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.

[2]

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

[3]

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

[4]

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

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[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.

[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.

[40]

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

[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.

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[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.

[48]

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

[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.

[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.

[51]

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

[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.

[53]

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

[54]

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

[55]

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

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.

[2]

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

[3]

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

[4]

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

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[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.

[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.

[40]

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

[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.

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[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.

[48]

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

[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.

[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.

[51]

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

[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.

[53]

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

[54]

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

[55]

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

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