November  2012, 17(8): 2713-2724. doi: 10.3934/dcdsb.2012.17.2713

Recent developments on wave propagation in 2-species competition systems

1. 

Department of Mathematics, Tamkang University, Tamsui, New Taipei City 25137, Taiwan, and MIMS, Meiji University, 1-1-1 Higashimita, Tamaku, Kawasaki 214-8571, Japan

2. 

Department of Mathematics, Tamkang University, Tamsui, New Taipei City 25137, Taiwan

Received  January 2011 Revised  September 2011 Published  July 2012

In this paper, we shall survey some recent results on the wave propagation in 2-species competition systems with Lotka-Volterra type nonlinearity. This includes systems with continuous and discrete diffusion (or migration). We are interested in both monostable case and bistable with strong competition case. Questions on minimal speed for the monostable case, uniqueness of wave speed and propagation failure in the bistable case, monotonicity and uniqueness of wave profile for both cases are addressed. Finally, we give some open problems on wave propagation in 2-species competition systems.
Citation: Jong-Shenq Guo, Chang-Hong Wu. Recent developments on wave propagation in 2-species competition systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2713-2724. doi: 10.3934/dcdsb.2012.17.2713
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in, 446 (1975), 5. Google Scholar

[2]

C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves,, Math. Proc. Cambridge Philos. Soc., 80 (1976), 315. Google Scholar

[3]

J. Bell, Some threshold results for models of myelinated nerves,, Math. Biosci., 54 (1981), 181. Google Scholar

[4]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions,, Arch. Rational Mech. Anal., 150 (1999), 281. Google Scholar

[5]

P. W. Bates, X. Chen and A. Chmaj, Travelling waves of bistable dynamics on a lattice,, SIAM J. Math. Anal., 35 (2003), 520. Google Scholar

[6]

J. W. Cahn, J. Mallet-Paret and E. Van Vleck, Traveling wave solutions of ODEs on a two-dimensional spacial lattice,, SIAM J. Appl. Math., 59 (1999), 455. Google Scholar

[7]

A. Carpio and L. L. Bonilla, Depinning transitions in discrete reaction-diffusion equations,, SIAM J. Appl. Math., 63 (2003), 1056. Google Scholar

[8]

A. Carpio, S. J. Chapman, S. P. Hastings and J. B. McLeod, Wave solutions for a discrete reaction-diffusion equation,, Eur. J. Appl. Math., 11 (2000), 399. Google Scholar

[9]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. Google Scholar

[10]

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. Google Scholar

[11]

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

[12]

X. Chen and J.-S. Guo, Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics,, Math. Ann., 326 (2003), 123. Google Scholar

[13]

X. Chen, J.-S. Guo and C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics,, Arch. Ration. Mech. Anal., 189 (2008), 189. Google Scholar

[14]

S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems,, J. Differential Equations, 149 (1998), 248. Google Scholar

[15]

C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model,, Indiana Univ. Math. J., 33 (1984), 319. Google Scholar

[16]

G. Fáth, Propagation failure of traveling waves in a discrete bistable medium,, Physica D, 116 (1998), 176. Google Scholar

[17]

S.-C. Fu, J.-S. Guo and S.-Y. Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations,, Nonl. Anal., 48 (2002), 1137. Google Scholar

[18]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach,, J. Differential Equations, 44 (1982), 343. Google Scholar

[19]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations,, Math. Ann., 335 (2006), 489. Google Scholar

[20]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system,, J. Dynam. Differential Equations, 23 (2011), 353. Google Scholar

[21]

J.-S. Guo and C.-H. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system,, Osaka J. Math., 45 (2008), 327. Google Scholar

[22]

J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice,, Tohoku Math. J. (2), 62 (2010), 17. Google Scholar

[23]

J.-S. Guo and C.-H. Wu, Front propagation for a two-dimensional periodic monostable lattice dynamical system,, Discrete Conti. Dyn. Syst., 26 (2010), 197. Google Scholar

[24]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models,, J. Differential Equations, 250 (2011), 3504. Google Scholar

[25]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models,, J. Differential Equations, 252 (2012), 4357. Google Scholar

[26]

J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system,, J. Differential Equations, 246 (2009), 3818. Google Scholar

[27]

D. Hankerson and B. Zinner, Wavefronts for a cooperative tridiagonal system of differential equations,, J. Dyn. Differential Equations, 5 (1993), 359. Google Scholar

[28]

Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models,, in, 1.2 (1989), 687. Google Scholar

[29]

Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model,, Bulletin of Math. Biology, 60 (1998), 435. Google Scholar

