Traveling Waves and Steady States of S-K-T Competition Model

Transcription

1 Traveling Waves and Steady States of S-K-T Competition Model with Cross diffusion Capital Normal University, Beijing, China (joint work with Wei-Ming Ni, Qian Xu, Xuefeng Wang and Yanxia Wu) 2015 KAIST CMC Mathematical Biology Conference on Cross Diffusion, Chemotaxis and Related Problems

2 Outlines Introduction Existence and stability of traveling waves with transition layers Existence and stability of steady states with boundary spike layers Existence and stability of nontrivial positive steady state

3 1. Introduction For investigating the spacial segregation of two competing species under intra- and inter-species pressure, in 1979, N. Shigesada, K. Kawasaki and E. Teramoto proposed a (S-K-T) competition model with self- and cross diffusion, which can be simply represented by u t = [(d 1 + ρ 1 u + γ 1 v)u] + u(a 1 b 1 u c 1 v), v t = [(d 2 + γ 2 u + ρ 2 v)v] + v(a 2 b 2 u c 2 v), (1) u n = v n = 0, x Ω, t > 0 d 1, d 2 (random diffusion rate): positive, ρ 1, ρ 2 (self-diffusion rate ): nonnegative, γ 1, γ 2 (cross-diffusion rate): nonnegative, a i, b i, c i, i = 1, 2: all positive constants.

4 Mathematical modelling of the SKT type of nonlinear diffusion Formulation of the nonlinear diffusion of u species: [(d 1 + ρ 1 u + γ 1 v)u] From macroscopic point of view: according to Fick s law, u t = J u + f(u, v), choosing dispersal flux J u = S(u, v), S(u, v) = u(d 1 + ρ 1 u + γ 1 v): species u moves from the higher density of S(u, v) to the lower density of S(u, v), due to the intra-species population pressure ( ρ 1 > 0) and inter-species population pressure ( γ 1 > 0).

5 From microscopic point of view : the selection of transition probability for the individuals of a species determines the form of the diffusion of the species. In one dim-space, let u i (t) = u(x i, t), v i (t) = v(x i, t), x i = ih, h > 0: mesh size. The corresponding microscopic scale model (continuous in time and spatially discrete model): du i dt = τ + i 1 u i 1 + τ i+1 u i+1 (τ + i + τ i )u i + f(u i, v i ), dv i dt = s + i 1 v i 1 + s i+1 v i+1 (s + i + s i )v i + g(u i, v i ), (2)

6 τ + i (τ i ), transition probability for individuals of species u per unit time for one-step jump, moving from x i to the neighboring site x i+1 (x i 1 ). In general: τ + i = τ(u i, v i, u i+1, v i+1 ). For the simplest case τ ± i = s ± i = α: positive constant, by passing limit h 0 and assuming αh 2 d (positive constant), the limiting model (diffusion limit) of the microscopic discrete model (2) becomes the macroscopic PDE model with random (linear) diffusion: u t = du xx + f(u, v), v t = dv xx + g(u, v).

7 The derivation of the SKT types of nonlinear diffusion terms from the microscopic model (2), by choosing the transition probability: τ + i = τ i = τ(u i, v i ) = (d 1 + ρ 1 u i + γ 1 v i )/h 2, s + i = s i = (d 2 + ρ 2 v i + γ 2 u i )/h 2, the transition probability of species u depends only on the desire of the individual to leave the point, and the repelling force is proportional to the value of d 1 + ρ 1 u + γ 1 v at the starting point.

8 Choosing other types of transition probability for a species ( or strategy of dispersal of a species ) induce different forms of diffusion terms or dispersal, e.g. if τ + i 1 = [d + γ(v i v i 1 )/2]/h 2, τ i+1 = [d + γ(v i v i+1 )/2]/h 2 ( species u can detect the difference of density of another species v before moving) and s ± i = α/h 2 (species v move randomly)

9 then the macroscopic PDE model of u and v becomes the typical Keller-Segel chemotactic model with diffusion: u t = d u γ (u v)] + f(u, v), v t = α v + g(u, v), α > 0, where γ (u v): called chemotactic term, another type of cross-diffusion term, γ: chemotactic sensitivity.

