September  2012, 32(9): 3303-3324. doi: 10.3934/dcds.2012.32.3303

Traveling waves of diffusive predator-prey systems: Disease outbreak propagation

1. 

Mprime Centre for Disease Modelling, York Institute for Health Research, Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada, Canada

2. 

Division of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069-7100, United States

Received  January 2012 Revised  March 2012 Published  April 2012

We study the traveling waves of reaction-diffusion equations for a diffusive SIR model. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Our proof is based on Schauder fixed point theorem and Laplace transform.
Citation: Xiang-Sheng Wang, Haiyan Wang, Jianhong Wu. Traveling waves of diffusive predator-prey systems: Disease outbreak propagation. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3303-3324. doi: 10.3934/dcds.2012.32.3303
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show all references

References:
[1]

Phys. Rev. E, 66 (2002), 011912, 5 pp. Google Scholar

[2]

Bull. Math. Biol., 65 (2003), 519-534. doi: 10.1016/S0092-8240(03)00013-2.  Google Scholar

[3]

Int. J. Math. Math. Sci., 23 (2000), 425-429. doi: 10.1155/S0161171200002696.  Google Scholar

[4]

Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 663-675. doi: 10.1017/S0308210500004054.  Google Scholar

[5]

Math. Proc. Camb. Phil. Soc., 80 (1976), 315-330. doi: 10.1017/S0305004100052944.  Google Scholar

[6]

J. Roy. Stat. Soc. A, 120 (1957), 48-70. doi: 10.2307/2342553.  Google Scholar

[7]

J. Math. Biol., 28 (1990), 529-565. doi: 10.1007/BF00164162.  Google Scholar

[8]

Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001.  Google Scholar

[9]

Math. Proc. Camb. Phil. Soc., 81 (1977), 431-433. doi: 10.1017/S0305004100053494.  Google Scholar

[10]

in "Nonlinear Phenomena in Mathematical Sciences" (ed. V. Lakshmikantham), Academic Press, New York, 1981. Google Scholar

[11]

Am. Nat., 160 (2002), 348-359. doi: 10.1086/341518.  Google Scholar

[12]

Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[13]

J. Math. Biol., 6 (1978), 109-130. doi: 10.1007/BF02450783.  Google Scholar

[14]

Nonlinear Analysis, 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9.  Google Scholar

[15]

Annales de la Faculté des Sciences de Toulouse Sér. 6, 10 (2001), 271-292.  Google Scholar

[16]

Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 251-273. doi: 10.3934/dcdsb.2007.7.251.  Google Scholar

[17]

Proc. R. Soc. Edin. Sect. A, 139 (2009), 459-482. doi: 10.1017/S0308210507000455.  Google Scholar

[18]

J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112.  Google Scholar

[19]

Trans. Amer. Math. Soc., 286 (1984), 557-594.  Google Scholar

[20]

J. Appl. Probab., 17 (1980), 839-845. doi: 10.2307/3212977.  Google Scholar

[21]

Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3043-3058. doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[22]

Ann. of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[23]

Math. Biosci., 128 (1995), 131-155. doi: 10.1016/0025-5564(94)00070-G.  Google Scholar

[24]

Nonlinear Anal. Real World Appl., 12 (2011), 52-68. doi: 10.1016/j.nonrwa.2010.05.035.  Google Scholar

[25]

Nature, 414 (2001), 716-723. doi: 10.1038/414716a.  Google Scholar

[26]

Nonlinear World, 1 (1994), 277-290.  Google Scholar

[27]

Math. Models Methods Appl. Sci., 5 (1995), 935-966. doi: 10.1142/S0218202595000504.  Google Scholar

[28]

Acta Mathematicae Applicatae Sinica Engl. Ser., 22 (2006), 243-256. doi: 10.1007/s10255-006-0300-0.  Google Scholar

[29]

J. Dynam. Differential Equations, 16 (2004), 745-765.  Google Scholar

[30]

Nonlinear Anal., 8 (1984), 851-856. doi: 10.1016/0362-546X(84)90107-X.  Google Scholar

[31]

J. Theor. Biol., 116 (1985), 377-393. doi: 10.1016/S0022-5193(85)80276-9.  Google Scholar

[32]

Proc. R. Soc. Lond. B, 115 (1927), 700-721. Google Scholar

[33]

