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Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 654539, 19 pages doi:10.1155/2009/654539 Research Article Global Behavior for a Diffusive Predator-Prey Model with Stage Structure and Nonlinear Density Restriction-II: The Case in R1 Rui Zhang,1, Ling Guo,1 and Shengmao Fu1 Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China Correspondence should be addressed to Shengmao Fu, fusm@nwnu.edu.cn Received April 2009; Accepted 31 August 2009 Recommended by Wenming Zou A Holling type III predator-prey model with self- and cross-population pressure is considered Using the energy estimate and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions to the model are dicussed In addition, global asymptotic stability of the positive equilibrium point for the model is proved by Lyapunov function Copyright q 2009 Rui Zhang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction This paper is a continuation of Part I In Section of Part I, using the energy estimate and bootstrap arguments, the global existence of solutions for a Holling type III cross-diffusion predator-prey model with stage-structure has been discussed when the space dimension be less than However, to obtain the L∞ estimate for the population density w of predator species, there is not cross-diffusion for w in Part I All diffusive predator-prey systems behave, more or less, in the same way, for both semilinear and cross-diffusive models, at least for small values of the cross diffusivities Consequently, all the available information for linear diffusive models is essential to realize the behavior of the most complicated cross-diffusive systems 2–17 In this paper, we consider the following cross-diffusion system: ut du α11 u2 α12 uv α13 uw vt dv α21 uv α22 v2 α23 vw xx xx βv − au − bu2 − cu3 − u − v, u2 w , u2 < x < 1, t > 0, Boundary Value Problems wt d3 w ux x, t vx x, t u x, u0 x , α31 uw α32 vw wx x, t v x, 0, v0 x , α33 w2 x xx − kw − γw2 αu2 w , u2 0, 1, t > 0, w x, w0 x , < x < 1, 1.1 where d, d3 , αij i, j 1, 2, , α, β, γ, a, b, c, and k are positive constants Also, d, d3 are linear diffusion coefficients of u, v, w, respectively, while αii i 1, 2, are referred as self-diffusion pressures, and αij i / j, i, j 1, 2, are cross-diffusion pressures If α12 α21 α23 α31 α32 0, then 1.1 reduces to the system 1.4 of Part I Recently, the work in 18–20 studied the existence, uniform boundedness, and uniform convergence of global solutions for the Lotka-Volterra cross-diffusion models without stage-structure in the case that the