American Institute of Mathematical Sciences

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August  2015, 20(6): 1663-1684. doi: 10.3934/dcdsb.2015.20.1663

Spreading speeds and traveling wave solutions in cooperative integral-differential systems

Changbing Hu 1, , Yang Kuang 2, , Bingtuan Li 1, and  Hao Liu 3,

1. 

Department of Mathematics, University of Louisville, Louisville, KY 40292
2. 

School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85281
3. 

School of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, United States

Received  November 2013 Revised  February 2015 Published  June 2015

We study a cooperative system of integro-differential equations. It is shown that the system in general has multiple spreading speeds, and when the linear determinacy conditions are satisfied all the spreading speeds are the same and equal to the spreading speed of the linearized system. The existence of traveling wave solutions is established via integral systems. It is shown that when the linear determinacy conditions are satisfied, if the unique spreading speed is not zero then it may be characterized as the slowest speed of a class of traveling wave solutions. Some examples are presented to illustrate the theoretical results.
Keywords: linear determinacy, Integral-differential system, spreading speed, traveling wave solution., integral system.
Mathematics Subject Classification: Primary: 45K05; Secondary: 92D2.
Citation: Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663
References:
[1]

S. Fedotov, Front propagation into an unstable state of reaction-transport systems,, Phys. Rev. Lett., 86 (2001), 926.  doi: 10.1103/PhysRevLett.86.926.  Google Scholar

[2]

Y. Jin and X. -Q. Zhao, Spatial dynamics of a periodic population model with dispersal,, Nonlinearity, 22 (2009), 1167.  doi: 10.1088/0951-7715/22/5/011.  Google Scholar

[3]

M. A. Lewis, B. Li and H. F. Weinberger, Spreading speeds and linear conjecture for two-species competition models,, J. Math. Biol., 45 (2002), 219.  doi: 10.1007/s002850200144.  Google Scholar

[4]

B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[5]

B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems,, J. Diff. Eqs., 252 (2012), 4842.  doi: 10.1016/j.jde.2012.01.018.  Google Scholar

[6]

B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems,, Nonlinearity, 24 (2011), 1759.  doi: 10.1088/0951-7715/24/6/004.  Google Scholar

[7]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Commun. Pure Appl. Math., 60 (2007), 1.  doi: 10.1002/cpa.20154.  Google Scholar

[8]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[9]

R. Lui, Biological growth and spread modeled by systems of recursions I. Mathematical theory,, Math. Biosci., 93 (1989), 269.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[10]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations,, SIAM Rev., 47 (2005), 749.  doi: 10.1137/050636152.  Google Scholar

[11]

V. Méndez, T. Pujol and J. Fort, Dispersal probability distributions and the wave-front speed problem,, Phys. Rev. E., 65 (2002), 1.   Google Scholar

[12]

K. Meyer and B. Li, A spatial model of plants with an age-Structured seed bank and juvenile stage,, SIAM. J. Appl. Math., 73 (2013), 1676.  doi: 10.1137/120880501.  Google Scholar

[13]

J. Medlock and M. Kot, Spreading disease: Integral-differential equations old and new,, Math. Biosci., 184 (2003), 201.  doi: 10.1016/S0025-5564(03)00041-5.  Google Scholar

[14]

D. Mollison, Dependence of epidemic and population velocities on basic parameters,, Math. Biosci., 107 (1991), 255.  doi: 10.1016/0025-5564(91)90009-8.  Google Scholar

[15]

S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal,, Bound. Value Probl., 2012 (2012).  doi: 10.1186/1687-2770-2012-120.  Google Scholar

[16]

Y.-J. Sun, W.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity,, Nonlinear Anal., 74 (2011), 814.  doi: 10.1016/j.na.2010.09.032.  Google Scholar

[17]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183.  doi: 10.1007/s002850200145.  Google Scholar

