Traveling wave solutions of a competitive recursion

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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

Abstract
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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 and 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.
[2]	X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
[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.
[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.
[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.
[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.
[7]	J. W. Evans, Nerve axon equations, I: Linear approximations,, Indiana Univ. Math. J., 21 (): 877. doi: 10.1512/iumj.1972.21.21071.
[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.
[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.
[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.
[11]	G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. doi: 10.1126/science.131.3409.1292.
[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.
[13]	M. A. Huston, "Biological Diversity: The Coexistence of Species on Changing Landscapes," Cambridge University Press, Cambridge, 1994.
[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.
[15]	M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436. doi: 10.1007/BF00173295.
[16]	A. W. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering," Mathematics and its Applications, Kluwer Academic Pub., Dordrecht, 1989.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator,, J. Math. Biol., 16 (): 199.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	K. Mischaikow and V. Hutson, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008. doi: 10.1137/0524059.
[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.
[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.
[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.
[39]	C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[52]	Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Foundations of Modern Mathematics Series, Science Press, Beijing, 1990.
[53]	B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.

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.
[2]	X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
[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.
[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.
[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.
[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.
[7]	J. W. Evans, Nerve axon equations, I: Linear approximations,, Indiana Univ. Math. J., 21 (): 877. doi: 10.1512/iumj.1972.21.21071.
[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.
[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.
[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.
[11]	G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. doi: 10.1126/science.131.3409.1292.
[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.
[13]	M. A. Huston, "Biological Diversity: The Coexistence of Species on Changing Landscapes," Cambridge University Press, Cambridge, 1994.
[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.
[15]	M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436. doi: 10.1007/BF00173295.
[16]	A. W. Leung, "Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering," Mathematics and its Applications, Kluwer Academic Pub., Dordrecht, 1989.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator,, J. Math. Biol., 16 (): 199.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	K. Mischaikow and V. Hutson, Traveling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008. doi: 10.1137/0524059.
[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.
[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.
[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.
[39]	C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[52]	Q. Ye and Z. Li, "Introduction to Reaction-Diffusion Equations," Foundations of Modern Mathematics Series, Science Press, Beijing, 1990.
[53]	B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.

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