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Traveling wave solutions of a competitive recursion
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000 |
2. | School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000 |
References:
[1] |
S. Ahmad, A. C. Lazer and A. Tineo, Traveling waves for a system of equations, Nonlinear Anal., 68 (2008), 3909-3912.
doi: 10.1016/j.na.2007.04.029. |
[2] |
X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. |
[3] |
X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.
doi: 10.1137/050627824. |
[4] |
X. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569. |
[5] |
J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Diff. Eqns. Appl., 10 (2004), 1139-1151. |
[6] |
P. J. Darlington, Competition, competitive repulsion, and coexistence, Proc. Nat. Acad. Sci. USA, 69 (1972), 3151-3155.
doi: 10.1073/pnas.69.11.3151. |
[7] |
J. W. Evans, Nerve axon equations, I: Linear approximations,, Indiana Univ. Math. J., 21 (): 877.
doi: 10.1512/iumj.1972.21.21071. |
[8] |
T. Faria and S. Trofimchuk, Nonmonotone traveling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. |
[9] |
P. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Arch. Ration. Mech. Anal., 65 (1977), 355-361.
doi: 10.1007/BF00250432. |
[10] |
C. E. Goulden and L. L. Hornig, Population oscillations and energy reserves in planktonic cladocera and their consequences to competition, Proc. Natl. Acad. Sci. USA, 77 (1980), 1716-1720.
doi: 10.1073/pnas.77.3.1716. |
[11] |
G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.
doi: 10.1126/science.131.3409.1292. |
[12] |
S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.
doi: 10.1137/070703016. |
[13] |
M. A. Huston, "Biological Diversity: The Coexistence of Species on Changing Landscapes," Cambridge University Press, Cambridge, 1994. |
[14] |
M. A. Huston and D. L. DeAngelis, Competition and coexistence: The effects of resource transport and supply rates, American Naturalist, 144 (1994), 954-977.
doi: 10.1086/285720. |
[15] |
M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436.
doi: 10.1007/BF00173295. |
[16] |
A. W. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering," Mathematics and its Applications, Kluwer Academic Pub., Dordrecht, 1989. |
[17] |
M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.
doi: 10.1007/s002850200144. |
[18] |
B. Li, Some remarks on traveling wave solutions in competition models, Discrete Contin. Dyn. Syst. Ser. B., 12 (2009), 389-399.
doi: 10.3934/dcdsb.2009.12.389. |
[19] |
B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.
doi: 10.1007/s00285-008-0175-1. |
[20] |
B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[21] |
W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.
doi: 10.1088/0951-7715/19/6/003. |
[22] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[23] |
G. Lin and W.-T. Li, Spreading speeds and traveling wavefronts for second order integrodifference equations, J. Math. Anal. Appl., 361 (2010), 520-532.
doi: 10.1016/j.jmaa.2009.07.035. |
[24] |
G. Lin, W.-T. Li and S. Ruan, Asymptotic stability of monostable wavefronts in discrete-time integral recursions, Sci. China Math., 53 (2010), 1185-1194.
doi: 10.1007/s11425-009-0123-6. |
[25] |
G. Lin, W.-T. Li and S. Ruan, Spreading speeds and traveling waves in competitive recursion systems, J. Math. Biol., 62 (2011), 162-201.
doi: 10.1007/s00285-010-0334-z. |
[26] |
R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator,, J. Math. Biol., 16 (): 199.
|
[27] |
R. Lui, A nonlinear integral operator arising from a model in population genetics. I. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937.
doi: 10.1137/0513064. |
[28] |
R. Lui, A nonlinear integral operator arising from a model in population genetics. II. Initial data with compact support, SIAM J. Math. Anal., 13 (1982), 938-953.
doi: 10.1137/0513065. |
[29] |
R. Lui, A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case, SIAM J. Math. Anal., 16 (1985), 1180-1206.
doi: 10.1137/0516087. |
[30] |
R. Lui, A nonlinear integral operator arising from a model in population genetics. IV. Clines, SIAM J. Math. Anal., 17 (1986), 152-168.