space dimension n In this paper, we consider mainly the existence and uniform boundedness of global solutions for the model 1.1 with nonlinear density restriction and stage-structure Moreover, global asymptotic stability of the positive equilibrium point for 1.1 is proved by an important lemma of 21 The proof is complete and complement the uniform convergence theorem in 18–20 Global Existence and Uniform Boundedness k For simplicity, denote | · |k,p · Wp 0,1 , | · |p · Lp 0,1 The local existence result of solutions to 1.1 is an immediate consequence of a series of papers 22, 23 by Amann Roughly speaking, if u0 , v0 , w0 ∈ Wp 0, , p > 1, then 1.1 has a unique nonnegative solution u, v, w ∈ C 0, T , Wp 0, C∞ 0, T , C∞ 0, , where T ≤ ∞ is the maximal existence time for the solution If u, v, w satisfies sup |u ·, t |1,p , |v ·, t |1,p , |w ·, t |1,p : < t < T < ∞, then T 2.1 2 ∞ If, in addition, u0 , v0 , w0 ∈ Wp 0, , then u, v, w ∈ C 0, ∞ , Wp 0, The main result in this section is as follows Theorem 2.1 Let u0 , v0 , w0 ∈ W2 0, , u, v, w is the unique nonnegative solution of 1.1 in its maximal existence interval 0, T Assume that 8α11 α21 α31 > α21 α2 13 α2 α31 , 12 8α12 α22 α32 > α32 α2 21 α2 α12 , 23 8α13 α23 α33 > α23 α2 31 α2 α13 32 2.2 Boundary Value Problems Then there exists t0 > and positive constants M, M which depend on d, d3 , αij i, j β, a, b, c, k, γ, α, such that sup |u ·, t |1,2 , |v ·, t |1,2 , |w ·, t |1,2 : t ∈ t0 , T ≤M, max{u x, t , v x, t , w x, t : ≤ x ≤ 1, t0 ≤ t < T } ≤ M, 1, 2, , 2.3 2.4 and T ∞ In particular, if d, d3 ≥ 1, d3 /d ∈ d, d , where d ≤ and d are positive constants, then M , M depend on d, d, but not depend on d, d3 ≥ The following Gagliardo-Nirenberg-type inequalities and corresponding corollary play an importance role in the proof of Theorem 2.1 Theorem 2.2 see 18 Let Ω ⊂ Rn be a bounded domain with ∂Ω ∈ Cm For every function m u ∈ Wr Ω , ≤ q, r ≤ ∞, the derivative Dj u ≤ j < m satisfies the inequality Dj u p ≤ C |Dm u|a |u|1−a r q |u|q , 2.5 provided one of the following three conditions is satisfied: r ≤ q, < n r − q /mrq < 1, or n r − q /mrq 1, and m − n/q is not a nonnegative integer, where 1/p j/n a 1/r − m/n − a /q, for all a ∈ j/m, , and the positive constant C depends on n, m, j, q, r, a Corollary 2.3 There exists a positive constant C such that |u|2 ≤ C |ux |1/3 |u|2/3 |u|1 , ∀u ∈ W2 0, , 2.6 |u|4 ≤ C |ux |1/2 |u|1/2 |u|1 , ∀u ∈ W2 0, , 2.7 |u|7/2 ≤ C |ux |10/21 |u|11/21 |ux |2 ≤ C |uxx |3/5 |u|2/5 |u|1 , |u|1 , ∀u ∈ W2 0, , ∀u ∈ W2 0, 2.8 2.9 For simplicity, denote that C is Sobolev embedding