格微分方程的行波解和整体解
|
摘要
格动力系统通常指离散空间上常微分方程的无穷维系统或者是差分方程的无穷维系统(如在D维空间中,由全体整数组成的格zD).一方面格动力系统来自于实际背景中,如在生物学、电路理论、材料学、图像处理及化学反应的很多实际问题的数学模型都可归纳为格动力系统.另一方面,它来自于偏微分方程的空间离散化.因此,对格动力系统的研究具有重要的理论和实际意义.本论文研究了格动力系统的行波解和整体解,其中整体解是指对所有的时间t∈R都有定义的解.
首先利用行波解在无穷远处指数衰减性的先验估计来构造合适的上、下解,应用上下解方法、比较原理得出具有对流项的双稳格微分方程在周期介质中的整体解.由于伴随着对流现象、空间的离散化以及周期介质的出现,两列沿着相反方向传播的行波有可能具有不同的传播速度(也就是说,c1≠c1),从而失去了两列波所具有的对称性,这就要求我们构造出不同于以往工作中所出现的上、下解.在存在性的基础上,进一步建立了整体解的唯一性和Lyapunov稳定性.
其次,考虑了二维格上单稳格动力系统的整体解.由于空间的离散化,最小波速c*的值依赖于波的传播方向θ.对于来自不同方向θ和θ'的行波解,其最小波速可能满足c*(θ)≠c*(θ'),因此,对于来自两个不同方向且具有不同的传播速度的有限多个行波解,其传播速度可能小于c*(θ)和c*(θ').当传播速度分别大于、小于或者等于c*(θ)和c*(θ')时,通过解一个以-k为初始时刻的初值问题序列并结合比较原理和最大值原理,给出了系统整体解的存在性和解对参数的连续依赖性.
在种群动力学中,格动力系统被用来描述在空间离散斑块环境下,物种种群的增长和入侵过程.二维格上具有静态阶段的反应扩散系统,它描述了种群个体在迁移和静止这两种状态之间的变换,其中种群中的一部分个体具有迁移性,另一部分个体则总是处于静止状态,并且只有物种处于迁移状态时是可再生的.在第四部分,通过结合与空间无关的解和具有不同传播速度和不同传播方向的行波解,给出系统的一个上界和下解,并利用比较原理建立了具有静态阶段的反应扩散系统的整体解.
最后,考虑了双稳型非局部积分微分方程的周期行波解.此类方程是利用卷积算子来描述扩散过程的反应方程,它可以从生物种群、空间生态和传染病等众多研究领域中得到.在双稳的假设条件下,运用基于上下解方法和比较原理的挤压技术证明了行波解的稳定性和唯一性,又由单调动力系统的理论得出了行波解的存在性.
Lattice dynamical systems usually refer to infinite systems of ordinary differ-ential equations on discrete space or infinite systems of difference equations (such as the D-dimensional integer lattice ZD). Such systems arise, on the one hand, from practical backgrounds, such as modeling in biology, electrical circuit theory, material science, image processing and chemical kinetics can be induced to lattice dynamical systems. On the other hand, they also arise as the spatial discretization of partial differential equations. Therefore, it is more meaningful and valuable in theory and practice to study such equations. In this thesis, we consider traveling wave solutions and entire solutions of lattice dynamical systems. Here, the entire solutions are defined in the whole space and for all time t∈R.
Our thesis firstly considers the entire solutions of a lattice reaction-diffusion-convection equation with bistable nonlinearity in periodic media. Using a priori estimates of the exponential decaying of the traveling wave solution at infinity, we can construct suitable sub-super solutions. Then the existence of entire solutions is obtained via sub-super solutions method and the comparison principle. With the appearing of the convection, the discretization of space and the periodic media, two traveling waves solutions with opposite directions may admit different speeds (that is, C1≠C1), and thus lose their symmetry, which requires us to construct the different sub-super solutions. On the basis of existence, uniqueness and Liapunov stability of entire solutions are established further.
Next, we study the entire solutions of a two-dimensional (2-D) lattice dy-namical system with monostable nonlinearity. Due to the discretization of space, c* depends on the directionθof the traveling wave. Traveling waves coming from different directionsθandθmay admit c*(θ)≠c*(θ). Therefore, for a limited number of traveling wave solutions with different propagation speeds from differ-ent directions, the propagation speed may be smaller than c*(θ) and c*(θ). While the propagation speeds are larger than, smaller than or equal to c*(θ) and c*(θ). by solving sequence of Cauchy problems starting at times -k with suitable initial conditions, and using the maximum principle and the comparison principle, we gain the existence of entire solutions and the continuous dependence of entire solutions on the parameters.
In the population dynamics, lattice dynamical system is used to describe pro-cesses of the growth and invasion of species in the space of discrete plaque environ-ment. A reaction-diffusion system with a quiescent stage on a 2D spatial lattice for a single-species population with two separate mobile and stationary states, de-scribes a species population of which the individuals alternate between mobile and stationary states, and only the mobile ones reproduce. In the third part, using the comparison principle with appropriate sub-solutions and upper estimates, some new entire solutions are constructed by combining spatially independent solutions and traveling fronts with different wave speeds and directions of propagation.
Finally, we consider the periodic traveling wave solutions of a nonlocal integro-differential equation with bistable nonlinearity. This kind of equation describes the diffusion process by the convolution operator. It can arise from the biological populations, ecological and infectious diseases and many other research areas. Under the bistable hypotheses, using the sub- and upper- solutions method, comparison principle and squeezing technique, we prove the stability and the uniqueness of periodic traveling wave solutions with phase shift. Then, we prove the existence of traveling wave solutions by the theory of monotone dynamical systems.
首先利用行波解在无穷远处指数衰减性的先验估计来构造合适的上、下解,应用上下解方法、比较原理得出具有对流项的双稳格微分方程在周期介质中的整体解.由于伴随着对流现象、空间的离散化以及周期介质的出现,两列沿着相反方向传播的行波有可能具有不同的传播速度(也就是说,c1≠c1),从而失去了两列波所具有的对称性,这就要求我们构造出不同于以往工作中所出现的上、下解.在存在性的基础上,进一步建立了整体解的唯一性和Lyapunov稳定性.
其次,考虑了二维格上单稳格动力系统的整体解.由于空间的离散化,最小波速c*的值依赖于波的传播方向θ.对于来自不同方向θ和θ'的行波解,其最小波速可能满足c*(θ)≠c*(θ'),因此,对于来自两个不同方向且具有不同的传播速度的有限多个行波解,其传播速度可能小于c*(θ)和c*(θ').当传播速度分别大于、小于或者等于c*(θ)和c*(θ')时,通过解一个以-k为初始时刻的初值问题序列并结合比较原理和最大值原理,给出了系统整体解的存在性和解对参数的连续依赖性.
