September  2019, 39(9): 5223-5262. doi: 10.3934/dcds.2019213

Asymptotic spreading speed for the weak competition system with a free boundary

1. 

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

2. 

School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

* Corresponding author: ydu@turing.une.edu.au

Received  August 2018 Revised  January 2019 Published  May 2019

Fund Project: The work is supported by the NSF of China (11671243, 11771262, 11572180), the Fundamental Research Funds for the Central Universities (GK201701001), and the Australian Research Council.

This paper is concerned with a diffusive Lotka-Volterra type competition system with a free boundary in one space dimension. Such a system may be used to describe the invasion of a new species into the habitat of a native competitor, and its long-time dynamical behavior can be described by a spreading-vanishing dichotomy. The main purpose of this paper is to determine the asymptotic spreading speed of the invading species when its spreading is successful, which involves two systems of traveling wave type equations.

Citation: Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213
References:
[1]

G. BuntingY. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.

[2]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[3]

W. Ding, Y. Du and X. Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. Henri Poincare Anal. Non Lineaire, in press (https://doi.org/10.1016/j.anihpc.2019.01.005).

[4]

Y. DuZ. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.

[5]

Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. Henri Poincare Anal. Non Lineaire, 32 (2015), 279-305.  doi: 10.1016/j.anihpc.2013.11.004.

[6]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  doi: 10.1137/110822608.

[7]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Cont. Dyn. Syst. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.

[8]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. European Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.

[9]

Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124.  doi: 10.1017/S0024610701002289.

[10]

Y. DuH. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063.

[11]

Y. DuH. Matsuzawa and M. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103 (2015), 741-787.  doi: 10.1016/j.matpur.2014.07.008.

[12]

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.

[13]

Y. Du and C.-H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), Art. 52, 36 pp. doi: 10.1007/s00526-018-1339-5.

[14]

B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318.  doi: 10.1007/BF00275063.

[15]

J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.

[16]

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.

[17]

Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.

[18]

B. LiH. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.

[19]

F. LiX. Liang and W. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: Ⅱ. Spreading speeds and semi-wave, J. Differential Equations, 261 (2016), 2403-2445.  doi: 10.1016/j.jde.2016.04.035.

[20]

W.-T. LiG. 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.

[21]

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

[22]

P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Inst. Math. Polish Acad. Sci. Zam., 190 (1979), 11-79. 

[23]

D. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.

[24]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.

[25]

Z. WangH. Nie and J. Wu, Existence and uniqueness of traveling waves for a reaction-diffusion model with general response functions, J. Math. Anal. Appl., 450 (2017), 406-426.  doi: 10.1016/j.jmaa.2017.01.017.

[26]

Z. WangH. Nie and J. Wu, Spatial propagation for a parabolic system with multiple species competing for single resource, Discrete Cont. Dyn. Syst. B, 24 (2019), 1785-1814.  doi: 10.3934/dcdsb.2018237.

[27]

C.-H. Wu, Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2441-2455.  doi: 10.3934/dcdsb.2013.18.2441.

[28]

C.-H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.

[29]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 20 (2008), 531-533.  doi: 10.1007/s10884-007-9090-1.

[30]

Y. Zhao and M. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.

show all references

References:
[1]

G. BuntingY. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.

[2]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[3]

W. Ding, Y. Du and X. Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. Henri Poincare Anal. Non Lineaire, in press (https://doi.org/10.1016/j.anihpc.2019.01.005).

[4]

Y. DuZ. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.

[5]

Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. Henri Poincare Anal. Non Lineaire, 32 (2015), 279-305.  doi: 10.1016/j.anihpc.2013.11.004.

[6]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  doi: 10.1137/110822608.

[7]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Cont. Dyn. Syst. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.

[8]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. European Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.

[9]

Y. Du and L. Ma, Logistic type equations on $\mathbb{R}^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124.  doi: 10.1017/S0024610701002289.

[10]

Y. DuH. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063.

[11]

Y. DuH. Matsuzawa and M. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103 (2015), 741-787.  doi: 10.1016/j.matpur.2014.07.008.

[12]

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.

[13]

Y. Du and C.-H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), Art. 52, 36 pp. doi: 10.1007/s00526-018-1339-5.

[14]

B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318.  doi: 10.1007/BF00275063.

[15]

J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.

[16]

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.

[17]

Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differential Equations, 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.

[18]

B. LiH. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.

[19]

F. LiX. Liang and W. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: Ⅱ. Spreading speeds and semi-wave, J. Differential Equations, 261 (2016), 2403-2445.  doi: 10.1016/j.jde.2016.04.035.

[20]

W.-T. LiG. 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.

[21]

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

[22]

P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Inst. Math. Polish Acad. Sci. Zam., 190 (1979), 11-79. 

[23]

D. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.

[24]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.

[25]

Z. WangH. Nie and J. Wu, Existence and uniqueness of traveling waves for a reaction-diffusion model with general response functions, J. Math. Anal. Appl., 450 (2017), 406-426.  doi: 10.1016/j.jmaa.2017.01.017.

[26]

Z. WangH. Nie and J. Wu, Spatial propagation for a parabolic system with multiple species competing for single resource, Discrete Cont. Dyn. Syst. B, 24 (2019), 1785-1814.  doi: 10.3934/dcdsb.2018237.

[27]

C.-H. Wu, Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2441-2455.  doi: 10.3934/dcdsb.2013.18.2441.

[28]

C.-H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.

[29]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 20 (2008), 531-533.  doi: 10.1007/s10884-007-9090-1.

[30]

Y. Zhao and M. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280.  doi: 10.1093/imamat/hxv035.

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