American Institute of Mathematical Sciences

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2013, 2013(special): 815-824. doi: 10.3934/proc.2013.2013.815

Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations

Aijun Zhang 1,

1. 

Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States

Received  September 2012 Revised  December 2012 Published  November 2013

Traveling wave solutions to a spatially periodic nonlocal/random mixed dispersal equation with KPP nonlinearity are studied. By constructions of super/sub solutions and comparison principle, we establish the existence of traveling wave solutions with all propagating speeds greater than or equal to the spreading speed in every direction. For speeds greater than the spreading speed, we further investigate their uniqueness and stability.
Keywords: nonlocal dispersal, principal eigenvalue, random dispersal, Fisher-KPP equation, spreading speed..
Mathematics Subject Classification: Primary: 34L15, 35K55, 35K57, 37L60, 45C05, 45G10, 45M20; Secondary: 92D2.
Citation: Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815
References:
[1]

P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl. , 332(1):428-440, 2007.

[2]

H. Berestycki, F. Hamel, and N. Nadirashvili, The speed of propagation for KPP type problems, I - Periodic framework, J. Eur. Math. Soc. 7 (2005), pp. 172-213.

[3]

H. Berestycki, F. Hamel, and N. Nadirashvili, The speed of propagation for KPP type problems, II - General domains, J. Amer. Math. Soc. 23 (2010), no. 1, pp. 1-34

[4]

H. Berestycki, F. Hamel, and L. Roques, Analysis of periodically fragmented environment model: I- Species persistence, J. Math. Biol. 51 (2005), 75-113.

[5]

H. Berestycki, F. Hamel, and L. Roques, Analysis of periodically fragmented environment model: II - Biological invasions and pulsating traveling fronts, J.Math. Pures Appl. 84 (2005), pp. 1101-1146.

[6]

E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006) 271291.

[7]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Diff. Eq., 184 (2002), no. 2, pp. 549-569.

[8]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl.185(3) (2006), pp. 461-485

[9]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations 249 (2010), 2921-2953.

[10]

J. Coville, J. Dávila, and S. Martínez, Pulsating waves for nonlocal dispersion and KPP nonlinearity, Preprint.

[11]

J. Coville, J. Dávila, and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal. 39 (2008), pp. 1693-1709.

[12]

J. Coville, J. Dávila, S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations 244 (2008), pp. 3080 - 3118.

[13]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Analysis 60 (2005), pp. 797 - 819.

[14]

J. Coville and L. Dupaigne, On a nonlocal equation arising in population dynamics, Proceedings of the Royal Society of Edinburgh, 137A (2007), 727-755.

[15]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7(1937), pp. 335-369.

[16]

M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), pp. 1282-1286.

[17]

J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009) 2138.

[18]

J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations 246 (2009), pp. 3818-3833.

[19]

F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. European Math. Soc. 13 (2011), 345-390

[20]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations", Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1981.

[21]

G. Hetzer, W. Shen, and A. Zhang, Effects of Spatial Variations and Dispersal Strategies on Principal Eigenvalues of Dispersal Operators and Spreading Speeds of Monostable Equations, Rocky Mountain Journal of Mathematics, 43 (2013), 489-513.

[22]

W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, Boundary value problems for functional-differential equations, 187-199, World Sci. Publ., River Edge, NJ, 1995.

[23]

V.Hutson, W.Shen and G.T.Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain Journal of Mathematics 38 (2008), pp. 1147-1175.

[24]

A. Kolmogorov, I. Petrowsky, and N.Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Bjul. Moskovskogo Gos. Univ., 1 (1937), pp. 1-26.

[25]

W.-T. Li, Y.-J. Sun, Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Analysis, Real World Appl., 11 (2010), no. 4, pp. 2302-2313.

[26]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), no. 1, pp. 1-40.

[27]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.

[28]

G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems, preprint.

[29]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., (9) 92 (2009), pp. 232-262.

[30]

J. Nolen, M. Rudd, and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), pp. 1-24.

[31]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), pp. 1217-1234.

[32]

S. Pan, W.-T. Li, and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 3150-3158.

[33]

A.Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations", Springer-Verlag New York Berlin Heidelberg Tokyo, 1983.

[34]

W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations, 235 (2007), pp. 262-297.

[35]

W. Shen and A. Zhang, Spreading Speeds for Monostable Equations with Nonlocal Dispersal in Space Periodic Habitats, Journal of Differential Equations 249 (2010), 747-795.

