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Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal
1. | Department of Mathematics, Auburn University, AL 36849-5310 |
2. | Department of Mathematical Sciences, University of Illinois Springeld, Springeld, IL 62703, United States |
3. | Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849-5310 |
References:
[1] |
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. |
[2] |
R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272. |
[3] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003. |
[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] |
Fengde Chen, Zhong Li and Xiangdong Xie, Permanence of a nonlinear integro-differential prey-competition model with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 2290-2297.
doi: 10.1016/j.cnsns.2007.05.022. |
[6] |
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. |
[7] |
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. Rational Mech. Anal., 187 (2007), 137-156
doi: 10.1007/s00205-007-0062-8. |
[8] |
C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44 (1984), 1112-1132.
doi: 10.1137/0144080. |
[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, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.
doi: 10.1007/s10231-005-0163-7. |
[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] |
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. |
[13] |
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol., 37 (1998), 61-83.
doi: 10.1007/s002850050120. |
[14] |
J. E. Furter and J. López-Gómez, On the existence and uniqueness of coexistence states for the Lotka-Volterra competition model with diffusion and spatially dependent coefficients, Nonlinear Analysis, Theory, Methods & Applicarions, 25 (1995), 363-398. |
[15] |
A. Hastings, Can spatial variation alone lead to selection for dispersal? Theor. Pop. Biol., 24 (1983), 244-251.
doi: 10.1016/0040-5809(83)90027-8. |
[16] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Springer-Verlag, New York, 1981. |
[17] |
G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34 (2002), 204-227.
doi: 10.1137/S0036141001390695. |
[18] |
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., ().
|
[19] |
V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.
doi: 10.1006/jdeq.2001.4157. |
[20] |
V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near a degenerate limit, SIAM J. Math. Anal., 35 (2003), 453-491.
doi: 10.1137/S0036141002402189. |
[21] |
V. Hutson, K. Mischaikow and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.
doi: 10.1007/s002850100106. |
[22] |
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. |
[23] |
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. |
[24] |
C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596. |
[25] |
A. Leung, Equilibria and stability for competing-species, reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl., 73 (1980), 204-218.
doi: 10.1016/0022-247X(80)90028-1. |
[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] |
G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems,, preprint., ().
|
[28] |
C. V. Pao, Coexistence and stability of a competition-diffusion system in population dynamics, J. Math. Anal. Appl., 83 (1981), 54-76.
doi: 10.1016/0022-247X(81)90246-8. |
[29] |
C. V. Pao, Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays, Nonlinear Anal. Real World Appl., 5 (2004), 91-104.
doi: 10.1016/S1468-1218(03)00018-X. |
[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. |
[31] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983. |
[32] |
S. Ruan, and X.-Q. Zhao, Persistence and extinction in two species reaction-diffusion systems with delays, J. Differential Equations, 156 (1999), 71-92.
doi: 10.1006/jdeq.1998.3599. |
[33] |
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. |
[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, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats,, Proceedings of the American Mathematical Society, ().
|
[36] |
Hal L. Smith, "Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995. |
[37] |
Joseph W.-H. So and J. S. Yu, Global attractivity for a population model with time delay, Proc. Amer. Math. Soc., 123 (1995), 2687-2694.
doi: 10.1090/S0002-9939-1995-1317052-5. |
[38] |
P. Takac, A short elementary proof of the Krein-Rutman theorem, Houston J. of Math., 20 (1994), 93-98. |
[39] |
X. H. Tang, D. M. Cao and X. Zou, Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay, J. Differential Equations, 228 (2006), 580-610.
doi: 10.1016/j.jde.2006.06.007. |
[40] |
X. H. Tang and X. Zou, Global attractivity of nonautonomous Lotka Volterra competition system without instantaneous negative feedbacks, J. Differential Equations, 192 (2003), 502-535.
doi: 10.1016/S0022-0396(03)00042-1. |
[41] |
J. Wu and X. Q. Zhao, Permanence and convergence in multi-species competition systems with delay, Proc. Amer. Math. Soc., 126 (1998), 1709-1714.
