二维格上年龄结构单种群模型双稳行波解
|
摘要
为了更加准确地描述研究对象的时空模式,人们在生物种群、空间生态、流行病学和材料学等领域导出了大量具有空间非局部作用的格微分系统。由于它们涉及空间的非局部作用,因而展现出比经典局部作用更复杂的动力学行为。因此对它们的研究具有重要的理论价值和现实意义。
本文研究了二维格上具有年龄结构和固定成熟周期的单种群模型的双稳行波解。该模型反应了空间扩散、非局部时滞及传播方向的共同作用。借助于相应的线性问题,建立了双稳行波解的存在性。通过构造不同的上下解并利用挤压技术得到了双稳行波解的平移唯一性和全局渐近稳定性。与以前的结果相比,本文还考虑了任意传播方向双稳行波解的对称性。这些结果也可以被推广到高维格上。
In order to more accurately describe the spatial-temporal patterns of the objects of study, people have derived many lattice differential systems with nonlocal effects in population biology, spatial ecology, materials and disease spread. Because they concern the spatial nonlocal effects, they show a more complex dynamic behavior than the classic. Therefore, research on such issues is of great theoretical value and practical significance.
This paper study the bistable traveling waves for a single species model with age structure and a fixed maturation period in a two-dimension lattice. This model reacts the joint effect of the diffusion dynamics, the nonlocal delayed effect and the direction of propagation. With the corresponding linear problem, we obtain the existence of bistable traveling waves. By constructing various pairs of super- and subsolutions and employing the squeezing technique to prove the uniqueness with phase shift and globally exponential stability of bistable traveling waves. Compared with previous results, this paper considers the symmetry of the bistable traveling waves with arbitraryθ. These results can be expended to the case with multi-dimension lattice.
本文研究了二维格上具有年龄结构和固定成熟周期的单种群模型的双稳行波解。该模型反应了空间扩散、非局部时滞及传播方向的共同作用。借助于相应的线性问题,建立了双稳行波解的存在性。通过构造不同的上下解并利用挤压技术得到了双稳行波解的平移唯一性和全局渐近稳定性。与以前的结果相比,本文还考虑了任意传播方向双稳行波解的对称性。这些结果也可以被推广到高维格上。
In order to more accurately describe the spatial-temporal patterns of the objects of study, people have derived many lattice differential systems with nonlocal effects in population biology, spatial ecology, materials and disease spread. Because they concern the spatial nonlocal effects, they show a more complex dynamic behavior than the classic. Therefore, research on such issues is of great theoretical value and practical significance.
This paper study the bistable traveling waves for a single species model with age structure and a fixed maturation period in a two-dimension lattice. This model reacts the joint effect of the diffusion dynamics, the nonlocal delayed effect and the direction of propagation. With the corresponding linear problem, we obtain the existence of bistable traveling waves. By constructing various pairs of super- and subsolutions and employing the squeezing technique to prove the uniqueness with phase shift and globally exponential stability of bistable traveling waves. Compared with previous results, this paper considers the symmetry of the bistable traveling waves with arbitraryθ. These results can be expended to the case with multi-dimension lattice.
引文
[1]V.S.Afraimovich,Class Notes on Lattice Dynamics,(1992).
[2]V.S.Afrainmovich,V.I.Nekorkin,Stable states in chain models of unbounded nonequilibrium media,Materaaticheskoe Modelirovauie,3(1991),65-77.
[3]D.G.Aronson,H.F.Weinberger,Multidimensional nonlinear diffusion arising in population genetics,Advances in Math.,30(1978),33-76.
[4]P.W.Bates,A.Chmaj,A discrete convolution model for phase transitions,Arch.Rational Mech.Anal.,150(1999),281-305.
[5]P.W.Bates,P.C.Fife,X.F.Wang,Traveling waves in a convolution model for phase transtions,Arch.Rational Mech.Anal.,138(1997),105-136.
[6]L.A.Bunimovich,Y.G.Sinai,Space-time chaos in couplerd map lattices,Nonlinearity,1(1988),491-518.
