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

[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

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

[1]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[2]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

[3]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[4]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030

[5]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[6]

Habib Ammari, Josselin Garnier, Vincent Jugnon. Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 389-417. doi: 10.3934/dcdss.2015.8.389

[7]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

[8]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[9]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[10]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

[11]

Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161

[12]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101

[13]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427

[14]

Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014

[15]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53

[16]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[17]

Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209

[18]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[19]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[20]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (339)
  • HTML views (293)
  • Cited by (0)

[Back to Top]