Breakdown of Pattern Formation in Activator-Inhibitor Systems and Unfolding of a Singular Equilibrium

Transcription

1 Breakdown of Pattern Formation in Activator-Inhibitor Systems and Unfolding of a Singular Equilibrium Izumi Takagi (Mathematical Institute, Tohoku University) joint work with Kanako Suzuki (Institute for International Advanced Interdisciplinary Research, Tohoku University) Partial Differential Equations in Mathematical Biology Bedlewo Conference Center, 13/sep/10

2 2/43 Plan of Talk Introduction 1. Boundedness of Solutions to IBVP 2. Breakdown of Pattern Formation 3. Possible Scenario

3 Introduction. Introduction 3/43 Diffusion-Driven Instability found by A. M. Turing in 1952: When two chemicals with different diffusion rates interact and diffuse, the spatially homogeneous state may become unstable, and as a result spatially nontrivial structure can be formed autonomously. Gierer and Meinhardt in 1972 developed Turing s idea and devised the following reaction-diffusion system which consists of a slowly diffusing activator and a rapidly diffusing inhibitor in order to simulate the transplantation experiment on hydra:

4 4/43 Introduction activator vs inhibitor

5 Introduction 5/43 Activator-Inhibitor System with Different Sources ([GM]) a t = D a 2 a µa + cρa2 x2 h + ρ 0ρ h t = D 2 h h x 2 νh + c ρ a 2 where D a, D h, c, c, ρ 0 are positive constants; µ(x), ν(x), ρ(x), ρ (x) are positive functions. The unknowns a = a(x, t) and h = h(x, t) denote the concentrations at point x and time t of chemicals called an activator and an inhibitor, respectively. It is postulated that a change in cells occurs at the place where the activator concentration is high. Let s take a look at two simulations.

6 6/43 Introduction movie1: covergence to a steady-state

7 Introduction 7/43 movie2: covergence to a limit cycle (1/3)

8 8/43 Introduction movie2: covergence to a limit cycle (2/3)

9 Introduction 9/43 movie2: covergence to a limit cycle (3/3)

10 10/43 Introduction Ω is a bounded domain in R n with smooth boundary Ω; ν = (ν 1,..., ν n ) is the unit outer normal to Ω.

11 Introduction 11/43 (GM) A t = A p ε2 Λ a A µ a A + ρ a H q (1 + κa p ) + σ a in Ω, τ H t = DΛ A r hh µ h H + ρ h H s + σ h in Ω, B a A = B h H = 0, on Ω A(x, 0) = A 0 (x), H(x, 0) = H 0 (x) in Ω. Λ a = n i,j=1 ( x i d (a) ij B a = n i,j=1 ν id (a) ij ) x j, Λ h = n i,j=1 x j, B h = n i,j=1 ν jd (h) ij ( x i d (h) ij x j. x j ), Here, Λ a and Λ h are uniformly strongly elliptic operators in Ω.

12 12/43 Introduction The coefficients ε, τ, D are positive constants whereas κ is a nonnegative constant. The basic production terms σ a = σ a (x), σ h = σ h (x) are nonnegative, and the cross-reaction rates ρ a = ρ a (x, A, H), ρ h = ρ h (x, A, H) are sufficiently smooth positive functions: 0 < ρ a C a, 0 < c h ρ h C h for x Ω, A R, H R. The decomposition rates µ a = µ a (x), µ h = µ h (x) are positive functions: 0 < k (a) 1 µ a (x) k (a) 2, 0 < k(h) 1 µ h (x) k (h) 2. The initial data u 0 (x), v 0 (x) are positive on Ω.

13 Introduction 13/43 The exponents p, q, r, s are assumed to satisfy (A) p > 1, q > 0, r > 0, s 0, p 1 q < r s + 1.

14 14/43 Introduction Nullclines in the Case of Homogeneous Media σ a > 0, σ h = 0 σ a = σ h = 0 f(a, H) = A + Ap H q + σ a = 0, g(a, H) = H + Ar H s + σ h = 0

15 Introduction 15/43 σ a > 0. σ h > 0 f(a, H) = A+ A p H q (1+κA p ) +σ a

16 16/43 Existence and Boundedness of Solutions 1. Existence of Solutions of the Initial-Boundary Value Problem To begin with, let us summarize the known results on the existence of solutions of the initial-boundary value problem (GM), to which many people contributed: F. Rothe [Rf], K. Masuda and K. Takahashi [MT] in 1980 s M. Lin, S. Chen and Y. Qin [LCQ] in 1990 s W.-M. Ni, K. Suzuki and I.T. [NST], H. Jiang [J], K. Suzuki and I.T. [ST] in 2000 s In the following Theorems A C, we assume (1.1) p 1 < r in addition to (A).

