doi: 10.3934/dcdsb.2021194
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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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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H. BerestyckiO. DiekmannC. 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.  Google Scholar

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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.  Google Scholar

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C.-C. ChenL.-C. HungM. MimuraM. 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.  Google Scholar

[7]

C.-C. ChenL.-C. HungM. 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.  Google Scholar

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[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.  Google Scholar

[10]

J.-S. GuoY. WangC.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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[16]

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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

[18]

B. LiH. 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.  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]

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.  Google Scholar

[21]

A. OkuboP. K. MainiM. 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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. BerestyckiO. DiekmannC. 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.  Google Scholar

[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.  Google Scholar

[6]

C.-C. ChenL.-C. HungM. MimuraM. 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.  Google Scholar

[7]

C.-C. ChenL.-C. HungM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J.-S. GuoY. WangC.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

B. LiH. 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.  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]

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.  Google Scholar

[21]

A. OkuboP. K. MainiM. 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.  Google Scholar

Figure 1.  The solution $ u(x, t) $ at $ t = 16, 36, 56 $ for two sets of parameters
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