Criteria for the existence of principal eigenvalues of time periodic cooperative linear systems with nonlocal dispersal
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- by Xiongxiong Bao and Wenxian Shen
- Proc. Amer. Math. Soc. 145 (2017), 2881-2894
- DOI: https://doi.org/10.1090/proc/13602
- Published electronically: February 21, 2017
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Abstract:
The current paper establishes criteria for the existence of principal eigenvalues of time periodic cooperative linear nonlocal dispersal systems with Dirichlet type, Neumann type or periodic type boundary conditions. It is shown that such a nonlocal dispersal system has a principal eigenvalue in the following cases: the nonlocal dispersal distance is sufficiently small; the spatial inhomogeneity satisfies a so-called vanishing condition; or the spatial inhomogeneity is nearly globally homogeneous. Moreover, it is shown that the principal eigenvalue of a time periodic cooperative linear nonlocal dispersal system (if it exists) is algebraically simple. A linear nonlocal dispersal system may not have a principal eigenvalue. The results established in the current paper extend those in literature for time independent or periodic nonlocal dispersal equations to time periodic cooperative nonlocal dispersal systems and will serve as a basic tool for the study of cooperative nonlinear systems with nonlocal dispersal.References
- D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 5–49. MR 0427837
- 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, DOI 10.1016/0001-8708(78)90130-5
- 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, DOI 10.1016/j.jmaa.2006.09.007
- Reinhard Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z. 197 (1988), no. 2, 259–272. MR 923493, DOI 10.1007/BF01215194
- Robert Stephen Cantrell and Chris Cosner, Spatial ecology via reaction-diffusion equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. MR 2191264, DOI 10.1002/0470871296
- 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, DOI 10.1016/j.na.2003.10.030
- Jerome Coville and Louis Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 4, 727–755. MR 2345778, DOI 10.1017/S0308210504000721
- 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, DOI 10.1137/060676854
- 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, DOI 10.1016/j.jde.2010.07.003
- Jérôme Coville, Juan Dávila, and Salomé Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), no. 2, 179–223. MR 3035974, DOI 10.1016/j.anihpc.2012.07.005
- 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, DOI 10.1007/s11856-009-0019-8
- Carmen Cortazar, Manuel Elgueta, Julio D. Rossi, and Noemi Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal. 187 (2008), no. 1, 137–156. MR 2358337, DOI 10.1007/s00205-007-0062-8
- Daniel Daners and Pablo Koch Medina, Abstract evolution equations, periodic problems and applications, Pitman Research Notes in Mathematics Series, vol. 279, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1204883
- Weiwei Ding and Xing Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media, SIAM J. Math. Anal. 47 (2015), no. 1, 855–896. MR 3313826, DOI 10.1137/140958141
- Paul Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, Springer, Berlin, 2003, pp. 153–191. MR 1999098
- 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, DOI 10.1016/j.jde.2008.04.015
- 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, DOI 10.57262/die/1356059743
- Georg Hetzer, Tung Nguyen, and Wenxian Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal. 11 (2012), no. 5, 1699–1722. MR 2911107, DOI 10.3934/cpaa.2012.11.1699
- Georg Hetzer, Wenxian Shen, and Aijun Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations, Rocky Mountain J. Math. 43 (2013), no. 2, 489–513. MR 3077838, DOI 10.1216/RMJ-2013-43-2-489
- Peter Hess, Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics Series, vol. 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR 1100011
- 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, DOI 10.1007/s00285-003-0210-1
- V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math. 17 (2006), no. 2, 221–232. MR 2266484, DOI 10.1017/S0956792506006462
- 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, DOI 10.1216/RMJ-2008-38-4-1147
- 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, DOI 10.3934/dcds.2010.26.551
- L. Kong, N. Rawal, and W. Shen, Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats, Math. Model. Nat. Phenom. 10 (2015), no. 6, 113–141. MR 3414250, DOI 10.1051/mmnp/201510609
- 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, DOI 10.1016/j.nonrwa.2009.07.005
- Shuxia Pan, Wan-Tong Li, and Guo Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys. 60 (2009), no. 3, 377–392. MR 2505409, DOI 10.1007/s00033-007-7005-y
- 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, DOI 10.1016/j.na.2009.12.008
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- Nar Rawal and Wenxian Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations 24 (2012), no. 4, 927–954. MR 3000610, DOI 10.1007/s10884-012-9276-z
- Nar Rawal, Wenxian Shen, and Aijun Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst. 35 (2015), no. 4, 1609–1640. MR 3285840, DOI 10.3934/dcds.2015.35.1609
- N. Shigesada and K. Kawasaki, Biological Invasions, Theory and Practice, Oxford University Press, New York, 1997.
- Wenxian Shen and Zhongwei Shen, Transition fronts in nonlocal Fisher-KPP equations in time heterogeneous media, Commun. Pure Appl. Anal. 15 (2016), no. 4, 1193–1213. MR 3503652, DOI 10.3934/cpaa.2016.15.1193
- W. Shen and Z. Shen, Existence, uniqueness and stability of transition fronts of nonlocal equations in time heterogeneous bistable media, submitted.
- W. Shen and Z. Shen, Regularity of transition fronts in nonlocal dispersal evolution equations, to appear in Journal of Dynamics and Differential Equations.
- W. Shen and Z. Shen, Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity, to appear in Discrete and Continuous Dynamical Systems-A.
- 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, DOI 10.1016/j.jde.2010.04.012
- Wenxian Shen and Aijun Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1681–1696. MR 2869152, DOI 10.1090/S0002-9939-2011-11011-6
- Wenxian Shen and Aijun Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal. 19 (2012), no. 3, 73–101. MR 2978683
- Wenxian Shen and Xiaoxia Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst. 35 (2015), no. 4, 1665–1696. MR 3285842, DOI 10.3934/dcds.2015.35.1665
- Wenxian Shen and Xiaoxia Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, J. Differential Equations 259 (2015), no. 12, 7375–7405. MR 3401600, DOI 10.1016/j.jde.2015.08.026
- Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817
Bibliographic Information
- Xiongxiong Bao
- Affiliation: School of Science, Chang’an University, Xi’an, Shaanxi 710064, People’s Republic of China
- MR Author ID: 1030409
- Email: baoxx2016@chd.edu.cn
- Wenxian Shen
- Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn University, Alabama 36849
- MR Author ID: 249920
- Email: wenxish@auburn.edu
- Received by editor(s): May 6, 2016
- Received by editor(s) in revised form: May 17, 2016
- Published electronically: February 21, 2017
- Communicated by: Yingfei Yi
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 2881-2894
- MSC (2010): Primary 35K55, 45C05, 45M15, 45G15, 47G20
- DOI: https://doi.org/10.1090/proc/13602
- MathSciNet review: 3637938