# American Institute of Mathematical Sciences

June  2022, 27(6): 3515-3532. doi: 10.3934/dcdsb.2021194

## Invasive speed for a competition-diffusion system with three species

 1 School of Mathematics and Physics, University of South China, Hengyang 421001, China 2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, A1C 5S7, Canada

* Corresponding author: Chunhua Ou

Received  December 2020 Revised  May 2021 Published  June 2022 Early access  July 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China grant (11626129 and 11801263), the Natural Science Foundation of Hunan Province grant (2018JJ3418); The second author is supported by the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 20B512); The third author is supported by the Canada NSERC discovery grant (RGPIN-2016-04709)

Competition stems from the fact that resources are limited. When multiple competitive species are involved with spatial diffusion, the dynamics becomes even complex and challenging. In this paper, we investigate the invasive speed to a diffusive three species competition system of Lotka-Volterra type. We first show that multiple species share a common spreading speed when initial data are compactly supported. By transforming the competitive system into a cooperative system, the determinacy of the invasive speed is studied by the upper-lower solution method. In our work, for linearly predicting the invasive speed, we concentrate on finding upper solutions only, and don't care about the existence of lower solutions. Similarly, for nonlinear selection of the spreading speed, we focus only on the construction of lower solutions with fast decay rate. This greatly develops and simplifies the ideas of past references in this topic.

Citation: Chaohong Pan, Hongyong Wang, Chunhua Ou. Invasive speed for a competition-diffusion system with three species. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3515-3532. doi: 10.3934/dcdsb.2021194
##### References:
 [1] A. Alhasanat and C. Ou, On a conjecture raised by Yuzo Hosono, J. Dyn. Diff. Equat., 31 (2019), 287-304.  doi: 10.1007/s10884-018-9651-5. [2] A. Alhasanat and C. Ou, Minimal-speed selection of traveling waves to the Lotka-Volterra competition model, J. Differential Equations, 266 (2019), 7357-7378.  doi: 10.1016/j.jde.2018.12.003. [3] A. Alhasanat and C. Ou, On the conjecture for the pushed wavefront to the diffusive Lotka-Volterra competition model, J. Math. Biol., 80 (2020), 1413-1422.  doi: 10.1007/s00285-020-01467-0. [4] H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5. [5] C.-H. Chang, C.-H. Hsu and T.-S. Yang, Traveling wavefronts for a Lotka-Volterra competition model with partially nonlocal interactions, Z. Angew. Math. Phys., 71 (2020), Paper No. 70, 18 pp. doi: 10.1007/s00033-020-1289-6. [6] C.-C. Chen, L.-C. Hung, M. Mimura, M. Tohma and D. Ueyama, Semi-exact equilibrium solutions for three-species competition-diffusion systems, Hiroshima Math. J., 43 (2013), 176-206.  doi: 10.32917/hmj/1372180511. [7] C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653-2669.  doi: 10.3934/dcdsb.2012.17.2653. [8] J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939. [9] J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc., 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556. [10] J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373. [11] Y. Hosono, The minimal speed of traveling fronts for diffusive Lotka-Volterra competition model, Bull. Math. Biol., 60 (1998), 435-448.  doi: 10.1006/bulm.1997.0008. [12] X. Hou and Y. Li, Traveling waves in a three species competition-cooperation system, Commun. Pur. Appl. Anal., 16 (2017), 1103-1120.  doi: 10.3934/cpaa.2017053. [13] W. Huang, Problem on minimum wave speed for Lotka-Volterra reaction-diffusion competition model, J. Dym. Diff. Equat., 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0. [14] W. Huang and M. Han, Non-linear determinacy of minimum wave speed for Lotka-Volterra competition model, J. Differential Equations, 251 (2011), 1549-1561.  doi: 10.1016/j.jde.2011.05.012. [15] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I. [16] Y. Kan-on and M. Mimura, Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math. Anal., 29 (1998), 1519-1536.  doi: 10.1137/S0036141097318328. [17] M. A. Lewis, B. 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. [18] 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. [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] M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition-diffusion system, Ecol. Complex., 21 (2015), 215-232.  doi: 10.1016/j.ecocom.2014.05.004. [21] A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in britain, P. Roy. Soc. Lond. B, Biol. Sci., 238 (1989), 113-125.  doi: 10.1098/rspb.1989.0070.

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##### References:
 [1] A. Alhasanat and C. Ou, On a conjecture raised by Yuzo Hosono, J. Dyn. Diff. Equat., 31 (2019), 287-304.  doi: 10.1007/s10884-018-9651-5. [2] A. Alhasanat and C. Ou, Minimal-speed selection of traveling waves to the Lotka-Volterra competition model, J. Differential Equations, 266 (2019), 7357-7378.  doi: 10.1016/j.jde.2018.12.003. [3] A. Alhasanat and C. Ou, On the conjecture for the pushed wavefront to the diffusive Lotka-Volterra competition model, J. Math. Biol., 80 (2020), 1413-1422.  doi: 10.1007/s00285-020-01467-0. [4] H. Berestycki, O. Diekmann, C. J. Nagelkerke and P. A. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2009), 399-429.  doi: 10.1007/s11538-008-9367-5. [5] C.-H. Chang, C.-H. Hsu and T.-S. Yang, Traveling wavefronts for a Lotka-Volterra competition model with partially nonlocal interactions, Z. Angew. Math. Phys., 71 (2020), Paper No. 70, 18 pp. doi: 10.1007/s00033-020-1289-6. [6] C.-C. Chen, L.-C. Hung, M. Mimura, M. Tohma and D. Ueyama, Semi-exact equilibrium solutions for three-species competition-diffusion systems, Hiroshima Math. J., 43 (2013), 176-206.  doi: 10.32917/hmj/1372180511. [7] C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653-2669.  doi: 10.3934/dcdsb.2012.17.2653. [8] J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939. [9] J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc., 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556. [10] J.-S. Guo, Y. Wang, C.-H. Wu and C.-C. Wu, The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwanese J. Math., 19 (2015), 1805-1829.  doi: 10.11650/tjm.19.2015.5373. [11] Y. Hosono, The minimal speed of traveling fronts for diffusive Lotka-Volterra competition model, Bull. Math. Biol., 60 (1998), 435-448.  doi: 10.1006/bulm.1997.0008. [12] X. Hou and Y. Li, Traveling waves in a three species competition-cooperation system, Commun. Pur. Appl. Anal., 16 (2017), 1103-1120.  doi: 10.3934/cpaa.2017053. [13] W. Huang, Problem on minimum wave speed for Lotka-Volterra reaction-diffusion competition model, J. Dym. Diff. Equat., 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0. [14] W. Huang and M. Han, Non-linear determinacy of minimum wave speed for Lotka-Volterra competition model, J. Differential Equations, 251 (2011), 1549-1561.  doi: 10.1016/j.jde.2011.05.012. [15] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I. [16] Y. Kan-on and M. Mimura, Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math. Anal., 29 (1998), 1519-1536.  doi: 10.1137/S0036141097318328. [17] M. A. Lewis, B. 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. [18] 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. [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] M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition-diffusion system, Ecol. Complex., 21 (2015), 215-232.  doi: 10.1016/j.ecocom.2014.05.004. [21] A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, On the spatial spread of the grey squirrel in britain, P. Roy. Soc. Lond. B, Biol. Sci., 238 (1989), 113-125.  doi: 10.1098/rspb.1989.0070.
The solution $u(x, t)$ at $t = 16, 36, 56$ for two sets of parameters
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