17 Existence and Boundedness of Solutions/Theorem A 17/43 Theorem A. ([MT]+[LCQ]) If σ a (x) 0, then the initialboundary value problem (GM) has a unique solution for all t > 0 and there exist positive constants r a, R a, r h, R h (r a < R a, r h < R h ) independent of the initial value (A 0 (x), H 0 (x)) such that r a lim inf min A(x, t) lim sup max A(x, t) R a, t + x Ω t + x Ω r h lim inf min H(x, t) lim sup max H(x, t) R h. t + x Ω t + x Ω

18 18/43 Existence and Boundedness of Solutions /Theorem B Theorem B. ([J], [ST]) If σ a (x) 0 and σ h (x) 0, then there exist positive constants R a, r h, R h initial value such that e k(a) 2 t min A 0 (x) min A(x, t) for all t 0, x Ω x Ω and lim sup t + max x Ω A(x, t) R a, independent of the r h lim inf min H(x, t) lim sup max H(x, t) R h. t + x Ω t + x Ω

19 Existence and Boundedness of Solutions / Theorem C 19/43 Theorem C. ([J], [ST]) If σ a (x) 0 and σ h (x) 0, then there exist positive constants λ, µ depending only on p, q, r, s, τ and a positive constant C depending on the initial value such that hold for all t > 0, x Ω. e k(a) 2 t min A 0 (x) A(x, t) Ce λt, x Ω e k(h) 2 t/τ min H 0 (x) H(x, t) Ce µt x Ω

20 20/43 Existence and Boundedness of Solutions /Blowup Results on the existence of global solutions appeared more than twenty years ago; see, e.g., [Rf], [MT]. In particular, [MT] proved the assertion of Theorem A under the condition (p 1)/r < N/(N + 2). It was [LCQ] that proved Theorem A, while Theorems B and C were obtained recently by [J], [S], [ST]. On the other hand, in the case of p 1 > r we have the following result. Proposition D. ([LCQ], [NST]) Assume that µ a, µ h, ρ a, ρ h are all positive constants and σ a (x) σ h (x) 0. If (1.2) p 1 > r then (GM) has solutions which blow up in finite time

21 Existence and Boundedness of Solutions / Summary 21/43 The case p 1 < r Basic production terms σ a (x) 0 σ a (x) 0, σ h (x) 0 σ a (x) 0, σ h (x) 0 Solutions are ultimately uniformly bounded. are ultimately uniformly bounded. may become unbounded. The case p 1 > r Some solutions blow up in finite time. p 1 is the self-activation index of the activator, whereas r is the cross-activation index of the activator.

22 22/43 Existence and Boundedness of Solutions / Summary Obviously, for the systematic study of global behavior of solutions of (GM), it is important to know the behavior of solutions of the following kinetic system: (K) du dt τ dv dt = u + up v q + σ a, = v + ur v s + σ h. Here we assume that σ a, σ h are both nonnegative constants. In this aspect, [NST] classified all the behavior of solution orbits in the case of σ a = 0 and σ h = 0 The case σ a > 0 is treated in an on-going project [NS].

23 Breakdown of Pattern Formation 23/43 2. Breakdown of Pattern Formation In some numerical simulations, it is observed that a solution starting from an almost uniform initial value develops localization in the activator concentration for a while, but it begins to oscillate and eventually converges uniformly to the trivial state u 0. We call this kind of phenomenon the collapse of patterns. In this section we would like to understand the mechanism behind the collapse of patterns and to know when it occurs. This section is based on the paper [ST4]. It is convenient to classify the basic production terms into four cases: Case I: σ a σ h 0; Case II: σ a 0 and σ h 0; Case III: σ a 0 and σ h 0; Case IV: σ a 0 and σ h 0.

24 24/43 Breakdown of Pattern Formation movie3: collapse of patterns (1/3)

25 Breakdown of Pattern Formation 25/43 movie3: collapse of patterns (2/3)

26 26/43 Breakdown of Pattern Formation movie3: collapse of patterns (3/3)

27 Collapse of Patterns / Theorem /43 Theorem 2.1. (Cases I and II) Assume that σ a (x) 0. If (2.1) τ > ( (2.2) min x Ω k (h) 2 q k (a) 1 (p 1), and H 0 (x)) q > k (a) C a (p 1) 2 q 1 (p 1) k(h) then the solution (A(x, t), H(x, t)) of (GM) satisfies τ ( p 1 max A 0 (x)), x Ω 0 < max x Ω A(x, t) Ce µ(a) 1 t, max H(x, t) Σ h,d (x) Ce µ(h) 1 t/τ, x Ω in which C is a positive constant depending on (A 0 (x), H 0 (x)), and u = Σ h,d (x) is the solution of the boundary value problem (2.3) DΛ h u µ h u + σ h (x) = 0 (x Ω), B h u = 0 (x Ω).