10 If γ > 0, the cross diffusion term γ (u v): called positive taxis, individuals of species u move in the direction of v, species u is attracted by species v, u: predator, bacteria, v: prey, nutrition, chemical attractant; If γ < 0: called negative taxis, species v is repelled by species u. In SKT competition model, the cross diffusion term in u equation γ 1 (uv) = γ 1 (u v)+γ 1 (v u) ( for γ 1 > 0): a special type of negative taxis.

11 In recent twenty years, SKT types of cross-diffusion models ( with a special type of negative taxis (uv) ) and various types of chemotactic PDE models ( with positive taxis (uψ(v) v) ) have been widely and deeply investigated: math modelling of some other biological ( ecological) models by these two types of cross-diffusion model, theoretical analysis and numerical simulation for the PDE models, aggregation phenomena and new pattern formation, finite time blow-up induced by cross-diffusion or chemotaxis. In the following we shall be focused on the theoretical analysis on the SKT competition model with cross-diffusion.

12 Without self diffusion and cross-diffusion, the SKT competition model becomes the typical Lotka-Volterra competition model with diffusion. L-V competition model with Neumann boundary condition has been widely and deeply investigated, the asymptotic behavior of solutions is nearly the same (except the case b 1 /b 2 < a 1 /a 2 < c 1 /c 2 ) as that for Lotka-Voltera ODE systems ( tending to some constant steady state eventually). Kishimoto and Weinberger (1984), Ω convex L-V competition model has no stable nonconstant steady states. Matano and Mimura (1983), Y.Kan-on and E.Yanagida (1993), for some types of nonconvex Ω there exist stable non-constant positive steady states (diffusion induce new pattern formation for some nonconvex domains.)

13 Existence and Stability of Traveling Waves for Lotka-Volterra competition models Traveling wave solution (u(x, t), v(x, t)) = (U(x ct), V(x ct)): planar wave solution. A lot of research work on the existence of several types of travelling fronts in one dimensional space for classical Lotka-Volterra competition models u t = d 1 u xx + (a 1 b 1 u c 1 v)u, x R, v t = d 2 v xx + (a 2 b 2 u c 2 v)v, x R,

14 Let A = a 1 /a 2, B = b 1 /b 2, C = c 1 /c 2. C < A < B: Tang and Fife (1984), existence of wave fronts connecting (0, 0) and (u, v ) with speed c c 0. B < A < C : Kanel and Zhou(1996), Fei and Carr (2004), existence of wave fronts connecting (u, v ) and (a 1 /b 1, 0) with speed c c +. max{b, C} < A and 0 < d 1 << 1: Hosono( 2002): existence of traveling waves with transition layers connecting (0, a 2 /c 2 ) and (a 1 /b 1, 0) for c > c.

15 For B < A < C(strong competition case), A. Gardner and Conley (1984), Y. Kan-on (1993,1997), existence and stability of waves with unique speed connecting (0, a 2 /c 2 ) and (a 1 /b 1, 0): competition exclude principle, sign of c > 0 determine which species invades successfully and win at end. c > 0: v species win; c < 0: u-species win; c = 0: coexist of two species. A. Gardner and Conley: topological degree method and comparison principle. Y. Kan-on: comparison principle, bifurcation theory and spectral analysis. No estimates on the sign of c.

16 2. Existence and Stability of Traveling Waves with Transition Layers for S-K-T model with cross-diffusion Consider the S-K-T competition model with cross-diffusion for small ɛ > 0 u t = ɛ 2 u xx + (a 1 b 1 u c 1 v)u v t = [(1 + γ 2 u)v] xx + (a 2 b 2 u c 2 v)v (3) W. and Ye Zhao (Science China 2010), If B < A < C, and γ 2 γ 0 > 0, then for small ɛ > 0, there exists a travelling wave (U c (x ct), V c (x ct)) connecting (0, a 2 /c 2 ) and (a 2 /b 2, 0) with a locally unique slow speed c = ɛc(ɛ) and both components of the wave have transition layers,

17 c(ɛ) c 0 as ɛ 0. c 0 : speed of front (transition layer) for the layer problem (in the stretched variable), sign of c 0 : sign of 0 h(β ) s(a 1 b 1 s c 1β 1+γ 2 s )ds.

