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November  2018, 23(9): 3901-3914. doi: 10.3934/dcdsb.2018116

Asymptotic spreading of time periodic competition diffusion systems

Key Laboratory of Applied Mathematics and Complex Systems, School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author: Guo Lin

Received  April 2017 Revised  November 2017 Published  April 2018

This paper deals with the asymptotic spreading of time periodic Lotka-Volterra competition diffusion systems, which formulates the coinvasion-coexistence process. By combining auxiliary systems with comparison principle, some results on asymptotic spreading are established. Our conclusions indicate that the coinvasions of two competitors may be successful, and the interspecific competitions slow the invasion speed of one species.

Citation: Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Mathematics, 446. Springer, Berlin, 446 (1975), 5-49.  Google Scholar

[2]

X. Bao and Z. C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.  doi: 10.1016/j.jde.2013.06.024.  Google Scholar

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L. J. DuW. T. Li and J. B. Wang, Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion, Math. Biosci. Eng., 14 (2017), 1187-1213.  doi: 10.3934/mbe.2017061.  Google Scholar

[4]

Y. DuM. Wang and M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.  doi: 10.1016/j.matpur.2016.06.005.  Google Scholar

[5]

J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226.  doi: 10.1016/j.jde.2010.01.009.  Google Scholar

[6]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin, 1979.  Google Scholar

[7]

P. C. Fife and M. Tang, Comparison principles for reaction-diffusion systems: Irregular comparison functions and applications to questions of stability and speed of propagation of disturbances, J. Differential Equations, 40 (1981), 168-185.  doi: 10.1016/0022-0396(81)90016-4.  Google Scholar

[8]

N. S. GoelS. C. Maitra and E. W. Montrol, On the Volterra and other nonlinear models of interacting populations, Revs. Mod. Phys., 43 (1971), 231-276.  doi: 10.1103/RevModPhys.43.231.  Google Scholar

[9]

J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.  Google Scholar

[10]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, in: Pitman Research Notes in Mathematics, Vol. 247, Longman Sci. Tech, Harlow, 1991.  Google Scholar

[11]

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

[12]

M. A. LewisB. 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

[13]

X. LiangY. Yi and X. Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.  Google Scholar

[14]

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

[15]

G. Lin, Spreading speeds of a Lotka-Volterra predator-prey system: the role of the predator, Nonlinear Anal., 74 (2011), 2448-2461.  doi: 10.1016/j.na.2010.11.046.  Google Scholar

[16]

G. Lin, W. J. Bo and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, submitted. Google Scholar

[17]

G. Lin and W. T. Li, Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689.  doi: 10.1017/S0956792512000198.  Google Scholar

[18]

G. LinW. 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

[19]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[20]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1973. Google Scholar

[21]

J. D. Murray, Mathematical Biology. Ⅰ. An Introduction, 3nd edition, 18, Springer-Verlag, New York, 2002.  Google Scholar

[22]

J. D. Murray, Mathematical Biology, Ⅱ. Spatial Models and Biomedical Applications, 3nd edition, 18, Springer-Verlag, New York, 2003.  Google Scholar

[23]

S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236.  doi: 10.1016/j.jmaa.2013.05.031.  Google Scholar

[24]

S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar

[25]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, UK, 1997, xiii+205 pp. Google Scholar

[26]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[27]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783.  doi: 10.1007/s00332-011-9099-9.  Google Scholar

[28]

M. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67(2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

H. F. WeinbergerM. 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

[33]

H. F. WeinbergerM. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222.  doi: 10.1007/s00285-007-0078-6.  Google Scholar

[34]

Y. Xia and M. Han, New conditions on the existence and stability of periodic solution in Lotka-Volterra's population system, SIAM J. Appl. Math., 69 (2009), 1580-1597.  doi: 10.1137/070702485.  Google Scholar

[35]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[36]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equations, Science Press, Beijing, 1990.  Google Scholar

[37]

T. YiY. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), 3538-3572.  doi: 10.1016/j.jde.2013.01.031.  Google Scholar

[38]

T. Yi and X. Zou, Asymptotic behavior, spreading speeds, and traveling waves of nonmonotone dynamical systems, SIAM J. Math. Anal., 47 (2015), 3005-3034.  doi: 10.1137/14095412X.  Google Scholar

[39]

X. Yu and X. Q. Zhao, A periodic reaction-advection-diffusion model for a stream population, J. Differential Equations, 258 (2015), 3037-3062.  doi: 10.1016/j.jde.2015.01.001.  Google Scholar