constant or other kind of absolute constant Aj , Bj , Cj are some positive constants which depend on αij i, j 1, 2, , β, a, b, c, k, γ, α Also, Kj are positive constants which depend on αij i, j 1, 2, , β, a, b, c, k, γ, α, d, d3 When d, d3 ≥ 1, Kj not depend on d, d3 , but on d, d Proof of Theorem 2.1 Step Estimate |u|1 , |v|1 ,|w|1 Firstly, taking integration of the first and second equations in 2.7 over the domain 0, , respectively, and combining the two integration equalities Boundary Value Problems linearly, we have d dt u β v dx ≤ −a a 1 βu − bu2 dx vdx 2.10 From Young inequality and Holder inequality, we can see ă d dt where C1 such that 1/4b β u a β v dx ≤ C1 − a/ a β a a u β a β v dx, 2.11 From which it follows that there exists a constant τ0 > 0, 1 udx, vdx ≤ M0 , t ≥ τ0 , 2.12 2C1 a β /a max{ a β −1 , 1} where M0 Secondly, taking integration of the third equations in 2.7 over domain 0, , we have d dt 1 wdx ≤ α − k wdx − γ wdx 2.13 This implies that there exists a constant τ0 > 0, such that wdx ≤ Let M1 max{M0 , 2|α − k| /γ}, τ1 t ≥ τ0 2.14 max{τ0 , τ0 } Then udx, 2|α − k| , γ vdx, wdx ≤ M1 , t ≥ τ1 2.15 Moreover, there exists a positive constant M1 which depends on β, a, b, c, k, γ, α and the L1 norm of u0 , v0 , w0 , such that 1 udx, vdx, 0 wdx ≤ M1 , t ≥ 2.15 Boundary Value Problems Step estimate |u|2 , |v|2 and |w|2 Multiplying the first three inequalities of Corollary 2.3 by u, v, w, respectively, and integrating over 0, , we have d dt u2 dx ≤ −d 1 d dt 2α11 u α12 v α13 w u2 x α12 uux vx α23 w vx α21 vux vx α13 uux wx dx β uvdx, v2 dx ≤ −d 1 vx dx − d dt u2 dx − x α21 u 2α22 v α23 vvx wx dx uvdx, w2 dx ≤ −d3 1 wx dx − α31 u α32 v 2α33 w wx α32 wvx wx dx α31 wux wx 2.16 min{d, d3 } By the above three inequalities and Young inequality, we have Let d d dt u2 v2 w2 dx ≤ −d u2 x vx wx dx − q ux , vx , wx dx β 1 α u2 v2 w2 dx, 2.17 where q ux , vx , wx 2α11 u α12 u α12 v α13 w u2 x α21 v ux vx α21 u α13 u 2α22 v α23 w vx α31 w ux wx α23 v α31 u α32 v 2α33 w wx α32 w vx wx 2.18 is quadratic form of ux , vx , wx It is not hard to verify that q ux , vx , wx is positive definite if 2.2 holds Moreover, if 2.2 holds, then d dt u2 v2 w2 d ≤ −d u2 x vx wx dx β α u2 v2 w2 dx 2.19 Now we proceed in the following two cases i It holds that t ≥ τ1 By 2.6 and 2.15 , we have u dx x ≥ 1/CM1 u dx − M1 , Boundary Value Problems and −d u2 x 2 wx dx ≤ 3d M1 − C2 d vx u2 v2 2 w2 dx 2.20 By 2.19 and 2.20 , we can see that 1 d dt u2 v2 w2 dx ≤ −C2 d u v w β dx 2.21 α u v w dx 3d M1 Thus, there exists positive constants τ2 > τ1 and M2 depending on d, d3 , β, a, b, c, k, γ, α, such that u2 dx, v2 dx, w2 dx ≤ M2 , t ≥ τ2 2.22 Since the zero point of the right-hand side in 2.21 can be estimated by positive constants independent of d , when d ≥ Thus M2 not depend on d ≥ ii t ≥ Repeating estimates in i by 2.9 , we can obtain that there exists a positive constant M2 depending on d, d3 , β, a, b, c, k, γ, α and the L1 , L2 -norm of u0 , v0 , w0 , such that u2 dx, v2 dx, w2 dx ≤ M2 , t ≥ 0, 2.22 t 2.23 when d ≥ 1, M1 is independent of d Step Estimate |ux |2 , |vx |2 ,|wx |2 Introduce the scaling that u denote η u , d1 v v , d1 w w , d1 d1 t, d3 /d, and redenote u, v, w, t by u, v, w, t, respectively Then 2.7 reduces to ut Pxx f u, v, w , < x < 1, t > 0, vt Qxx g u, v, w , < x < 1, t > 0, wt Rxx h u, v, w , < x < 1, t > 0, ux x, t vx x, t u x, u0 x , wx x, t v x, 0, v0 x , x 2.24 0, 1, t > 0, w x, w0 x , < x < 1, Boundary Value Problems where P u α11 u2 α12 uv α13 uw, Q v α21 uv α22 v2 α23 vw, R ηw α31 uw α32 vw βd−1 v − ad−1 u − bu2 − cdu3 − du2 w/ d2 u2 , g u, v, w d−1 u − v , α33 w2 , f u, v, w −1 2 2 αdu w/ d u We still proceed in following two cases h u, v, w −kd w − rw ∗ i It holds that t ≥ τ2 dτ2 From 2.15 and 2.22 , we can easily obtain that 1 udx, 1 vdx, wdx ≤ M1 d−1 , u2 dx, 0 v2 dx, 2.25 w2 dx ≤ M2 d−2 , |P |1 , |Q|1 , |R|1 ≤ DK1 d−1 , η M2 d−2 , D max{M1 , α11 α12 α13 , α21 α22 α23 , α31 α32 α33 } where K1 Multiply the first three equations in 2.24 by Pt , Qt , Rt and integrate them over 0, , respectively, then adding up the three new equations, we have y t ≤− u2 dx − t vt dx − η 1 wt dx − q ut , vt , wt dx 1 2α11 u α12 v α13 w ut f α12 uvt f α13 uwt f dx 2.26 α21 vut g α21 u α31 wut h 2α22 v α32 wvt h α23 w vt g α23 vwt g dx η α31 u α32 v 2α33 w wt h dx, where y P Qx R2 dx It is not hard to verify by 2.4 that there exists a positive x x constant C3 depending only on αij i, j 1, 2, , such that q ut , vt , wt ≥ C3 u v u2 t w vt wt 2.27 Thus, y t ≤− u2 dx − t vt dx − η wt dx − C3 u v 1 u2 t w α21 u vt wt dx 1 2α11 u α12 v α13 w ut fdx 2α22 v α23 w vt gdx 1 η α31 u α32 v 2α33 w wt hdx α21 vut gdx α12 uvt fdx α23 vwt gdx α13 uwt fdx α31 wut hdx 2.28 1 α32 wvt hdx Boundary Value Problems Using Young inequality, Holder inequality and 2.24 , we can obtain the following estimates: ă 1 u3 dx ≤ 1 u4 dx ≤ u5 dx ≤ 1 uvdx ≤ u vdx ≤ 4/5 1/2 1/5 u4 vdx ≤ 1 2/5 u ut dx ≤ 1 1 1 1 uvut dx ≤ u2 vut dx ≤ 3/10 4/5 u7 dx , 0 0 u dx u7 dx u2 vdx u4 vdx 2 v dx u7 u dx 2/5 3/5 M2 d−6/5 u dx 1 4/5 −8/5 M d 2 M1 d−1 2 1/5 u7 dx u7 v7 dx, uu2 dx, t uu2 dx t 2/5 M2 d−4/5 ≤ 1/5 1 u dx 0 3/5 u dx 0 uu2 dx, t uu2 dx, t