[30]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations,, Discete Contin. Dyn. Syst. Ser. B, 3 (2003), 79. Google Scholar

[31]

M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system,, Japan J. Indust. Appl. Math., 15 (1998), 223. Google Scholar

[32]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations,, SIAM J. Math. Anal., 26 (1995), 340. Google Scholar

[33]

Y. Kan-on, Existence of standing waves for competition-diffusion equations,, Japan J. Indust. Appl. Math., 13 (1996), 117. Google Scholar

[34]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonlinear Anal., 28 (1997), 145. Google Scholar

[35]

Y. Kan-on, Instability of stationary solutions for a Lotka-Volterra competition model with diffusion,, J. Math. Anal. Appl., 208 (1997), 158. Google Scholar

[36]

Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations,, Japan J. Indust. Appl. Math., 13 (1996), 343. Google Scholar

[37]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells,, SIAM J. Appl. Math., 47 (1987), 556. Google Scholar

[38]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de laquantité de matiére et son applicatioŋ á un probléme biologique,, Bull. Univ. Moskov. Ser. Internat., 1 (1937), 1. Google Scholar

[39]

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. Google Scholar

[40]

B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82. Google Scholar

[41]

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. Google Scholar

[42]

S. Ma and X.-Q. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations,, Discrete Contin. Dyn. Syst., 21 (2008), 259. Google Scholar

[43]

R. S. MacKay and J.-A. Sepulchre, Multistability in networks of weakly coupled bistable units,, Physica D, 82 (1995), 243. Google Scholar

[44]

J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems,, J. Dyn. Diff. Eq., 11 (1999), 49. Google Scholar

[45]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217. Google Scholar

[46]

J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989). Google Scholar

[47]

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. doi: 10.1098/rspb.1989.0070. Google Scholar

[48]

S.-F. Shieh, Horseshoes for coupled discrete nonlinear Schrödinger equations,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3077226. 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. Google Scholar

[50]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of the linear determinancy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. Google Scholar

[51]

C.-C. Wu, Existence of traveling wavefront for discrete bistable competition model,, Discrete Contin. Dyn. Syst. Ser. B., 16 (2011), 973. Google Scholar

[52]

J. Wu and X. Zou, Asymtotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations,, J. Differential Equations, 135 (1997), 315. Google Scholar

[53]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equations,, SIAM J. Math. Anal., 22 (1991), 1016. Google Scholar

[54]

B. Zinner, Existence of traveling wavefronts for the discrete Nagumo equations,, J. Differential Equations, 96 (1992), 1. Google Scholar

[55]

B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discreteFisher's equation,, J. Differential Equations, 105 (1993), 46. Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in, 446 (1975), 5. Google Scholar

[2]

C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves,, Math. Proc. Cambridge Philos. Soc., 80 (1976), 315. Google Scholar

[3]

J. Bell, Some threshold results for models of myelinated nerves,, Math. Biosci., 54 (1981), 181. Google Scholar

[4]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions,, Arch. Rational Mech. Anal., 150 (1999), 281. Google Scholar

[5]

P. W. Bates, X. Chen and A. Chmaj, Travelling waves of bistable dynamics on a lattice,, SIAM J. Math. Anal., 35 (2003), 520. Google Scholar

[6]

J. W. Cahn, J. Mallet-Paret and E. Van Vleck, Traveling wave solutions of ODEs on a two-dimensional spacial lattice,, SIAM J. Appl. Math., 59 (1999), 455. Google Scholar

[7]

A. Carpio and L. L. Bonilla, Depinning transitions in discrete reaction-diffusion equations,, SIAM J. Appl. Math., 63 (2003), 1056. Google Scholar

[8]

A. Carpio, S. J. Chapman, S. P. Hastings and J. B. McLeod, Wave solutions for a discrete reaction-diffusion equation,, Eur. J. Appl. Math., 11 (2000), 399. Google Scholar

[9]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433. Google Scholar

[10]

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. Google Scholar

[11]

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

[12]

X. Chen and J.-S. Guo, Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics,, Math. Ann., 326 (2003), 123. Google Scholar

[13]

X. Chen, J.-S. Guo and C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics,, Arch. Ration. Mech. Anal., 189 (2008), 189. Google Scholar

[14]

S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems,, J. Differential Equations, 149 (1998), 248. Google Scholar

[15]

C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model,, Indiana Univ. Math. J., 33 (1984), 319. Google Scholar

[16]

G. Fáth, Propagation failure of traveling waves in a discrete bistable medium,, Physica D, 116 (1998), 176. Google Scholar

