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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

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