Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

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
Posted: September 2, 2011
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.

References

Bibliography

1. D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740 (80a:35013), http://dx.doi.org/10.1016/0001-8708(78)90130-5
2. Peter W. Bates and Guangyu Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl. 332 (2007), no. 1, 428–440. MR 2319673 (2008h:35019), http://dx.doi.org/10.1016/j.jmaa.2006.09.007
3. Henri Berestycki, François Hamel, and Nikolai Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 2, 173–213. MR 2127993 (2005k:35186), http://dx.doi.org/10.4171/JEMS/26
4. Henri Berestycki, François Hamel, and Nikolai Nadirashvili, The speed of propagation for KPP type problems. II. General domains, J. Amer. Math. Soc. 23 (2010), no. 1, 1–34. MR 2552247 (2010k:35260), http://dx.doi.org/10.1090/S0894-0347-09-00633-X
5. Henri Berestycki, François Hamel, and Lionel Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl. (9) 84 (2005), no. 8, 1101–1146 (English, with English and French summaries). MR 2155900 (2006d:35123), http://dx.doi.org/10.1016/j.matpur.2004.10.006
6. Reinhard Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z. 197 (1988), no. 2, 259–272. MR 923493 (89b:47057), http://dx.doi.org/10.1007/BF01215194
7. Emmanuel Chasseigne, Manuela Chaves, and Julio D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9) 86 (2006), no. 3, 271–291 (English, with English and French summaries). MR 2257732 (2007e:35279), http://dx.doi.org/10.1016/j.matpur.2006.04.005
8. Carmen Cortazar, Manuel Elgueta, and Julio D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel J. Math. 170 (2009), 53–60. MR 2506317 (2010e:35197), http://dx.doi.org/10.1007/s11856-009-0019-8
9. Jérôme Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl. (4) 185 (2006), no. 3, 461–485. MR 2231034 (2007e:35156), http://dx.doi.org/10.1007/s10231-005-0163-7
10. Jérôme Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations 249 (2010), no. 11, 2921–2953. MR 2718672 (2011h:47092), http://dx.doi.org/10.1016/j.jde.2010.07.003
11. Jérôme Coville and Louis Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal. 60 (2005), no. 5, 797–819. MR 2113158 (2005k:35213), http://dx.doi.org/10.1016/j.na.2003.10.030
12. Jérôme Coville, Juan Dávila, and Salomé Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal. 39 (2008), no. 5, 1693–1709. MR 2377295 (2009d:45013), http://dx.doi.org/10.1137/060676854
13. Paul C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal. 65 (1977), no. 4, 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. Jorge García-Melián and Julio D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations 246 (2009), no. 1, 21–38. MR 2467013 (2009j:35150), http://dx.doi.org/10.1016/j.jde.2008.04.015
16. M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, and G. T. Vickers, Non-local dispersal, Differential Integral Equations 18 (2005), no. 11, 1299–1320. MR 2174822 (2006m:35033)
17. François Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity, J. Math. Pures Appl. (9) 89 (2008), no. 4, 355–399 (English, with English and French summaries). MR 2401143 (2009g:35132), http://dx.doi.org/10.1016/j.matpur.2007.12.005
18. Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 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. Jianhua Huang and Wenxian Shen, Speeds of spread and propagation of KPP models in time almost and space periodic media, SIAM J. Appl. Dyn. Syst. 8 (2009), no. 3, 790–821. MR 2533625 (2010h:35192), http://dx.doi.org/10.1137/080723259
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, World Sci. Publ., River Edge, NJ, 1995, pp. 187–199. MR 1375475 (97a:35112)
22. V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math. 17 (2006), no. 2, 221–232. MR 2266484 (2007j:35087), http://dx.doi.org/10.1017/S0956792506006462
23. V. Hutson, S. Martinez, K. Mischaikow, and G. T. Vickers, The evolution of dispersal, J. Math. Biol. 47 (2003), no. 6, 483–517. MR 2028048 (2004j:92074), http://dx.doi.org/10.1007/s00285-003-0210-1
24. V. Hutson, W. Shen, and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain J. Math. 38 (2008), no. 4, 1147–1175. MR 2436718 (2009g:47197), http://dx.doi.org/10.1216/RMJ-2008-38-4-1147
25. Yoshinori Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka J. Math. 13 (1976), no. 1, 11–66. MR 0422875 (54 #10861)
26. Chiu-Yen Kao, Yuan Lou, and Wenxian Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst. 26 (2010), no. 2, 551–596. MR 2556498 (2011a:35253), http://dx.doi.org/10.3934/dcds.2010.26.551
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. Xing Liang and Xiao-Qiang Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math. 60 (2007), no. 1, 1–40. MR 2270161 (2007i:37144), http://dx.doi.org/10.1002/cpa.20154
29. Xing Liang and Xiao-Qiang Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal. 259 (2010), no. 4, 857–903. MR 2652175, http://dx.doi.org/10.1016/j.jfa.2010.04.018
30. Xing Liang, Yingfei Yi, and Xiao-Qiang Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations 231 (2006), no. 1, 57–77. MR 2287877 (2008a:47110), http://dx.doi.org/10.1016/j.jde.2006.04.010
31. Wan-Tong Li, Yu-Juan Sun, and Zhi-Cheng Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2302–2313. MR 2661901 (2011e:35185), http://dx.doi.org/10.1016/j.nonrwa.2009.07.005
32. Roger Lui, Biological growth and spread modeled by systems of recursions. I.\ Mathematical theory, Math. Biosci. 93 (1989), no. 2, 269–295. MR 984281 (90g:92069), http://dx.doi.org/10.1016/0025-5564(89)90026-6
33. G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems, preprint.
34. James Nolen, Matthew Rudd, and Jack Xin, Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dyn. Partial Differ. Equ. 2 (2005), no. 1, 1–24. MR 2142338 (2006h:35129)
35. James Nolen and Jack Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete Contin. Dyn. Syst. 13 (2005), no. 5, 1217–1234. MR 2166666 (2006e:35175), http://dx.doi.org/10.3934/dcds.2005.13.1217
36. Shuxia Pan, Wan-Tong Li, and Guo Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal. 72 (2010), no. 6, 3150–3158. MR 2580167 (2010m:35538), http://dx.doi.org/10.1016/j.na.2009.12.008
37. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
38. D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 0435602 (55 #8561)
39. Wenxian Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5125–5168. MR 2657675 (2011h:35162), http://dx.doi.org/10.1090/S0002-9947-10-04950-0
40. W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations 235 (2007), no. 1, 262–297. MR 2309574 (2008d:35091), http://dx.doi.org/10.1016/j.jde.2006.12.015
41. Wenxian Shen and Aijun Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations 249 (2010), no. 4, 747–795. MR 2652153 (2012c:45023), http://dx.doi.org/10.1016/j.jde.2010.04.012
42. W. Shen and A. Zhang, Traveling wave solutions of monostable equations with nonlocal dispersal in space periodic habitats, submitted.
43. Kōhei Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ. 18 (1978), no. 3, 453–508. MR 509494 (80g:35016)
44. H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982), no. 3, 353–396. MR 653463 (83f:35019), http://dx.doi.org/10.1137/0513028
45. Hans F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 (2002), no. 6, 511–548. MR 1943224 (2004b:92043a), http://dx.doi.org/10.1007/s00285-002-0169-3

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|