doi: 10.1137/0517015. |
[31] |
S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Diff. Eqns., 19 (2007), 391-436.
doi: 10.1007/s10884-006-9065-7. |
[32] |
S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190. |
[33] |
S. Ma and X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. |
[34] |
M. Mei, J. W.-H. So, M. Y. Li and S. S. P. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinburgh Sect. A, 134 (2004), 579-594.
doi: 10.1017/S0308210500003358. |
[35] |
K. Mischaikow and V. Hutson, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008.
doi: 10.1137/0524059. |
[36] |
A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in Britain, Proc. R. Soc. Lond. B, 238 (1989), 113-125.
doi: 10.1098/rspb.1989.0070. |
[37] |
S. Pan, Traveling wave solutions in delayed diffusion systems via a cross iteration scheme, Nonlinear Anal. RWA, 10 (2009), 2807-2818.
doi: 10.1016/j.nonrwa.2008.08.007. |
[38] |
S. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Anal. RWA, 12 (2011), 535-544.
doi: 10.1016/j.nonrwa.2010.06.038. |
[39] |
C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992. |
[40] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[41] |
H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, AMS, Providence, RI, 1995. |
[42] |
H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
doi: 10.1137/S0036141098346785. |
[43] |
M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.
doi: 10.1007/BF00283257. |
[44] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, AMS, Providence, RI, 1994. |
[45] |
Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. |
[46] |
Z.-C. Wang, W.-T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Diff. Eqns., 20 (2008), 573-607.
doi: 10.1007/s10884-008-9103-8. |
[47] |
M.-H. Wang, M. Kot and M. G. Neubert, Integrodifference equations, Allee effects, and invasions, J. Math. Biol., 44 (2002), 150-168.
doi: 10.1007/s002850100116. |
[48] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[49] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
[50] |
H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions, J. Math. Biol., 57 (2008), 387-411.
doi: 10.1007/s00285-008-0168-0. |
[51] |
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.
doi: 10.1007/s002850200145. |
[52] |
Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Foundations of Modern Mathematics Series, Science Press, Beijing, 1990. |
[53] |
B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020.
doi: 10.1137/0522066. |
show all references
References:
[1] |
S. Ahmad, A. C. Lazer and A. Tineo, Traveling waves for a system of equations, Nonlinear Anal., 68 (2008), 3909-3912.
doi: 10.1016/j.na.2007.04.029. |
[2] |
X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. |
[3] |
X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258.
doi: 10.1137/050627824. |
[4] |
X. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569. |
[5] |
J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Diff. Eqns. Appl., 10 (2004), 1139-1151. |
[6] |
P. J. Darlington, Competition, competitive repulsion, and coexistence, Proc. Nat. Acad. Sci. USA, 69 (1972), 3151-3155.
doi: 10.1073/pnas.69.11.3151. |
[7] |
J. W. Evans, Nerve axon equations, I: Linear approximations,, Indiana Univ. Math. J., 21 (): 877.
doi: 10.1512/iumj.1972.21.21071. |
[8] |
T. Faria and S. Trofimchuk, Nonmonotone traveling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376. |
[9] |
P. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Arch. Ration. Mech. Anal., 65 (1977), 355-361.
doi: 10.1007/BF00250432. |
[10] |
C. E. Goulden and L. L. Hornig, Population oscillations and energy reserves in planktonic cladocera and their consequences to competition, Proc. Natl. Acad. Sci. USA, 77 (1980), 1716-1720.
doi: 10.1073/pnas.77.3.1716. |
[11] |
G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297.
doi: 10.1126/science.131.3409.1292. |
[12] |
S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.
doi: 10.1137/070703016. |
[13] |
M. A. Huston, "Biological Diversity: The Coexistence of Species on Changing Landscapes," Cambridge University Press, Cambridge, 1994. |
[14] |
M. A. Huston and D. L. DeAngelis, Competition and coexistence: The effects of resource transport and supply rates, American Naturalist, 144 (1994), 954-977.
doi: 10.1086/285720. |
[15] |
M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436.
doi: 10.1007/BF00173295. |
[16] |
A. W. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering," Mathematics and its Applications, Kluwer Academic Pub., Dordrecht, 1989. |
[17] |
M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.
