# American Institute of Mathematical Sciences

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

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

 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.
Citation: Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete and 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), 926-929. doi: 10.1103/PhysRevLett.86.926. [2] Y. Jin and X. -Q. Zhao, Spatial dynamics of a periodic population model with dispersal, Nonlinearity, 22 (2009), 1167-1189. doi: 10.1088/0951-7715/22/5/011. [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), 219-233. doi: 10.1007/s002850200144. [4] B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008. [5] B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems, J. Diff. Eqs., 252 (2012), 4842-4861. doi: 10.1016/j.jde.2012.01.018. [6] B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems, Nonlinearity, 24 (2011), 1759-1776. doi: 10.1088/0951-7715/24/6/004. [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), 1-40. doi: 10.1002/cpa.20154. [8] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903. doi: 10.1016/j.jfa.2010.04.018. [9] R. Lui, Biological growth and spread modeled by systems of recursions I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6. [10] F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772. doi: 10.1137/050636152. [11] V. Méndez, T. Pujol and J. Fort, Dispersal probability distributions and the wave-front speed problem, Phys. Rev. E., 65 (2002), 041109/1-041109/6. [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), 1676-1702. doi: 10.1137/120880501. [13] J. Medlock and M. Kot, Spreading disease: Integral-differential equations old and new, Math. Biosci., 184 (2003), 201-222. doi: 10.1016/S0025-5564(03)00041-5. [14] D. Mollison, Dependence of epidemic and population velocities on basic parameters, Math. Biosci., 107 (1991), 255-287. doi: 10.1016/0025-5564(91)90009-8. [15] S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal, Bound. Value Probl., 2012 (2012), 11pp. doi: 10.1186/1687-2770-2012-120. [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), 814-826. doi: 10.1016/j.na.2010.09.032. [17] H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. [18] H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6. [19] Z.-X. Yu and R. Yuan, Travelling wave solutions in nonlocal reactiondiffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406. [20] L. Zhang and B. Li, Traveling waves in an integro-differential competition model, Discrete and Continuous Dynamical Systems-Series B, 17 (2012), 417-428. doi: 10.3934/dcdsb.2012.17.417.

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##### References:
 [1] S. Fedotov, Front propagation into an unstable state of reaction-transport systems, Phys. Rev. Lett., 86 (2001), 926-929. doi: 10.1103/PhysRevLett.86.926. [2] Y. Jin and X. -Q. Zhao, Spatial dynamics of a periodic population model with dispersal, Nonlinearity, 22 (2009), 1167-1189. doi: 10.1088/0951-7715/22/5/011. [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), 219-233. doi: 10.1007/s002850200144. [4] B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008. [5] B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems, J. Diff. Eqs., 252 (2012), 4842-4861. doi: 10.1016/j.jde.2012.01.018. [6] B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems, Nonlinearity, 24 (2011), 1759-1776. doi: 10.1088/0951-7715/24/6/004. [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), 1-40. doi: 10.1002/cpa.20154. [8] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903. doi: 10.1016/j.jfa.2010.04.018. [9] R. Lui, Biological growth and spread modeled by systems of recursions I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6. [10] F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772. doi: 10.1137/050636152. [11] V. Méndez, T. Pujol and J. Fort, Dispersal probability distributions and the wave-front speed problem, Phys. Rev. E., 65 (2002), 041109/1-041109/6. [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), 1676-1702. doi: 10.1137/120880501. [13] J. Medlock and M. Kot, Spreading disease: Integral-differential equations old and new, Math. Biosci., 184 (2003), 201-222. doi: 10.1016/S0025-5564(03)00041-5. [14] D. Mollison, Dependence of epidemic and population velocities on basic parameters, Math. Biosci., 107 (1991), 255-287. doi: 10.1016/0025-5564(91)90009-8. [15] S. Pan and G. Lin, Invasion traveling wave solutions of a competitive system with dispersal, Bound. Value Probl., 2012 (2012), 11pp. doi: 10.1186/1687-2770-2012-120. [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), 814-826. doi: 10.1016/j.na.2010.09.032. [17] H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. [18] H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6. [19] Z.-X. Yu and R. Yuan, Travelling wave solutions in nonlocal reactiondiffusion systems with delays and applications, ANZIAM J., 51 (2009), 49-66. doi: 10.1017/S1446181109000406. [20] L. Zhang and B. Li, Traveling waves in an integro-differential competition model, Discrete and Continuous Dynamical Systems-Series B, 17 (2012), 417-428. doi: 10.3934/dcdsb.2012.17.417.
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