18 Stability of Travelling waves: W. and Ye Zhao, applying Evan s function and topological index method ( the first Chern number) and detailed spectral analysis, the travelling waves with transition layers is locally asymptotically stable with shift if ɛ is sufficiently small. Remark 3. For B < A < C case, if γ 2 > 0 is small, no results on the existence of travelling waves connecting connecting (0, a 2 /c 2 ) and (a 2 /b 2, 0). It seems that for γ 2 = 0 and γ 2 > 0 small enough, the structure of the waves are different from the cases as γ 2 γ2 0.

19 Existence of traveling waves with transition layers for the degenerate S-K-T system with cross-diffusion For the degenerate S-K-T competition models, u t = ɛ 2 (u 2 ) xx + (a 1 b 1 u c 1 v)u v t = [(1 + γ 2 u)v] xx + (a 2 b 2 u c 2 v)v (4)

20 Yanxia Wu and W. [CPAA, 2012]: using analytic singular perturbation method and center manifold theorem inspired by Hosono s work on degenerate reaction diffusion system), for B < A < C case, if the cross-diffusion rate γ 2 γ 0 > 0 and for small ɛ > 0, obtained the existence of travelling waves (U ɛ (x ct), V ɛ (x ct)) connecting (0, a 2 /c 2 ) and (a 2 /b 2, 0) with a locally unique slow speed c = ɛc(ɛ), where c(ɛ) c = 0 as ɛ 0, and both components of the waves have transition layers, c is the speed of the front for the layer problem. If c < 0, component U(z) 0 for z l(ɛ), U(z) > 0 for z > l(ɛ): weak but continuous wave. If c > 0, the wave is smooth.

21 Remark 4: Different from non-degenerate systems, for the degenerate S-K-T competition system without cross-diffusion or when the cross diffusion rate is small, there are no results on the existence of travelling waves (including the cases B < A < C) = for the degenerate systems the appearance of cross-diffusion may induce new wave phenomena, however the stability of the wave is still an open problem.

22 Existence and Stability of Nontrivial Positive Steady States For the Neumann boundary value problem some theoretical and numerical results: the cross-diffusion in S-K-T competition system induce some new pattern formation. Stationary S-K-T competition model with cross-diffusion d 1 [(1 + γ 1 v)u] + u(a 1 b 1 u c 1 v) = 0, x Ω, d 2 v + v(a 2 b 2 u c 2 v) = 0, x Ω, u n = v n = 0, x Ω.

23 Mimura (1981), Mimura-Kawasaki(1980), Y. Mimura,Y. Nishiura, Tesei,& Tsujikawa (1984), Y.Kan-on (1993),...,Y.Lou & W-M. Ni (1996,1999) Lou-Ni-Yatsutani(2004),..., W (2002, Dirichet B.C), W.(2005), W. & Q. Xu (2011), Kuto-Yamada(2004-, competition model, predator-prey model,...) Ni-W-Xu(2014), Lou-Ni-Yatsutani(2015)

24 Consider the boundary value problem in one dimensional space: 0 = [(d 1 + γ 1 v)u] xx + u(a 1 b 1 u c 1 v) = 0, 0 < x < 1, 0 = d 2 v xx + v(a 2 b 2 u c 2 v) = 0, 0 < x < 1, u x = v x = 0, x = 0, 1, (5) 1984, Y.Mimura,Y. Nishiura, Tesei, etc.: existence of positive steady states with interior or boundary transition layers for the cross-diffusion system with small d 2, large enough d 1 and some γ 1 > 1, under some abstract assumptions; ( covering the case when B < A < C (strong competition) or A > max{b, C}; if γ 1 is large enough. )

25 1993,Y.Kan-on, the stability/instability of steady states with interior transition layer, by SLEP method, the steady states are stable under some additional abstract assumptions on A, B, C and γ 1 cross-diffusion induce new pattern formation.