在种群动力学中,格动力系统被用来描述在空间离散斑块环境下,物种种群的增长和入侵过程.二维格上具有静态阶段的反应扩散系统,它描述了种群个体在迁移和静止这两种状态之间的变换,其中种群中的一部分个体具有迁移性,另一部分个体则总是处于静止状态,并且只有物种处于迁移状态时是可再生的.在第四部分,通过结合与空间无关的解和具有不同传播速度和不同传播方向的行波解,给出系统的一个上界和下解,并利用比较原理建立了具有静态阶段的反应扩散系统的整体解.
最后,考虑了双稳型非局部积分微分方程的周期行波解.此类方程是利用卷积算子来描述扩散过程的反应方程,它可以从生物种群、空间生态和传染病等众多研究领域中得到.在双稳的假设条件下,运用基于上下解方法和比较原理的挤压技术证明了行波解的稳定性和唯一性,又由单调动力系统的理论得出了行波解的存在性.
Lattice dynamical systems usually refer to infinite systems of ordinary differ-ential equations on discrete space or infinite systems of difference equations (such as the D-dimensional integer lattice ZD). Such systems arise, on the one hand, from practical backgrounds, such as modeling in biology, electrical circuit theory, material science, image processing and chemical kinetics can be induced to lattice dynamical systems. On the other hand, they also arise as the spatial discretization of partial differential equations. Therefore, it is more meaningful and valuable in theory and practice to study such equations. In this thesis, we consider traveling wave solutions and entire solutions of lattice dynamical systems. Here, the entire solutions are defined in the whole space and for all time t∈R.
Our thesis firstly considers the entire solutions of a lattice reaction-diffusion-convection equation with bistable nonlinearity in periodic media. Using a priori estimates of the exponential decaying of the traveling wave solution at infinity, we can construct suitable sub-super solutions. Then the existence of entire solutions is obtained via sub-super solutions method and the comparison principle. With the appearing of the convection, the discretization of space and the periodic media, two traveling waves solutions with opposite directions may admit different speeds (that is, C1≠C1), and thus lose their symmetry, which requires us to construct the different sub-super solutions. On the basis of existence, uniqueness and Liapunov stability of entire solutions are established further.
Next, we study the entire solutions of a two-dimensional (2-D) lattice dy-namical system with monostable nonlinearity. Due to the discretization of space, c* depends on the directionθof the traveling wave. Traveling waves coming from different directionsθandθmay admit c*(θ)≠c*(θ). Therefore, for a limited number of traveling wave solutions with different propagation speeds from differ-ent directions, the propagation speed may be smaller than c*(θ) and c*(θ). While the propagation speeds are larger than, smaller than or equal to c*(θ) and c*(θ). by solving sequence of Cauchy problems starting at times -k with suitable initial conditions, and using the maximum principle and the comparison principle, we gain the existence of entire solutions and the continuous dependence of entire solutions on the parameters.
In the population dynamics, lattice dynamical system is used to describe pro-cesses of the growth and invasion of species in the space of discrete plaque environ-ment. A reaction-diffusion system with a quiescent stage on a 2D spatial lattice for a single-species population with two separate mobile and stationary states, de-scribes a species population of which the individuals alternate between mobile and stationary states, and only the mobile ones reproduce. In the third part, using the comparison principle with appropriate sub-solutions and upper estimates, some new entire solutions are constructed by combining spatially independent solutions and traveling fronts with different wave speeds and directions of propagation.
Finally, we consider the periodic traveling wave solutions of a nonlocal integro-differential equation with bistable nonlinearity. This kind of equation describes the diffusion process by the convolution operator. It can arise from the biological populations, ecological and infectious diseases and many other research areas. Under the bistable hypotheses, using the sub- and upper- solutions method, comparison principle and squeezing technique, we prove the stability and the uniqueness of periodic traveling wave solutions with phase shift. Then, we prove the existence of traveling wave solutions by the theory of monotone dynamical systems.
引文
[1]Alikakos, N., Bates, P., Chen, X. Periodic traveling waves and locating oscillating patterns in multidimensional domains[J]. Trans. Amer. Math. Soc.1999,351:2777-2805.
[2]Aronson, D.G., Weinberger, H.F. Multidimensional nonlinear difusions arising in pop-ulation genetics[J]. Adv. Math.1978,30:33-76.
[3]Bates, P.W., Chen, F.X. Periodic traveling waves for nonlocal integro-differential model[J]. Electron. J. Diff. Eqns.1999,26:1-19.
[4]Bates, P.W., Chen, X., Chmaj, A.J.J. Traveling waves of bistable dynamics on a lattice[J]. SIAM J. Math. Anal.2003,35:520-546.
[5]Bates, P.W., Chmaj, A. A discrete convolution model for phase transitions[J]. Arch. Rational. Mech. Anal.1999,150:281-305.
[6]Bates, P.W., Fife, P.C., Ren, X.F., Wang, X.F. Traveling waves in a convolution model for phase transtions[J]. Arch. Rational. Mech. Anal.1997,138:105-136.
[7]Bell, J., Conser, C. Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons[J]. Quart. Appl. Math. 1984,42:1-44.
[8]Berestycki, H., Hamel, F. Front propagation in periodic excitable media[J]. Comm. Pure Appl. Math.2002,55:949-1032.
[9]Berestycki, H., Hamel, F., Roques, L. Analysis of the periodically fragmented en-vironment model:I-Influence of periodic heterogeneous on species persistence[J]. J. Math. Biol.2005,51:75-113.
[10]Berestycki, H., Hamel, F.. Roques, L. Analysis of the periodically fragmented en-vironment model:II-Biological invasions and pulsating traveling fronts[J]. J. Math. Pures Appl.2005,84:1101-1146.
[11]Bodnar, M., Velazquez, J.J.L. An integro-differential equation arising as a limit of individual cell-based models[J]. J. Differential Equations.2006,222:341-380.
[12]Bramson, M. Convergence of solutions of the Kolmogorov equation to travelling waves[M]. Memoirs Amer. Math. Soc.44. Providence 1983.
[13]Bunimovich, L.A., Sinai, Y.G. Space-time chaos in couplerd map lattices[J]. Nonlin-earity 1988,1:491-518.
[14]Cahn, J.W. Theory of crystal growth and interface motion in crystalline materials[J]. Acta. Metall.1960,8:554-562.
[15]Cahn, J.W., Mallet-Paret, J., van Vleck, E.S. Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice[J]. SIAM J. Appl. Math.1999,59:455-493.
[16]Carr, J., Chmaj, A. Uniqueness of travelling waves for nonlocal monostable equa-tions[J]. Proc. Amer. Math. Soc.2004,132:2433-2439.
[17]Carrillo, C., Fife, P., Spatial effects in discrete generation population models[J]. J. Math. Biol.2005,50:161-188.
[18]Chen, X. Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equation[J]. Adv. Differential Equations 1997,2:125-160.
[19]Chen, X., Fu, S.C., Guo, J.S. Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices[J]. SIAM J. Math. Anal.2006,38:233-258.
[20]Chen, X., Guo, J.S. Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations[J]. J. Differential Equations 2002,184:549-569.