28 28/43 Collapse of Patterns / Theorem 2.1 τ > k (h) 2 q k (a) 1 (p 1) Collapse occurs for initial data contained in the gray region.

29 Collapse of Patterns / σ a 0 29/43 Theorem 2.2. (Case II) Assume that σ a 0 and σ h 0. Let δ h = min x Ω Σ h,d (x), (A 0 (x), H 0 (x)) satisfies min {( (δ h /Γ h ) k(h) 2 /k (h) 1 min > C a k (a) 1 Γ h = max x Ω Σ h,d (x). If the initial data x Ω ( p 1 max A 0 (x)), x Ω H 0 (x)) q, ( ) } q δ h k (h) 1 /k(h) 2 then (A(x, t), H(x, t)) converges exponentially to (0, Σ h,d (x)) uniformly on Ω as t +.

30 30/43 Collapse of Patterns / σ a 0 Theorem 2.3. (Almost decoupled stationary patterns) Assume that σ a (x) 0, σ h (x) 0. Let min x Ω σ a (x) > γ a ( maxx Ω σ a (x) ) p for some positive constant γ a if 0 < r < 1. If max x Ω σ a (x) is sufficiently small, then there exists a stationary solution (A (x), H (x)) of (GM) which satisfies A Σ a,ε C σ a p, H Σ h,d C σ a r, where C is a positive constant and Σ a,ε, Σ h,d are solutions of ε 2 Λ a Σ a,ε µ a Σ a,ε + σ a = 0, and DΛ h Σ h,d µ h Σ h,d + σ h = 0 subject to the boundary conditions B a Σ a,ε = 0, B h Σ h,d = 0, respectively. Furthermore, this stationary solution is asymptotically stable.

31 Collapse of Patterns / σ a 0 31/43 To treat the case σ a 0 and σ h 0, we need an algebraic observation: Consider the equation k (a) 1 ξ + C a (min x Ω Σ h,d (x)) q ξp + σ a = 0. If σ a > 0 is sufficiently small (depending on min Σ h,d ), then this equation has exactly two positive roots 0 < κ < K and they satisfy κ = σ a K = k (a) 1 as σ a 0. + O( σ a p ), { k (a) 1 (min x Ω Σ h,d(x)) q C a } 1/(p 1) σ a (1 + o(1)) (p 1)k (a) 1

32 32/43 Collapse of Patterns / σ a 0 Theorem 2.4. (Case III) Under the same assumptions as in Theorem 2.3, if the initial data (A 0 (x), H 0 (x)) satisfies max x Ω A 0 (x) < K and H 0 (x) max x Ω Σ h,d (x), then max( A(x, t) A (x) + H(x, t) H (x) ) Ce γt x Ω for all t 0. Here, (A (x), H (x)) is the almost decoupled stationary pattern given by Theorem 2.3; C and γ are positive constants depending also on (A 0 (x), H 0 (x)).

33 Remarks. Collapse of Patterns / Remarks 33/43 (i) A precise definition of the almost decoupled pattern: stationary solution (A(x), H(x)) is called an almost decoupled pattern if µ a (x) + ρ a (A(x), H(x), x) A(x)p 1 H(x) q < 0, µ h (x) + ρ h (A(x), H(x), x) A(x)r H(x) s+1 < 0 for all x Ω. (ii) If σ a 0 and σ h 0, then there is no almost decoupled pattern. Hence, patterns never collapse in Case IV. A

34 34/43 Collapse of Patterns / Remarks (iii) In the case where σ h (x) 0, the condition on the initial data does not contain τ. On the other hand, we have Lemma. Let σ a (x) σ h (x) 0. If a solution (A(x, t), H(x, t)) of (GM) converges to (0, 0) as t + uniformly on Ω and satisfies µ a A(x, t) + ρ a A p then τ [k (h) 1 q]/[k(a) 1 (p 1)]. 0 for all x Ω, t > 0, Hq (iv) It was Professor Niro Yanagihara who, more than thirty years ago, found a solution of (GM) such that u(x, t) 0, v(x, t) 0 as t + in the case where both of ρ a, ρ h are constants, σ a (x) 0, σ h (x) 0, κ = 0, and (p, q, r, s) = (2, 1, 2, 0).

