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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 and 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.

[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., to appear.

[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, to appear.

[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., to appear.

[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, to appear.

[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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