[21]Chen, X., Guo, J.S. Uniqueness and existence of traveling waves for discrete quasi-linear monostable dynamics[J]. Math. Ann.2003,326:123-146.
[22]Chen, X., Guo, J.S. Existence and uniqueness of entire solutions for a reaction-diffusion equation[J]. J. Differential Equations 2005,212:62-84.
[23]Chen, X., Guo, J.S. Hirokazu Ninomiya, Entire solutions of reaction-diffusion equa-tions with balanced bistable nonlinearities[J]. Proc. Roy. Soc. Edinburgh Sect.2006, A 136:1207-1237.
[24]Chen, X., Guo, J.S., Wu, C.C. Traveling waves in discrete Periodic media for bistable dynamics[J]. Arch. Rational Mech. Anal.2008,189:189-236.
[25]Cheng, C.P., Li, W.T., Wang, Z.C. Spreading speeds and traveling waves for a delayed population model with stage structure on a two-dimensional spatial lattice[J]. IMA J. Appl. Math.2008.73:592-018.
[26]Cheng, C.P., Li, W.T., Wang, Z.C. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice[J]. Discrete Cont. Dyn. Systems Ser. B 2010,13:559-575.
[27]Chow, S.N. Lattice Dynamical Systemes[M]. in:J.W.Macki, P.Zecca (Eds.), Dynam-ical systems, Lecture Notes in Mathematics, vol.1822, Berlin:Springer.2003, pp. 1-102.
[28]Chow, S.N., Mallet-Paret, J. Pattern formation and spatial chaos in lattice dynamical systems, Ⅰ and Ⅱ[J]. IEEE Trans. Circuits Systems 1995,42:746-756.
[29]Chow, S.N., Mallet-Paret, J., Shen, W. Traveling waves in lattice dynamical sys-tems[J]. J. Differential Equations 1998,149:248-291.
[30]Chow, S.N., Shen, W. Stability and bifurcation of traveling wave solutions in coupled map lattices[J]. Dynamices Systems and Applications 1995,4:1-26.
[31]Chua, l.O., Roska, T. The CNN paradigm, IEEE Trans. Circuits Syst.1993,40: 147-156.
[32]Chua, L.O., Yang, L. Cellular neural networks:Theorey[J]. IEEE Trans. Circuits Syst.1998,35:1257-1272.
[33]Chua, L.O., Yang, L. Cellular neural networks:Applications[J]. IEEE Trans. Circuits Syst.1998,35:1273-1290.
[34]Cortazar, C., Elgueta, M., Rossi, J.D., A non-local diffusion equation whose solutions develop a free boundary[J]. Annales Henri Poincare 2005,6:269-281.
[35]Coville, J. Travelling waves in a nonlocal reaction diffusion equation with ignition nonlinearity[Ph.D. Thesis]. Paris:Universit'e Pierre et Marie Curie.2003.
[36]Coville. J.. Dupaigne, L. Travelling fronts solutions in integrodifferential equations[J]. C R. Acal. Sci. Paris 2003,337:25-30.
[37]Coville, J.. Dupaigne, L. Propagation speed of travelling fronts in nonlocal reaction-diffusion equat.ions[J]. Nonlinear Anal.2005,60:797-819.
[38]Coville. J., Dupaigne, L. On a nonlocal reaction diffusion equation arising in popula-tion dynamics[J]. Proc. Roy. Soc. Edinburgh Sect.2007.137A:1-29.
[39]Ei, S.I. The motion of weakly interacting pulses in reaction-diffusion systems[J]. J. Dynam. Differential Equations 2002,14:85-136.
[40]Ei, S.I., Mimura M., Nagayama, M. Pulse-pulse interaction in reaction-diffusion sys-tems[J]. Physica D 2002,165:176-198.
[41]Evans, Lawrence C. Partial Differential Equations[M]. American Mathematical Soci-ety 1998.
[42]Fang, J., Zhao, X.Q. Bistable traveling waves for monotone semiflows with applica-tions. arXiv:1102.4556v1 [math.AP] 22Feb 2011.
[43]Freidlin, M.I. Limit theorems for large deviations and reaction-diffusion equations[J]. Ann. Probab.1985,13:639-675.
[44]Fu, S.C., Guo, J.S., Shieh, S.Y. Traveling wave solutions for some discrete quasilinear parabolic equations [J]. Nonlinear Analysis:Theorey, Methods and Applications 2002, 48:1137-1149.
[45]Fukao, Y., Motita, Y., Ninomiya, H. Some entire solutions of the Allen-Cahn equa-tion[J]. Taiwanese J. Math.2004,8:15-32.
[46]Gartner, J., Freidlin, M.I. On the propagation of concentration waves in periodic and random media[J]. Soviet Math. Dokl.1979,20:1282-1286.
[47]Gilboa, G., Osher, S. Nonlocal linear regularization and supervised segmentation [J]. SIAM Multiscale Modeling and Simulation.2007,6:595-630.
[48]Guo, Y.-J.L. Entire solutions for a disctrte diffusive equation[J]. J. Math. Anal. Appl. 2008,347:450-458.
[49]Guo, J.S., Hamel, F. Front propagation for discrete periodic monostable equations[J]. Math. Ann.2006,335:489-525.
[50]Guo, J.S., Morita, Y. Entire solutions of reaction-diffusion equations and an applica-tion to discrete diffusive equations[J]. Discrete Cont. Dyn. Systems 2005,12:193-212.
[51]Guo, J.S., Wu, C.C. Uniqueness and stability of traveling waves for periodic monos-table lattice dynamical system [J]. J. Differentical Equations 2009,246:3818-3833.
[52]Guo. J.S., Wu. C.H. Exisrence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system[J]. Osaka J. Math.2008.45:327-346.
[53]Guo, J.S., Wu, C.H. Front propagation for a two-dimensional periodic monostable lattice dynamical system[J]. Discrete Contin. Dyn. Syst. 2010, 26: 197-223.
[54]Guo J.S., Wu, C.H. Entire solutions for a two-component competition system in a lattice[J]. Tohoku Math. J. 2010, 62: 17-28
[55]Hadeler, K.P., Lewis, M.A. Spatial dynamics of the diffusive logistic equation with a sedentary compartment [J]. Canad. Appl. Math. Quart. 2002, 10: 473-499.
[56]Hamel, F., Nadirashvili, N. Entire solutions of the KPP Equation[J]. Comm. Pure Appl. Math. 1999, LII: 1255-1276.
[57]Hamel, F., Nadirashvili, N. Travelling fronts and entire solutions of the Fisher-KPP equation in RN[J]. Arch. Rational Mech. Anal. 2001, 157: 91-163.
[58]Huang, J., Lu, G. Travelling wave solutions in delayed lattice dynamical system[J]. Chinese Ann. Math. Ser. A 25 (2004), pp. 153-164.
[59]Huang, J., Lu, G., Zou, X. Existence of travelling wave fronts of delayed lattice differential equations[J]. J. Math. Anal. Appl. 298 (2000), pp. 538-558.