[7]J.W.Cahn,Theory of crystal qrowth and interface motion in crystalline materials,Acta Metalluurgica,8(1960),554-562.
[8]J.W.Cahn,J.Mallet-Paret,E.S.van Vleck,Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice,SIAM J.Appl.Math.,59(1998)455-493.
[9]X.Chen,Existence,uniqueness,and asymptotic stability of traveling waves in nonlocal evolution equation,Adv.Differential Equations,2(1997),125-160.
[10]X.Chen,J.S.Guo,Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations,J.Differetial Equations,184(2002),549-569.
[11]X.Chen,J.S.Guo,Uniqueness and existence of traveling waves of discrete quasilinear monostable dynamics,Math.Ann.,326(2003),123-146.
[12]C.P.Cheng,W.T.Li,Z.C.Wang,Spreading speeds and traveling waves in a delayed population model with stage structure on a two-dimensional spatial lattice,2007,submitted.
[13]S.N.Chow,Lattice Dynamical Systems.Dynamical systems(J.W.Macki and P.Zecca Eds.).Lecture Notes in Mathematics,vol.1822.Berlin:Springer,(2003),1-102.
[14]S.N.Chow,J.Mallet-Paret,W.Shen,Stability and bifurcation of traveling wave solution in coupled map lattices,Dynam.Systems Appl.,4(1994),1-26.
[15]S.N.Chow,J.Mallet-Paret,W.Shen,Traveling waves in lattice dynamical systems,J.Differential Equations,149(1998),248-291.
[16]S.N.Chow,W.Shen,Stability and bifurcation of traveling wave solutions in coupled map lattices,Dynamices Systems and Applications,4(1995),1-26.
[17]H.E.Cook,D.de Fontaine,J.E.Hilliard,A model for diffusion on cubic lattices and its application to the early stages of ordering,Acta Metallurgica,17(1969),765-773.
[18]G.B.Ermentrout,Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators,SIAM J.Appl.Math.,52(1992),1665-1687.
[19]T.Erneux,G.Nicolis,Propagation waves in discrete bistable reaction-diffusion systems,Physica D,67(1993),237-244.
[20]G.Fath,Propagation failure of traveling waves in a discrete bistable medium,Physica D,116(1998),176-190.
[21]W.J.Firth,Optical memory and spatial chaos,Phys.Rev.Lett.,61(1988),329-332.
[22]S.A.Gourley,S.Ruan,Dynamics of the diffusive Nicholson's blowflies equation with distributed delays,Proc.Royal Soc.Edinburgh Sect.A,130(2000),1275-1291.
[23]W.S.C.Gurney,S.P.Blythe,R.M.Nisbet,Nicholson's blowflies revisited,Nature,287(1990),17-21.
[24]D.Hankerson,B.Zinner,Wavefronts for a cooperative tridiagonal system of differential equations,J.Dyn.Differential Equations,5(1993),359-373.
[25]M.Hillert,A solid-solution model for inhomogeneous systems,Acta Metallurgica,9(1961),525-535.
[26]J.P.Keener,Propagation and its failure in coupled systems of discrete excitable cells,SIAM J.Appl.Math.,22(1987),556-572.
[27]N.Kopell,G.B.Ermentrout,T.L.Williams,On chains of oscillators forced at one end,SIAM J.Appl.Math.,51(1991),1397-1417.
[28]Y.Kyrychko,S.A.Gourley,M.V.Bartuccelli,Dynamics of a stage-structured population model on an isolated finite lattice,SIAM J.Math.Anal.,37(2006),1688-1708.
[29]J.P.Laplante,T.Emeux,Propagation failure in arrays of coupled bistable chemical reactors,J.Phys.Chem.,96(1992),4931-4934.
[30]G.Lin,W.T.Li,Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays,J.Differential Equations,in press.
[31]W.T.Li,Y.H.Fan,Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson's blowflies model,J.Comput.Appl.Math.,201(2007),55-68.
[32]W.T.Li,S.Ruan,Z.C.Wang,On the diffusive Nicholson's blowflies equation with nonlocal delay,J.Nonlinear Sci.,17(2007),505:525.