[40]

G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005.  Google Scholar

[41]

G. Zhao and S. Ruan, Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations, 257 (2014), 1078-1147.  doi: 10.1016/j.jde.2014.05.001.  Google Scholar

[42]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, NewYork, 2003.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Mathematics, 446. Springer, Berlin, 446 (1975), 5-49.  Google Scholar

[2]

X. Bao and Z. C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.  doi: 10.1016/j.jde.2013.06.024.  Google Scholar

[3]

L. J. DuW. T. Li and J. B. Wang, Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion, Math. Biosci. Eng., 14 (2017), 1187-1213.  doi: 10.3934/mbe.2017061.  Google Scholar

[4]

Y. DuM. Wang and M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.  doi: 10.1016/j.matpur.2016.06.005.  Google Scholar

[5]

J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226.  doi: 10.1016/j.jde.2010.01.009.  Google Scholar

[6]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin, 1979.  Google Scholar

[7]

P. C. Fife and M. Tang, Comparison principles for reaction-diffusion systems: Irregular comparison functions and applications to questions of stability and speed of propagation of disturbances, J. Differential Equations, 40 (1981), 168-185.  doi: 10.1016/0022-0396(81)90016-4.  Google Scholar

[8]

N. S. GoelS. C. Maitra and E. W. Montrol, On the Volterra and other nonlinear models of interacting populations, Revs. Mod. Phys., 43 (1971), 231-276.  doi: 10.1103/RevModPhys.43.231.  Google Scholar

[9]

J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.  Google Scholar

[10]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, in: Pitman Research Notes in Mathematics, Vol. 247, Longman Sci. Tech, Harlow, 1991.  Google Scholar

[11]

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

[12]

M. A. LewisB. 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

[13]

X. LiangY. Yi and X. Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.  Google Scholar

[14]

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

[15]

G. Lin, Spreading speeds of a Lotka-Volterra predator-prey system: the role of the predator, Nonlinear Anal., 74 (2011), 2448-2461.  doi: 10.1016/j.na.2010.11.046.  Google Scholar

[16]

G. Lin, W. J. Bo and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, submitted. Google Scholar

[17]

G. Lin and W. T. Li, Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689.  doi: 10.1017/S0956792512000198.  Google Scholar

[18]

G. LinW. 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

[19]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[20]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1973. Google Scholar

[21]

J. D. Murray, Mathematical Biology. Ⅰ. An Introduction, 3nd edition, 18, Springer-Verlag, New York, 2002.  Google Scholar

[22]

J. D. Murray, Mathematical Biology, Ⅱ. Spatial Models and Biomedical Applications, 3nd edition, 18, Springer-Verlag, New York, 2003.  Google Scholar

[23]

S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236.  doi: 10.1016/j.jmaa.2013.05.031.  Google Scholar

[24]

S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar

[25]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, UK, 1997, xiii+205 pp. Google Scholar

[26]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[27]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783.  doi: 10.1007/s00332-011-9099-9.  Google Scholar

[28]

M. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys., 67(2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

H. F. WeinbergerM. 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

[33]

H. F. WeinbergerM. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222.  doi: 10.1007/s00285-007-0078-6.  Google Scholar

[34]

Y. Xia and M. Han, New conditions on the existence and stability of periodic solution in Lotka-Volterra's population system, SIAM J. Appl. Math., 69 (2009), 1580-1597.  doi: 10.1137/070702485.  Google Scholar

[35]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296.  Google Scholar

[36]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equations, Science Press, Beijing, 1990.  Google Scholar

[37]

T. YiY. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), 3538-3572.  doi: 10.1016/j.jde.2013.01.031.  Google Scholar

[38]

T. Yi and X. Zou, Asymptotic behavior, spreading speeds, and traveling waves of nonmonotone dynamical systems, SIAM J. Math. Anal., 47 (2015), 3005-3034.  doi: 10.1137/14095412X.  Google Scholar

[39]

X. Yu and X. Q. Zhao, A periodic reaction-advection-diffusion model for a stream population, J. Differential Equations, 258 (2015), 3037-3062.  doi: 10.1016/j.jde.2015.01.001.  Google Scholar

[40]

G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005.  Google Scholar

[41]

G. Zhao and S. Ruan, Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations, 257 (2014), 1078-1147.  doi: 10.1016/j.jde.2014.05.001.  Google Scholar

[42]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, NewYork, 2003.  Google Scholar

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