[60]Hudson, W., Zinner, B. Existence of traveling waves for a generalized discrete Fisher's equation[J]. Commun. Appl. Nonlin. Anal. 1994, 1: 23-46.
[61]W. Hudson, B. Zinner, Existence of traveling waves for reaction-dissusion equa-tion of Fisher type in periodic mediap[M]. Boundary Value Problems for Functional-Differential Equations (Ed. Henderson J.) World Scientific 1995, pp. 187-199.
[62]Hsu, C.H., Lin, S.S. Existence and multiplicity of traveling waves in a lattice dynam-ical system[J]. J. Differential Equations 2000, 164: 431-450.
[63]Kawahara, T., Tanaka, M. Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation[J]. Physics letters 1983, 97A(8): 311-314.
[64]Keener, J.P. Propagation and its failure to coupled systems of discrete excitable cells[J]. SIAM J. Appl. Math. 1987, 47: 556-572.
[65]Kindermann, S., Osher, S., Jones, P.W. Deblurring and denoising of images by non-local functionals[J]. SIAM Multiscale Model. Simul. 2005, 4: 1091-1115.
[66]Kopell. N., Ermentrout. G.B., Williams. T.L. On chains of oscillators forced at one end[J]. SIAM J. Appl. Math. 1991, 51: 1397-1417.
[67]Kyrychko, Y., Gourley, S.A., Bartuccelli, M.V. Dynamics of a stage-structured popu-lation model on an isolated finite lattice[J]. SIAM J. Math.Anal.2006,37:1688-1708.
[68]Laplante, J.P., Emeux, T. Propagation failure in arrays of coupled bistable chemical reactors[J]. J. Phys. Chem.1992,96:4931-4934.
[69]Lewis, M.A., Schmitz, G. Biological invasion of an organism with separate mobile and stationary states:modeling and analysis[J]. Forma 1996,11:1-25.
[70]Li, W.T., Liu, N.W., Wang, Z.C. Entire solutions in reaction-advection-diffusion equations in cylinders[J]. J. Math. Pures Appl.2008,90:492-504.
[71]Li, W.T., Sun, Y.J., Wang, Z.C. Entire solutions in the Fisher-KPP equation with nonlocal dispersal[J]. Nonlinear Anal.2010,11:2302-2313.
[72]Li, W.T., Wang, Z.C, Wu, J. Entire solutions in monostable reaction-diffusion equa-tions with delayed nonlinearity[J]. J. Differential Equations 2008,245:102-129.
[73]Liang, X., Zhao, X.Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications [J]. Comm. Pure Appl. Math.2007,60:1-40.
[74]Lin, G., Li, W.T. Traveling waves in delayed lattice dynamical systems with competi-tion interactions[J]. Nonlinear Analysis:Real Word Applications 2010,11:3666-3679.
[75]Lin, G., Li, W.T., Pan, S.X. Travelling wavefronts in delayed lattice dynamical sys-tems with global interaction [J]. Journal of Difference Equations and Applications 2010,16:1429-1446.
[76]Liu, N.W., Li, W.T. Pulsating type entire slutions in monostable reaction-advection-diffusion equations in periodic excitable media[J]. Nonlinear Analysis:Theory, Meth-ods Applications 2012,75:1869-1880.
[77]Liu, N.W., Li, W.T. Entire solutions in bistable reaction-advection-diffusion equa-tions in heterogeneous media [J]. Sci. China Math.2010,53:1775-1786.
[78]Liu. N.W., Li, W.T., Wang, Z.C. Entire solutions of reaction-adverction-diffusion equations with bistable nonlinearity in cylinders[J]. J. Differential Equations 2009, 246:4249-4267.
[79]Ma. S., Liao, X.X., Wu, J.H. Traveling wave solutions for planar lattice differential steins with applications to nenral networks[J], J. Differential Equations 2002.182: 269-297.
[80]Ma, S., Weng, P., Zou, X. Asymptotic speeds of propagation and traveling wavefront in a non-local delayed lattice differential equation[J]. Nonlinear Analysis TMA 2006, 65:1858-1890.
[81]Ma, S., Zou, X. Propagation and its failure in a lattice delayed differential equation with global interaction[J]. J.Differential Equations 2005,212:129-190.
[82]Ma, S., Zou, X. Existence, uniqueness and stability of traveling wavefronts in a dis-crete reaction-diffusion monostable equation with delay,[J]. J. Differential Equations 2005,217:54-87.
[83]Mallet-Paret, J. The Fredholm alternative for functional differential equations of mixed type[J]. J. Dynam. Differential Equations 1999,11:1-47.
[84]Mallet-Paret, J. The global structure of traveling waves in spatially discrete dynamical systems[J]. J. Dynam. Differential Equations 1999,11:49-127.
[85]Mogilner, A., Edelstein-Keshet, L. A non-local model for a swarm[J]. J. Math. Biol. 1999,38:534-570.
[86]Morita, Y., Mimoto, Y. Collision and collapse of layer in a ID scalar reaction-diffusion equation[J]. Physica D 2000,140:151-170.
[87]Morita, Y., Ninomiya, H. Entire solutions with merging fronts to raction-diffusion equations[J]. J. Dynam. Differential Equations 2006,18:841-861.
[88]Morita, Y., Tachibana, K. An entire solution to the Lotka-Volterra competition-diffusion equations[J]. SIAM J. Math. Anal.2009,40:2217-2240.
[89]Murray, J.D. Mathematical Biology[M]. New York:Springer.2003.
[90]Nakamura, K.I. Effective speed of traveling wavefronts in periodic inhomogeneous media, Proceedings Workshop " Nonlinear Partial Differential Equations and Related Topics", Joint Research Center for science and Technology of Ryukoku University, 1999. pp.53-60.
[91]Schumacher, K. Travelling-front solutions for integro-differential equations, I[J]. J. Reine Angew. Math.1980,316:54-70.
[92]Shen, W. Traveling waves in time almost periodic structures governed by bistable nonlinearities:I. Stability and uniqueness[J]. J. Differential Equations 1999.159: 1-54.
[93]Shen, W.Traveling waves in time almost periodic structures governed by bistable nonlinearities:Ⅱ. Existence[J]. J. Differential Equations 1999,159:55-101.
[94]Shi, Z.X., Li, W.T., Cheng, C.P. Stability and uniqueness of traveling wavefronts in a two-dimensional lattice differential equation with delay [J]. Appl. Math. Comput. 2009,208:484-494.
[95]Shigesada, N., Kawasaki, K. Biological invasions:theory and practice[M]. Oxford: Oxford Series in Ecology and Evolution Oxford University Press.1997.
[96]Shigesada, N., Kawasaki, K., Teramoto, E. Traveling periodic waves in heterogeneous environments[J]. Theor. Popul. Biol.1986,30:143-160.
[97]Shorrocks, B., Swingland, I.R. Living in a Patch Environment[M]. New York:Oxford University Press.1990.
