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 & Continuous Dynamical Systems - A, 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. doi: 10.1137/S0036141004443968. Google Scholar

[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. doi: 10.1016/j.jmaa.2006.09.007. Google Scholar

[3]

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

[4]

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

[5]

F. Chen, Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity,, Discrete Contin. Dyn. Syst. 24 (2009), 24 (2009), 659. doi: 10.3934/dcds.2009.24.659. Google Scholar

[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. doi: 10.1007/s00205-007-0062-8. Google Scholar

[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. doi: 10.1007/s11856-009-0019-8. Google Scholar

[8]

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

[9]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,, J. Differential Equations, 249 (2010), 2921. doi: 10.1016/j.jde.2010.07.003. Google Scholar

[10]

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

[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. doi: 10.1137/060676854. Google Scholar

[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. doi: 10.1073/pnas.72.3.780. Google Scholar

[13]

D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators,, The Clarendon Press Oxford University Press, (1987). Google Scholar

[14]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, 19 (1998). Google Scholar

[15]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, Trends in nonlinear analysis, (2003), 153. Google Scholar

[16]

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

[17]

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

[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. doi: 10.3934/cpaa.2012.11.1699. Google Scholar

[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). Google Scholar

[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. doi: 10.1216/RMJ-2013-43-2-489. Google Scholar

[21]

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

[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. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar

[23]

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

[24]

C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete and Continuous Dynamical Systems, 17 (2012), 2047. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar

[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. doi: 10.4310/MAA.2011.v18.n4.a5. Google Scholar

[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. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar

[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. doi: 10.1090/S0002-9947-2010-04931-1. Google Scholar

[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. doi: 10.1016/j.jmaa.2011.07.033. Google Scholar

[29]

P. Meyre-Nieberg, Banach Lattices,, Springer-Verlag, (1991). doi: 10.1007/978-3-642-76724-1. Google Scholar

[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, 72 (2010), 3150. doi: 10.1016/j.na.2009.12.008. Google Scholar

[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. doi: 10.1007/s10884-012-9276-z. Google Scholar

[32]

W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators,, J. Differential Equations, 235 (2007), 262. doi: 10.1016/j.jde.2006.12.015. Google Scholar

[33]

W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations,, submitted., (). Google Scholar

[34]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, Journal of Differential Equations 249 (2010), 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[35]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations,, Communications on Applied Nonlinear Analysis, 19 (2012), 73. Google Scholar

[36]

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

show all references

References:
[1]

P. Bates and F. Chen, Spectral analysis of traveling waves for nonlocal evolution equations,, SIAM J. Math. Anal., 38 (2006), 116. doi: 10.1137/S0036141004443968. Google Scholar

[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. doi: 10.1016/j.jmaa.2006.09.007. Google Scholar

[3]

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

[4]

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

[5]

F. Chen, Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity,, Discrete Contin. Dyn. Syst. 24 (2009), 24 (2009), 659. doi: 10.3934/dcds.2009.24.659. Google Scholar

[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. doi: 10.1007/s00205-007-0062-8. Google Scholar

[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. doi: 10.1007/s11856-009-0019-8. Google Scholar

[8]

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

[9]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,, J. Differential Equations, 249 (2010), 2921. doi: 10.1016/j.jde.2010.07.003. Google Scholar

[10]

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

[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. doi: 10.1137/060676854. Google Scholar

[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. doi: 10.1073/pnas.72.3.780. Google Scholar

[13]

D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators,, The Clarendon Press Oxford University Press, (1987). Google Scholar

[14]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, 19 (1998). Google Scholar

[15]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, Trends in nonlinear analysis, (2003), 153. Google Scholar

[16]

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

[17]

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

[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. doi: 10.3934/cpaa.2012.11.1699. Google Scholar

[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). Google Scholar

[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. doi: 10.1216/RMJ-2013-43-2-489. Google Scholar

[21]

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

[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. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar

[23]

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

[24]

C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete and Continuous Dynamical Systems, 17 (2012), 2047. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar

[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. doi: 10.4310/MAA.2011.v18.n4.a5. Google Scholar

[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. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar

[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. doi: 10.1090/S0002-9947-2010-04931-1. Google Scholar

[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. doi: 10.1016/j.jmaa.2011.07.033. Google Scholar

[29]

P. Meyre-Nieberg, Banach Lattices,, Springer-Verlag, (1991). doi: 10.1007/978-3-642-76724-1. Google Scholar

[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, 72 (2010), 3150. doi: 10.1016/j.na.2009.12.008. Google Scholar

[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. doi: 10.1007/s10884-012-9276-z. Google Scholar

[32]

W. Shen and G. T. Vickers, Spectral theory for general nonautonomous/random dispersal evolution operators,, J. Differential Equations, 235 (2007), 262. doi: 10.1016/j.jde.2006.12.015. Google Scholar

[33]

W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations,, submitted., (). Google Scholar

[34]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, Journal of Differential Equations 249 (2010), 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[35]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations,, Communications on Applied Nonlinear Analysis, 19 (2012), 73. Google Scholar

[36]

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

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