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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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, Lecture Notes in Math. 840 Springer-Verlag, Berlin, 1981.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[32]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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, Lecture Notes in Math. 840 Springer-Verlag, Berlin, 1981.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[32]

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

[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.

[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.

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