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The efficient approximation of coherent pairs in non-autonomous dynamical systems
Monotone traveling waves for delayed Lotka-Volterra competition systems
1. | Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China |
2. | Centre for Disease Modelling and Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada |
References:
[1] |
M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts: DiekmannKaper theory of a nonlinear convolution equation re-visited, Math. Ann., online press. |
[2] |
H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.
doi: 10.1088/0951-7715/22/12/002. |
[3] |
N. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.
doi: 10.1137/0150099. |
[4] |
J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applicaitons, J. Diff. Eqs., 248 (2010), 2199-2226.
doi: 10.1016/j.jde.2010.01.009. |
[5] |
J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.
doi: 10.1088/0951-7715/24/11/002. |
[6] |
T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261.
doi: 10.1098/rspa.2005.1554. |
[7] |
T. Faria and J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks, J. Diff. Eqs., 244 (2008), 1049-1079.
doi: 10.1016/j.jde.2007.12.005. |
[8] |
T. Faria and S. Trofimchuk, Positive travelling fronts for reaction-diffusion systems with distributed delay, Nonlinearity, 23 (2010), 2457-2481.
doi: 10.1088/0951-7715/23/10/006. |
[9] |
T. Faria, Stability and extinction for Lotka-Volterra systems with infinite delay, J. Dynam. Diff. Eqs., 22 (2010), 299-324.
doi: 10.1007/s10884-010-9166-1. |
[10] |
G. Friesecke, Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Diff. Eqs., 5 (1993), 89-103.
doi: 10.1007/BF01063736. |
[11] |
A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Diff. Eqs., 250 (2011), 1767-1787.
doi: 10.1016/j.jde.2010.11.011. |
[12] |
S. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.
doi: 10.1007/s002850000047. |
[13] |
J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Diff. Eqs., 23 (2011), 353-363.
doi: 10.1007/s10884-011-9214-5. |
[14] |
W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Diff. Eqs., 251 (2011), 1549-1561.
doi: 10.1016/j.jde.2011.05.012. |
[15] |
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. |
[16] |
Y. Kuang and H. Smith, Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks, J. Austral. Math. Soc. Ser. B, 34 (1993), 471-494.
doi: 10.1017/S0334270000009036. |
[17] |
Y. Kuang and H. Smith, Global stability for infinite delay Lotka-Volterra type systems, J. Diff. Eqs., 103 (1993), 221-246.
doi: 10.1006/jdeq.1993.1048. |
[18] |
Y. Kuang, Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. Diff. Eqs., 119 (1995), 503-532.
doi: 10.1006/jdeq.1995.1100. |
[19] |
M. K. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Diff. Eqs., 249 (2010), 728-745.
doi: 10.1016/j.jde.2010.04.017. |
[20] |
M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.
doi: 10.1007/s002850200144. |
[21] |
W.-T. Li, G. Lin and S. Ruan, Existence of traveling 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] |
G. Lin, W.-T. Li, and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Dis. Cont. Dyn. Syst. - Series B, 13 (2010), 393-414.
doi: 10.3934/dcdsb.2010.13.393. |
[23] |
B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[24] |
B. Li, M. Lewis and H. 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. |
[25] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[26] |
R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.
doi: 10.1016/0025-5564(89)90026-6. |
[27] |
J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Diff. Eqs., 11 (1999), 1-47.
doi: 10.1023/A:1021889401235. |
[28] |
R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[29] |
G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris, 349 (2011), 553-557.
doi: 10.1016/j.crma.2011.03.008. |
[30] |
C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125.
doi: 10.1137/050638011. |
[31] |
C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Diff. Eqs., 235 (2007), 219-261.
doi: 10.1016/j.jde.2006.12.010. |
[32] |
H. Smith, Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Rev., 30 (1988), 87-113.
doi: 10.1137/1030003. |
[33] |
H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Math. Surveys and Monographs, 41, American Mathematical Society, Providence, R.I., 1995. |
[34] |
H. Smith and H. Thieme, Monotone semiflows in scalar non-quasi-monotone functional-differential equations, J. Math. Anal. Appl., 150 (1990), 289-306.
doi: 10.1016/0022-247X(90)90105-O. |
[35] |
H. Smith and H. Thieme, Strongly order preserving semiflows generated by functional-differential equations, J. Diff. Eqs., 93 (1991), 332-363.
doi: 10.1016/0022-0396(91)90016-3. |
[36] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translation of Mathematical Monographs, 140, Amer. math. Soc., Providence, RI, 1994. |
[37] |
H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Non. Sci., 21 (2011), 747-783.
doi: 10.1007/s00332-011-9099-9. |
[38] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[39] |
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. |
[40] |
J. Wu and X.-Q. Zhao, Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations, J. Diff. Eqs., 186 (2002), 470-484.
doi: 10.1016/S0022-0396(02)00012-8. |
[41] |
J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Diff. Eqs., 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
show all references
References:
[1] |
M. Aguerrea, C. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts: DiekmannKaper theory of a nonlinear convolution equation re-visited, Math. Ann., online press. |
[2] |
H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.