uu2 dx, t vu2 dx ≤ t vu2 dx t , v7 dx, 4/5 M2 d−8/5 ≤ , 0 uu2 dx t u6 vdx ≤ ≤ 1/5 4/5 M2 d−8/5 1/2 uu2 dx ≤ t u dx 1/10 udx ≤ v dx 1/2 v7 dx ≤ u4 ut dx ≤ , ≤ M2 d−2 , u dx u2 vdx 3/5 u7 dx 1/2 v2 dx u7 dx u ut dx ≤ 0 1/5 ≤ M2 d−2/5 u dx u dx , 1/5 uut dx ≤ 1 u6 vdx ≤ 2/5 ≤ M2 d−4/5 u2 dx 2/5 u7 dx 2/5 u2 dx , u vdx ≤ 1 u2 dx 3/5 ≤ M2 d−6/5 u dx 1 u7 dx 0 1 1/5 u7 dx 3/5 u2 dx 3/5 u7 dx 4/5 ≤ M2 d−8/5 0 1 1 2/5 u7 dx u6 dx ≤ 4/5 u2 dx 0 1 0 1/5 u7 dx 4/5 M2 d−8/5 1/5 u7 dx vu2 dx, t Boundary Value Problems ≤ u3 vut dx ≤ 1 4/5 M2 d−8/5 1 1/5 u6 vdx 0 vu2 dx ≤ t 14 u7 dx v7 dx 14 u7 u7 v7 dx 0 vu2 dx, t vu2 dx t 2.29 Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of 2.28 , we have − − −η u2 dx ≤ − t 2 vt dx ≤ − η 2 wt dx ≤ − 0 Pxx dx Qxx dx R2 dx xx f dx ≤ β2 d−2 f dx, g dx, η h2 dx, a2 d−2 v2 dx u2 dx 2abd−1 u3 dx 2ac 2bd−1 uwdx β 1 g dx ≤ d−2 η u2 u3 wdx 3/5 M2 d−13/5 2b a 1/5 u7 dx 2/5 u7 dx u dx c 0 2kd−1 γη w2 dx ≤ η k2 γ α2 M2 d−4 3/5 ηM2 d−6/5 1/5 M2 d8/5 4/5 u dx w3 dx γ 2η 4/5 −13/5 2kγηM2 d1 2/5 w dx 2.30 v2 dx ≤ 2M2 d−4 , α2 u6 dx 0 h2 dx ≤ d−2 η k2 3/5 c2 d4 w2 dx u2 wdx 3/5 M2 d−6/5 2/5 2bcM2 d1/5 u5 dx 2a M2 d−4 b2 2ac d−2 u4 dx 1 2bcd2 0 ≤ a2 u4 dx 2ad−2 b2 w4 dx 1/5 w7 dx , 10 Boundary Value Problems Thus − u2 dx − t ≤− 1 1 vt dx − η Pxx dx − C5 d−1 1 0 wt dx Qxx dx − R2 dx xx η M2 d−4 C4 1/5 4/5 η M2 d−8/5 2/5 C7 M2 d1/5 η u7 w7 dx 3/5 η M2 d−6/5 C6 3/5 1/5 C8 M2 d8/5 u7 dx 2/5 u5 w7 dx 4/5 u7 dx 2.31 For the other terms on the right-hand side of 2.28 , we have 1 ut fdx ≤ βd−1 ad−1 ut vdx cd ≤ wut dx a2 M1 d 3/5 u dx 2α11 ad−1 uut vdx 0 2α2 b 11 α2 β2 11 α2 c2 11 u dx 1 uu2 dx t βd−1 u3 ut dx 1 0 vu2 dx t wu2 dx, t u2 ut dx 2α11 dc 2α11 d−1 u4 ut dx uut wdx a2 0 d2 7 1/5 b2 4/5 −8/5 M2 d −3 uut fdx ≤ 2α11 βd−1 ≤ u2 ut dx d−1 u3 ut dx c2 2/5 6/5 M2 d 2α11 b 0 β2 uut dx 4/5 M2 d−18/5 u7 dx 1/5 u7 dx α2 b2 11 uu2 dx t vu2 dx t 2/5 M2 d−4/5 wu2 dx, t 3/5 u7 dx Boundary Value Problems α12 11 vut fdx ≤ α12 βd−1 α12 ad−1 ut v2 dx α12 dc vwut dx α2 ≤ 12 b2 a d 3α2 12 α13 −2 α2 β2 12 1 c2 d2 u dx 1/5 v dx u7 α13 d−1 u3 wut dx b2 /2 α21 b2 ≤ 1 v7 dx u7 w7 dx wu2 dx, t 2 uvt vvt dx, uvvt dx uvvt dx α22 vvt dx 1/5 u dx 2α22 d−1 uvt vvt dx, v2 vt dx ⎡ 4/5 M2 d−18/5 ⎣ uwvt dx ⎤ ⎦ w7 dx M1 d−3 vvt dx ≤ wvt gdx ≤ α23 d−1 1/5 α21 d−1 u2 vt dx 1 d−1 4/5 M2 d−18/5 vvt gdx ≤ 2α22 d−1 ≤ 0 α23 1/5 α2 21 u dx 0 uvt gdx ≤ α21 d−1 1/5 uvt dx 2α22 c2 d2 u2 wut dx w2 ut dx 4/5 M2 d−8/5 ⎡ α2 13 4/5 M2 d−18/5 ⎣β vt gdx ≤ d−1 α2 a2 d−2 ≤ 13 α2 b 13 uwut dx 3α2 13 vu2 dx, t 1 v7 dx α13 ad−1 vwut dx α13 dc 1/5 4/5 M2 d−8/5 4/5 M2 d−18/5 b2 wut fdx ≤ α13 βd−1 u2 vut dx α12 d−1 u3 vut dx α2 b 12 uvut dx α23 d−1 1/5 u7 dx 1 vwvt dx 1/5 v7 dx ⎤ ⎦ vvt dx, 12 ⎡ ≤ η α23 vvt dx k α k α 3α2 γ 31 14 α 3α2 32 γ 14 v7 ≤ k α2 33 1 α12 u7 dx 1/5 ⎤ ⎦ w7 dx 1 2 wwt dx, vw2 wt dx ⎡ γ2 4/5 M2 d−8/5 ⎣ 1 w7 dx 1 v7 dx 2α33 γ 1/5 ⎤ ⎦ w7 dx 1 2 vwt dx w2 wt dx 1/5 wwt dx, w3 wt dx 4/5 M2 d−18/5 1/5 α2 γ 33 w dx 2/5 M2 d−4/5 3/5 w dx wwt dx, uvt fdx ≤ α12 βd−1 uvvt dx α12 ad−1 α12 cd α2 β2 12 u4 vt dx a2 u2 vt dx α12 d−1 ≤ 0 α 1/5 k d−1 α33 wwt hdx ≤ α α32 γ vwwt dx k d−2 wwt dx, uw2 wt dx uwt dx α2 ≤ 32 2α33 1 w7 dx 0 0 k d−1 α32 vwt hdx ≤ α η ⎡ γ2 4/5 M2 d−8/5 ⎣ u7 w dx k d−2 1/5 α31 γ uwwt dx α2 ≤ 31 2 wvt dx, w2 wt dx α32 γη γ 2 4/5 −8/5 η M2 d 2 −3 k d−1 α31 uwt hdx ≤ α ⎦ η d M1 1/5 v7 dx wwt dx 0 ≤ 1/5 u7 dx wt hdx ≤ d−1 η α α31 4/5 M2 d−18/5 ⎣ Boundary Value Problems ⎤ α12 b u3 vt dx uwvt dx 4/5 M2 d−18/5 1/5 u7 dx α2 b2 2/5 −4/5 12 M2 d 3/5 u7 dx Boundary Value Problems 13 α2 c2 12 d α13 u7 dx α2 β2 13 α21 a2 1 u7 dx uvut dx uwt dx 1/5 α23 d−1 uvwt dx k d−1 k d−1 α2 1 3/5 u dx wwt dx, vu2 dx, t v2 wt dx 1 1/5 1 u7 dx 1/5 v7 dx w2 ut dx α31 γ ⎤ ⎦ vwt dx, w3 ut dx 1/5 w7 dx 3/5 w dx 0 w2 vt dx α32 γ wu2 dx, t w3 vt dx k2 α2 32 4/5 M2 d−18/5 α2 γ 2/5 −4/5 32 M2 d ≤ ⎤ ⎦ v dx α2 2/5 −4/5 31 γ M2 d 2 vwt dx 1/5 k α2 31 4/5 M2 d−18/5 wvt hdx ≤ α32 α α2 b2 2/5 −4/5 13 M2 d v2 ut dx α 1 u dx 1 u dx α21 d−1 α2 4/5 ≤ 23 M2 d−18/5 ⎣ α32 ⎡ ≤ u3 wt dx 1/5 wut hdx ≤ α31 α α13 b uwwt dx 4/5 M2 d−18/5 ⎡ α2 4/5 −18/5 ⎣ 21 ≤ M2 d 2 wvt dx , 0 vwt gdx ≤ α23 d−1 u2 wt dx α13 d−1 u4 wt dx vut gdx ≤ α21 d−1 vvt dx 0 α2 c2 13 d 1 ≤ α31 uvt dx α13 ad−1 uvwt dx α3 cd 13 uwt fdx ≤ α13 βd−1 α23 1 1/5 w7 dx 3/5 w dx 0 wvt dx 2.32 14 Boundary Value Problems Thus 1 α12 v 2α11 α13 ut fdx α21 u 2α22 v α23 w vt gdx 1 α31 u η α32 v 2α33 w wt hdx α13 uwt fdx 1 α21 vut gdx u v C11 4/5 M2 d−8/5 d C9 wt dx η η2 2.33 1/5 M1 d−3 u v 7 w dx d 3/5 u v 7 w dx d2 −2 α32 wvt hdx 0 4/5 M2 d−8/5 C12 vt C10 u2 t w α31 wut hdx 1 α23 vwt gdx ≤λ α12 uvt fdx u7 v7 w7 dx, where λ is a positive constant Note by 2.8 and 2.9 4/3 B1 K1 d−4/3 − 2 K1 d−2 |Pxx |2 2 Pxx dx − that |P |7/2 7/2 ≤ C |Px |5/3 |P |11/6 |P |7/2 , |Px |10/3 ≤ , and Qxx dx − η Choose a small enough number −4/3 R2 dx ≤ −B2 1, η K1 d4/3 y 5/3 xx η 2.34 > 0, such that λ < C3 According to 2.28 – 2.34 , we have −4/3 y t ≤ −A1 1, η K1 y5/3 A5 K5 y K1 d−2 2/3 A6 K6 y 5/6 A2 K2 y1/6 A3 K3 y1/3 A4 K4 y1/2 2.35 A7 K7 , dPx dQx dRx dx where y However, 2.35 implies that there exist positive constants τ3 > and M3 depending on d, d3 , αij i, j 1, 2, , β, a, b, c, k, γ, α, such that dPx dx, dQx dx, dRx