[98]Smith, H.L., Thiema, H. Strongle order preserving semiflows generated by functional differential equations[J]. J. Differential Equations 1991,93:332-363.
[99]Sun, Y.J., Li, W.T., Wang, Z.C. Entire solutions in nonlocal dispersal equations with bistable nonlinearity [J]. J. Differential Equations 2011,251:551-581.
[100]Taylar, J.E., Cahn, J.W., Handwerker, C.A. Geometric models for crystal growth[J]. Acta Metall Mater 1992,40:1443-1474.
[101]Volpert A.I., Volpert, V.A. Traveling wave solutions of parabolic systems[M]. Transl. Math. Monogr., vol.140, Amer. Math. Soc, Providence, RI.1994.
[102]Wang, M.X., Lv, G.Y. Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays[J]. Nonlinearity 2010,23:1609-1630.
[103]Wang, Q.R., Zhao, X.Q. Spreading speed and traveling waves for the diffusive logistic equation with a sedentary compartment [J]. Dynamics of Continuous, Discrete and Impulsive System 2006,13A:231-246.
[104]Wang, Z.C, Li, W.T.. Ruan, S. Entire solutions in bistable reaction-diffusion equa-tions with nonlocal delayed nonlinearity [J]. Trans. Amer. Math. Soc.2009,361:2047-2084.
[105]Wang. Z.C., Li, W.T., Ruan. S. Entire solutions in delayed lattice differential equa-tions with monostable; nonlinearity [J]. SIAM.1. Math. Anal.2009.40:23922420.
[106]Wang, Z.C., Li, W.T., Ruan, S. Entire solutions in Lattice delayed differential equa-tions with global interaction:bistable case[J]. submitted.
[107]Wang, Z.C., Li, W.T., Wu, J. Entire solutions in delayed lattice differential equations with monostable nonlinearity[J]. SIAM J. Math. Anal.2009,40:2392-2420.
[108]Weinberger, H. F. Some deterministic models for the spread of genetic and other alterations[M]. In Jager, W., Rost, H., and Tautu, P. (eds.), Biological Growth and Spread, Springer Lecture Notes in Biomathematics, Vol.38,1980. pp.320-349.
[109]Weinberger, H. F. Long time behavior of a class of biological models[J]. SIAM J. Math. Anal.1982,13:353-396.
[110]Weinberger, H.F. On spreading speeds and traveling waves for growth and migration in periodic habitat [J]. J. Math. Biol.2002,45:511-548.
[111]Weinberger, H.F., Lewis, M.A., Li, B. Analysis of linear determincy for spread in cooperative models[J]. J. Math. Biol.2002,45:183-218.
[112]Weng, P., Huang, H., Wu, J. Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction [J]. IMA J. Appl. Math. 2003,68:409-439.
[113]Wu, J., Zou, X. Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations[J]. J. Differential Equations 1997, 135:315-357.
[114]Wu, S.L. Entire Solutions in a Bistable Reaction-Diffusion System Modeling Man-Environment-Man Epidemics [J]. Nonlinear Analysis:RWA 2012,13:1991-2005.
[115]Xin, X. Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity[J]. J. Dyn. Differ. Equ.1991,3:541-573.
[116]Xin, X. Existence of plame fronts in covective-diffusive periodic media[J] Arch. Ra-tion. Mech. Anal.1992,121:205-233.
[117]Xin, X. Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media[J]. J. Stat. Phys.1993.73:893-925.
[118]Xin. J.X. Existence of multidimensional traveling waves in tranport of reactive so-lutes through periodic porous media[J]. Arch. Ration. Mech. Anal.1994.128:75-103.
[119]Xin, J.X. Front propagation in heterogeneous media[J]. SIAM Rev.2000,42:161-230.
[120]Yagisita, H. Backward global solutions characterizing annihilation dynamics of trav-elling fronts[J]. Publ. Res. Inst. Math. Sci.2003,39:117-164.
[121]Zhang, G.B., Li, W.T., Lin, G. Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure[J]. Math. Computers Modelling 2009,49: 1021-1029.
[122]Zhang, K. Zhao, X.Q. Asymptotic behaviour of a reaction-diffusion model with a quiescent stage[J]. Proc. R. Soc. Lond.2007,463A:1029-1043.
[123]Zhang, P.A., Li, W.T. Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage[J]. Nonlinear Anal. TMA 2010,72:2178-2189.
[124]Zhao, H.Q., Wu, S.L. Wave propagation for a reaction-diffusion model with a qui-escent stage on a 2D spatial lattice[J]. Nonlinear Analysis:Real World Applications 2011,12:1178-1191.
[125]Zinner, B. Stability of traveling wavefronts for the discrete Nagumo equation[J]. SIAM J. Math. Anal.1991,22:1016-1020.
[126]Zinner, B. Existence of traveling wavefront solutions for the discrete Nagumo equa-tion[J]. J. Differential Equations 1992,96:1-27.
[127]Zinner, B., Harris, G., Hudson, W. Traveling wavefronts for the discrete Fisher's equation[J]. J. Differential Equations 1993,105:46-62.
[128]Zou, X. Traveling waves fronts in spatially discrete reaction-diffusion equations on higher dimensional lattice, in:Proceedings of the Third Mississippi State Conference on Difference Equation and Computational Simulations, Mississippi State, MS,1997, Electron[J]. J. Differ. Equ. Conf.1997,1:211-221.
[2]Aronson, D.G., Weinberger, H.F. Multidimensional nonlinear difusions arising in pop-ulation genetics[J]. Adv. Math.1978,30:33-76.
[3]Bates, P.W., Chen, F.X. Periodic traveling waves for nonlocal integro-differential model[J]. Electron. J. Diff. Eqns.1999,26:1-19.
[4]Bates, P.W., Chen, X., Chmaj, A.J.J. Traveling waves of bistable dynamics on a lattice[J]. SIAM J. Math. Anal.2003,35:520-546.
[5]Bates, P.W., Chmaj, A. A discrete convolution model for phase transitions[J]. Arch. Rational. Mech. Anal.1999,150:281-305.
[6]Bates, P.W., Fife, P.C., Ren, X.F., Wang, X.F. Traveling waves in a convolution model for phase transtions[J]. Arch. Rational. Mech. Anal.1997,138:105-136.
[7]Bell, J., Conser, C. Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons[J]. Quart. Appl. Math. 1984,42:1-44.
[8]Berestycki, H., Hamel, F. Front propagation in periodic excitable media[J]. Comm. Pure Appl. Math.2002,55:949-1032.
[9]Berestycki, H., Hamel, F., Roques, L. Analysis of the periodically fragmented en-vironment model:I-Influence of periodic heterogeneous on species persistence[J]. J. Math. Biol.2005,51:75-113.
[10]Berestycki, H., Hamel, F.. Roques, L. Analysis of the periodically fragmented en-vironment model:II-Biological invasions and pulsating traveling fronts[J]. J. Math. Pures Appl.2005,84:1101-1146.