doi: 10.1088/0951-7715/22/12/002. |
[3] |
N. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.
doi: 10.1137/0150099. |
[4] |
J. Fang and X.-Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applicaitons, J. Diff. Eqs., 248 (2010), 2199-2226.
doi: 10.1016/j.jde.2010.01.009. |
[5] |
J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.
doi: 10.1088/0951-7715/24/11/002. |
[6] |
T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261.
doi: 10.1098/rspa.2005.1554. |
[7] |
T. Faria and J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks, J. Diff. Eqs., 244 (2008), 1049-1079.
doi: 10.1016/j.jde.2007.12.005. |
[8] |
T. Faria and S. Trofimchuk, Positive travelling fronts for reaction-diffusion systems with distributed delay, Nonlinearity, 23 (2010), 2457-2481.
doi: 10.1088/0951-7715/23/10/006. |
[9] |
T. Faria, Stability and extinction for Lotka-Volterra systems with infinite delay, J. Dynam. Diff. Eqs., 22 (2010), 299-324.
doi: 10.1007/s10884-010-9166-1. |
[10] |
G. Friesecke, Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Diff. Eqs., 5 (1993), 89-103.
doi: 10.1007/BF01063736. |
[11] |
A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Diff. Eqs., 250 (2011), 1767-1787.
doi: 10.1016/j.jde.2010.11.011. |
[12] |
S. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.
doi: 10.1007/s002850000047. |
[13] |
J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Diff. Eqs., 23 (2011), 353-363.
doi: 10.1007/s10884-011-9214-5. |
[14] |
W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Diff. Eqs., 251 (2011), 1549-1561.
doi: 10.1016/j.jde.2011.05.012. |
[15] |
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. |
[16] |
Y. Kuang and H. Smith, Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks, J. Austral. Math. Soc. Ser. B, 34 (1993), 471-494.
doi: 10.1017/S0334270000009036. |
[17] |
Y. Kuang and H. Smith, Global stability for infinite delay Lotka-Volterra type systems, J. Diff. Eqs., 103 (1993), 221-246.
doi: 10.1006/jdeq.1993.1048. |
[18] |
Y. Kuang, Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. Diff. Eqs., 119 (1995), 503-532.
doi: 10.1006/jdeq.1995.1100. |
[19] |
M. K. Kwong and C. Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Diff. Eqs., 249 (2010), 728-745.
doi: 10.1016/j.jde.2010.04.017. |
[20] |
M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.
doi: 10.1007/s002850200144. |
[21] |
W.-T. Li, G. Lin and S. Ruan, Existence of traveling 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] |
G. Lin, W.-T. Li, and M. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Dis. Cont. Dyn. Syst. - Series B, 13 (2010), 393-414.
doi: 10.3934/dcdsb.2010.13.393. |
[23] |
B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.
doi: 10.1016/j.mbs.2005.03.008. |
[24] |
B. Li, M. Lewis and H. 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. |
[25] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[26] |
R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.
doi: 10.1016/0025-5564(89)90026-6. |
[27] |
J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Diff. Eqs., 11 (1999), 1-47.
doi: 10.1023/A:1021889401235. |
[28] |
R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[29] |
G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Acad. Sci. Paris, 349 (2011), 553-557.
doi: 10.1016/j.crma.2011.03.008. |
[30] |
C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125.
doi: 10.1137/050638011. |
[31] |
C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Diff. Eqs., 235 (2007), 219-261.
doi: 10.1016/j.jde.2006.12.010. |
[32] |
H. Smith, Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Rev., 30 (1988), 87-113.
doi: 10.1137/1030003. |
[33] |
H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Math. Surveys and Monographs, 41, American Mathematical Society, Providence, R.I., 1995. |
[34] |
H. Smith and H. Thieme, Monotone semiflows in scalar non-quasi-monotone functional-differential equations, J. Math. Anal. Appl., 150 (1990), 289-306.
doi: 10.1016/0022-247X(90)90105-O. |
[35] |
H. Smith and H. Thieme, Strongly order preserving semiflows generated by functional-differential equations, J. Diff. Eqs., 93 (1991), 332-363.
doi: 10.1016/0022-0396(91)90016-3. |
[36] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translation of Mathematical Monographs, 140, Amer. math. Soc., Providence, RI, 1994. |
[37] |
H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Non. Sci., 21 (2011), 747-783.
doi: 10.1007/s00332-011-9099-9. |
[38] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[39] |
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. |
[40] |
J. Wu and X.-Q. Zhao, Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations, J. Diff. Eqs., 186 (2002), 470-484.
doi: 10.1016/S0022-0396(02)00012-8. |
[41] |
J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Diff. Eqs., 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
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