dx ≤ M3 , t ≥ τ3 2.36 Boundary Value Problems 15 When d, d3 ≥ 1, η ∈ d, d , the coefficients of 2.35 can be estimated by constants depending on d, d, but not on d, d3 Thus, when d, d3 ≥ 1, η ∈ d, d , M3 depends on αij i, j 1, 2, , β, a, b, c, k, γ, α,d, d, and is irrelevant to d, d3 ≥ Since ⎛ Px ⎞ ⎛ ⎜ ⎟ ⎜Qx ⎟ ⎝ ⎠ Rx P u Pv Pw ⎞⎛ ux ⎞ ⎜ ⎟⎜ ⎟ ⎜Qu Qv Qw ⎟⎜ vx ⎟, ⎝ ⎠⎝ ⎠ Ru Rv Rw wx 2.37 similar to 2.26 in 24 , we have |dux | |dvx | |dwx | ≤ D |dPx | |dQx | |dRx | , < x < 1, t > 0, 2.38 where D is a positive constant only depending on η, αij i, j 1, 2, Scaling back with 2.22 to original variable u, v, w, t and combining 2.36 , 2.38 , there exist positive constants τ3 > and M3 depending on d, d3 , αij i, j 1, 2, , β, a, b, c, k, γ, α, such that u2 dx, x vx dx, wx dx ≤ M3 , t ≥ τ3 2.39 In addition, when d, d3 ≥ 1, η ∈ d, d , M3 is dependent of d, d, but independent of d, d3 ≥ ii It holds that t ≥ Replacing M1 , M2 with M1 , M2 in 2.24 – 2.34 , we can obtain that there exists a positive constant M3 depending on d, d3 , αij i, j 1, 2, , β, a, b, c, k, γ, α and the W2 -norm of u0 , v0 , w0 such that u2 dx, x vx dx, wx dx ≤ M3 , t ≥ 2.39 When d, d3 ≥ 1, η ∈ d, d , M3 is dependent of d, d, but independent of d, d3 ≥ Concluding from 2.15 , 2.22 , 2.39 , and Sobolev embedding theorem, there exists a positive constants t0 > 0, M, M depending on d, d3 , αij i, j 1, 2, , β, a, b, c, k, γ, α, such that 2.3 and 2.4 are satisfied Furthermore, when d, d3 ≥ 1, η ∈ d, d and the time t is large enough, M, M are dependent of αij i, j 1, 2, , β, a, b, c, k, γ, α, d, d, but independent of d, d3 ≥ Similarly, according to 2.15 , 2.22 , 2.39 , we can see that there exists a positive 1, 2, , β, a, b, c, k, γ, α and the initial functions constant M depending on d, d3 , αij i, j u0 , v0 , w0 , such that |u ·, t |1,2 , |v ·, t |1,2 , |w ·, t |1,2 ≤ M , t ≥ 2.40 16 Boundary Value Problems When d, d3 ≥ 1, η ∈ d, d , M is dependent of d, d, but independent of d, d3 Thus T This completes proof of Theorem 2.1 ∞ Global Stability From , we know that if α > k, β−a−c where p1 9b2 E∗ u∗ , v∗ , w∗ k < m0 , α−k β > a, √ b b p1 ≤ 8c 24c H 24 β − a c2 √ , 4c β − a − c − b p1 3b2 24c β − a − c ≥ , then 1.1 has the unique position equilibrium point Theorem 3.1 Assume that all conditions in Theorem 2.1 and H are satisfied Assume further that a β bu∗ cu∗2 > √ u∗2 u∗2 u∗4 , ∗ w d d3 > α23 M2 αβ β α32 w∗ α α13 M2 β 1 α12 α β 3.1 α31 w∗ α α21 γ > , α u∗2 2α11 M d α12 M α13 M α21 M d M2 w∗ d3 α31 M 2α22 M α23 M α32 M 3.2 2α33 M hold, where M is the positive constant in 2.4 Then the unique