[36]

W. Shen and A. Zhang, Stationary Solutions and Spreading Speeds of Nonlocal Monostable Equations in Space Periodic Habitats, Proceedings of the American Mathematical Society, Volume 140, Number 5, (2012), 1681-1696.

[37]

W. Shen and A. Zhang, Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations , Communications on Applied Nonlinear Analysis, Volume 19(2012), Number 3, 73-101.

[38]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), pp. 511-548.

show all references

References:
[1]

P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl. , 332(1):428-440, 2007.

[2]

H. Berestycki, F. Hamel, and N. Nadirashvili, The speed of propagation for KPP type problems, I - Periodic framework, J. Eur. Math. Soc. 7 (2005), pp. 172-213.

[3]

H. Berestycki, F. Hamel, and N. Nadirashvili, The speed of propagation for KPP type problems, II - General domains, J. Amer. Math. Soc. 23 (2010), no. 1, pp. 1-34

[4]

H. Berestycki, F. Hamel, and L. Roques, Analysis of periodically fragmented environment model: I- Species persistence, J. Math. Biol. 51 (2005), 75-113.

[5]

H. Berestycki, F. Hamel, and L. Roques, Analysis of periodically fragmented environment model: II - Biological invasions and pulsating traveling fronts, J.Math. Pures Appl. 84 (2005), pp. 1101-1146.

[6]

E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006) 271291.

[7]

X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Diff. Eq., 184 (2002), no. 2, pp. 549-569.

[8]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl.185(3) (2006), pp. 461-485

[9]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations 249 (2010), 2921-2953.

[10]

J. Coville, J. Dávila, and S. Martínez, Pulsating waves for nonlocal dispersion and KPP nonlinearity, Preprint.

[11]

J. Coville, J. Dávila, and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal. 39 (2008), pp. 1693-1709.

[12]

J. Coville, J. Dávila, S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations 244 (2008), pp. 3080 - 3118.

[13]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Analysis 60 (2005), pp. 797 - 819.

[14]

J. Coville and L. Dupaigne, On a nonlocal equation arising in population dynamics, Proceedings of the Royal Society of Edinburgh, 137A (2007), 727-755.

[15]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7(1937), pp. 335-369.

[16]

M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), pp. 1282-1286.

[17]

J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009) 2138.

[18]

J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations 246 (2009), pp. 3818-3833.

[19]

F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. European Math. Soc. 13 (2011), 345-390

[20]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations", Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1981.

[21]

G. Hetzer, W. Shen, and A. Zhang, Effects of Spatial Variations and Dispersal Strategies on Principal Eigenvalues of Dispersal Operators and Spreading Speeds of Monostable Equations, Rocky Mountain Journal of Mathematics, 43 (2013), 489-513.

[22]

W. Hudson and B. Zinner, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, Boundary value problems for functional-differential equations, 187-199, World Sci. Publ., River Edge, NJ, 1995.

[23]

V.Hutson, W.Shen and G.T.Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain Journal of Mathematics 38 (2008), pp. 1147-1175.

[24]

A. Kolmogorov, I. Petrowsky, and N.Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Bjul. Moskovskogo Gos. Univ., 1 (1937), pp. 1-26.

[25]

W.-T. Li, Y.-J. Sun, Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Analysis, Real World Appl., 11 (2010), no. 4, pp. 2302-2313.

[26]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), no. 1, pp. 1-40.

[27]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.

[28]

G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems, preprint.

[29]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., (9) 92 (2009), pp. 232-262.

[30]

J. Nolen, M. Rudd, and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), pp. 1-24.

[31]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), pp. 1217-1234.

[32]

S. Pan, W.-T. Li, and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 3150-3158.

[33]

A.Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations", Springer-Verlag New York Berlin Heidelberg Tokyo, 1983.

[34]

W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations, 235 (2007), pp. 262-297.

[35]

W. Shen and A. Zhang, Spreading Speeds for Monostable Equations with Nonlocal Dispersal in Space Periodic Habitats, Journal of Differential Equations 249 (2010), 747-795.

[36]

W. Shen and A. Zhang, Stationary Solutions and Spreading Speeds of Nonlocal Monostable Equations in Space Periodic Habitats, Proceedings of the American Mathematical Society, Volume 140, Number 5, (2012), 1681-1696.

[37]

W. Shen and A. Zhang, Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations , Communications on Applied Nonlinear Analysis, Volume 19(2012), Number 3, 73-101.

[38]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), pp. 511-548.

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