doi: 10.1090/S0002-9939-98-04522-5. |
[42] |
J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
[43] |
X.-Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16. Springer-Verlag, New York, 2003. |
[44] |
L. Zhou and C. V. Pao, Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear Anal., 6 (1982), 1163-1184.
doi: 10.1016/0362-546X(82)90028-1. |
show all references
References:
[1] |
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. |
[2] |
R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272. |
[3] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003. |
[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] |
Fengde Chen, Zhong Li and Xiangdong Xie, Permanence of a nonlinear integro-differential prey-competition model with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 2290-2297.
doi: 10.1016/j.cnsns.2007.05.022. |
[6] |
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. |
[7] |
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. Rational Mech. Anal., 187 (2007), 137-156
doi: 10.1007/s00205-007-0062-8. |
[8] |
C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44 (1984), 1112-1132.
doi: 10.1137/0144080. |
[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, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann. Mat. Pura Appl., 185 (2006), 461-485.
doi: 10.1007/s10231-005-0163-7. |
[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] |
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. |
[13] |
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol., 37 (1998), 61-83.
doi: 10.1007/s002850050120. |
[14] |
J. E. Furter and J. López-Gómez, On the existence and uniqueness of coexistence states for the Lotka-Volterra competition model with diffusion and spatially dependent coefficients, Nonlinear Analysis, Theory, Methods & Applicarions, 25 (1995), 363-398. |
[15] |
A. Hastings, Can spatial variation alone lead to selection for dispersal? Theor. Pop. Biol., 24 (1983), 244-251.
doi: 10.1016/0040-5809(83)90027-8. |
[16] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Springer-Verlag, New York, 1981. |
[17] |
G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34 (2002), 204-227.
doi: 10.1137/S0036141001390695. |
[18] |
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., ().
|
[19] |
V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.
doi: 10.1006/jdeq.2001.4157. |
[20] |
V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near a degenerate limit, SIAM J. Math. Anal., 35 (2003), 453-491.
doi: 10.1137/S0036141002402189. |
[21] |
V. Hutson, K. Mischaikow and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.
doi: 10.1007/s002850100106. |
[22] |
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. |
[23] |
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. |
[24] |
C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596. |
[25] |
A. Leung, Equilibria and stability for competing-species, reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl., 73 (1980), 204-218.
doi: 10.1016/0022-247X(80)90028-1. |
[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] |
G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems,, preprint., ().
|
[28] |
C. V. Pao, Coexistence and stability of a competition-diffusion system in population dynamics, J. Math. Anal. Appl., 83 (1981), 54-76.
doi: 10.1016/0022-247X(81)90246-8. |
[29] |
C. V. Pao, Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays, Nonlinear Anal. Real World Appl., 5 (2004), 91-104.
doi: 10.1016/S1468-1218(03)00018-X. |
[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. |
[31] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983. |
[32] |
S. Ruan, and X.-Q. Zhao, Persistence and extinction in two species reaction-diffusion systems with delays, J. Differential Equations, 156 (1999), 71-92.
doi: 10.1006/jdeq.1998.3599. |
[33] |
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. |
[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, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats,, Proceedings of the American Mathematical Society, ().
|
[36] |
Hal L. Smith, "Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995. |
[37] |
Joseph W.-H. So and J. S. Yu, Global attractivity for a population model with time delay, Proc. Amer. Math. Soc., 123 (1995), 2687-2694.
doi: 10.1090/S0002-9939-1995-1317052-5. |
[38] |
P. Takac, A short elementary proof of the Krein-Rutman theorem, Houston J. of Math., 20 (1994), 93-98. |
[39] |
X. H. Tang, D. M. Cao and X. Zou, Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay, J. Differential Equations, 228 (2006), 580-610.
doi: 10.1016/j.jde.2006.06.007. |
[40] |
X. H. Tang and X. Zou, Global attractivity of nonautonomous Lotka Volterra competition system without instantaneous negative feedbacks, J. Differential Equations, 192 (2003), 502-535.
doi: 10.1016/S0022-0396(03)00042-1. |
[41] |
J. Wu and X. Q. Zhao, Permanence and convergence in multi-species competition systems with delay, Proc. Amer. Math. Soc., 126 (1998), 1709-1714.
doi: 10.1090/S0002-9939-98-04522-5. |
[42] |
J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
[43] |
X.-Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16. Springer-Verlag, New York, 2003. |
[44] |
L. Zhou and C. V. Pao, Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear Anal., 6 (1982), 1163-1184.
doi: 10.1016/0362-546X(82)90028-1. |
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