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March  2015, 20(2): 385-395. doi: 10.3934/dcdsb.2015.20.385

Asymptotic behavior for a reaction-diffusion population model with delay

Keng Deng 1, and  Yixiang Wu 2,

1. 

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
2. 

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, United States

Received  May 2014 Revised  August 2014 Published  January 2015

In this paper, we study a reaction-diffusion population model with time delay. We establish a comparison principle for coupled upper/lower solutions and prove the existence/uniqueness result for the model. We then show the global asymptotic behavior of the model.
Keywords: existence-uniqueness, Population model with delay, comparison principle, asymptotic behavior..
Mathematics Subject Classification: 35A01, 35B40, 35K57, 92D2.
Citation: Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385
References:
[1]

S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133. doi: 10.1016/j.jde.2006.08.015.  Google Scholar

[2]

M. V. Bartuccelli and S. A. Gourley, A Population model with time-dependent delay, Math. Comput. Modelling, 26 (1997), 13-30. doi: 10.1016/S0895-7177(97)00237-9.  Google Scholar

[3]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.  Google Scholar

[4]

K. Deng, On a nonlocal reaction-diffusion population model, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 65-73. doi: 10.3934/dcdsb.2008.9.65.  Google Scholar

[5]

S. A. Gourley and N. F. Briton, On a modified Volterra population equation with diffusion, Nonlinear Anal., 21 (1993), 389-395. doi: 10.1016/0362-546X(93)90082-4.  Google Scholar

[6]

Y. Kyrychko, S. A. Gourley and M. V. Bartuccelli, Comparison and convergence to equlibrium in a nonlocal delayed reaction-diffusion model on an infinite domain, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 1015-1026. doi: 10.3934/dcdsb.2005.5.1015.  Google Scholar

[7]

R. Laister, Global asymptotic behavior in some functional parabolic equations, Nonlinear Anal., 50 (2002), 347-361. doi: 10.1016/S0362-546X(01)00766-0.  Google Scholar

[8]

C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125. doi: 10.1137/050638011.  Google Scholar

[9]

M. Protter and H. Weinberger, Maximum Priciples in Differential Equations, Prentice-Hall Inc, New Jersey, 1967.  Google Scholar

[10]

R. Redlinger, Existence theorems for semilinear parabolic systems with functionals, Nonlinear Anal., 8 (1984), 667-682. doi: 10.1016/0362-546X(84)90011-7.  Google Scholar

[11]

R. Redlinger, On Volterra's population equation with diffusion, SIAM J. Math. Anal., 16 (1985), 135-142. doi: 10.1137/0516008.  Google Scholar

[12]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Royal Soc. Edinburgh. Ser. A, 134 (2004), 991-1011. doi: 10.1017/S0308210500003590.  Google Scholar

[13]

A. Schiaffino, On a diffusion Volterra equation, Nonlinear Anal., 3 (1979), 595-600. doi: 10.1016/0362-546X(79)90088-9.  Google Scholar

[14]

Z-C. Wang, W-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025.  Google Scholar

[15]

J. Wu and X-Q. Zhao, Permanence and convergence in multi-species competition systems with delay, Trans. Amer. Math. Soc., 126 (1998), 1709-1714. doi: 10.1090/S0002-9939-98-04522-5.  Google Scholar

show all references

References:
[1]

S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133. doi: 10.1016/j.jde.2006.08.015.  Google Scholar

[2]

M. V. Bartuccelli and S. A. Gourley, A Population model with time-dependent delay, Math. Comput. Modelling, 26 (1997), 13-30. doi: 10.1016/S0895-7177(97)00237-9.  Google Scholar

[3]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.  Google Scholar

[4]

K. Deng, On a nonlocal reaction-diffusion population model, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 65-73. doi: 10.3934/dcdsb.2008.9.65.  Google Scholar

[5]

S. A. Gourley and N. F. Briton, On a modified Volterra population equation with diffusion, Nonlinear Anal., 21 (1993), 389-395. doi: 10.1016/0362-546X(93)90082-4.  Google Scholar

[6]

Y. Kyrychko, S. A. Gourley and M. V. Bartuccelli, Comparison and convergence to equlibrium in a nonlocal delayed reaction-diffusion model on an infinite domain, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 1015-1026. doi: 10.3934/dcdsb.2005.5.1015.  Google Scholar

[7]

R. Laister, Global asymptotic behavior in some functional parabolic equations, Nonlinear Anal., 50 (2002), 347-361. doi: 10.1016/S0362-546X(01)00766-0.  Google Scholar

[8]

C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125. doi: 10.1137/050638011.  Google Scholar

[9]

M. Protter and H. Weinberger, Maximum Priciples in Differential Equations, Prentice-Hall Inc, New Jersey, 1967.  Google Scholar

[10]

R. Redlinger, Existence theorems for semilinear parabolic systems with functionals, Nonlinear Anal., 8 (1984), 667-682. doi: 10.1016/0362-546X(84)90011-7.  Google Scholar

[11]

R. Redlinger, On Volterra's population equation with diffusion, SIAM J. Math. Anal., 16 (1985), 135-142. doi: 10.1137/0516008.  Google Scholar

[12]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Royal Soc. Edinburgh. Ser. A, 134 (2004), 991-1011. doi: 10.1017/S0308210500003590.  Google Scholar

[13]

A. Schiaffino, On a diffusion Volterra equation, Nonlinear Anal., 3 (1979), 595-600. doi: 10.1016/0362-546X(79)90088-9.  Google Scholar

[14]

Z-C. Wang, W-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025.  Google Scholar

[15]

J. Wu and X-Q. Zhao, Permanence and convergence in multi-species competition systems with delay, Trans. Amer. Math. Soc., 126 (1998), 1709-1714. doi: 10.1090/S0002-9939-98-04522-5.  Google Scholar

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