June  2020, 40(6): 3451-3466. doi: 10.3934/dcds.2020047

The sign of traveling wave speed in bistable dynamics

1. 

Department of Mathematics, Tamkang University, Tamsui, New Taipei City 25137, Taiwan

2. 

Faculty of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192, Japan

3. 

Department of Mathematics, Josai University, Sakado, 350-0295, Japan

4. 

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan

* Corresponding author: Jong-Shenq Guo

Received  March 2019 Revised  July 2019 Published  October 2019

Fund Project: This work was partially supported by the Ministry of Science and Technology of Taiwan under the grants 105-2115-M-032-003-MY3 and MOST 108-2636-M-024-001 and by JSPS KAKENHI Grant Numbers JP15K04996 and JP18K03412

We are concerned with the sign of traveling wave speed in bistable dynamics. This question is related to which species wins the competition in multiple species competition models. It is well-known that the wave speed is unique for traveling wave connecting two stable states. In this paper, we first review some known results on the sign of wave speed in bistable two species competition models. Then we derive rigorously the sign of bistable wave speed for a special three species competition model describing the competition in two different circumstances: (1) two species are weak competitors and one species is a strong competitor; (2) three species are very strong competitors. It is interesting to observe that, under certain conditions on the parameters, two weaker competitors can wipe out the strongest competitor.

Citation: Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047
References:
[1]

E. O. AlzahraniF. A. Davidson and N. Dodds, Travelling waves in near-degenerate bistable competition models, Math. Model. Nat. Phenom., 5 (2010), 13-35.  doi: 10.1051/mmnp/20105502.  Google Scholar

[2]

G. BuntingY. H. 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

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[4]

E. C. M. CrooksE. N. DancerD. HilhorstM. Mimura and H. Ninomiya, Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions, Nonlinear Analysis: Real World Applications, 5 (2004), 645-665.  doi: 10.1016/j.nonrwa.2004.01.004.  Google Scholar

[5]

E. N. DancerD. HilhorstM. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115.  doi: 10.1017/S0956792598003660.  Google Scholar

[6]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffsive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar

[7]

J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc. (JEMS), 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556.  Google Scholar

[8]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8.  Google Scholar

[9]

L. Girardin and G. Nadin, Travelling waves for diffusive and strongly competitive systems: Relative motility and invasion speed, European J. Appl. Math., 26 (2015), 521-534.  doi: 10.1017/S0956792515000170.  Google Scholar

[10]

J.-S. Guo and Y.-C. Lin, The sign of the wave speed for the Lotka-Volterra competition-diffusion system, Comm. Pure Appl. Anal., 12 (2013), 2083-2090.  doi: 10.3934/cpaa.2013.12.2083.  Google Scholar

[11]

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

[12]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533.  doi: 10.1016/j.jde.2010.12.004.  Google Scholar

[13]

J.-S. Guo and C.-C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036.  Google Scholar

[14]

J.-S. GuoK.-I. NakamuraT. Ogiwara and C.-C. Wu, Stability and uniqueness of traveling waves for a discrete bistable 3-species competition system, J. Math. Anal. Appl., 472 (2019), 1534-1550.  doi: 10.1016/j.jmaa.2018.12.007.  Google Scholar

[15]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.  Google Scholar

[16]

Y. Kan-on, Existence of standing waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 117-133.  doi: 10.1007/BF03167302.  Google Scholar

[17]

Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349.  doi: 10.1007/BF03167252.  Google Scholar

[18]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Analysis, 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[19]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.  doi: 10.32917/hmj/1206124686.  Google Scholar

[20]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.  doi: 10.1007/BF00283257.  Google Scholar

[21]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715.  Google Scholar

[22]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994.  Google Scholar

show all references

References:
[1]

E. O. AlzahraniF. A. Davidson and N. Dodds, Travelling waves in near-degenerate bistable competition models, Math. Model. Nat. Phenom., 5 (2010), 13-35.  doi: 10.1051/mmnp/20105502.  Google Scholar

[2]

G. BuntingY. H. 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

[3]

C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitve reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018.  Google Scholar

[4]

E. C. M. CrooksE. N. DancerD. HilhorstM. Mimura and H. Ninomiya, Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions, Nonlinear Analysis: Real World Applications, 5 (2004), 645-665.  doi: 10.1016/j.nonrwa.2004.01.004.  Google Scholar

[5]

E. N. DancerD. HilhorstM. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system, European J. Appl. Math., 10 (1999), 97-115.  doi: 10.1017/S0956792598003660.  Google Scholar

[6]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffsive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089.  Google Scholar

[7]

J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc. (JEMS), 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556.  Google Scholar

[8]

R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8.  Google Scholar

[9]

L. Girardin and G. Nadin, Travelling waves for diffusive and strongly competitive systems: Relative motility and invasion speed, European J. Appl. Math., 26 (2015), 521-534.  doi: 10.1017/S0956792515000170.  Google Scholar

[10]

J.-S. Guo and Y.-C. Lin, The sign of the wave speed for the Lotka-Volterra competition-diffusion system, Comm. Pure Appl. Anal., 12 (2013), 2083-2090.  doi: 10.3934/cpaa.2013.12.2083.  Google Scholar

[11]

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

[12]

J.-S. Guo and C.-H. Wu, Wave propagation for a two-component lattice dynamical system arising in strong competition models, J. Differential Equations, 250 (2011), 3504-3533.  doi: 10.1016/j.jde.2010.12.004.  Google Scholar

[13]

J.-S. Guo and C.-C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036.  Google Scholar

[14]

J.-S. GuoK.-I. NakamuraT. Ogiwara and C.-C. Wu, Stability and uniqueness of traveling waves for a discrete bistable 3-species competition system, J. Math. Anal. Appl., 472 (2019), 1534-1550.  doi: 10.1016/j.jmaa.2018.12.007.  Google Scholar

[15]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.  Google Scholar

[16]

Y. Kan-on, Existence of standing waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 117-133.  doi: 10.1007/BF03167302.  Google Scholar

[17]

Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349.  doi: 10.1007/BF03167252.  Google Scholar

[18]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Analysis, 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[19]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270.  doi: 10.32917/hmj/1206124686.  Google Scholar

[20]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73 (1980), 69-77.  doi: 10.1007/BF00283257.  Google Scholar

[21]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715.  Google Scholar

[22]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994.  Google Scholar

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