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January  2012, 17(1): 173-189. doi: 10.3934/dcdsb.2012.17.173

Traveling wave solutions of a competitive recursion

Guo Lin 1, and  Wan-Tong Li 2,

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000
2. 

School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received  April 2010 Revised  March 2011 Published  October 2011

This paper is concerned with the traveling wave solutions of a competitive recursion. By using a cross iteration scheme, we first establish the existence of traveling wave solutions, which are the invasion waves of two competitive invaders. These wave solutions are useful in understanding the long time behavior of solution of the corresponding Cauchy type problem where the initial distribution is a perturbation of the wave profile of a traveling wave solution that may be nonmonotone.
Keywords: competition system., Comparison principle, cross iteration.
Mathematics Subject Classification: Primary: 37L15, 39A11; Secondary: 92D2.
Citation: Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173
References:
[1]

S. Ahmad, A. C. Lazer and A. Tineo, Traveling waves for a system of equations, Nonlinear Anal., 68 (2008), 3909-3912. doi: 10.1016/j.na.2007.04.029.  Google Scholar

[2]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.  Google Scholar

[3]

X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258. doi: 10.1137/050627824.  Google Scholar

[4]

X. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  Google Scholar

[5]

J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Diff. Eqns. Appl., 10 (2004), 1139-1151.  Google Scholar

[6]

P. J. Darlington, Competition, competitive repulsion, and coexistence, Proc. Nat. Acad. Sci. USA, 69 (1972), 3151-3155. doi: 10.1073/pnas.69.11.3151.  Google Scholar

[7]

J. W. Evans, Nerve axon equations, I: Linear approximations,, Indiana Univ. Math. J., 21 (): 877.  doi: 10.1512/iumj.1972.21.21071.  Google Scholar

[8]

T. Faria and S. Trofimchuk, Nonmonotone traveling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.  Google Scholar

[9]

P. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Arch. Ration. Mech. Anal., 65 (1977), 355-361. doi: 10.1007/BF00250432.  Google Scholar

[10]

C. E. Goulden and L. L. Hornig, Population oscillations and energy reserves in planktonic cladocera and their consequences to competition, Proc. Natl. Acad. Sci. USA, 77 (1980), 1716-1720. doi: 10.1073/pnas.77.3.1716.  Google Scholar

[11]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. doi: 10.1126/science.131.3409.1292.  Google Scholar

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S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016.  Google Scholar

[13]

M. A. Huston, "Biological Diversity: The Coexistence of Species on Changing Landscapes," Cambridge University Press, Cambridge, 1994. Google Scholar

[14]

M. A. Huston and D. L. DeAngelis, Competition and coexistence: The effects of resource transport and supply rates, American Naturalist, 144 (1994), 954-977. doi: 10.1086/285720.  Google Scholar

[15]

M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436. doi: 10.1007/BF00173295.  Google Scholar

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A. W. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering," Mathematics and its Applications, Kluwer Academic Pub., Dordrecht, 1989.  Google Scholar

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M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.  Google Scholar

[18]

B. Li, Some remarks on traveling wave solutions in competition models, Discrete Contin. Dyn. Syst. Ser. B., 12 (2009), 389-399. doi: 10.3934/dcdsb.2009.12.389.  Google Scholar

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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), 323-338. doi: 10.1007/s00285-008-0175-1.  Google Scholar

[20]

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[21]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003.  Google Scholar

[22]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.  Google Scholar

[23]

G. Lin and W.-T. Li, Spreading speeds and traveling wavefronts for second order integrodifference equations, J. Math. Anal. Appl., 361 (2010), 520-532. doi: 10.1016/j.jmaa.2009.07.035.  Google Scholar

[24]

G. Lin, W.-T. Li and S. Ruan, Asymptotic stability of monostable wavefronts in discrete-time integral recursions, Sci. China Math., 53 (2010), 1185-1194. doi: 10.1007/s11425-009-0123-6.  Google Scholar

[25]

G. Lin, W.-T. Li and S. Ruan, Spreading speeds and traveling waves in competitive recursion systems, J. Math. Biol., 62 (2011), 162-201. doi: 10.1007/s00285-010-0334-z.  Google Scholar

[26]

R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator,, J. Math. Biol., 16 (): 199.   Google Scholar

[27]

R. Lui, A nonlinear integral operator arising from a model in population genetics. I. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937. doi: 10.1137/0513064.  Google Scholar

[28]