doi: 10.1007/s002850200144. |
[18] |
B. Li, Some remarks on traveling wave solutions in competition models, Discrete Contin. Dyn. Syst. Ser. B., 12 (2009), 389-399.
doi: 10.3934/dcdsb.2009.12.389. |
[19] |
B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.
doi: 10.1007/s00285-008-0175-1. |
[20] |
B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[21] |
W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.
doi: 10.1088/0951-7715/19/6/003. |
[22] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[23] |
G. Lin and W.-T. Li, Spreading speeds and traveling wavefronts for second order integrodifference equations, J. Math. Anal. Appl., 361 (2010), 520-532.
doi: 10.1016/j.jmaa.2009.07.035. |
[24] |
G. Lin, W.-T. Li and S. Ruan, Asymptotic stability of monostable wavefronts in discrete-time integral recursions, Sci. China Math., 53 (2010), 1185-1194.
doi: 10.1007/s11425-009-0123-6. |
[25] |
G. Lin, W.-T. Li and S. Ruan, Spreading speeds and traveling waves in competitive recursion systems, J. Math. Biol., 62 (2011), 162-201.
doi: 10.1007/s00285-010-0334-z. |
[26] |
R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator,, J. Math. Biol., 16 (): 199.
|
[27] |
R. Lui, A nonlinear integral operator arising from a model in population genetics. I. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937.
doi: 10.1137/0513064. |
[28] |
R. Lui, A nonlinear integral operator arising from a model in population genetics. II. Initial data with compact support, SIAM J. Math. Anal., 13 (1982), 938-953.
doi: 10.1137/0513065. |
[29] |
R. Lui, A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case, SIAM J. Math. Anal., 16 (1985), 1180-1206.
doi: 10.1137/0516087. |
[30] |
R. Lui, A nonlinear integral operator arising from a model in population genetics. IV. Clines, SIAM J. Math. Anal., 17 (1986), 152-168.
doi: 10.1137/0517015. |
[31] |
S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Diff. Eqns., 19 (2007), 391-436.
doi: 10.1007/s10884-006-9065-7. |
[32] |
S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190. |
[33] |
S. Ma and X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. |
[34] |
M. Mei, J. W.-H. So, M. Y. Li and S. S. P. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinburgh Sect. A, 134 (2004), 579-594.
doi: 10.1017/S0308210500003358. |
[35] |
K. Mischaikow and V. Hutson, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008.
doi: 10.1137/0524059. |
[36] |
A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in Britain, Proc. R. Soc. Lond. B, 238 (1989), 113-125.
doi: 10.1098/rspb.1989.0070. |
[37] |
S. Pan, Traveling wave solutions in delayed diffusion systems via a cross iteration scheme, Nonlinear Anal. RWA, 10 (2009), 2807-2818.
doi: 10.1016/j.nonrwa.2008.08.007. |
[38] |
S. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Anal. RWA, 12 (2011), 535-544.
doi: 10.1016/j.nonrwa.2010.06.038. |
[39] |
C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992. |
[40] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[41] |
H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, AMS, Providence, RI, 1995. |
[42] |
H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.
doi: 10.1137/S0036141098346785. |
[43] |
M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.
doi: 10.1007/BF00283257. |
[44] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, AMS, Providence, RI, 1994. |
[45] |
Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. |
[46] |
Z.-C. Wang, W.-T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Diff. Eqns., 20 (2008), 573-607.
doi: 10.1007/s10884-008-9103-8. |
[47] |
M.-H. Wang, M. Kot and M. G. Neubert, Integrodifference equations, Allee effects, and invasions, J. Math. Biol., 44 (2002), 150-168.
doi: 10.1007/s002850100116. |
[48] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[49] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
[50] |
H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions, J. Math. Biol., 57 (2008), 387-411.
doi: 10.1007/s00285-008-0168-0. |
[51] |
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.
doi: 10.1007/s002850200145. |
[52] |
Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Foundations of Modern Mathematics Series, Science Press, Beijing, 1990. |
[53] |
B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020.
doi: 10.1137/0522066. |
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