April  2015, 35(4): 1609-1640. doi: 10.3934/dcds.2015.35.1609

Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats

1. 

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, United States

2. 

Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849

3. 

Department of Mathematics, Drexel University, Philadelphia, PA 19014, United States

Received  July 2013 Revised  June 2014 Published  November 2014

This paper is devoted to the investigation of spatial spreading speeds and traveling wave solutions of monostable evolution equations with nonlocal dispersal in time and space periodic habitats. It has been shown in an earlier work by the first two authors of the current paper that such an equation has a unique time and space periodic positive stable solution $u^*(t,x)$. In this paper, we show that such an equation has a spatial spreading speed $c^*(\xi)$ in the direction of any given unit vector $\xi$. A variational characterization of $c^*(\xi)$ is given. Under the assumption that the nonlocal dispersal operator associated to the linearization of the monostable equation at the trivial solution $0$ has a principal eigenvalue, we also show that the monostable equation has a continuous periodic traveling wave solution connecting $u^*(\cdot,\cdot)$ and $0$ propagating in any given direction of $\xi$ with speed $c>c^*(\xi)$.
Citation: Nar Rawal, Wenxian Shen, Aijun Zhang. Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1609-1640. doi: 10.3934/dcds.2015.35.1609
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 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007.

[2]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Func. Anal., 255 (2008), 2146-2189. doi: 10.1016/j.jfa.2008.06.030.

[3]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, I - Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213. doi: 10.4171/JEMS/26.

[4]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, II - General domains, J. Amer. Math. Soc., 23 (2010), 1-34. doi: 10.1090/S0894-0347-09-00633-X.

[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), 1101-1146. doi: 10.1016/j.matpur.2004.10.006.

[6]

R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272. doi: 10.1007/BF01215194.

[7]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005.

[8]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Annali di Matematica, 185(3) (2006), 461-485. doi: 10.1007/s10231-005-0163-7.

[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. doi: 10.1016/j.jde.2010.07.003.

[10]

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), 1693-1709. doi: 10.1137/060676854.

[11]

J. Coville, J. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223. doi: 10.1016/j.anihpc.2012.07.005.

[12]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Analysis, 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030.

[13]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153-191, Springer, Berlin, 2003.

[14]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 335-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[15]

M. Freidlin, On wave front propagation in periodic media. In: Stochastic analysis and applications, ed. M. Pinsky, Advances in probablity and related topics, 7 (1984), 147-166.

[16]

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

[17]

J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015.

[18]

M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320.

[19]

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

[20]

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. doi: 10.1216/RMJ-2013-43-2-489.

[21]

J. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space peirodic media, SIAM J. Appl. Dynam. Syst., 8 (2009), 790-821. doi: 10.1137/080723259.

[22]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1.

[23]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-Local dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.

[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), 1-26.

[25]

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.

[26]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903. doi: 10.1016/j.jfa.2010.04.018.

[27]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Anal., 92 (2009), 232-262. doi: 10.1016/j.matpur.2009.04.002.

[28]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653. doi: 10.1016/j.matpur.2012.05.005.

[29]

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), 1-24. doi: 10.4310/DPDE.2005.v2.n1.a1.

[30]

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), 1217-1234. doi: 10.3934/dcds.2005.13.1217.

[31]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York Berlin Heidelberg Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1.

[32]

N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, Journal of Dynamics and Differential Equations, 24 (2012), 927-954. doi: 10.1007/s10884-012-9276-z.

[33]

W. Shen, Variational principle for spatial spreading speeds and generalized propgating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168. doi: 10.1090/S0002-9947-10-04950-0.

[34]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93.

[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. doi: 10.1016/j.jde.2010.04.012.

[36]

W. Shen and A. Zhang, Traveling wave solutions of monostable equations with nonlocal dispersal in space periodic habitats, Communications on Applied Nonlinear Analysis, 19 (2012), 73-101.

[37]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6.

[38]

H. F. Weinberger, Long-time behavior of a class of biology 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.

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 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007.

[2]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Func. Anal., 255 (2008), 2146-2189. doi: 10.1016/j.jfa.2008.06.030.

[3]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, I - Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213. doi: 10.4171/JEMS/26.

[4]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, II - General domains, J. Amer. Math. Soc., 23 (2010), 1-34. doi: 10.1090/S0894-0347-09-00633-X.

[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), 1101-1146. doi: 10.1016/j.matpur.2004.10.006.

[6]

R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272. doi: 10.1007/BF01215194.

[7]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005.

[8]

J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Annali di Matematica, 185(3) (2006), 461-485. doi: 10.1007/s10231-005-0163-7.

[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. doi: 10.1016/j.jde.2010.07.003.

[10]

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), 1693-1709. doi: 10.1137/060676854.

[11]

J. Coville, J. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179-223. doi: 10.1016/j.anihpc.2012.07.005.

[12]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Analysis, 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030.

[13]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153-191, Springer, Berlin, 2003.

[14]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 335-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[15]

M. Freidlin, On wave front propagation in periodic media. In: Stochastic analysis and applications, ed. M. Pinsky, Advances in probablity and related topics, 7 (1984), 147-166.

[16]

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

[17]

J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015.

[18]

M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320.

[19]

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

[20]

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. doi: 10.1216/RMJ-2013-43-2-489.

[21]

J. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space peirodic media, SIAM J. Appl. Dynam. Syst., 8 (2009), 790-821. doi: 10.1137/080723259.

[22]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1.

[23]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-Local dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.

[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), 1-26.

[25]

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.

[26]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903. doi: 10.1016/j.jfa.2010.04.018.

[27]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Anal., 92 (2009), 232-262. doi: 10.1016/j.matpur.2009.04.002.

[28]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653. doi: 10.1016/j.matpur.2012.05.005.

[29]

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), 1-24. doi: 10.4310/DPDE.2005.v2.n1.a1.

[30]

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), 1217-1234. doi: 10.3934/dcds.2005.13.1217.

[31]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag New York Berlin Heidelberg Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1.

[32]

N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, Journal of Dynamics and Differential Equations, 24 (2012), 927-954. doi: 10.1007/s10884-012-9276-z.

[33]

W. Shen, Variational principle for spatial spreading speeds and generalized propgating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168. doi: 10.1090/S0002-9947-10-04950-0.

[34]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93.

[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. doi: 10.1016/j.jde.2010.04.012.

[36]

W. Shen and A. Zhang, Traveling wave solutions of monostable equations with nonlocal dispersal in space periodic habitats, Communications on Applied Nonlinear Analysis, 19 (2012), 73-101.

[37]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6.

[38]

H. F. Weinberger, Long-time behavior of a class of biology 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.

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