R. Lui, A nonlinear integral operator arising from a model in population genetics. II. Initial data with compact support, SIAM J. Math. Anal., 13 (1982), 938-953. doi: 10.1137/0513065.  Google Scholar

[29]

R. Lui, A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case, SIAM J. Math. Anal., 16 (1985), 1180-1206. doi: 10.1137/0516087.  Google Scholar

[30]

R. Lui, A nonlinear integral operator arising from a model in population genetics. IV. Clines, SIAM J. Math. Anal., 17 (1986), 152-168. doi: 10.1137/0517015.  Google Scholar

[31]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Diff. Eqns., 19 (2007), 391-436. doi: 10.1007/s10884-006-9065-7.  Google Scholar

[32]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  Google Scholar

[33]

S. Ma and X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  Google Scholar

[34]

M. Mei, J. W.-H. So, M. Y. Li and S. S. P. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinburgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358.  Google Scholar

[35]

K. Mischaikow and V. Hutson, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008. doi: 10.1137/0524059.  Google Scholar

[36]

A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in Britain, Proc. R. Soc. Lond. B, 238 (1989), 113-125. doi: 10.1098/rspb.1989.0070.  Google Scholar

[37]

S. Pan, Traveling wave solutions in delayed diffusion systems via a cross iteration scheme, Nonlinear Anal. RWA, 10 (2009), 2807-2818. doi: 10.1016/j.nonrwa.2008.08.007.  Google Scholar

[38]

S. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Anal. RWA, 12 (2011), 535-544. doi: 10.1016/j.nonrwa.2010.06.038.  Google Scholar

[39]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992.  Google Scholar

[40]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[41]

H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, AMS, Providence, RI, 1995.  Google Scholar

[42]

H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785.  Google Scholar

[43]

M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257.  Google Scholar

[44]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, AMS, Providence, RI, 1994.  Google Scholar

[45]

Z.-C. Wang, W.-T. 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.  Google Scholar

[46]

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

[47]

M.-H. Wang, M. Kot and M. G. Neubert, Integrodifference equations, Allee effects, and invasions, J. Math. Biol., 44 (2002), 150-168. doi: 10.1007/s002850100116.  Google Scholar

[48]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.  Google Scholar

[49]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.  Google Scholar

[50]

H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions, J. Math. Biol., 57 (2008), 387-411. doi: 10.1007/s00285-008-0168-0.  Google Scholar

[51]

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.  Google Scholar

[52]

Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Foundations of Modern Mathematics Series, Science Press, Beijing, 1990.  Google Scholar

[53]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.  Google Scholar

show all references

References:
[1]

S. Ahmad, A. C. Lazer and A. Tineo, Traveling waves for a system of equations, Nonlinear Anal., 68 (2008), 3909-3912. doi: 10.1016/j.na.2007.04.029.  Google Scholar

[2]

X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.  Google Scholar

[3]

X. Chen, S.-C. Fu and J.-S. Guo, Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices, SIAM J. Math. Anal., 38 (2006), 233-258. doi: 10.1137/050627824.  Google Scholar

[4]

X. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  Google Scholar

[5]

J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Diff. Eqns. Appl., 10 (2004), 1139-1151.  Google Scholar

[6]

P. J. Darlington, Competition, competitive repulsion, and coexistence, Proc. Nat. Acad. Sci. USA, 69 (1972), 3151-3155. doi: 10.1073/pnas.69.11.3151.  Google Scholar

[7]

J. W. Evans, Nerve axon equations, I: Linear approximations,, Indiana Univ. Math. J., 21 (): 877.  doi: 10.1512/iumj.1972.21.21071.  Google Scholar

[8]

T. Faria and S. Trofimchuk, Nonmonotone traveling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.  Google Scholar

[9]

P. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Arch. Ration. Mech. Anal., 65 (1977), 355-361. doi: 10.1007/BF00250432.  Google Scholar

[10]

C. E. Goulden and L. L. Hornig, Population oscillations and energy reserves in planktonic cladocera and their consequences to competition, Proc. Natl. Acad. Sci. USA, 77 (1980), 1716-1720. doi: 10.1073/pnas.77.3.1716.  Google Scholar

[11]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. doi: 10.1126/science.131.3409.1292.  Google Scholar

[12]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016.  Google Scholar

[13]

M. A. Huston, "Biological Diversity: The Coexistence of Species on Changing Landscapes," Cambridge University Press, Cambridge, 1994. Google Scholar

[14]