J. Roy. Stat. Soc. A, 120 (1957), 64-67. Google Scholar

[34]

in "Mathematics and Computer Science in Biology and Medicine," Medical Research Council, London, (1965), 213-225. Google Scholar

[35]

Bull. Math. Biol., 42 (1980), 397-429.  Google Scholar

[36]

Physica A, 272 (1999), 206-222. doi: 10.1016/S0378-4371(99)00284-8.  Google Scholar

[37]

Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003.  Google Scholar

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[39]

Comm. Pure Appl. Math., 60 (2007), 1-40; Erratum: 61 (2008), 137-138, MR2361306.  Google Scholar

[40]

J. Differential Equations, 171 (2001), 294-314.  Google Scholar

[41]

Math. Biosci., 215 (2008), 64-77. doi: 10.1016/j.mbs.2008.05.008.  Google Scholar

[42]

Adv. Appl. Prob., 4 (1972), 233-257. doi: 10.2307/1425997.  Google Scholar

[43]

J. Roy. Stat. Soc. B, 39 (1977), 283-326.  Google Scholar

[44]

Internat. J. Systems Sci., 38 (2007), 699-707. doi: 10.1080/00207720701596417.  Google Scholar

[45]

J. Theor. Biol., 156 (1992), 327-348. doi: 10.1016/S0022-5193(05)80679-4.  Google Scholar

[46]

Proc. R. Soc. Lond. B, 229 (1986), 111-150. doi: 10.1098/rspb.1986.0078.  Google Scholar

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Int. Journal of Math. Analysis, 2 (2008), 1083-1088.  Google Scholar

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Math. Surveys Monogr., 102, Amer. Math. Soc., Providence, RI, 2003.  Google Scholar

[49]

J. Appl. Probab., 18 (1981), 715-720. doi: 10.2307/3213325.  Google Scholar

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in "Mathematics for Life Science and Medicine" (eds. Y. Iwasa, K. Sato and Y. Takeuchi), Biol. Med. Phys. Biomed. Eng., Springer, Berlin, (2007), 97-122.  Google Scholar

[51]

in "Spatial Ecology," Chapman & Hall/CRC, Boca Raton, FL, (2009), 293-316. Google Scholar

[52]

Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991-1011. doi: 10.1017/S0308210500003590.  Google Scholar

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Math. Med. Biol., 28 (2011), 165-183. doi: 10.1093/imammb/dqq016.  Google Scholar

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Mathematical Modelling of Natural Phenomena, 3 (2008), 28-47. doi: 10.1051/mmnp:2008069.  Google Scholar

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Nonlinear Anal. Real World Appl., 5 (2004), 895-909. doi: 10.1016/j.nonrwa.2004.05.001.  Google Scholar

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Bull. Math. Biol., 67 (2005), 509-528. doi: 10.1016/j.bulm.2004.08.005.  Google Scholar

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J. Math. Biol., 8 (1979), 173-187. doi: 10.1007/BF00279720.  Google Scholar

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J. Differential Equations, 195 (2003), 430-470.  Google Scholar

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Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[60]

J. Differential Equations, 247 (2009), 887-905.  Google Scholar

[61]

X.-S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics,, submitted., ().   Google Scholar

[62]

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377.  Google Scholar

[63]

J. Math. Anal. Appl., 385 (2012), 683-692. doi: 10.1016/j.jmaa.2011.06.084.  Google Scholar

[64]

J. Differential Equations, 229 (2006), 270-296.  Google Scholar

[65]

J. Math. Anal. Appl., 365 (2010), 729-741. doi: 10.1016/j.jmaa.2009.11.028.  Google Scholar

[66]

Discrete Contin. Dyn. Syst., 23 (2009), 561-569. doi: 10.3934/dcds.2009.23.561.  Google Scholar

[67]

J. Dynamics and Differential Equations, 16 (2004), 679-707.  Google Scholar

[68]

Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 1043-1056. doi: 10.3934/dcdsb.2005.5.1043.  Google Scholar

[69]

PLoS ONE, 6 (2011), e21128. doi: 10.1371/journal.pone.0021128.  Google Scholar

[70]

Nonlinear Anal. Real World Appl., 12 (2011), 1223-1234. doi: 10.1016/j.nonrwa.2010.09.017.  Google Scholar

[71]

Nonlinearity, 21 (2008), 97-112. doi: 10.1088/0951-7715/21/1/005.  Google Scholar

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Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128. doi: 10.3934/dcdsb.2004.4.1117.  Google Scholar

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