March  2011, 31(1): 1-23. doi: 10.3934/dcds.2011.31.1

Monostable wavefronts in cooperative Lotka-Volterra systems with nonlocal delays

1. 

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

2. 

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

3. 

Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250

Received  January 2010 Revised  April 2011 Published  June 2011

This paper is concerned with traveling wavefronts in a Lotka-Volterra model with nonlocal delays for two cooperative species. By using comparison principle, some existence and nonexistence results are obtained. If the wave speed is larger than a threshold which can be formulated in terms of basic parameters, we prove the asymptotic stability of traveling wavefronts by the spectral analysis method together with squeezing technique.
Citation: Guo Lin, Wan-Tong Li, Shigui Ruan. Monostable wavefronts in cooperative Lotka-Volterra systems with nonlocal delays. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 1-23. doi: 10.3934/dcds.2011.31.1
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (ed. J. A. Goldstein), Lecture Notes in Mathematics, 446, Springer, Berlin, (1975), 5-49.

[2]

F. van den Bosch, J. A. J. Metz and O. Diekmann, The velocity of spatial population expansion, J. Math. Biol., 28 (1990), 529-565. doi: 10.1007/BF00164162.

[3]

N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.

[4]

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

[5]

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.

[6]

T. Faria, W. Huang and J. Wu, Traveling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261. doi: 10.1098/rspa.2005.1554.

[7]

K. Gopalsamy, Pursuit-evasion wave trains in prey-predator systems with diffusionally coupled delays, Bull. Math. Biol., 42 (1980), 871-887.

[8]

S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822. doi: 10.1137/S003614100139991.

[9]

S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in "Nonlinear Dynamics and Evolution Equations" (eds. H. Brunner, X. Q. Zhao and X. Zou), Fields Inst. Commun., 48, AMS, Providence, RI, (2006), 137-200.

[10]

J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466. doi: 10.1016/S0022-247X(02)00135-X.

[11]

W. Huang, Uniqueness of the bistable traveling wave for mutualist species, J. Dynam. Diff. Eqns., 13 (2001), 147-183. doi: 10.1023/A:1009048616476.

[12]

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.

[13]

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.

[14]

W.-T. Li and Z. Wang, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays, Z. Angew. Math. Phys., 58 (2007), 571-591. doi: 10.1007/s00033-006-5125-4.

[15]

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.

[16]

G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513. doi: 10.1016/j.jde.2007.10.019.

[17]

G. Lin, W.-T. Li and M. Ma, Traveling wave solutions in delayed reaction-diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393.

[18]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.

[19]

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.

[20]

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. doi: 10.1016/j.jde.2004.07.014.

[21]

S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. doi: 10.1016/j.jde.2005.05.004.

[22]

M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026.

[23]

M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020.

[24]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

[25]

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

[26]

C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 235 (2007), 219-261. doi: 10.1016/j.jde.2006.12.010.

[27]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

[28]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine," Biol. Med. Phys. Biomed. Eng., Springer, Berlin, (2007), 97-122.

[29]

S. Ruan and J. Wu, Reaction-diffusion equations with infinite delay, Canad. Appl. Math. Quart., 2 (1994), 485-550.

[30]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. R. Soc. Edinburgh Sect. A, 134 (2004), 991-1011. doi: 10.1017/S0308210500003590.

[31]

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.

[32]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Fundamental Principles of Mathematical Sciences, 258, Springer-Verlag, New York, 1994.

[33]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[34]

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.

[35]

Z. Wang, W.-T. Li and S. Ruan, Travelling wave fronts of reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.

[36]

Z. 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. doi: 10.1016/j.jde.2007.03.025.

[37]

Z. 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.

[38]

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.

[39]

J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996.

[40]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diff. Eqns., 13 (2001), 651-687; Erratum, 20 (2008), 531-533.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (ed. J. A. Goldstein), Lecture Notes in Mathematics, 446, Springer, Berlin, (1975), 5-49.

[2]

F. van den Bosch, J. A. J. Metz and O. Diekmann, The velocity of spatial population expansion, J. Math. Biol., 28 (1990), 529-565. doi: 10.1007/BF00164162.

[3]

N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.

[4]

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

[5]

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.

[6]

T. Faria, W. Huang and J. Wu, Traveling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261. doi: 10.1098/rspa.2005.1554.

[7]

K. Gopalsamy, Pursuit-evasion wave trains in prey-predator systems with diffusionally coupled delays, Bull. Math. Biol., 42 (1980), 871-887.

[8]

S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822. doi: 10.1137/S003614100139991.

[9]

S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in "Nonlinear Dynamics and Evolution Equations" (eds. H. Brunner, X. Q. Zhao and X. Zou), Fields Inst. Commun., 48, AMS, Providence, RI, (2006), 137-200.

[10]

J. Huang and X. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays, J. Math. Anal. Appl., 271 (2002), 455-466. doi: 10.1016/S0022-247X(02)00135-X.

[11]

W. Huang, Uniqueness of the bistable traveling wave for mutualist species, J. Dynam. Diff. Eqns., 13 (2001), 147-183. doi: 10.1023/A:1009048616476.

[12]

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.

[13]

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.

[14]

W.-T. Li and Z. Wang, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays, Z. Angew. Math. Phys., 58 (2007), 571-591. doi: 10.1007/s00033-006-5125-4.

[15]

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.

[16]

G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513. doi: 10.1016/j.jde.2007.10.019.

[17]

G. Lin, W.-T. Li and M. Ma, Traveling wave solutions in delayed reaction-diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393.

[18]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.

[19]

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.

[20]

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. doi: 10.1016/j.jde.2004.07.014.

[21]

S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87. doi: 10.1016/j.jde.2005.05.004.

[22]

M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026.

[23]

M. Mei, C.-K. Lin, C.-T. Lin and J. W.-H. So, Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529. doi: 10.1016/j.jde.2008.12.020.

[24]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

[25]

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

[26]

C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 235 (2007), 219-261. doi: 10.1016/j.jde.2006.12.010.

[27]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

[28]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine," Biol. Med. Phys. Biomed. Eng., Springer, Berlin, (2007), 97-122.

[29]

S. Ruan and J. Wu, Reaction-diffusion equations with infinite delay, Canad. Appl. Math. Quart., 2 (1994), 485-550.

[30]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. R. Soc. Edinburgh Sect. A, 134 (2004), 991-1011. doi: 10.1017/S0308210500003590.

[31]

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.

[32]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Fundamental Principles of Mathematical Sciences, 258, Springer-Verlag, New York, 1994.

[33]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[34]

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.

[35]

Z. Wang, W.-T. Li and S. Ruan, Travelling wave fronts of reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.

[36]

Z. 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. doi: 10.1016/j.jde.2007.03.025.

[37]

Z. 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.

[38]

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.

[39]

J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996.

[40]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diff. Eqns., 13 (2001), 651-687; Erratum, 20 (2008), 531-533.

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