# American Institute of Mathematical Sciences

April  2015, 35(4): 1665-1696. doi: 10.3934/dcds.2015.35.1665

## On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications

 1 Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849 2 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, United States

Received  August 2013 Revised  June 2014 Published  November 2014

This paper is to investigate the dependence of the principal spectrum points of nonlocal dispersal operators on underlying parameters and to consider its applications. In particular, we study the effects of the spatial inhomogeneity, the dispersal rate, and the dispersal distance on the existence of the principal eigenvalues, the magnitude of the principal spectrum points, and the asymptotic behavior of the principal spectrum points of nonlocal dispersal operators with Dirichlet type, Neumann type, and periodic boundary conditions in a unified way. We also discuss the applications of the principal spectral theory of nonlocal dispersal operators to the asymptotic dynamics of two species competition systems with nonlocal dispersal operators.
Citation: Wenxian Shen, Xiaoxia Xie. On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1665-1696. doi: 10.3934/dcds.2015.35.1665
##### References:
 [1] P. Bates and F. Chen, Spectral analysis of traveling waves for nonlocal evolution equations, SIAM J. Math. Anal., 38 (2006), 116-126. doi: 10.1137/S0036141004443968. [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), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [3] R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z. ,197 (1988), 259-272. doi: 10.1007/BF01215194. [4] 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. [5] F. Chen, Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity, Discrete Contin. Dyn. Syst. 24 (2009), 659-673. doi: 10.3934/dcds.2009.24.659. [6] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156. doi: 10.1007/s00205-007-0062-8. [7] 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), 53-60. doi: 10.1007/s11856-009-0019-8. [8] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Annali di Matematica, 185 (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 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. [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), 1693-1709. doi: 10.1137/060676854. [12] M. D. Donsker and S. R. S. Varadhan, On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Nat. Acad. Sci. USA, 72 (1975), 780-783. doi: 10.1073/pnas.72.3.780. [13] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, The Clarendon Press Oxford University Press, New York, 1987. [14] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, Rhode Island, 1998. [15] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153-191, Springer, Berlin, 2003. [16] 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. [17] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320. [18] G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Communications on Pure and Applied Analysis, 11 (2012), 1699-1722. doi: 10.3934/cpaa.2012.11.1699. [19] G. Hetzer, T. Nguyen and W. Shen, Effects of small variation of the reproduction rate in a two species competition model, Electron. J. Differential Equations , (2010), No. 160, 17 pp. [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] 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. [22] 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), 1147-1175. doi: 10.1216/RMJ-2008-38-4-1147. [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] C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 2047-2072. doi: 10.3934/dcdsb.2012.17.2047. [25] L. Kong and W. Shen, Positive stationary solutions and spreading speeds of KPP equations in locally spatially inhomogeneous media, Methods and Applications of Analysis, 18 (2011), 427-456. doi: 10.4310/MAA.2011.v18.n4.a5. [26] W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Analysis, 11 (2010), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005. [27] X. Liang, X. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, Trans. Amer. Math. Soc., 362 (2010), 5605-5633. doi: 10.1090/S0002-9947-2010-04931-1. [28] G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems, J. Math. Anal. Appl., 385 (2012), 1094-1106. doi: 10.1016/j.jmaa.2011.07.033. [29] P. Meyre-Nieberg, Banach Lattices, Springer-Verlag, 1991. doi: 10.1007/978-3-642-76724-1. [30] 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), 3150-3158. doi: 10.1016/j.na.2009.12.008. [31] N. Rawal and W. 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), 927-954. doi: 10.1007/s10884-012-9276-z. [32] W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations, 235 (2007), 262-297. doi: 10.1016/j.jde.2006.12.015. [33] W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations,, submitted., (). [34] 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. [35] W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Communications on Applied Nonlinear Analysis, 19 (2012), 73-101. [36] W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. AMS, 140 (2012), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6.

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##### References:
 [1] P. Bates and F. Chen, Spectral analysis of traveling waves for nonlocal evolution equations, SIAM J. Math. Anal., 38 (2006), 116-126. doi: 10.1137/S0036141004443968. [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), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [3] R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z. ,197 (1988), 259-272. doi: 10.1007/BF01215194. [4] 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. [5] F. Chen, Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity, Discrete Contin. Dyn. Syst. 24 (2009), 659-673. doi: 10.3934/dcds.2009.24.659. [6] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156. doi: 10.1007/s00205-007-0062-8. [7] 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), 53-60. doi: 10.1007/s11856-009-0019-8. [8] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Annali di Matematica, 185 (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 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. [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), 1693-1709. doi: 10.1137/060676854. [12] M. D. Donsker and S. R. S. Varadhan, On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Nat. Acad. Sci. USA, 72 (1975), 780-783. doi: 10.1073/pnas.72.3.780. [13] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, The Clarendon Press Oxford University Press, New York, 1987. [14] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, Rhode Island, 1998. [15] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153-191, Springer, Berlin, 2003. [16] 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. [17] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320. [18] G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Communications on Pure and Applied Analysis, 11 (2012), 1699-1722. doi: 10.3934/cpaa.2012.11.1699. [19] G. Hetzer, T. Nguyen and W. Shen, Effects of small variation of the reproduction rate in a two species competition model, Electron. J. Differential Equations , (2010), No. 160, 17 pp. [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] 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. [22] 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), 1147-1175. doi: 10.1216/RMJ-2008-38-4-1147. [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] C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 2047-2072. doi: 10.3934/dcdsb.2012.17.2047. [25] L. Kong and W. Shen, Positive stationary solutions and spreading speeds of KPP equations in locally spatially inhomogeneous media, Methods and Applications of Analysis, 18 (2011), 427-456. doi: 10.4310/MAA.2011.v18.n4.a5. [26] W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Analysis, 11 (2010), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005. [27] X. Liang, X. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, Trans. Amer. Math. Soc., 362 (2010), 5605-5633. doi: 10.1090/S0002-9947-2010-04931-1. [28] G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems, J. Math. Anal. Appl., 385 (2012), 1094-1106. doi: 10.1016/j.jmaa.2011.07.033. [29] P. Meyre-Nieberg, Banach Lattices, Springer-Verlag, 1991. doi: 10.1007/978-3-642-76724-1. [30] 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), 3150-3158. doi: 10.1016/j.na.2009.12.008. [31] N. Rawal and W. 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), 927-954. doi: 10.1007/s10884-012-9276-z. [32] W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators, J. Differential Equations, 235 (2007), 262-297. doi: 10.1016/j.jde.2006.12.015. [33] W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations,, submitted., (). [34] 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. [35] W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Communications on Applied Nonlinear Analysis, 19 (2012), 73-101. [36] W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. AMS, 140 (2012), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6.
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