M. A. Huston and D. L. DeAngelis, Competition and coexistence: The effects of resource transport and supply rates, American Naturalist, 144 (1994), 954-977. doi: 10.1086/285720.  Google Scholar

[15]

M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436. doi: 10.1007/BF00173295.  Google Scholar

[16]

A. W. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering," Mathematics and its Applications, Kluwer Academic Pub., Dordrecht, 1989.  Google Scholar

[17]

M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.  Google Scholar

[18]

B. Li, Some remarks on traveling wave solutions in competition models, Discrete Contin. Dyn. Syst. Ser. B., 12 (2009), 389-399. doi: 10.3934/dcdsb.2009.12.389.  Google Scholar

[19]

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), 323-338. doi: 10.1007/s00285-008-0175-1.  Google Scholar

[20]

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

[21]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003.  Google Scholar

[22]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.  Google Scholar

[23]

G. Lin and W.-T. Li, Spreading speeds and traveling wavefronts for second order integrodifference equations, J. Math. Anal. Appl., 361 (2010), 520-532. doi: 10.1016/j.jmaa.2009.07.035.  Google Scholar

[24]

G. Lin, W.-T. Li and S. Ruan, Asymptotic stability of monostable wavefronts in discrete-time integral recursions, Sci. China Math., 53 (2010), 1185-1194. doi: 10.1007/s11425-009-0123-6.  Google Scholar

[25]

G. Lin, W.-T. Li and S. Ruan, Spreading speeds and traveling waves in competitive recursion systems, J. Math. Biol., 62 (2011), 162-201. doi: 10.1007/s00285-010-0334-z.  Google Scholar

[26]

R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator,, J. Math. Biol., 16 (): 199.   Google Scholar

[27]

R. Lui, A nonlinear integral operator arising from a model in population genetics. I. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937. doi: 10.1137/0513064.  Google Scholar

[28]

R. Lui, A nonlinear integral operator arising from a model in population genetics. II. Initial data with compact support, SIAM J. Math. Anal., 13 (1982), 938-953. doi: 10.1137/0513065.  Google Scholar

[29]

R. Lui, A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case, SIAM J. Math. Anal., 16 (1985), 1180-1206. doi: 10.1137/0516087.  Google Scholar

[30]

R. Lui, A nonlinear integral operator arising from a model in population genetics. IV. Clines, SIAM J. Math. Anal., 17 (1986), 152-168. doi: 10.1137/0517015.  Google Scholar

[31]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Diff. Eqns., 19 (2007), 391-436. doi: 10.1007/s10884-006-9065-7.  Google Scholar

[32]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  Google Scholar

[33]

S. Ma and X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  Google Scholar

[34]

M. Mei, J. W.-H. So, M. Y. Li and S. S. P. Shen, Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion, Proc. Royal Soc. Edinburgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358.  Google Scholar

[35]

K. Mischaikow and V. Hutson, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008. doi: 10.1137/0524059.  Google Scholar

[36]

A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in Britain, Proc. R. Soc. Lond. B, 238 (1989), 113-125. doi: 10.1098/rspb.1989.0070.  Google Scholar

[37]

S. Pan, Traveling wave solutions in delayed diffusion systems via a cross iteration scheme, Nonlinear Anal. RWA, 10 (2009), 2807-2818. doi: 10.1016/j.nonrwa.2008.08.007.  Google Scholar

[38]

S. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Anal. RWA, 12 (2011), 535-544. doi: 10.1016/j.nonrwa.2010.06.038.  Google Scholar

[39]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992.  Google Scholar

[40]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[41]

H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, AMS, Providence, RI, 1995.  Google Scholar

[42]

H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785.  Google Scholar

[43]

M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257.  Google Scholar

[44]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, AMS, Providence, RI, 1994.  Google Scholar

[45]

Z.-C. Wang, W.-T. 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.  Google Scholar

[46]

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

[47]

M.-H. Wang, M. Kot and M. G. Neubert, Integrodifference equations, Allee effects, and invasions, J. Math. Biol., 44 (2002), 150-168. doi: 10.1007/s002850100116.  Google Scholar

[48]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.  Google Scholar

[49]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.  Google Scholar

[50]

H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions, J. Math. Biol., 57 (2008), 387-411. doi: 10.1007/s00285-008-0168-0.  Google Scholar

[51]

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.  Google Scholar

[52]

Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Foundations of Modern Mathematics Series, Science Press, Beijing, 1990.  Google Scholar

[53]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.  Google Scholar

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