Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats

Authors:
Wenxian Shen and Aijun Zhang

Journal:
Proc. Amer. Math. Soc. **140** (2012), 1681-1696

MSC (2010):
Primary 45C05, 45G10, 45M20, 47G10, 92D25

DOI:
https://doi.org/10.1090/S0002-9939-2011-11011-6

Published electronically:
September 2, 2011

MathSciNet review:
2869152

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equations with symmetric convolution kernels are established in the authors' earlier work for the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth. The above conditions guarantee the existence of principal eigenvalues of nonlocal dispersal operators associated to linearized equations at the trivial solution. In general, a nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we extend our earlier results to general spatially periodic nonlocal monostable equations. As a consequence, it is seen that the spatial spreading feature is generic for monostable equations with nonlocal dispersal.

**1.**D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,*Adv. Math.***30**(1978), pp. 33-76. MR**511740 (80a:35013)****2.**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), pp. 428-440. MR**2319673 (2008h:35019)****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), pp. 172-213. MR**2127993 (2005k:35186)****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), no. 1, pp. 1-34. MR**2552247 (2010k:35260)****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. MR**2155900 (2006d:35123)****6.**Reinhard Bürger, Perturbations of positive semigroups and applications to population genetics,*Math. Z.***197**(1988), pp. 259-272. MR**923493 (89b:47057)****7.**E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,*J. Math. Pures Appl.***86**(2006), 271-291. MR**2257732 (2007e:35279)****8.**C. Cortazar, M. Elgueta, and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions,*Israel J. of Math.***170**(2009), pp. 53-60. MR**2506317 (2010e:35197)****9.**J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation,*Annali di Matematica***185(3)**(2006), pp. 461-485. MR**2231034 (2007e:35156)****10.**J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,*J. Differential Equations***249**(2010), pp. 2921-2953. MR**2718672****11.**J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations,*Nonlinear Analysis***60**(2005), pp. 797-819. MR**2113158 (2005k:35213)****12.**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. MR**2377295 (2009d:45013)****13.**P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,*Arch. Ration. Mech. Anal.***65**(1977), pp. 335-361. MR**0442480 (56:862)****14.**R. Fisher, The wave of advance of advantageous genes,*Ann. of Eugenics***7**(1937), pp. 335-369.**15.**J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,*J. Differential Equations***246**(2009), pp. 21-38. MR**2467013 (2009j:35150)****16.**M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, and G. T. Vickers, Non-local dispersal,*Differential Integral Equations***18**(2005), pp. 1299-1320. MR**2174822 (2006m:35033)****17.**F. Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity,*J. Math. Pures Appl.***89**(2008), pp. 355-399. MR**2401143 (2009g:35132)****18.**D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math.**840**, Springer-Verlag, Berlin, 1981. MR**610244 (83j:35084)****19.**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*, to appear.**20.**J. Huang and W. Shen, Speeds of spread and propagation for KPP models in time almost and space periodic media,*SIAM J. Appl. Dynam. Syst.***8**(2009), pp. 790-821. MR**2533625 (2010h:35192)****21.**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. MR**1375475 (97a:35112)****22.**V. Hutson and M. Grinfeld, Non-local dispersal and bistability,*Euro. J. Appl. Math*.**17**(2006), pp. 221-232. MR**2266484 (2007j:35087)****23.**V. Hutson, S. Martinez, K. Mischaikow, and G.T. Vickers, The evolution of dispersal,*J. Math. Biol.***47**(2003), pp. 483-517. MR**2028048 (2004j:92074)****24.**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. MR**2436718 (2009g:47197)****25.**Y. Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type,*Osaka J. Math.***13**(1976), pp. 11-66. MR**0422875 (54:10861)****26.**C.-Y. Kao, Y. Lou, and W. Shen, Random dispersal vs. non-local dispersal,*Discrete and Continuous Dynamical Systems***26**(2010), no. 2, pp. 551-596. MR**2556498 (2011a:35253)****27.**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.**28.**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. MR**2270161 (2007i:37144)****29.**X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,*Journal of Functional Analysis***259**(2010), no. 4, pp. 857-903. MR**2652175****30.**X. Liang, Y. Yi, and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems,*J. Diff. Eq.***231**(2006), no. 1, pp. 57-77. MR**2287877 (2008a:47110)****31.**W.-T. Li, Y.-J. Sun, and 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. MR**2661901****32.**R. Lui, Biological growth and spread modeled by systems of recursions, I. Mathematical Theory,*Math. Biosciences***93**(1989), pp. 269-312. MR**984281 (90g:92069)****33.**G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems, preprint.**34.**J. Nolen, M. Rudd, and J. Xin, Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds,*Dynamics of PDE***2**(2005), pp. 1-24. MR**2142338 (2006h:35129)****35.**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. MR**2166666 (2006e:35175)****36.**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), pp. 3150-3158. MR**2580167 (2010m:35538)****37.**A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1983. MR**710486 (85g:47061)****38.**D. H. Sattinger, On the stability of waves of nonlinear parabolic systems,*Advances in Math.***22**(1976), pp. 312-355. MR**0435602 (55:8561)****39.**W. Shen, Variational principle for spatial spreading speeds and generalized propagating speeds in time almost and space periodic KPP models,*Trans. Amer. Math. Soc.***362**(2010), pp. 5125-5168. MR**2657675****40.**W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators,*J. Differential Equations***235**(2007), pp. 262-297. MR**2309574 (2008d:35091)****41.**W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,*Journal of Differential Equations***249**(2010), pp. 747-795. MR**2652153****42.**W. Shen and A. Zhang, Traveling wave solutions of monostable equations with nonlocal dispersal in space periodic habitats, submitted.**43.**K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time,*J. Math. Kyoto Univ.***183**(1978), pp. 453-508. MR**509494 (80g:35016)****44.**H. F. Weinberger, Long-time behavior of a class of biology models,*SIAM J. Math. Anal.***13**(1982), pp. 353-396. MR**653463 (83f:35019)****45.**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. MR**1943224 (2004b:92043a)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
45C05,
45G10,
45M20,
47G10,
92D25

Retrieve articles in all journals with MSC (2010): 45C05, 45G10, 45M20, 47G10, 92D25

Additional Information

**Wenxian Shen**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849

Email:
wenxish@auburn.edu

**Aijun Zhang**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849

Email:
zhangai@auburn.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-11011-6

Keywords:
Monostable equation,
nonlocal dispersal,
random dispersal,
periodic habitat,
spreading speed,
principal eigenvalue,
principal eigenfunction,
variational principle.

Received by editor(s):
September 17, 2010

Received by editor(s) in revised form:
January 17, 2011

Published electronically:
September 2, 2011

Additional Notes:
This work was partially supported by NSF grant DMS–0907752

Communicated by:
Yingfei Yi

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.