[33]S.Ma,Y.R.Duan,Asymptotic stability of traveling waves in a discrete convolution model for phase transitions,J.Math.Anal.Appl.,308(2005),240-256.
[34]S.Ma,X.Zou,Propagation and its failure in a lattice delayed differential equation with global interaction,J.Differential Equations,212(2005),129-190.
[35]S.Ma,X.Zou,Existence,uniqueness and stability of traveling waves in a discrete reactiondiffusion monostable equation with delay,J.Differential Equations,217(2005),54-87.
[36]S.Ma,J.H.Wu,Existence uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equatiion,J.Dynam.Differential Equations,19(2007),391-436.
[37]J.Mallet-Paret,The Fredholm alternative for functional-differential equations of mixed type,J.Dynam.Differential equations,11(1999),1-47.
[38]J.Mallet-Paret,The global structure of traveling waves in spatially discrete dynamical systems,J.Dynam.Differential Equations,11(1999),49-127.
[39]R.H.Martin,H.L.Smith,Abstract functional differential equations and reaction-diffusion systems,Trans.Amer.Math.Soc.,321(1990),1-44.
[40]J.A.J.Metz,O.Diekmann,The Dynamics of Physiologically Structured Populations,New York:Springer,1986.
[41]A.J.Nicholson,The self adjustment of populations to change,Cold Spring Harb.Syrup.Quant.Biol.,22(1957),153-173.
[42]M.A.Pozio,Behavior of solutions of some abstract functional differential equations and application to predator-prey dynamics,Nonlinear Analysis TMA,4(1980),917-938.
[43]M.A.Pozio,Some conditions for global asymptotic stability of equilibra of integrodifferential equations,J.Math.Anal.Appl.,95(1983),501-527.
[44]M.H.Protter,H.F.Weinberger,"Maximum Principle in Differential Equations",Englewood Cliffs,NJ:Prentice-Hall,1967.
[45]S.Puri,R.C.Desai,R.Kapral,Coupled map model with a conserved order parameter,Physica D,50(1991),207-230.
[46]T.Roska,L.O.Chua,The CNN universal machine:an analogic array computer,IEEE Trans.Circuits Syst.,40(1993),163-173.
[47]H.L.Smith,X.Q.Zhao,Global asymptotic stability of traveling waves in delayed reactiondiffusion equation,SIAM J.Math.Anal.,31(2000),514-534.
[48]J.W.H.So,J.Wu,Y.Yang,Numerical Hopf bifurcation analysis on the diffusive Nicholson's blowflies equation,Appl.Math.Comput.,111(2000),53-69.
[49]J.W.H.So,Y.Yang,Dirichlet problem for the diffusive Nicholson's blowflies equation,J.Differential Equations,150(1998),317-348.
[50]J.W.H.So,X.Zou,Traveling waves for the diffusive Nicholson's blowflies equation,Appl.Math.Comput.,122(2001),385-392.
[51]P.Thiran,K.R.Crounse,L.O.Chua,M.Hasler,Pattern formation properties of autonomous celluar neural networks,IEEE Trans.Circuits Syst.,42(1995),757-774.
[52]I.Waller,R.Kapral,Spatial and temporal structure in systems of coupled nonlinear oscillarors,Phy.Rev.A,30(1984),2047-2055.
[53]Z.C.Wang,W.T.Li,S.Ruan,Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays,J.Differential Equations,222(2006),185-232.
[54]Z.C.Wang,W.T.Li,S.Ruan,Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay,J.Differential Equations,238(2007),153-200.
[55]Z.C.Wang,W.T.Li,S.Ruan,Traveling fronts in monostable equations with nonlocal delayed effects,J.Dynam.Differential Equations,in press.
[56]J.Wei,M.Li,Hopf bifurcation analysis in a delayed Nicholson blowflies equation,Nonlinear Analysis TMA,60(2005),1351-1367.
[57]J.Wu,"Theory and Applications of Partial Functional Differential Equations",Springer-Verlag,New York,1996.
