# American Institute of Mathematical Sciences

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

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

 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.
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) (2007), 428. Google Scholar [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), 7 (2005), 172. Google Scholar [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), 23 (2010), 1. Google Scholar [4] H. Berestycki, F. Hamel, and L. Roques, Analysis of periodically fragmented environment model: I- Species persistence,, J. Math. Biol. 51 (2005), 51 (2005), 75. Google Scholar [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), 84 (2005), 1101. Google Scholar [6] E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl., 86 (2006). Google Scholar [7] X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations,, J. Diff. Eq., 184 (2002), 549. Google Scholar [8] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation,, Ann. Mat. Pura Appl.185(3) (2006), 185(3) (2006), 461. Google Scholar [9] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,, J. Differential Equations 249 (2010), 249 (2010), 2921. Google Scholar [10] J. Coville, J. Dávila, and S. Martínez, Pulsating waves for nonlocal dispersion and KPP nonlinearity,, Preprint., (). Google Scholar [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), 39 (2008), 1693. Google Scholar [12] J. Coville, J. Dávila, S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations 244 (2008), 244 (2008). Google Scholar [13] J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations,, Nonlinear Analysis 60 (2005), 60 (2005). Google Scholar [14] J. Coville and L. Dupaigne, On a nonlocal equation arising in population dynamics,, Proceedings of the Royal Society of Edinburgh, 137A (2007), 727. Google Scholar [15] R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 335. Google Scholar [16] M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media,, Soviet Math. Dokl., 20 (1979), 1282. Google Scholar [17] J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,, J. Differential Equations, 246 (2009). Google Scholar [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), 246 (2009), 3818. Google Scholar [19] F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts,, J. European Math. Soc. 13 (2011), 13 (2011), 345. Google Scholar [20] D. Henry, "Geometric Theory of Semilinear Parabolic Equations",, Lecture Notes in Math. 840, 840 (1981). Google Scholar [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. Google Scholar [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, (1995), 187. Google Scholar [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), 38 (2008), 1147. Google Scholar [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), 1. Google Scholar [25] W.-T. Li, Y.-J. Sun, Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Analysis, 11 (2010), 2302. Google Scholar [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), 1. Google Scholar [27] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, Journal of Functional Analysis, 259 (2010), 857. Google Scholar [28] G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems,, preprint., (). Google Scholar [29] G. Nadin, Traveling fronts in space-time periodic media,, J. Math. Pures Appl., (9) 92 (2009), 232. Google Scholar [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), 1. Google Scholar [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), 1217. Google Scholar [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, 72 (2010), 3150. Google Scholar [33] A.Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations",, Springer-Verlag New York Berlin Heidelberg Tokyo, (1983). Google Scholar [34] W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators,, J. Differential Equations, 235 (2007), 262. Google Scholar [35] W. Shen and A. Zhang, Spreading Speeds for Monostable Equations with Nonlocal Dispersal in Space Periodic Habitats,, Journal of Differential Equations 249 (2010), 249 (2010), 747. Google Scholar [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, (2012), 1681. Google Scholar [37] W. Shen and A. Zhang, Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations ,, Communications on Applied Nonlinear Analysis, (2012), 73. Google Scholar [38] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. Google Scholar

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##### 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) (2007), 428. Google Scholar [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), 7 (2005), 172. Google Scholar [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), 23 (2010), 1. Google Scholar [4] H. Berestycki, F. Hamel, and L. Roques, Analysis of periodically fragmented environment model: I- Species persistence,, J. Math. Biol. 51 (2005), 51 (2005), 75. Google Scholar [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), 84 (2005), 1101. Google Scholar [6] E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl., 86 (2006). Google Scholar [7] X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations,, J. Diff. Eq., 184 (2002), 549. Google Scholar [8] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation,, Ann. Mat. Pura Appl.185(3) (2006), 185(3) (2006), 461. Google Scholar [9] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,, J. Differential Equations 249 (2010), 249 (2010), 2921. Google Scholar [10] J. Coville, J. Dávila, and S. Martínez, Pulsating waves for nonlocal dispersion and KPP nonlinearity,, Preprint., (). Google Scholar [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), 39 (2008), 1693. Google Scholar [12] J. Coville, J. Dávila, S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations 244 (2008), 244 (2008). Google Scholar [13] J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations,, Nonlinear Analysis 60 (2005), 60 (2005). Google Scholar [14] J. Coville and L. Dupaigne, On a nonlocal equation arising in population dynamics,, Proceedings of the Royal Society of Edinburgh, 137A (2007), 727. Google Scholar [15] R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 335. Google Scholar [16] M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media,, Soviet Math. Dokl., 20 (1979), 1282. Google Scholar [17] J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,, J. Differential Equations, 246 (2009). Google Scholar [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), 246 (2009), 3818. Google Scholar [19] F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts,, J. European Math. Soc. 13 (2011), 13 (2011), 345. Google Scholar [20] D. Henry, "Geometric Theory of Semilinear Parabolic Equations",, Lecture Notes in Math. 840, 840 (1981). Google Scholar [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. Google Scholar [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, (1995), 187. Google Scholar [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), 38 (2008), 1147. Google Scholar [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), 1. Google Scholar [25] W.-T. Li, Y.-J. Sun, Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Analysis, 11 (2010), 2302. Google Scholar [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), 1. Google Scholar [27] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, Journal of Functional Analysis, 259 (2010), 857. Google Scholar [28] G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems,, preprint., (). Google Scholar [29] G. Nadin, Traveling fronts in space-time periodic media,, J. Math. Pures Appl., (9) 92 (2009), 232. Google Scholar [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), 1. Google Scholar [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), 1217. Google Scholar [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, 72 (2010), 3150. Google Scholar [33] A.Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations",, Springer-Verlag New York Berlin Heidelberg Tokyo, (1983). Google Scholar [34] W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators,, J. Differential Equations, 235 (2007), 262. Google Scholar [35] W. Shen and A. Zhang, Spreading Speeds for Monostable Equations with Nonlocal Dispersal in Space Periodic Habitats,, Journal of Differential Equations 249 (2010), 249 (2010), 747. Google Scholar [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, (2012), 1681. Google Scholar [37] W. Shen and A. Zhang, Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations ,, Communications on Applied Nonlinear Analysis, (2012), 73. Google Scholar [38] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. Google Scholar
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