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April  2015, 35(4): 1743-1765. doi: 10.3934/dcds.2015.35.1743

Traveling waves of a mutualistic model of mistletoes and birds

Chuncheng Wang 1, , Rongsong Liu 2, , Junping Shi 3, and  Carlos Martinez del Rio 4,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China
2. 

Department of Mathematics and Department of Zoology and Physiology, University of Wyoming, Laramie, WY, 82071
3. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795
4. 

Department of Zoology and Physiology, University of Wyoming, Laramie, WY, 82071, United States

Received  July 2013 Revised  September 2014 Published  November 2014

The existences of an asymptotic spreading speed and traveling wave solutions for a diffusive model which describes the interaction of mistletoe and bird populations with nonlocal diffusion and delay effect are proved by using monotone semiflow theory. The effects of different dispersal kernels on the asymptotic spreading speeds are investigated through concrete examples and simulations.
Keywords: reaction-diffusion, nonlocal, traveling wave., asymptotic spreading speed, Model of mistletoes and birds.
Mathematics Subject Classification: 92D25, 92D40, 35K5.
Citation: Chuncheng Wang, Rongsong Liu, Junping Shi, Carlos Martinez del Rio. Traveling waves of a mutualistic model of mistletoes and birds. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1743-1765. doi: 10.3934/dcds.2015.35.1743
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[2]

J. E. Aukema, Vectors, viscin, and viscaceae: Mistletoes as parasites, mutualists, and resources, Front. Ecol. Environ., 1 (2003), 212-219.

[3]

J. E. Aukema and C. M. del Rio, Where does a fruit-eating bird deposit mistletoe seeds? seed deposition patterns and an experiment, Ecology, 83 (2002), 3489-3496. doi: 10.2307/3072097.

[4]

J. Fang, J. Wei and X. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction diffusion system, J. Differential Equations, 245 (2008), 2749-2770. doi: 10.1016/j.jde.2008.09.001.

[5]

J. Fang and X. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dynam. Differential Equations, 21 (2009), 663-680. doi: 10.1007/s10884-009-9152-7.

[6]

J. Fang and X. Zhao, Traveling waves for monotone semiflows with weak compactness,, SIAM J. Math. Anal., (). 

[7]

C. Gosper, C. D. Stansbury and G. Vivian-Smith, Seed dispersal of fleshy-fruited invasive plants by birds: Contributing factors and management options, Diversity and Distributions, 11 (2005), 549-558. doi: 10.1111/j.1366-9516.2005.00195.x.

[8]

B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems, J. Differential Equations, 252 (2012), 4842-4861. doi: 10.1016/j.jde.2012.01.018.

[9]

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.

[10]

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.

[11]

X. Liang and X. 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.

[12]

X. Liang and X. 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.

[13]

G. Lin, W. Li and S. Ruan, Monostable wavefronts in cooperative Lotka-Volterra systems with nonlocal delays, Discrete Contin. Dyn. Syst., 31 (2011), 1-23. doi: 10.3934/dcds.2011.31.1.

[14]

R. Liu, C. M. del Rio and J. Wu, Spatiotemporal variation of mistletoes: A dynamic modeling approach, Bull. Math. Biol., 73 (2011), 1794-1811. doi: 10.1007/s11538-010-9592-6.

[15]

N. Reid, Coevolution of mistletoes and frugivorous birds, Australian Journal of Ecology, 16 (1991), 457-469. doi: 10.1111/j.1442-9993.1991.tb01075.x.

[16]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, vol. 41 of Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 1995.

[17]

C. Wang, R. Liu, J. Shi and C. M. del Rio, Spatiotemporal mutualistic model of mistletoes and birds, J. Math. Biol., 68 (2014), 1479-1520. doi: 10.1007/s00285-013-0664-8.

[18]

Z. Wang, W. 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.

[19]

Z. Wang, W. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607. doi: 10.1007/s10884-008-9103-8.

[20]

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.

[21]

P. Weng and X. Zhao, Spreading speed and traveling waves for a multi-type {SIS} epidemic model, J. Differential Equations, 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020.

[22]

S. Wu, Y. Sun and S. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity, Discrete Contin. Dyn. Syst., 33 (2013), 921-946. doi: 10.3934/dcds.2013.33.921.

[23]

X. Zhao, Dynamical System in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[2]

J. E. Aukema, Vectors, viscin, and viscaceae: Mistletoes as parasites, mutualists, and resources, Front. Ecol. Environ., 1 (2003), 212-219.

[3]

J. E. Aukema and C. M. del Rio, Where does a fruit-eating bird deposit mistletoe seeds? seed deposition patterns and an experiment, Ecology, 83 (2002), 3489-3496. doi: 10.2307/3072097.

[4]

J. Fang, J. Wei and X. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction diffusion system, J. Differential Equations, 245 (2008), 2749-2770. doi: 10.1016/j.jde.2008.09.001.

[5]

J. Fang and X. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dynam. Differential Equations, 21 (2009), 663-680. doi: 10.1007/s10884-009-9152-7.

[6]

J. Fang and X. Zhao, Traveling waves for monotone semiflows with weak compactness,, SIAM J. Math. Anal., (). 

[7]

C. Gosper, C. D. Stansbury and G. Vivian-Smith, Seed dispersal of fleshy-fruited invasive plants by birds: Contributing factors and management options, Diversity and Distributions, 11 (2005), 549-558. doi: 10.1111/j.1366-9516.2005.00195.x.

[8]

B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems, J. Differential Equations, 252 (2012), 4842-4861. doi: 10.1016/j.jde.2012.01.018.

[9]

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.

[10]

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.

[11]

X. Liang and X. 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.

[12]

X. Liang and X. 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.

[13]

G. Lin, W. Li and S. Ruan, Monostable wavefronts in cooperative Lotka-Volterra systems with nonlocal delays, Discrete Contin. Dyn. Syst., 31 (2011), 1-23. doi: 10.3934/dcds.2011.31.1.

[14]

R. Liu, C. M. del Rio and J. Wu, Spatiotemporal variation of mistletoes: A dynamic modeling approach, Bull. Math. Biol., 73 (2011), 1794-1811. doi: 10.1007/s11538-010-9592-6.

[15]

N. Reid, Coevolution of mistletoes and frugivorous birds, Australian Journal of Ecology, 16 (1991), 457-469. doi: 10.1111/j.1442-9993.1991.tb01075.x.

[16]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, vol. 41 of Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 1995.

[17]

C. Wang, R. Liu, J. Shi and C. M. del Rio, Spatiotemporal mutualistic model of mistletoes and birds, J. Math. Biol., 68 (2014), 1479-1520. doi: 10.1007/s00285-013-0664-8.

[18]

Z. Wang, W. 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.

[19]

Z. Wang, W. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607. doi: 10.1007/s10884-008-9103-8.

[20]

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.

[21]

P. Weng and X. Zhao, Spreading speed and traveling waves for a multi-type {SIS} epidemic model, J. Differential Equations, 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020.

[22]

S. Wu, Y. Sun and S. Liu, Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity, Discrete Contin. Dyn. Syst., 33 (2013), 921-946. doi: 10.3934/dcds.2013.33.921.

[23]

X. Zhao, Dynamical System in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.

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