[11]Bodnar, M., Velazquez, J.J.L. An integro-differential equation arising as a limit of individual cell-based models[J]. J. Differential Equations.2006,222:341-380.
[12]Bramson, M. Convergence of solutions of the Kolmogorov equation to travelling waves[M]. Memoirs Amer. Math. Soc.44. Providence 1983.
[13]Bunimovich, L.A., Sinai, Y.G. Space-time chaos in couplerd map lattices[J]. Nonlin-earity 1988,1:491-518.
[14]Cahn, J.W. Theory of crystal growth and interface motion in crystalline materials[J]. Acta. Metall.1960,8:554-562.
[15]Cahn, J.W., Mallet-Paret, J., van Vleck, E.S. Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice[J]. SIAM J. Appl. Math.1999,59:455-493.
[16]Carr, J., Chmaj, A. Uniqueness of travelling waves for nonlocal monostable equa-tions[J]. Proc. Amer. Math. Soc.2004,132:2433-2439.
[17]Carrillo, C., Fife, P., Spatial effects in discrete generation population models[J]. J. Math. Biol.2005,50:161-188.
[18]Chen, X. Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equation[J]. Adv. Differential Equations 1997,2:125-160.
[19]Chen, X., Fu, S.C., Guo, J.S. Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices[J]. SIAM J. Math. Anal.2006,38:233-258.
[20]Chen, X., Guo, J.S. Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations[J]. J. Differential Equations 2002,184:549-569.
[21]Chen, X., Guo, J.S. Uniqueness and existence of traveling waves for discrete quasi-linear monostable dynamics[J]. Math. Ann.2003,326:123-146.
[22]Chen, X., Guo, J.S. Existence and uniqueness of entire solutions for a reaction-diffusion equation[J]. J. Differential Equations 2005,212:62-84.
[23]Chen, X., Guo, J.S. Hirokazu Ninomiya, Entire solutions of reaction-diffusion equa-tions with balanced bistable nonlinearities[J]. Proc. Roy. Soc. Edinburgh Sect.2006, A 136:1207-1237.
[24]Chen, X., Guo, J.S., Wu, C.C. Traveling waves in discrete Periodic media for bistable dynamics[J]. Arch. Rational Mech. Anal.2008,189:189-236.
[25]Cheng, C.P., Li, W.T., Wang, Z.C. Spreading speeds and traveling waves for a delayed population model with stage structure on a two-dimensional spatial lattice[J]. IMA J. Appl. Math.2008.73:592-018.
[26]Cheng, C.P., Li, W.T., Wang, Z.C. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice[J]. Discrete Cont. Dyn. Systems Ser. B 2010,13:559-575.
[27]Chow, S.N. Lattice Dynamical Systemes[M]. in:J.W.Macki, P.Zecca (Eds.), Dynam-ical systems, Lecture Notes in Mathematics, vol.1822, Berlin:Springer.2003, pp. 1-102.
[28]Chow, S.N., Mallet-Paret, J. Pattern formation and spatial chaos in lattice dynamical systems, Ⅰ and Ⅱ[J]. IEEE Trans. Circuits Systems 1995,42:746-756.
[29]Chow, S.N., Mallet-Paret, J., Shen, W. Traveling waves in lattice dynamical sys-tems[J]. J. Differential Equations 1998,149:248-291.
[30]Chow, S.N., Shen, W. Stability and bifurcation of traveling wave solutions in coupled map lattices[J]. Dynamices Systems and Applications 1995,4:1-26.
[31]Chua, l.O., Roska, T. The CNN paradigm, IEEE Trans. Circuits Syst.1993,40: 147-156.
[32]Chua, L.O., Yang, L. Cellular neural networks:Theorey[J]. IEEE Trans. Circuits Syst.1998,35:1257-1272.
[33]Chua, L.O., Yang, L. Cellular neural networks:Applications[J]. IEEE Trans. Circuits Syst.1998,35:1273-1290.
[34]Cortazar, C., Elgueta, M., Rossi, J.D., A non-local diffusion equation whose solutions develop a free boundary[J]. Annales Henri Poincare 2005,6:269-281.
[35]Coville, J. Travelling waves in a nonlocal reaction diffusion equation with ignition nonlinearity[Ph.D. Thesis]. Paris:Universit'e Pierre et Marie Curie.2003.
[36]Coville. J.. Dupaigne, L. Travelling fronts solutions in integrodifferential equations[J]. C R. Acal. Sci. Paris 2003,337:25-30.
[37]Coville, J.. Dupaigne, L. Propagation speed of travelling fronts in nonlocal reaction-diffusion equat.ions[J]. Nonlinear Anal.2005,60:797-819.
[38]Coville. J., Dupaigne, L. On a nonlocal reaction diffusion equation arising in popula-tion dynamics[J]. Proc. Roy. Soc. Edinburgh Sect.2007.137A:1-29.
[39]Ei, S.I. The motion of weakly interacting pulses in reaction-diffusion systems[J]. J. Dynam. Differential Equations 2002,14:85-136.
[40]Ei, S.I., Mimura M., Nagayama, M. Pulse-pulse interaction in reaction-diffusion sys-tems[J]. Physica D 2002,165:176-198.
[41]Evans, Lawrence C. Partial Differential Equations[M]. American Mathematical Soci-ety 1998.
[42]Fang, J., Zhao, X.Q. Bistable traveling waves for monotone semiflows with applica-tions. arXiv:1102.4556v1 [math.AP] 22Feb 2011.
[43]Freidlin, M.I. Limit theorems for large deviations and reaction-diffusion equations[J]. Ann. Probab.1985,13:639-675.
[44]Fu, S.C., Guo, J.S., Shieh, S.Y. Traveling wave solutions for some discrete quasilinear parabolic equations [J]. Nonlinear Analysis:Theorey, Methods and Applications 2002, 48:1137-1149.
[45]Fukao, Y., Motita, Y., Ninomiya, H. Some entire solutions of the Allen-Cahn equa-tion[J]. Taiwanese J. Math.2004,8:15-32.
[46]Gartner, J., Freidlin, M.I. On the propagation of concentration waves in periodic and random media[J]. Soviet Math. Dokl.1979,20:1282-1286.
[47]Gilboa, G., Osher, S. Nonlocal linear regularization and supervised segmentation [J]. SIAM Multiscale Modeling and Simulation.2007,6:595-630.
[48]Guo, Y.-J.L. Entire solutions for a disctrte diffusive equation[J]. J. Math. Anal. Appl. 2008,347:450-458.
[49]Guo, J.S., Hamel, F. Front propagation for discrete periodic monostable equations[J]. Math. Ann.2006,335:489-525.
[50]Guo, J.S., Morita, Y. Entire solutions of reaction-diffusion equations and an applica-tion to discrete diffusive equations[J]. Discrete Cont. Dyn. Systems 2005,12:193-212.
