d * 1 . By (103), we have T 0 (d 1 ) < 0. Thus, for all i ∈ N 0 , we have T i (d 1 ) < 0. Solving D i (d 1 ) = 0, we have the set of critical values of (d 1 , d 2 ), given by the hyperbolic curves C i , with i ∈ N := N 0 \{0} : Suppose that λ i , i ∈ N, is the simple eigenvalue of −∆. Following [6], we call B := ∞ i=1 C i the bifurcation set with respect to (u * , v * ), and denote by B 0 be the countable set of intersection points of two curves of {C i } ∞ i=1 , and B = B\B 0 .