26 4. Existence and Stability of Steady States with Spike Layers 0 = [(d 1 + γ 1 v)u] xx + u(a 1 b 1 u c 1 v) = 0, 0 < x < 1, 0 = d 2 [(1 + γ 2 u)v] xx + v(a 2 b 2 u c 2 v) = 0, 0 < x < 1, u x = v x = 0, x = 0, 1, (6) Y.Lou and W-M.Ni [JDE,1996,1999]: existence and nonexistence of non-constant positive solutions of (6), esp. positive steady states with boundary spike layer in 1-dim when one of the cross-diffusion parameter (γ 1 is large enough ( cross-diffusion may induce other new types of pattern formation)

27 Y.Lou & W-M. Ni[JDE, 1999]: For 1-dim case, esp. for the cases Case I : Case II : 1 2 (B + C) < A < 1 4 B + 3 C, B < C, B C < A < 1 (B + C), C < B, 2 existence of positive spiky steady states (with uniformly bounded height of the boundary spike layers) near the positive constant s.s.(u, v ), when γ 2 = 0, d 2 > 0 is small, both γ 1 and γ 1 /d 1 are large enough.

28 3. Existence of positive steady states near (a 1 /b 1, 0) with boundary spike layers: Y.Lou and W-M. Ni [JDE 1999] Case III: A > B, existence of positive spiky steady states near (a 1 /b 1, 0) with boundary spike layers, when γ 2 0 and small d 2 > 0, and d 1 and γ 1 /d 1 are large enough, both u(x) and v(x) have bounded boundary spike layers.

29 Cases (I) and (II): Case(III):

30 Instability of Spiky Steady States W. [JDE, 2005]: Instability of spiky steady states ( near (u, v )) for most of Cases I and II. L. Wang, W and Q. Xu ( preprint), Instability of spiky steady states (near (a 1 /b 1, 0)) for Case III. Method of the proof of instability for three cases: spectral analysis for two types of limiting system, detailed singular or regular perturbation estimates based on the detailed spiky structure of the steady states. Spectral results: Existence of unstable eigenvalues for the shadow and the cross-diffusion system near some positive numbers. The location and the estimates of the unstable eigenvalues are different

31 Existence and nonexistence of the positive steady states for the shadow system in 1-dim Lou, Ni and Yostsutani [DCDSA, 2004]: for 1-dim case, complete and nearly optimal results on the existence/nonexistence of monotone positive steady states for the following limiting system (shadow system) of the cross-diffusion system as γ 1 and γ 1 /d 1 d 2 v xx + v(a 2 c 2 v) b 2 τ = 0, x (0, 1), v (a 1 b 1τ v c 1v) dx = 0, v x (0) = v x (1) = 0. (7)

32 Some existence/non-existence results in [LNY,2004]: (I) For any d 2 a 2 /π 2, the shadow system has no positive non-constant solution. (II) If A (B + C)/2 and B < C, or if A B > C, then for every d 2 (0, a 2 /π 2 ), the shadow system has a positive solution (v, τ) with v being strictly increasing in (0, 1). (III) If B > A > (B + C)/2 and B > C, then for every ( d 2 0, (2A B C)a 2 ), the shadow system has a positive solution (B C)π 2 (v, τ) with v being strictly increasing in (0, 1). All the solutions can be represented by elliptic functions.

33 The results in (II)-(III) also imply that the shadow system (7) has nontrivial positive solutions for the case A > (B + C)/2 when B < C, or A > (B + 3C)/4 when B > C, if d 2 > 0 is small; and for the case A > B, or A = B when B > C; if a 2 /π 2 d 2 > 0 is small. (8)

34 Esp. for the cases:(b + 3C)/4 < A, if B < C or (B + C)/2 < A, for C < B, there exist a special type of positive spiky steady states (not near constant steady states) for shadow system as γ 1 +, γ 1 /d 1 + for small d 2 > 0. a 2 2 As d 2 0, τ(d 2 ) τ = 3 16 b 2 c 2, and V(x, d 2 ) has a boundary spike layer at x = 0 satisfying V(0; d 2 ) 0, V(x; d 2 ) 3 a 2 4 c 2, x (0, 1] as d 2 0, here (τ, 3 a 2 4 c 2 ) is not a constant steady states to the shadow system.

