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Interaction between water and plants: Rich dynamics in a simple model
Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations
1. | School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China |
2. | School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China |
In this paper, we mainly discuss the existence and asymptotic stability of traveling fronts for the nonlocal evolution equations. With the monostable assumption, we obtain that there exists a constant $c^*>0$, such that the equation has no traveling fronts for $0<c<c^*$ and a traveling front for each c ≥ c*. For $c>c^*$, we will further show that the traveling front is globally asymptotic stable and is unique up to translation. If we applied to some differential equations or integro-differential equations, our results recover and/or complement a number of existing ones.
References:
[1] |
D. G. Aronson and H. F. Weinberger,
Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Goldstein J, ed. Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, Berlin: Springer-Verlag, 466 (1975), 5-49.
|
[2] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
P. W. Bates, P. C. Fife, X. Ren and X. Wang,
Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.
doi: 10.1007/s002050050037. |
[4] |
N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology Academic Press, San Diego, 1986. |
[5] |
N. F. 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. |
[6] |
X. Cabré and J. M. Roquejoffre,
The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722.
doi: 10.1007/s00220-013-1682-5. |
[7] |
J. Carr and A. Chmaj,
Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
doi: 10.1090/S0002-9939-04-07432-5. |
[8] |
X. Chen,
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[9] |
X. Chen and J. S. Guo,
Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.
doi: 10.1006/jdeq.2001.4153. |
[10] |
J. Coville,
On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.
doi: 10.1007/s10231-005-0163-7. |
[11] |
J. Coville, J. Dávila and S. Martínez,
Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.
doi: 10.1016/j.jde.2007.11.002. |
[12] |
J. Coville and L. Dupaigne,
Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.
doi: 10.1016/j.na.2003.10.030. |
[13] |
J. Coville and L. Dupaigne,
On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755.
doi: 10.1017/S0308210504000721. |
[14] |
J. Fang and X. Zhao,
Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.
doi: 10.1088/0951-7715/24/11/002. |
[15] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems Springer Verlag, New York, 1979. |
[16] |
P. C. Fife and J. B. Mcleod,
The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.
doi: 10.1007/BF00250432. |
[17] |
R. A. Fisher,
The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[18] |
F. Hamel and L. Ryzhik,
On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.
doi: 10.1088/0951-7715/27/11/2735. |
[19] |
A. Kolmogorov, I. Petrovskii and N. Piskunov,
A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskow. Gos. Univ, 1 (1937), 1-26.
|
[20] |
S. Ma and X. Zou,
Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.
doi: 10.1016/j.jde.2005.05.004. |
[21] |
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. |
[22] |
J. Medlock and M. Kot,
Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.
doi: 10.1016/S0025-5564(03)00041-5. |
[23] |
M. Mei, C. H. Ou and X. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Appl. Math. , 42 (2010), 2762-2790; Erratum: Erratum: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal. , 44 (2012), 538-540.
doi: 10.1137/110850633. |
[24] |
J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. |
[25] |
S. Pan, W. Li and G. Lin,
Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.
doi: 10.1007/s00033-007-7005-y. |
[26] |
S. Pan, W. Li and G. Lin,
Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.
doi: 10.1016/j.na.2009.12.008. |
[27] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer Science & Business Media, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[28] |
S. Ruan and D. Xiao,
Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991-1011.
doi: 10.1017/S0308210500003590. |
[29] |
D. H. Sattinger,
On the stability of waves of nonlinear parabolic systems, Adv. in Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[30] |
K. W. Schaaf,
Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.
doi: 10.2307/2000859. |
[31] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems American Mathematical Soc. , Providence, RI, 1994. |
[32] |
Z. Wang, W. 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.
doi: 10.1016/j.jde.2007.03.025. |
[33] |
Z. Wang, W. Li and S. Ruan,
Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.
doi: 10.1007/s10884-008-9103-8. |
[34] |
Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations Science Press, Beijing, 1990. |
[35] |
L. Zhang,
Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations, 197 (2004), 162-196.