[51]Guo, J.S., Wu, C.C. Uniqueness and stability of traveling waves for periodic monos-table lattice dynamical system [J]. J. Differentical Equations 2009,246:3818-3833.
[52]Guo. J.S., Wu. C.H. Exisrence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system[J]. Osaka J. Math.2008.45:327-346.
[53]Guo, J.S., Wu, C.H. Front propagation for a two-dimensional periodic monostable lattice dynamical system[J]. Discrete Contin. Dyn. Syst. 2010, 26: 197-223.
[54]Guo J.S., Wu, C.H. Entire solutions for a two-component competition system in a lattice[J]. Tohoku Math. J. 2010, 62: 17-28
[55]Hadeler, K.P., Lewis, M.A. Spatial dynamics of the diffusive logistic equation with a sedentary compartment [J]. Canad. Appl. Math. Quart. 2002, 10: 473-499.
[56]Hamel, F., Nadirashvili, N. Entire solutions of the KPP Equation[J]. Comm. Pure Appl. Math. 1999, LII: 1255-1276.
[57]Hamel, F., Nadirashvili, N. Travelling fronts and entire solutions of the Fisher-KPP equation in RN[J]. Arch. Rational Mech. Anal. 2001, 157: 91-163.
[58]Huang, J., Lu, G. Travelling wave solutions in delayed lattice dynamical system[J]. Chinese Ann. Math. Ser. A 25 (2004), pp. 153-164.
[59]Huang, J., Lu, G., Zou, X. Existence of travelling wave fronts of delayed lattice differential equations[J]. J. Math. Anal. Appl. 298 (2000), pp. 538-558.
[60]Hudson, W., Zinner, B. Existence of traveling waves for a generalized discrete Fisher's equation[J]. Commun. Appl. Nonlin. Anal. 1994, 1: 23-46.
[61]W. Hudson, B. Zinner, Existence of traveling waves for reaction-dissusion equa-tion of Fisher type in periodic mediap[M]. Boundary Value Problems for Functional-Differential Equations (Ed. Henderson J.) World Scientific 1995, pp. 187-199.
[62]Hsu, C.H., Lin, S.S. Existence and multiplicity of traveling waves in a lattice dynam-ical system[J]. J. Differential Equations 2000, 164: 431-450.
[63]Kawahara, T., Tanaka, M. Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation[J]. Physics letters 1983, 97A(8): 311-314.
[64]Keener, J.P. Propagation and its failure to coupled systems of discrete excitable cells[J]. SIAM J. Appl. Math. 1987, 47: 556-572.
[65]Kindermann, S., Osher, S., Jones, P.W. Deblurring and denoising of images by non-local functionals[J]. SIAM Multiscale Model. Simul. 2005, 4: 1091-1115.
[66]Kopell. N., Ermentrout. G.B., Williams. T.L. On chains of oscillators forced at one end[J]. SIAM J. Appl. Math. 1991, 51: 1397-1417.
[67]Kyrychko, Y., Gourley, S.A., Bartuccelli, M.V. Dynamics of a stage-structured popu-lation model on an isolated finite lattice[J]. SIAM J. Math.Anal.2006,37:1688-1708.
[68]Laplante, J.P., Emeux, T. Propagation failure in arrays of coupled bistable chemical reactors[J]. J. Phys. Chem.1992,96:4931-4934.
[69]Lewis, M.A., Schmitz, G. Biological invasion of an organism with separate mobile and stationary states:modeling and analysis[J]. Forma 1996,11:1-25.
[70]Li, W.T., Liu, N.W., Wang, Z.C. Entire solutions in reaction-advection-diffusion equations in cylinders[J]. J. Math. Pures Appl.2008,90:492-504.
[71]Li, W.T., Sun, Y.J., Wang, Z.C. Entire solutions in the Fisher-KPP equation with nonlocal dispersal[J]. Nonlinear Anal.2010,11:2302-2313.
[72]Li, W.T., Wang, Z.C, Wu, J. Entire solutions in monostable reaction-diffusion equa-tions with delayed nonlinearity[J]. J. Differential Equations 2008,245:102-129.
[73]Liang, X., Zhao, X.Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications [J]. Comm. Pure Appl. Math.2007,60:1-40.
[74]Lin, G., Li, W.T. Traveling waves in delayed lattice dynamical systems with competi-tion interactions[J]. Nonlinear Analysis:Real Word Applications 2010,11:3666-3679.
[75]Lin, G., Li, W.T., Pan, S.X. Travelling wavefronts in delayed lattice dynamical sys-tems with global interaction [J]. Journal of Difference Equations and Applications 2010,16:1429-1446.
[76]Liu, N.W., Li, W.T. Pulsating type entire slutions in monostable reaction-advection-diffusion equations in periodic excitable media[J]. Nonlinear Analysis:Theory, Meth-ods Applications 2012,75:1869-1880.
[77]Liu, N.W., Li, W.T. Entire solutions in bistable reaction-advection-diffusion equa-tions in heterogeneous media [J]. Sci. China Math.2010,53:1775-1786.
[78]Liu. N.W., Li, W.T., Wang, Z.C. Entire solutions of reaction-adverction-diffusion equations with bistable nonlinearity in cylinders[J]. J. Differential Equations 2009, 246:4249-4267.
[79]Ma. S., Liao, X.X., Wu, J.H. Traveling wave solutions for planar lattice differential steins with applications to nenral networks[J], J. Differential Equations 2002.182: 269-297.
[80]Ma, S., Weng, P., Zou, X. Asymptotic speeds of propagation and traveling wavefront in a non-local delayed lattice differential equation[J]. Nonlinear Analysis TMA 2006, 65:1858-1890.
[81]Ma, S., Zou, X. Propagation and its failure in a lattice delayed differential equation with global interaction[J]. J.Differential Equations 2005,212:129-190.
[82]Ma, S., Zou, X. Existence, uniqueness and stability of traveling wavefronts in a dis-crete reaction-diffusion monostable equation with delay,[J]. J. Differential Equations 2005,217:54-87.
[83]Mallet-Paret, J. The Fredholm alternative for functional differential equations of mixed type[J]. J. Dynam. Differential Equations 1999,11:1-47.
[84]Mallet-Paret, J. The global structure of traveling waves in spatially discrete dynamical systems[J]. J. Dynam. Differential Equations 1999,11:49-127.
[85]Mogilner, A., Edelstein-Keshet, L. A non-local model for a swarm[J]. J. Math. Biol. 1999,38:534-570.
[86]Morita, Y., Mimoto, Y. Collision and collapse of layer in a ID scalar reaction-diffusion equation[J]. Physica D 2000,140:151-170.
[87]Morita, Y., Ninomiya, H. Entire solutions with merging fronts to raction-diffusion equations[J]. J. Dynam. Differential Equations 2006,18:841-861.
[88]Morita, Y., Tachibana, K. An entire solution to the Lotka-Volterra competition-diffusion equations[J]. SIAM J. Math. Anal.2009,40:2217-2240.