[58]P.X.Weng,H.Huang,J.Wu,Asymptotic speed of propagation of wave fronts in a lattice differential equation with global interaction,IMA J.Appl.Math.,68(2003),409-439.
[59]J.Wu,X.Zou,Patterns of sustained oscillations in neural networks with delayed interactions,Appl.Math.Comput.,73(1995),55-76.
[60]S.L.Wu,W.T.Li,Global asymptotic stability of bistable traveling fronts in reactiondiffusion systems and their applications to population models,Chaos,Solitons(?)Fractals,in press.
[61]Y.Yang,J.W.H.So,Dynamics for the diffusive Nicholson's blowflies equation,Discrete.Contin.Dynam.Systems Added Vol.,Ⅱ(1998),333-352.
[62]G.Zhang,Y.Wang,Minimal wave speed in a diffusive epidemic model with temporal delay,Appl.Math.Comput.,188(2007),275-280.
[63]B.Zinner,Stability of traveling,wavefronts for the discrete Nagumo equation,SIAM J.Math.Anal.,22(1991),1016-1020.
[64]B.Zinner,G.Harris,W.Hudson,Traveling wavefronts for the discrete Fisher's equation,J.Differential Equations,105(1993),46-62.
[65]X.Zou,Traveling wave fronts in spatially discrete reaction-diffusion equations on higher dimensional lattices,in:Proceedings of the Third Mississippi State Conference on Difference Equations and Computational Simulations,Mississippi State,MS,1997,Electron.J.Differ.Equ.Conf.,1(1997),211-221.
[2]V.S.Afrainmovich,V.I.Nekorkin,Stable states in chain models of unbounded nonequilibrium media,Materaaticheskoe Modelirovauie,3(1991),65-77.
[3]D.G.Aronson,H.F.Weinberger,Multidimensional nonlinear diffusion arising in population genetics,Advances in Math.,30(1978),33-76.
[4]P.W.Bates,A.Chmaj,A discrete convolution model for phase transitions,Arch.Rational Mech.Anal.,150(1999),281-305.
[5]P.W.Bates,P.C.Fife,X.F.Wang,Traveling waves in a convolution model for phase transtions,Arch.Rational Mech.Anal.,138(1997),105-136.
[6]L.A.Bunimovich,Y.G.Sinai,Space-time chaos in couplerd map lattices,Nonlinearity,1(1988),491-518.
[7]J.W.Cahn,Theory of crystal qrowth and interface motion in crystalline materials,Acta Metalluurgica,8(1960),554-562.
[8]J.W.Cahn,J.Mallet-Paret,E.S.van Vleck,Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice,SIAM J.Appl.Math.,59(1998)455-493.
[9]X.Chen,Existence,uniqueness,and asymptotic stability of traveling waves in nonlocal evolution equation,Adv.Differential Equations,2(1997),125-160.
[10]X.Chen,J.S.Guo,Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations,J.Differetial Equations,184(2002),549-569.
[11]X.Chen,J.S.Guo,Uniqueness and existence of traveling waves of discrete quasilinear monostable dynamics,Math.Ann.,326(2003),123-146.
[12]C.P.Cheng,W.T.Li,Z.C.Wang,Spreading speeds and traveling waves in a delayed population model with stage structure on a two-dimensional spatial lattice,2007,submitted.
[13]S.N.Chow,Lattice Dynamical Systems.Dynamical systems(J.W.Macki and P.Zecca Eds.).Lecture Notes in Mathematics,vol.1822.Berlin:Springer,(2003),1-102.
[14]S.N.Chow,J.Mallet-Paret,W.Shen,Stability and bifurcation of traveling wave solution in coupled map lattices,Dynam.Systems Appl.,4(1994),1-26.
[15]S.N.Chow,J.Mallet-Paret,W.Shen,Traveling waves in lattice dynamical systems,J.Differential Equations,149(1998),248-291.
[16]S.N.Chow,W.Shen,Stability and bifurcation of traveling wave solutions in coupled map lattices,Dynamices Systems and Applications,4(1995),1-26.
