# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213
##### References:
 [1] G. Bunting, Y. 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. Google Scholar [2] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Google Scholar [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).Google Scholar [4] Y. Du, Z. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] Y. Du, H. 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. Google Scholar [11] Y. Du, H. 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. Google Scholar [12] Y. Du, M. 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 [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. Google Scholar [14] B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318. doi: 10.1007/BF00275063. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] B. Li, H. 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. Google Scholar [19] F. Li, X. 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. Google Scholar [20] 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. Google Scholar [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. Google Scholar [22] P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Inst. Math. Polish Acad. Sci. Zam., 190 (1979), 11-79. Google Scholar [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. Google Scholar [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. Google Scholar [25] Z. Wang, H. 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. Google Scholar [26] Z. Wang, H. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### References:
 [1] G. Bunting, Y. 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. Google Scholar [2] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Google Scholar [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).Google Scholar [4] Y. Du, Z. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] Y. Du, H. 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. Google Scholar [11] Y. Du, H. 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. Google Scholar [12] Y. Du, M. 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 [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. Google Scholar [14] B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318. doi: 10.1007/BF00275063. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] B. Li, H. 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. Google Scholar [19] F. Li, X. 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. Google Scholar [20] 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. Google Scholar [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. Google Scholar [22] P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems, Inst. Math. Polish Acad. Sci. Zam., 190 (1979), 11-79. Google Scholar [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. Google Scholar [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. Google Scholar [25] Z. Wang, H. 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. Google Scholar [26] Z. Wang, H. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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