doi: 10.1016/S0022-0396(03)00170-0. |
[36] |
X. Zhao and W. Wang,
Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128.
doi: 10.3934/dcdsb.2004.4.1117. |
show all references
References:
[1] |
D. G. Aronson and H. F. Weinberger,
Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Goldstein J, ed. Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, Berlin: Springer-Verlag, 466 (1975), 5-49.
|
[2] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
P. W. Bates, P. C. Fife, X. Ren and X. Wang,
Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal., 138 (1997), 105-136.
doi: 10.1007/s002050050037. |
[4] |
N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology Academic Press, San Diego, 1986. |
[5] |
N. F. 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. |
[6] |
X. Cabré and J. M. Roquejoffre,
The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722.
doi: 10.1007/s00220-013-1682-5. |
[7] |
J. Carr and A. Chmaj,
Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
doi: 10.1090/S0002-9939-04-07432-5. |
[8] |
X. Chen,
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
[9] |
X. Chen and J. S. Guo,
Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.
doi: 10.1006/jdeq.2001.4153. |
[10] |
J. Coville,
On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.
doi: 10.1007/s10231-005-0163-7. |
[11] |
J. Coville, J. Dávila and S. Martínez,
Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.
doi: 10.1016/j.jde.2007.11.002. |
[12] |
J. Coville and L. Dupaigne,
Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.
doi: 10.1016/j.na.2003.10.030. |
[13] |
J. Coville and L. Dupaigne,
On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755.
doi: 10.1017/S0308210504000721. |
[14] |
J. Fang and X. Zhao,
Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.
doi: 10.1088/0951-7715/24/11/002. |
[15] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems Springer Verlag, New York, 1979. |
[16] |
P. C. Fife and J. B. Mcleod,
The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.
doi: 10.1007/BF00250432. |
[17] |
R. A. Fisher,
The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[18] |
F. Hamel and L. Ryzhik,
On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.
doi: 10.1088/0951-7715/27/11/2735. |
[19] |
A. Kolmogorov, I. Petrovskii and N. Piskunov,
A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskow. Gos. Univ, 1 (1937), 1-26.
|
[20] |
S. Ma and X. Zou,
Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.
doi: 10.1016/j.jde.2005.05.004. |
[21] |
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. |
[22] |
J. Medlock and M. Kot,
Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.
doi: 10.1016/S0025-5564(03)00041-5. |
[23] |
M. Mei, C. H. Ou and X. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Appl. Math. , 42 (2010), 2762-2790; Erratum: Erratum: Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal. , 44 (2012), 538-540.
doi: 10.1137/110850633. |
[24] |
J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. |
[25] |
S. Pan, W. Li and G. Lin,
Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.
doi: 10.1007/s00033-007-7005-y. |
[26] |
S. Pan, W. Li and G. Lin,
Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal., 72 (2010), 3150-3158.
doi: 10.1016/j.na.2009.12.008. |
[27] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer Science & Business Media, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[28] |
S. Ruan and D. Xiao,
Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991-1011.
doi: 10.1017/S0308210500003590. |
[29] |
D. H. Sattinger,
On the stability of waves of nonlinear parabolic systems, Adv. in Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[30] |
K. W. Schaaf,
Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.
doi: 10.2307/2000859. |
[31] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems American Mathematical Soc. , Providence, RI, 1994. |
[32] |
Z. Wang, W. 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.
doi: 10.1016/j.jde.2007.03.025. |
[33] |
Z. Wang, W. Li and S. Ruan,
Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.
doi: 10.1007/s10884-008-9103-8. |
[34] |
Q. Ye and Z. Li, Introduction to Reaction-Diffusion Equations Science Press, Beijing, 1990. |
[35] |
L. Zhang,
Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations, 197 (2004), 162-196.
doi: 10.1016/S0022-0396(03)00170-0. |
[36] |
X. Zhao and W. Wang,
Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128.
doi: 10.3934/dcdsb.2004.4.1117. |
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