[17]

S.-C. Fu, J.-S. Guo and S.-Y. Shieh, Traveling wave solutions for some discrete quasilinear parabolic equations,, Nonl. Anal., 48 (2002), 1137. Google Scholar

[18]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach,, J. Differential Equations, 44 (1982), 343. Google Scholar

[19]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations,, Math. Ann., 335 (2006), 489. Google Scholar

[20]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system,, J. Dynam. Differential Equations, 23 (2011), 353. Google Scholar

[21]

J.-S. Guo and C.-H. Wu, Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system,, Osaka J. Math., 45 (2008), 327. Google Scholar

[22]

J.-S. Guo and C.-H. Wu, Entire solutions for a two-component competition system in a lattice,, Tohoku Math. J. (2), 62 (2010), 17. Google Scholar

[23]

J.-S. Guo and C.-H. Wu, Front propagation for a two-dimensional periodic monostable lattice dynamical system,, Discrete Conti. Dyn. Syst., 26 (2010), 197. Google Scholar

[24]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models,, J. Differential Equations, 250 (2011), 3504. Google Scholar

[25]

J.-S. Guo and C.-H. Wu, Traveling wave front for a two-component lattice dynamical system arising in competition models,, J. Differential Equations, 252 (2012), 4357. Google Scholar

[26]

J.-S. Guo and C.-C. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system,, J. Differential Equations, 246 (2009), 3818. Google Scholar

[27]

D. Hankerson and B. Zinner, Wavefronts for a cooperative tridiagonal system of differential equations,, J. Dyn. Differential Equations, 5 (1993), 359. Google Scholar

[28]

Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competitive models,, in, 1.2 (1989), 687. Google Scholar

[29]

Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model,, Bulletin of Math. Biology, 60 (1998), 435. Google Scholar

[30]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations,, Discete Contin. Dyn. Syst. Ser. B, 3 (2003), 79. Google Scholar

[31]

M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system,, Japan J. Indust. Appl. Math., 15 (1998), 223. Google Scholar

[32]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations,, SIAM J. Math. Anal., 26 (1995), 340. Google Scholar

[33]

Y. Kan-on, Existence of standing waves for competition-diffusion equations,, Japan J. Indust. Appl. Math., 13 (1996), 117. Google Scholar

[34]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonlinear Anal., 28 (1997), 145. Google Scholar

[35]

Y. Kan-on, Instability of stationary solutions for a Lotka-Volterra competition model with diffusion,, J. Math. Anal. Appl., 208 (1997), 158. Google Scholar

[36]

Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations,, Japan J. Indust. Appl. Math., 13 (1996), 343. Google Scholar

[37]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells,, SIAM J. Appl. Math., 47 (1987), 556. Google Scholar

[38]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de laquantité de matiére et son applicatioŋ á un probléme biologique,, Bull. Univ. Moskov. Ser. Internat., 1 (1937), 1. Google Scholar

[39]

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. Google Scholar

[40]

B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82. Google Scholar

[41]

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. Google Scholar

[42]

S. Ma and X.-Q. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations,, Discrete Contin. Dyn. Syst., 21 (2008), 259. Google Scholar

[43]

R. S. MacKay and J.-A. Sepulchre, Multistability in networks of weakly coupled bistable units,, Physica D, 82 (1995), 243. Google Scholar

[44]

J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems,, J. Dyn. Diff. Eq., 11 (1999), 49. Google Scholar

[45]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217. Google Scholar

[46]

J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989). Google Scholar

[47]

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. doi: 10.1098/rspb.1989.0070. Google Scholar

[48]

S.-F. Shieh, Horseshoes for coupled discrete nonlinear Schrödinger equations,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3077226. 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. Google Scholar

[50]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of the linear determinancy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. Google Scholar

[51]

C.-C. Wu, Existence of traveling wavefront for discrete bistable competition model,, Discrete Contin. Dyn. Syst. Ser. B., 16 (2011), 973. Google Scholar

[52]

J. Wu and X. Zou, Asymtotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations,, J. Differential Equations, 135 (1997), 315. Google Scholar

[53]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equations,, SIAM J. Math. Anal., 22 (1991), 1016. Google Scholar

[54]

B. Zinner, Existence of traveling wavefronts for the discrete Nagumo equations,, J. Differential Equations, 96 (1992), 1. Google Scholar

[55]

B. Zinner, G. Harris and W. Hudson, Traveling wavefronts for the discreteFisher's equation,, J. Differential Equations, 105 (1993), 46. Google Scholar

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