November  2018, 23(9): 3799-3816. doi: 10.3934/dcdsb.2018080

Coexistence and extinction in Time-Periodic Volterra-Lotka type systems with nonlocal dispersal

1. 

Department of Mathematical Sciences, University of Illinois Springfield, Springfield, IL 62703, USA

2. 

Department of Mathematics, Hampton University, Hampton, VA 23668, USA

Received  August 2016 Revised  July 2017 Published  March 2018

This paper deals with coexistence and extinction of time periodic Volterra-Lotka type competing systems with nonlocal dispersal. Such issues have already been studied for time independent systems with nonlocal dispersal and time periodic systems with random dispersal, but have not been studied yet for time periodic systems with nonlocal dispersal. In this paper, the relations between the coefficients representing Malthusian growths, self regulations and competitions of the two species have been obtained which ensure coexistence and extinction for the time periodic Volterra-Lotka type system with nonlocal dispersal. The underlying environment of the Volterra-Lotka type system under consideration has either hostile surroundings, or non-flux boundary, or is spatially periodic.

Citation: Tung Nguyen, Nar Rawal. Coexistence and extinction in Time-Periodic Volterra-Lotka type systems with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3799-3816. doi: 10.3934/dcdsb.2018080
References:
[1]

S. Ahmad and A. Lazer, Asymptotic behavior of solutions of periodic competition diffusion system, Nonlinear Anal., 13 (1989), 263-284.  doi: 10.1016/0362-546X(89)90054-0.  Google Scholar

[2]

X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.  Google Scholar

[3]

X. BaoW. Li and W. Shen, Traveling wave solutions of Lotka-Volterra competition systems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590-8637.  doi: 10.1016/j.jde.2016.02.032.  Google Scholar

[4]

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

[5]

R. S. Cantrell and C. Cosner, Spatial Energy Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK., 2003.  Google Scholar

[6]

E. ChasseigneM. 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

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C. CortazarM. 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

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C. CortazarM. Elgueta, ManuelJ. 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.   Google Scholar

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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

[10]

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.  Google Scholar

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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

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P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003.  Google Scholar

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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.  doi: 10.1016/0362-546X(94)00139-9.  Google Scholar

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M. GrinfeldG. HinesV. HutsonK. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320.   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

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840 Springer-Verlag, Berlin, 1981.  Google Scholar

[17]

G. HetzerT. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar

[18]

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

[19]

S. HsuH. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[20]

V. HutsonY. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[21]

V. HutsonS. MartinezK. 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

[22]

V. HutsonK. Mischaikow and P. Polacik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.  doi: 10.1007/s002850100106.  Google Scholar

[23]

C.-Y. KaoY. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072.  doi: 10.3934/dcdsb.2012.17.2047.  Google Scholar

[24]

C.-Y. KaoY. Lou and W. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.   Google Scholar

[25]

L. KongN. Rawal and W. Shen, Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats, Math. Model. Nat. Phenom., 10 (2015), 113-141.  doi: 10.1051/mmnp/201510609.  Google Scholar

[26]

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

[27]

W.-T. LiL. Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.   Google Scholar

[28]

C. V. Pao, Coexistence and statbility 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]

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

[30]

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.  Google Scholar

[31]

W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, J. Differential Equations, 259 (2015), 7375-7405.  doi: 10.1016/j.jde.2015.08.026.  Google Scholar

[32]

P. Zhao, Asymptotic Dynamics of Competition Systems with Immigration and/or Time Periodic Dependence, PhD dissertation, Auburn University, 2015. Google Scholar

[33]

X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495.   Google Scholar

[34]

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]

S. Ahmad and A. Lazer, Asymptotic behavior of solutions of periodic competition diffusion system, Nonlinear Anal., 13 (1989), 263-284.  doi: 10.1016/0362-546X(89)90054-0.  Google Scholar

[2]

X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.  Google Scholar

[3]

X. BaoW. Li and W. Shen, Traveling wave solutions of Lotka-Volterra competition systems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590-8637.  doi: 10.1016/j.jde.2016.02.032.  Google Scholar

[4]

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

[5]

R. S. Cantrell and C. Cosner, Spatial Energy Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK., 2003.  Google Scholar

[6]

E. ChasseigneM. 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

[7]

C. CortazarM. 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

[8]

C. CortazarM. Elgueta, ManuelJ. 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.   Google Scholar

[9]

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

[10]

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.  Google Scholar

[11]

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

[12]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, 153-191, Springer, Berlin, 2003.  Google Scholar

[13]

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.  doi: 10.1016/0362-546X(94)00139-9.  Google Scholar

[14]

M. GrinfeldG. HinesV. HutsonK. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320.   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, Lecture Notes in Math. 840 Springer-Verlag, Berlin, 1981.  Google Scholar

[17]

G. HetzerT. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar

[18]

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

[19]

S. HsuH. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[20]

V. HutsonY. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[21]

V. HutsonS. MartinezK. 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

[22]

V. HutsonK. Mischaikow and P. Polacik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.  doi: 10.1007/s002850100106.  Google Scholar

[23]

C.-Y. KaoY. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072.  doi: 10.3934/dcdsb.2012.17.2047.  Google Scholar

[24]

C.-Y. KaoY. Lou and W. Shen, Random dispersal vs. non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.   Google Scholar

[25]

L. KongN. Rawal and W. Shen, Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats, Math. Model. Nat. Phenom., 10 (2015), 113-141.  doi: 10.1051/mmnp/201510609.  Google Scholar

[26]

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

[27]

W.-T. LiL. Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.   Google Scholar

[28]

C. V. Pao, Coexistence and statbility 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]

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

[30]

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.  Google Scholar

[31]

W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, J. Differential Equations, 259 (2015), 7375-7405.  doi: 10.1016/j.jde.2015.08.026.  Google Scholar

[32]

P. Zhao, Asymptotic Dynamics of Competition Systems with Immigration and/or Time Periodic Dependence, PhD dissertation, Auburn University, 2015. Google Scholar

[33]

X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495.   Google Scholar

[34]

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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