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September  2012, 11(5): 1699-1722. doi: 10.3934/cpaa.2012.11.1699

Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal

Georg Hetzer 1, , Tung Nguyen 2, and  Wenxian Shen 3,

1. 

Department of Mathematics, Auburn University, AL 36849-5310
2. 

Department of Mathematical Sciences, University of Illinois Spring eld, Spring eld, IL 62703, United States
3. 

Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849-5310

Received  January 2011 Revised  June 2011 Published  March 2012

Coexistence and extinction for two species Volterra-Lotka competition systems with nonlocal dispersal are investigated in this paper. Sufficient conditions in terms of diffusion, reproduction, self-limitation, and competition rates are established for existence, uniqueness, and stability of coexistence states as well as for the extinction of one species. The focus is on environments with hostile surroundings. In this case, our results correspond to those for random dispersal under Dirichlet boundary conditions. Similar results hold for environments with non-flux boundary and for periodic environments, which correspond to those for random dispersal under Neumann boundary conditions and periodic boundary conditions, respectively.
Keywords: extinction, comparison principle., principal eigenvalue, nonlocal dispersal, coexistence, Volterra-Lotka competition model.
Mathematics Subject Classification: 35K55, 47G20, 47N20, 92D25, 92D4.
Citation: Georg Hetzer, Tung Nguyen, Wenxian Shen. Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1699-1722. doi: 10.3934/cpaa.2012.11.1699
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.  Google Scholar

[2]

R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 2921-2953. doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[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.  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), 1693-1709. doi: 10.1137/060676854.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Springer-Verlag, New York, 1981.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596.  Google Scholar

[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.  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), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[27]

G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  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, Methods & Applications, 72 (2010), 3150-3158.  Google Scholar

[31]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  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), 747-795. doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[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, ().   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[38]

P. Takac, A short elementary proof of the Krein-Rutman theorem, Houston J. of Math., 20 (1994), 93-98.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 2921-2953. doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[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.  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), 1693-1709. doi: 10.1137/060676854.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Springer-Verlag, New York, 1981.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596.  Google Scholar

[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.  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), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[27]

G. Lv and M. Wang, Existence and stability of traveling wave fronts for nonlocal delayed reaction diffusion systems,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  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, Methods & Applications, 72 (2010), 3150-3158.  Google Scholar

[31]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  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), 747-795. doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[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, ().   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[38]

P. Takac, A short elementary proof of the Krein-Rutman theorem, Houston J. of Math., 20 (1994), 93-98.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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