[17]H.E.Cook,D.de Fontaine,J.E.Hilliard,A model for diffusion on cubic lattices and its application to the early stages of ordering,Acta Metallurgica,17(1969),765-773.
[18]G.B.Ermentrout,Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators,SIAM J.Appl.Math.,52(1992),1665-1687.
[19]T.Erneux,G.Nicolis,Propagation waves in discrete bistable reaction-diffusion systems,Physica D,67(1993),237-244.
[20]G.Fath,Propagation failure of traveling waves in a discrete bistable medium,Physica D,116(1998),176-190.
[21]W.J.Firth,Optical memory and spatial chaos,Phys.Rev.Lett.,61(1988),329-332.
[22]S.A.Gourley,S.Ruan,Dynamics of the diffusive Nicholson's blowflies equation with distributed delays,Proc.Royal Soc.Edinburgh Sect.A,130(2000),1275-1291.
[23]W.S.C.Gurney,S.P.Blythe,R.M.Nisbet,Nicholson's blowflies revisited,Nature,287(1990),17-21.
[24]D.Hankerson,B.Zinner,Wavefronts for a cooperative tridiagonal system of differential equations,J.Dyn.Differential Equations,5(1993),359-373.
[25]M.Hillert,A solid-solution model for inhomogeneous systems,Acta Metallurgica,9(1961),525-535.
[26]J.P.Keener,Propagation and its failure in coupled systems of discrete excitable cells,SIAM J.Appl.Math.,22(1987),556-572.
[27]N.Kopell,G.B.Ermentrout,T.L.Williams,On chains of oscillators forced at one end,SIAM J.Appl.Math.,51(1991),1397-1417.
[28]Y.Kyrychko,S.A.Gourley,M.V.Bartuccelli,Dynamics of a stage-structured population model on an isolated finite lattice,SIAM J.Math.Anal.,37(2006),1688-1708.
[29]J.P.Laplante,T.Emeux,Propagation failure in arrays of coupled bistable chemical reactors,J.Phys.Chem.,96(1992),4931-4934.
[30]G.Lin,W.T.Li,Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays,J.Differential Equations,in press.
[31]W.T.Li,Y.H.Fan,Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson's blowflies model,J.Comput.Appl.Math.,201(2007),55-68.
[32]W.T.Li,S.Ruan,Z.C.Wang,On the diffusive Nicholson's blowflies equation with nonlocal delay,J.Nonlinear Sci.,17(2007),505:525.
[33]S.Ma,Y.R.Duan,Asymptotic stability of traveling waves in a discrete convolution model for phase transitions,J.Math.Anal.Appl.,308(2005),240-256.
[34]S.Ma,X.Zou,Propagation and its failure in a lattice delayed differential equation with global interaction,J.Differential Equations,212(2005),129-190.
[35]S.Ma,X.Zou,Existence,uniqueness and stability of traveling waves in a discrete reactiondiffusion monostable equation with delay,J.Differential Equations,217(2005),54-87.
[36]S.Ma,J.H.Wu,Existence uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equatiion,J.Dynam.Differential Equations,19(2007),391-436.
[37]J.Mallet-Paret,The Fredholm alternative for functional-differential equations of mixed type,J.Dynam.Differential equations,11(1999),1-47.
[38]J.Mallet-Paret,The global structure of traveling waves in spatially discrete dynamical systems,J.Dynam.Differential Equations,11(1999),49-127.
[39]R.H.Martin,H.L.Smith,Abstract functional differential equations and reaction-diffusion systems,Trans.Amer.Math.Soc.,321(1990),1-44.
[40]J.A.J.Metz,O.Diekmann,The Dynamics of Physiologically Structured Populations,New York:Springer,1986.
[41]A.J.Nicholson,The self adjustment of populations to change,Cold Spring Harb.Syrup.Quant.Biol.,22(1957),153-173.
[42]M.A.Pozio,Behavior of solutions of some abstract functional differential equations and application to predator-prey dynamics,Nonlinear Analysis TMA,4(1980),917-938.