35

36 It is naturally expected and guessed that for the original cross-diffusion system, for d 2 > 0 small, γ 1 and γ 1 /d 1 large enough, there should exist some positive large spiky steady states satisfying (U d1,γ 1 (x), V d1,γ 1 (x)) as γ 1, γ 1 /d 1. ( ) τ(d2 ) V(x, d 2 ), V(x, d 2), x [0, 1],

37

38 W. and Q. Xu [DCDS, 2011], applying different approach: (special perturbation method, finding explicit approximation solution, detailed integral estimates): detailed spiky structure of the spiky steady states for shadow system and the existence of large spiky steady states for non-shadow system.

39 W. and Q. Xu, DCDSA 2011: Theorem ( Existence and fast-slow structure of the spiky steady states for the shadow system) Assume A > B+3C 4, then for small enough d 2 > 0 the shadow system has a positive spike layer solution (τ(d 2 ), ψ(x, d 2 )) (τ ε, ψ ε (x, τ ε )) satisfying a 2 2 τ ε = 3 + ε 3 2 τ0 + O(ε), 16 b 2 c 2 ψ ε (x, τ ε ) = 3 a 2 2 4b 2c 2 τ ε 2c 2 sech 2 ( ) x 2ε + a 2+ a 2 2 4b 2c 2 τ ε 2c 2 + O(e 1 ε ),

40 W. and Q. Xu, (2011 DCDS A): Theorem(Existence of large spiky steady states for the original cross-diffusion systems) Assume A > B+3C 4, then for each fixed small d 2 > 0, there exists large enough α such that if both α γ 1 d 1 > α and γ 1 > α hold, then there exists a non-constant positive spiky steady state (u α,γ1 (x), v α,γ1 (x)) of the non-shadow system satisfying (u α,γ1 (x), v α,γ1 (x)) ( ) τ(d2 ) ψ(x, d 2 ), ψ(x, d 2), as γ 1, γ 1 /d 1.

41 Stability of large spiky steady states for the shadow system and the cross-diffusion model Some numerical simulations (S. Yotsutani ) on the eigenvalue problems and our recent theoretical analysis ( Xuefeng Wang and W. ) imply the large spiky steady states are stable. Main difficulties in spectral analysis: spectral analysis on a linear differential operator (not self-adjoint) with several nonlocal terms, and the weights of nonlocal terms have spiky structure and singularity when d 0 0, the standard regular or singular perturbation argument can t be applied directly, the eigenvalues with the largest real part may be not real.

42 4. Existence and stability of nontrivial positive steady states for d 2 near a 2 /π 2 Consider the shadow system: d 2 v xx + v(a 2 c 2 v) b 2 ξ = 0, x (0, 1), v (a 1 b 1ξ v c 1v) dx = 0, v x (0) = v x (1) = 0.

43 Lou, Ni and Yotsutani, (DCDSA 2004): For A > B, or A = B and B > C, there exists positive solution (ξ(d 2 ), ψ(x, d 2 )) satisfying the following asymptotic behavior (ξ(d 2 ), ψ(x, d 2 )) (0, 0), in [0, 1]; ξ(d 2 ) ψ(x, d 2 ) a 2 b 2 as d 2 a 2 π , x [0, 1], 1 B/A cos(πx)

44 W-M. Ni, W. and Q. Xu, (DCDSA, 2014): More detailed structure of the steady states for the shadow system for d 2 near a 2 π 2, by applying different method of proof ( after some transformation, applying detailed and special perturbation method based on Lyapunov-Schmidt decomposition and similar local bifurcation argument).