[89]Murray, J.D. Mathematical Biology[M]. New York:Springer.2003.
[90]Nakamura, K.I. Effective speed of traveling wavefronts in periodic inhomogeneous media, Proceedings Workshop " Nonlinear Partial Differential Equations and Related Topics", Joint Research Center for science and Technology of Ryukoku University, 1999. pp.53-60.
[91]Schumacher, K. Travelling-front solutions for integro-differential equations, I[J]. J. Reine Angew. Math.1980,316:54-70.
[92]Shen, W. Traveling waves in time almost periodic structures governed by bistable nonlinearities:I. Stability and uniqueness[J]. J. Differential Equations 1999.159: 1-54.
[93]Shen, W.Traveling waves in time almost periodic structures governed by bistable nonlinearities:Ⅱ. Existence[J]. J. Differential Equations 1999,159:55-101.
[94]Shi, Z.X., Li, W.T., Cheng, C.P. Stability and uniqueness of traveling wavefronts in a two-dimensional lattice differential equation with delay [J]. Appl. Math. Comput. 2009,208:484-494.
[95]Shigesada, N., Kawasaki, K. Biological invasions:theory and practice[M]. Oxford: Oxford Series in Ecology and Evolution Oxford University Press.1997.
[96]Shigesada, N., Kawasaki, K., Teramoto, E. Traveling periodic waves in heterogeneous environments[J]. Theor. Popul. Biol.1986,30:143-160.
[97]Shorrocks, B., Swingland, I.R. Living in a Patch Environment[M]. New York:Oxford University Press.1990.
[98]Smith, H.L., Thiema, H. Strongle order preserving semiflows generated by functional differential equations[J]. J. Differential Equations 1991,93:332-363.
[99]Sun, Y.J., Li, W.T., Wang, Z.C. Entire solutions in nonlocal dispersal equations with bistable nonlinearity [J]. J. Differential Equations 2011,251:551-581.
[100]Taylar, J.E., Cahn, J.W., Handwerker, C.A. Geometric models for crystal growth[J]. Acta Metall Mater 1992,40:1443-1474.
[101]Volpert A.I., Volpert, V.A. Traveling wave solutions of parabolic systems[M]. Transl. Math. Monogr., vol.140, Amer. Math. Soc, Providence, RI.1994.
[102]Wang, M.X., Lv, G.Y. Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays[J]. Nonlinearity 2010,23:1609-1630.
[103]Wang, Q.R., Zhao, X.Q. Spreading speed and traveling waves for the diffusive logistic equation with a sedentary compartment [J]. Dynamics of Continuous, Discrete and Impulsive System 2006,13A:231-246.
[104]Wang, Z.C, Li, W.T.. Ruan, S. Entire solutions in bistable reaction-diffusion equa-tions with nonlocal delayed nonlinearity [J]. Trans. Amer. Math. Soc.2009,361:2047-2084.
[105]Wang. Z.C., Li, W.T., Ruan. S. Entire solutions in delayed lattice differential equa-tions with monostable; nonlinearity [J]. SIAM.1. Math. Anal.2009.40:23922420.
[106]Wang, Z.C., Li, W.T., Ruan, S. Entire solutions in Lattice delayed differential equa-tions with global interaction:bistable case[J]. submitted.
[107]Wang, Z.C., Li, W.T., Wu, J. Entire solutions in delayed lattice differential equations with monostable nonlinearity[J]. SIAM J. Math. Anal.2009,40:2392-2420.
[108]Weinberger, H. F. Some deterministic models for the spread of genetic and other alterations[M]. In Jager, W., Rost, H., and Tautu, P. (eds.), Biological Growth and Spread, Springer Lecture Notes in Biomathematics, Vol.38,1980. pp.320-349.
[109]Weinberger, H. F. Long time behavior of a class of biological models[J]. SIAM J. Math. Anal.1982,13:353-396.
[110]Weinberger, H.F. On spreading speeds and traveling waves for growth and migration in periodic habitat [J]. J. Math. Biol.2002,45:511-548.
[111]Weinberger, H.F., Lewis, M.A., Li, B. Analysis of linear determincy for spread in cooperative models[J]. J. Math. Biol.2002,45:183-218.
[112]Weng, P., Huang, H., Wu, J. Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction [J]. IMA J. Appl. Math. 2003,68:409-439.
[113]Wu, J., Zou, X. Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations[J]. J. Differential Equations 1997, 135:315-357.
[114]Wu, S.L. Entire Solutions in a Bistable Reaction-Diffusion System Modeling Man-Environment-Man Epidemics [J]. Nonlinear Analysis:RWA 2012,13:1991-2005.
[115]Xin, X. Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity[J]. J. Dyn. Differ. Equ.1991,3:541-573.
[116]Xin, X. Existence of plame fronts in covective-diffusive periodic media[J] Arch. Ra-tion. Mech. Anal.1992,121:205-233.
[117]Xin, X. Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media[J]. J. Stat. Phys.1993.73:893-925.
[118]Xin. J.X. Existence of multidimensional traveling waves in tranport of reactive so-lutes through periodic porous media[J]. Arch. Ration. Mech. Anal.1994.128:75-103.
[119]Xin, J.X. Front propagation in heterogeneous media[J]. SIAM Rev.2000,42:161-230.
[120]Yagisita, H. Backward global solutions characterizing annihilation dynamics of trav-elling fronts[J]. Publ. Res. Inst. Math. Sci.2003,39:117-164.
[121]Zhang, G.B., Li, W.T., Lin, G. Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure[J]. Math. Computers Modelling 2009,49: 1021-1029.
[122]Zhang, K. Zhao, X.Q. Asymptotic behaviour of a reaction-diffusion model with a quiescent stage[J]. Proc. R. Soc. Lond.2007,463A:1029-1043.
[123]Zhang, P.A., Li, W.T. Monotonicity and uniqueness of traveling waves for a reaction-diffusion model with a quiescent stage[J]. Nonlinear Anal. TMA 2010,72:2178-2189.
[124]Zhao, H.Q., Wu, S.L. Wave propagation for a reaction-diffusion model with a qui-escent stage on a 2D spatial lattice[J]. Nonlinear Analysis:Real World Applications 2011,12:1178-1191.
[125]Zinner, B. Stability of traveling wavefronts for the discrete Nagumo equation[J]. SIAM J. Math. Anal.1991,22:1016-1020.
[126]Zinner, B. Existence of traveling wavefront solutions for the discrete Nagumo equa-tion[J]. J. Differential Equations 1992,96:1-27.
[127]Zinner, B., Harris, G., Hudson, W. Traveling wavefronts for the discrete Fisher's equation[J]. J. Differential Equations 1993,105:46-62.
[128]Zou, X. Traveling waves fronts in spatially discrete reaction-diffusion equations on higher dimensional lattice, in:Proceedings of the Third Mississippi State Conference on Difference Equation and Computational Simulations, Mississippi State, MS,1997, Electron[J]. J. Differ. Equ. Conf.1997,1:211-221.