[43]M.A.Pozio,Some conditions for global asymptotic stability of equilibra of integrodifferential equations,J.Math.Anal.Appl.,95(1983),501-527.
[44]M.H.Protter,H.F.Weinberger,"Maximum Principle in Differential Equations",Englewood Cliffs,NJ:Prentice-Hall,1967.
[45]S.Puri,R.C.Desai,R.Kapral,Coupled map model with a conserved order parameter,Physica D,50(1991),207-230.
[46]T.Roska,L.O.Chua,The CNN universal machine:an analogic array computer,IEEE Trans.Circuits Syst.,40(1993),163-173.
[47]H.L.Smith,X.Q.Zhao,Global asymptotic stability of traveling waves in delayed reactiondiffusion equation,SIAM J.Math.Anal.,31(2000),514-534.
[48]J.W.H.So,J.Wu,Y.Yang,Numerical Hopf bifurcation analysis on the diffusive Nicholson's blowflies equation,Appl.Math.Comput.,111(2000),53-69.
[49]J.W.H.So,Y.Yang,Dirichlet problem for the diffusive Nicholson's blowflies equation,J.Differential Equations,150(1998),317-348.
[50]J.W.H.So,X.Zou,Traveling waves for the diffusive Nicholson's blowflies equation,Appl.Math.Comput.,122(2001),385-392.
[51]P.Thiran,K.R.Crounse,L.O.Chua,M.Hasler,Pattern formation properties of autonomous celluar neural networks,IEEE Trans.Circuits Syst.,42(1995),757-774.
[52]I.Waller,R.Kapral,Spatial and temporal structure in systems of coupled nonlinear oscillarors,Phy.Rev.A,30(1984),2047-2055.
[53]Z.C.Wang,W.T.Li,S.Ruan,Traveling wave fronts of reaction-diffusion systems with spatio-temporal delays,J.Differential Equations,222(2006),185-232.
[54]Z.C.Wang,W.T.Li,S.Ruan,Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay,J.Differential Equations,238(2007),153-200.
[55]Z.C.Wang,W.T.Li,S.Ruan,Traveling fronts in monostable equations with nonlocal delayed effects,J.Dynam.Differential Equations,in press.
[56]J.Wei,M.Li,Hopf bifurcation analysis in a delayed Nicholson blowflies equation,Nonlinear Analysis TMA,60(2005),1351-1367.
[57]J.Wu,"Theory and Applications of Partial Functional Differential Equations",Springer-Verlag,New York,1996.
[58]P.X.Weng,H.Huang,J.Wu,Asymptotic speed of propagation of wave fronts in a lattice differential equation with global interaction,IMA J.Appl.Math.,68(2003),409-439.
[59]J.Wu,X.Zou,Patterns of sustained oscillations in neural networks with delayed interactions,Appl.Math.Comput.,73(1995),55-76.
[60]S.L.Wu,W.T.Li,Global asymptotic stability of bistable traveling fronts in reactiondiffusion systems and their applications to population models,Chaos,Solitons(?)Fractals,in press.
[61]Y.Yang,J.W.H.So,Dynamics for the diffusive Nicholson's blowflies equation,Discrete.Contin.Dynam.Systems Added Vol.,Ⅱ(1998),333-352.
[62]G.Zhang,Y.Wang,Minimal wave speed in a diffusive epidemic model with temporal delay,Appl.Math.Comput.,188(2007),275-280.
[63]B.Zinner,Stability of traveling,wavefronts for the discrete Nagumo equation,SIAM J.Math.Anal.,22(1991),1016-1020.
[64]B.Zinner,G.Harris,W.Hudson,Traveling wavefronts for the discrete Fisher's equation,J.Differential Equations,105(1993),46-62.
[65]X.Zou,Traveling wave fronts in spatially discrete reaction-diffusion equations on higher dimensional lattices,in:Proceedings of the Third Mississippi State Conference on Difference Equations and Computational Simulations,Mississippi State,MS,1997,Electron.J.Differ.Equ.Conf.,1(1997),211-221.