45 Theorem (Existence and detailed structure of positive steady state for the shadow system) Assume A > B, there exist smooth s(d 2 ), ξ 1 (s) and ψ 1 (s, x)) satisfying ξ 1 (0) = 0, ψ(0, x) 0, s( a 2 π 2 ) = 0, and s ( a 2 π 2 ) = a 2π 2 2b 2 c 2 < 0; and for small a 2 π 2 d 2 > 0 the shadow system has a nontrivial positive steady state (ξ 0 d 2, ψ 0 d 2 (x)) ξ 0 d 2 = s + sξ 1 (s(d 2 )), s = s(d 2 ), ψ 0 d 2 (x) = s b 2 a 2 (1 1 B/A cos(πx)) + sψ 1 (s, x).

46 Outline of the proof Consider the steady state shadow system: 1 ξ ψ (a 1 b 1ξ ψ c 1ψ) dx = 0, 0 d 2 ψ xx + ψ(a 2 c 2 ψ) b 2 ξ = 0, x (0, 1), ψ x (0) = ψ x (1) = 0. Multiplying the first equation by ξ, we have with H(ξ, ψ(x), d 2 ) = H(ξ, ψ(x), d 2 ) = 0, (9) 1 0 ξ 2 ( ) ξ a 1 b 1 dx c 1 ξ 2 ψ(x) ψ(x) d 2 ψ xx (x) + ψ(x)(a 2 c 2 ψ(x)) b 2 ξ,

47 For s = 0, substituting ξ(s) = s + sξ 1 (s, d 2 ), ψ(s, x) = s b 2 a 2 (1 1 B/A cos(πx)) + sψ 1 (s, d 2, x) into system (9), then for s = 0 system (9) becomes 0 = H(ξ, ψ(x), d 2 ) = s H(s, d 2, ξ 1, ψ 1 (x)). Here H(s, d 2, ξ 1, ψ 1 (x)) can be defined for s = 0, and it can be proved that zero is a simple eigenvalue of the linearized operator of H(0, a 2 /π 2, ξ 1, ψ 1 (x)) around (ξ 1, ψ 1 ) = (0, 0).

48 By applying detailed analysis based on Lyapunov-Schimidt decomposition method and Implicit function theorem, we can prove the existence of d 2 = d 2 (s) C 2 ([ δ, δ]) and (ξ 1, ψ 1 ) = (ξ 1 (s, d 2 ), ψ 1 (s, d 2, x)) C 1 ([ δ, δ] [a 2 /π 2 δ, a 2 /π 2 + δ], X 0 ), which solve the system and d 2 (0) = 2b 2c 2 a 2 π 2 < 0. H(s, d 2, ξ 1, ψ 1 (x)) = 0, for d 2 = d 2 (s).

49 Theorem (Stability of positive steady state for the shadow system) Assume A > B and a 2 π 2 d 2 > 0 is small enough, then the nontrivial positive steady state (ξ 0 d 2, ψ 0 d 2 (x)) to the shadow system is asymptotically stable in R H 1 (0, 1).

50 Method of proof of stability for the shadow system: Spectral perturbation argument on an equivalent linearized system after some transformation, zero is the principle eigenvalue of the limiting system as d 2 a 2 /π 2 ; Applying spectral perturbation argument via Lyapunov-Schmidt decomposition method to show there exists a simple real eigenvalue λ(s) near zero for small s > 0 and the sign of λ(s) determines the stability/instability of positive steady state, applying similar spectral argument used in local bifurcation theory to get the sign of λ (0), which is the same as the sign of d 2 (0) < 0.

