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Global stability of a fivedimensional model with immune responses and delay
Traveling wave solutions in an integrodifferential competition model
1.  Institute of Applied Mathematics, College of Science, Northwest A & F University, Yangling, Shaanxi 712100, China 
2.  Department of Mathematics, University of Louisville, Louisville, KY 40292 
References:
[1] 
D. Aronson, The asymptotic speed of propagation of a simple epidemic, in "Nonlinear Diffusion" (eds. W. Fitzgibbon and H. Walker) (NSFCBMS Regional Conf. Nonlinear Diffusion Equations, Univ. Houston, Houton, TX, 1976), Res. Notes Math., 14, Pitman, London, (1977), 123. 
[2] 
P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transition, Arch. Rational. Mech. Anal., 138 (1997), 105136. 
[3] 
S. Fedotov, Front propagation into an unstable state of reactiontransport systems, Phys. Rev. Lett., 86 (2001), 926929. doi: 10.1103/PhysRevLett.86.926. 
[4] 
M. A. Lewis, B. Li and H. F. Weinberger, Spreading speeds and linear determinancy for twospecies competition models, J. Math. Biol., 45 (2002), 219233. doi: 10.1007/s002850200144. 
[5] 
B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323338. doi: 10.1007/s0028500801751. 
[6] 
W.T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reactiondiffusion systems with application to diffusioncompetition systems, Nonlinearity, 19 (2006), 12531273. doi: 10.1088/09517715/19/6/003. 
[7] 
F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749772. doi: 10.1137/050636152. 
[8] 
B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 8298. doi: 10.1016/j.mbs.2005.03.008. 
[9] 
V. Méndez, T. Pujol and J. Fort, Dispersal probability distributions and the wavefront speed problem, Phys. Rev. E., 65 (2002), 041109/1041109/6. 
[10] 
K. Müller, "Investigations on the Organic Drift in North Swedish Streams," Tech. Report 34, Institute of Freshwater Research, Drottningholm, Sweden, 1954. 
[11] 
K. Müller, Stream drift as a chronobiological phenomenon in running water ecosystems, Ann. Rev. Eco. Sys., 5 (1974), 309323. doi: 10.1146/annurev.es.05.110174.001521. 
[12] 
J. D. Murray, "Mathematical Biology I: An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, SpringerVerlag, New York, 2002. 
[13] 
J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, 18, SpringerVerlag, New York, 2003. 
[14] 
A. Nilsson, Coleoptera from a Malaisetrap placed across a coastal stream in northern Sweden, Fauna Norrlandica, 3 (1981), 19. 
[15] 
A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, SpringerVerlag, New York, 2001. 
[16] 
W. D. Pearson and R. H. Kramer, Drift and production of two aquatic insects in a mountain stream, Ecol. Monogr., 42 (1972), 365385. doi: 10.2307/1942214. 
[17] 
T. Roos, Studies on upstream migration in adult streamdwelling insects,, Inst. Freshwater Res. Drottningholm, (): 167. 
[18] 
W. Rudin, "Functional Analysis," Second edition, International Series in Pure and Applied Mathematics, McGrawHill, Inc., New York, 1991. 
[19] 
N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford University Press, Oxford, 1997. 
[20] 
M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 6977. doi: 10.1007/BF00283257. 
[21] 
D. Tilman and P. Kareiva, "Spatial Ecology," Princeton University Press, Princeton, New Jersey, 1997. 
[22] 
D. Volkov and R. Lui, Spreading speed and travelling wave solutions of a partially sedentary population, IMA. J. Appl. Math., 72 (2007), 801816. doi: 10.1093/imamat/hxm025. 
[23] 
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183218. doi: 10.1007/s002850200145. 
[24] 
L. Zhang, "Traveling Wave Solutions and Periodic Solutions for Several Classes of Nonlinear Population Models," Ph.D thesis, Sichuan University, 2011. 
show all references
References:
[1] 
D. Aronson, The asymptotic speed of propagation of a simple epidemic, in "Nonlinear Diffusion" (eds. W. Fitzgibbon and H. Walker) (NSFCBMS Regional Conf. Nonlinear Diffusion Equations, Univ. Houston, Houton, TX, 1976), Res. Notes Math., 14, Pitman, London, (1977), 123. 
[2] 
P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transition, Arch. Rational. Mech. Anal., 138 (1997), 105136. 
[3] 
S. Fedotov, Front propagation into an unstable state of reactiontransport systems, Phys. Rev. Lett., 86 (2001), 926929. doi: 10.1103/PhysRevLett.86.926. 
[4] 
M. A. Lewis, B. Li and H. F. Weinberger, Spreading speeds and linear determinancy for twospecies competition models, J. Math. Biol., 45 (2002), 219233. doi: 10.1007/s002850200144. 
[5] 
B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323338. doi: 10.1007/s0028500801751. 
[6] 
W.T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reactiondiffusion systems with application to diffusioncompetition systems, Nonlinearity, 19 (2006), 12531273. doi: 10.1088/09517715/19/6/003. 
[7] 
F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749772. doi: 10.1137/050636152. 
[8] 
B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 8298. doi: 10.1016/j.mbs.2005.03.008. 
[9] 
V. Méndez, T. Pujol and J. Fort, Dispersal probability distributions and the wavefront speed problem, Phys. Rev. E., 65 (2002), 041109/1041109/6. 
[10] 
K. Müller, "Investigations on the Organic Drift in North Swedish Streams," Tech. Report 34, Institute of Freshwater Research, Drottningholm, Sweden, 1954. 
[11] 
K. Müller, Stream drift as a chronobiological phenomenon in running water ecosystems, Ann. Rev. Eco. Sys., 5 (1974), 309323. doi: 10.1146/annurev.es.05.110174.001521. 
[12] 
J. D. Murray, "Mathematical Biology I: An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, SpringerVerlag, New York, 2002. 
[13] 
J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, 18, SpringerVerlag, New York, 2003. 
[14] 
A. Nilsson, Coleoptera from a Malaisetrap placed across a coastal stream in northern Sweden, Fauna Norrlandica, 3 (1981), 19. 
[15] 
A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, SpringerVerlag, New York, 2001. 
[16] 
W. D. Pearson and R. H. Kramer, Drift and production of two aquatic insects in a mountain stream, Ecol. Monogr., 42 (1972), 365385. doi: 10.2307/1942214. 
[17] 
T. Roos, Studies on upstream migration in adult streamdwelling insects,, Inst. Freshwater Res. Drottningholm, (): 167. 
[18] 
W. Rudin, "Functional Analysis," Second edition, International Series in Pure and Applied Mathematics, McGrawHill, Inc., New York, 1991. 
[19] 
N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford University Press, Oxford, 1997. 
[20] 
M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 6977. doi: 10.1007/BF00283257. 
[21] 
D. Tilman and P. Kareiva, "Spatial Ecology," Princeton University Press, Princeton, New Jersey, 1997. 
[22] 
D. Volkov and R. Lui, Spreading speed and travelling wave solutions of a partially sedentary population, IMA. J. Appl. Math., 72 (2007), 801816. doi: 10.1093/imamat/hxm025. 
[23] 
H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183218. doi: 10.1007/s002850200145. 
[24] 
L. Zhang, "Traveling Wave Solutions and Periodic Solutions for Several Classes of Nonlinear Population Models," Ph.D thesis, Sichuan University, 2011. 
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