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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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[42]

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

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[42]

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

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