51 Outline of the proof Consider the eigenvalue problem of the linearized system of the shadow system around steady state (ξ 0 d 2 (s), ψ 0 d 2 (s, x)) for small a 2 d π 2 2 or for small s : ξ 1 0 b0,d 2 11 (x) dx b0,d 2 12 (x)ψ(x)dx = λξ 1 0 a0,d 2 11 (x) dx + λ 1 0 a0,d 2 12 (x)ψ(x) dx, d 2 ψ xx (x) + b 0,d 2 21 (x)ξ + b0,d 2 22 (x)ψ(x) = λψ(x), x (0, 1), ψ x (0) = ψ x (1) = 0, ξ = constant; (10)

52 Multiplying the first equation of (10) by ξ 0 d 2, (10) becomes L s ξ ψ(x) = λd s ξ ψ(x). (11) As s 0 the eigenvalue problem of (11) can be reduced to the following limiting eigenvalue problem 1 b 1 ξ = λξ λl 2 (x)dx + l 1 (x)dx λ ( a1 l 2 (x) + 2b 1 l 3 (x) ) ψ(x)dx l 2 (x)ψ(x)dx, a 2 π 2 b 2 ξ + a 2 ψ = λψ(x), x (0, 1), ψ x (0) = ψ x (1) = 0; l(x) = b 2 a 2 (1 1 B/A cos(πx)) (12)

53 which can be simply written as L 0 ξ = λd 0 ξ ψ(x) ψ(x), (13) Lemma 6.1 Assume A > B, then there exists small ε 0 > 0 such that zero is the unique and a simple eigenvalue of D 1 0 L 0 in {λ C Reλ ε 0 }, with the eigenspace spanned by (1, l(x)), l(x) = b 2 a 2 (1 1 B/A cos(πx)).

54 By applying spectral perturbation argument and Lyapunov-Schimitz decomposition method, we can prove Lemma 6.2 For small δ, δ 0 > 0 there exist a unique eigenvalue λ(s) C 1 [ δ 0, δ 0 ] of Ds 1 L s located in {λ C, Re λ δ}, which is real and λ(0) = 0, with an eigenfunction (1 + ζ(s), l(x) + φ(s, x)) satisfying ζ(0) = 0, φ(0, x) = 0.

55 By detailed estimates and similar argument as in bifurcation theory, it can be further proved that λ (0) = π 2 d 2(0) = 2b 2c 2 a 2 < 0, a which proves that for small 2 d π 2 2 > 0, the unique eigenvalue λ(s) (near zero) of Ds 1 L s is negative = asymptotic stability of the steady states when a 2 π 2 d 2 > 0 is small.

56 Theorem (Existence of positive steady state for the original cross-diffusion system) Assume A > B. For small a 2 π 2 d 2 > 0, if both ρ 12 and ρ 12 /d 1 are large enough, then the cross-diffusion system has a nontrivial positive steady state (u(x), v(x)) satisfying ) (u(x), v(x)) ( ξ 0 d2 ψ 0 d 2 (x), ψ0 d 2 (x), as ρ 12, ρ 12 /d 1.

57

58 Theorem (Stability of positive steady state for the cross-diffusion system) Assume A > B. For each fixed d 2 ( a 2 π 2 δ, a 2 π 2 ) with small δ > 0, if both ρ 12 and ρ 12 /d 1 are large enough, then the nontrivial positive steady state (u(x), v(x)) of the cross-diffusion system is asymptotically stable in H 1 (R) H 1 (R).

59 Method of proof of the existence result: perturbation argument via appropriate decomposition and using the detailed structure of the steady states for the shadow system; Method of proof of the stability result: spectral perturbation argument via appropriate decomposition and some uniform spectral estimates.

60 For higher dimension case n 4, recent work of Y. Lou, W-M.Ni, S. Yotsutani DCDS 2015): similar results on existence and stability for the shadow and the original cross-diffusion system for d 2 near the critical value d 0 = a 2 /λ 1 (λ 1 = π 2, n = 1, ω = (0, 1)), by applying a simplified and different argument.

61 Open Problem The Existence of positive steady states for SKT model in 1-dim case for the original cross-diffusion system, when d 2 is not near 0, or a 2 /π 2 The stability/instability of other types of positive steady states for SKT model in 1-dim case for the shadow system, esp. when d 2 is not near 0, or a 2 /π 2. The existence and structure of nontrivial positive steady state for SKT shadow system in higher dimension for d 2 not near d 0.

62 Thank You for Your Attention!