35 Collapse of Patterns / Remarks 35/43 From a view point of the possibility of collapse, the results may be summarized as in the table below: Cases Basic production terms Collapse Case I σ a (x) 0, σ h (x) 0 occurs for τ > q/(p 1). Case II σ a (x) 0, σ h (x) 0 occurs (for any τ > 0). Case III σ a (x) 0, σ h (x) 0 occurs (if σ a is small) for any τ > 0. Case IV σ a (x) 0, σ h (x) 0 never occurs.

36 36/43 Collapse of Patterns / Summary Activator-Inhibitor System with Different Sources A t = ε2 A A + A2 H + σ a τ H t = D H H + A2 + σ h A ν = H ν = 0 on Ω A(x, 0) = A 0 (x), H(x, 0) = H 0 (x) f(a, H) = A + A2 H + σ a, g(a, H) = H + A 2 + σ h.

37 Collapse of Patterns / Summary 37/43 Inhibitor-Dominant Strips: a) Case I, b) Case III, c) Case II a) b) c1) c2)

38 38/43 Collapse of Patterns / Possible Scenario 3. Possible Scenario. Summing up, the breakdown of pattern formation seems to contain the following three ingredients: Destabilization of the constant stationary solution by diffusioninduced instability (Turing instability) local property Existence of an unstable periodic solution (or a spiral-out mechanism ) that amplifies disturbances global property Existence of an (almost) decoupled stationary pattern Or, more precisely, it is an orbit connecting the unstable constant stationary solution with the almsot decoupled stationary pattern (in the case σ h 0).

39 Collapse of Patterns / Idea of Proof 39/43 4. Idea of Proof. To prove the theorems we follow the approach due to Wu and Li [WL] and make use of the following two lemmas: Lemma 4.1. If H 0 (x) Σ h,d (x), then H(x, t) max{min x Ω H 0 (x)e k(h) 2 t/τ, Σ h,d (x)}. Lemma 4.2. Let w(t) = min x Ω H 0 (x)e k(h) 2 t/τ, and let U(t) be the solution of the initial value problem du dt = U p k(a) 1 U + C a w(t) q + σ a (t > 0), U(0) = max x Ω A 0 (x). Then A(x, t) U(t) for all x Ω and t 0 in the maximal existence interval of U(t).

40 40/43 Collapse of Patterns / References References. [GM] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin) 12 (1972), [J] H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete and Continuous Dynamical Systems 14 (2006), [LCQ] M. Lin, S. Chen and Y. Qin, Boundedness and blow up for the general activator-inhibitor model, Acta Math. Appl. Sinica 11 (1995), [MT] K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math. 4 (1987),

41 Collapse of Patterns / References 41/43 [NS] W.-M. Ni and K. Suzuki, in preparation. [NST] W.-M. Ni, K. Suzuki and I. Takagi, The dynamics of a kinetic activator-inhibitor system, J. Differential Equations 229 (2006), [Rf] F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Math. 1072, Springer, [S] K. Suzuki, Existence and behavior of solutions to a reactiondiffusion system, Dessertation, Tohoku University, [ST] K. Suzuki and I. Takagi, On the role of source terms in some activator-inhibitor systems modeling biological pattern formation, Advanced Studies in Pure Mathematics: Asymptotic Analysis and Singularities 47-2 (2007) [ST2] K. Suzuki and I. Takagi, Behavior of solutions to an activator-

42 42/43 Collapse of Patterns / References inhibitor system with basic production terms, COE Lect. Note 14, Kyushu Univ. Global COE Program, Fukuoka, 2009, [ST3] K. Suzuki and I. Takagi, Collapse of patterns and effect of basic production terms in some reaction-diffusion systems, Current Advances in Nonlinear Analysis and Related Topics, , GAKUTO Internat. Ser. Math. Sci. Appl., 32 (2010). [ST4] K. Suzuki and I. Takagi, On the role of basic production terms in an activator-inhibitor system modeling biological pattern formation, to appear in Funkcialaj Ekvacioj. [T] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London, Ser. B 237 (1952), [WL] J. Wu and Y. Li, Classical global solutions for the activatorinhibitor model, Acta Math. Appl. Sinica 13 (1990)

43 Collapse of Patterns / References 43/43 [Y] N. Yanagihara, private communications. [bedlewo100913r.tex]; revised on September 17, 2010