positive equilibrium point E∗ of 1.1 is globally asymptotically stable Remark 3.2 Since M is independent of d, d3 in the case of d, d3 ≥ 1, 3.2 is always satisfied if d and d3 are big enough Proof Define the Lyapunov function H u, v, w 2β u − u∗ dx v − v∗ dx α w − w∗ − w∗ ln w dx w∗ 3.3 Boundary Value Problems 17 ≡ Let u, v, w be any solution of 1.1 with initial functions u0 x , v0 x , w0 x ≥ / From the strong maximum principle for parabolic equations, it is not hard to verify that u, v, w > for t > Thus dH ≤− dt d β 2α11 u d3 α α31 u α13 u β − 21 u − u∗ β α32 v w − w∗ d 2α33 w a u∗ b u α23 v 2 α23 w vx 2α22 v α12 u β α21 v ux vx w∗ α32 vx wx dx α w uu∗ c u2 dx − γ − α u∗2 α21 u w∗ w w2 x w∗ α31 ux wx α w u u∗ − u∗2 u2 −2 − − α13 w u2 x α12 v w u u∗ u2 u∗2 u∗2 3.4 v − v∗ dx dx The first integrand in the right hand of the above inequality is positive definite if ∗ w d αβ 2α11 u w2 > α12 v α12 u β α23 vw β α13 uw β α13 w d α13 u α α21 v α32 w∗ α 2α22 v α31 u α23 v w∗ α31 α w α23 w d3 w∗ w βα32 α32 v 2α33 w α31 w∗ α ∗ w α12 u α β α21 u 2α11 u d α12 v α13 w 3.5 d α21 u 2α22 v α23 w d3 α31 u α32 v 2α33 w α21 v From the maximum-norm estimate in Theorem 2.1, 3.2 is a sufficient condition of 3.5 Thus when 3.1 holds, there exists a positive constant δ such that dH u, v, w ≤ −δ dt u − u∗ v − v∗ w − w∗ dx 3.6 18 Boundary Value Problems By integration by parts, Holder inequality and the maximum-norm estimate in ă v v w − w∗ dx is bounded from Theorem 2.1, we can see that d/dt u − u∗ above According to Lemma 3.1 in and 3.6 , we obtain |u ·, t − u∗ |2 −→ 0, |v ·, t − v∗ |2 −→ 0, |w ·, t − w∗ |2 −→ 0, t −→ ∞ 3.7 Using Gagliardo-Nirenberg inequalities, we have |u ·, t |∞ ≤ C|u|1/2 |u|1/2 Thus 1,2 |u ·, t − u∗ |∞ −→ 0, |v ·, t − v∗ |∞ −→ 0, |w ·, t − w∗ |∞ −→ 0, t −→ ∞ 3.8 That is, u, v, w converges uniformly to E∗ Since H u, v, w is decreasing for t > 0, E∗ is globally asymptotically stable Acknowledgments This work has been partially supported by the China National Natural Science Foundation no 10871160 , the NSF of Gansu Province no 096RJZA118 , the Scientific Research Fund of Gansu Provincial Education Department, and the NWNU-KJCXGC-03-47 Foundation References R Zhang, L Guo, and S Fu, “Global behavior for a diffusive predator-prey model with stage structure and nonlinear density restriction—I: the case in Rn ,” Boundary Value Problems, vol 2009, Article ID 378763, p 27, 2009 J Blat and K J Brown, “Bifurcation of steady-state solutions in predator-prey and competition systems,” Proceedings of the Royal 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