[18]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems,, J. Math. Biol., 55 (2007), 207.  doi: 10.1007/s00285-007-0078-6.  Google Scholar

[19]

Z.-X. Yu and R. Yuan, Travelling wave solutions in nonlocal reactiondiffusion systems with delays and applications,, ANZIAM J., 51 (2009), 49.  doi: 10.1017/S1446181109000406.  Google Scholar

[20]

L. Zhang and B. Li, Traveling waves in an integro-differential competition model,, Discrete and Continuous Dynamical Systems-Series B, 17 (2012), 417.  doi: 10.3934/dcdsb.2012.17.417.  Google Scholar

show all references

References:
[1]

S. Fedotov, Front propagation into an unstable state of reaction-transport systems,, Phys. Rev. Lett., 86 (2001), 926.  doi: 10.1103/PhysRevLett.86.926.  Google Scholar

[2]

Y. Jin and X. -Q. Zhao, Spatial dynamics of a periodic population model with dispersal,, Nonlinearity, 22 (2009), 1167.  doi: 10.1088/0951-7715/22/5/011.  Google Scholar

[3]

M. A. Lewis, B. Li and H. F. Weinberger, Spreading speeds and linear conjecture for two-species competition models,, J. Math. Biol., 45 (2002), 219.  doi: 10.1007/s002850200144.  Google Scholar

[4]

B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[5]

B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems,, J. Diff. Eqs., 252 (2012), 4842.  doi: 10.1016/j.jde.2012.01.018.  Google Scholar

[6]

B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems,, Nonlinearity, 24 (2011), 1759.  doi: 10.1088/0951-7715/24/6/004.  Google Scholar

[7]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Commun. Pure Appl. Math., 60 (2007), 1.  doi: 10.1002/cpa.20154.  Google Scholar

[8]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[9]

R. Lui, Biological growth and spread modeled by systems of recursions I. Mathematical theory,, Math. Biosci., 93 (1989), 269.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[10]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations,, SIAM Rev., 47 (2005), 749.  doi: 10.1137/050636152.  Google Scholar

[11]

V. Méndez, T. Pujol and J. Fort, Dispersal probability distributions and the wave-front speed problem,, Phys. Rev. E., 65 (2002), 1.   Google Scholar

[12]

K. Meyer and B. Li, A spatial model of plants with an age-Structured seed bank and juvenile stage,, SIAM. J. Appl. Math., 73 (2013), 1676.  doi: 10.1137/120880501.  Google Scholar

[13]

J. Medlock and M. Kot, Spreading disease: Integral-differential equations old and new,, Math. Biosci., 184 (2003), 201.  doi: 10.1016/S0025-5564(03)00041-5.  Google Scholar

[14]

D. Mollison, Dependence of epidemic and population velocities on basic parameters,, Math. Biosci., 107 (1991), 255.  doi: 10.1016/0025-5564(91)90009-8.  Google Scholar

[15]

S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal,, Bound. Value Probl., 2012 (2012).  doi: 10.1186/1687-2770-2012-120.  Google Scholar

[16]

Y.-J. Sun, W.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity,, Nonlinear Anal., 74 (2011), 814.  doi: 10.1016/j.na.2010.09.032.  Google Scholar

[17]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183.  doi: 10.1007/s002850200145.  Google Scholar

[18]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems,, J. Math. Biol., 55 (2007), 207.  doi: 10.1007/s00285-007-0078-6.  Google Scholar

[19]

Z.-X. Yu and R. Yuan, Travelling wave solutions in nonlocal reactiondiffusion systems with delays and applications,, ANZIAM J., 51 (2009), 49.  doi: 10.1017/S1446181109000406.  Google Scholar

[20]

L. Zhang and B. Li, Traveling waves in an integro-differential competition model,, Discrete and Continuous Dynamical Systems-Series B, 17 (2012), 417.  doi: 10.3934/dcdsb.2012.17.417.  Google Scholar

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