June  2017, 22(4): 1341-1360. doi: 10.3934/dcdsb.2017065

Competition in periodic media:Ⅰ-Existence of pulsating fronts

Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France

Received  May 2016 Revised  December 2016 Published  February 2017

This paper is concerned with the existence of pulsating front solutions in space-periodic media for a bistable two-species competition-diffusion Lotka-Volterra system. Considering highly competitive systems, a simple-high frequency or small amplitudes" algebraic sufficient condition for the existence of pulsating fronts is stated. This condition is in fact sufficient to guarantee that all periodic coexistence states vanish and become unstable as the competition becomes large enough.

Citation: Léo Girardin. Competition in periodic media:Ⅰ-Existence of pulsating fronts. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1341-1360. doi: 10.3934/dcdsb.2017065
References:
[1]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.

[2]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. i. species persistence, J. Math. Biol., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.

[3]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. ii. biological invasions and pulsating travelling fronts, J. Math. Pures Appl. (9), 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.

[4]

H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math., 68 (2015), 1014-1065.  doi: 10.1002/cpa.21536.

[5]

M. ContiS. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560.  doi: 10.1016/j.aim.2004.08.006.

[6]

E. C. M. CrooksE. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36. 

[7]

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

[8]

E. N. Dancer and Y. H. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations, 114 (1994), 434-475.  doi: 10.1006/jdeq.1994.1156.

[9]

E. N. Dancer and Z. M. Guo, Some remarks on the stability of sign changing solutions, Tohoku Math. J. (2), 47 (1995), 199-225.  doi: 10.2748/tmj/1178225592.

[10]

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.

[11]

E. N. DancerK. Wang and Z. Zhang, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005.  doi: 10.1090/S0002-9947-2011-05488-7.

[12]

E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, J. Differential Equations, 182 (2002), 470-489.  doi: 10.1006/jdeq.2001.4102.

[13]

D. G. de Figueiredo and E. Mitidieri, Maximum principles for linear elliptic systems, Rend. Istit. Mat. Univ. Trieste, 22 (1990), 36-66. 

[14]

W. Ding, F. Hamel and X. -Q. Zhao, Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat, ArXiv e-prints, arXiv: 1408.0723 [math. AP].

[15]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.

[16]

J. Fang, X. Yu and X. -Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, ArXiv e-prints, arXiv: 1504.03788 [math. AP].

[17]

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.

[18]

J. E. Furter and J. López-Gómez, On the existence and uniqueness of coexistence states for the lotka-volterra competition model with diffusion and spatially dependent coefficients, Nonlinear Anal., 25 (1995), 363-398.  doi: 10.1016/0362-546X(94)00139-9.

[19]

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.

[20]

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.

[21]

L. Girardin and G. Nadin, Competition in periodic media: Ⅱ-Segregative limit of pulsating fronts and "Unity is not Strength"-type result, ArXiv e-prints, arXiv: 1611.03237 [math. AP].

[22]

J.-S. Guo and C.-H. Wu, Recent developments on wave propagation in 2-species competition systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2713-2724.  doi: 10.3934/dcdsb.2012.17.2713.

[23]

V. HutsonY. Lou and K. Mischaikow, Spatial heterogeneity of resources versus lotka-volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.

[24]

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.

[25]

X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces, Trans. Amer. Math. Soc., 278 (1983), 21-55.  doi: 10.2307/1999300.

[26]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl. (9), 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.

[27]

G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation, European J. Appl. Math., 22 (2011), 169-185.  doi: 10.1017/S0956792511000027.

[28]

C.-V. Pao, Coexistence and stability of a competition-diffusion system in population dynamics, J. Math. Anal. Appl., 83 (1981), 54-76.  doi: 10.1016/0022-247X(81)90246-8.

[29]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[30]

X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, Journal of Dynamics and Differential Equations, (2015), 1-26.  doi: 10.1007/s10884-015-9426-1.

[31]

A. Zlatos, Existence and non-existence of transition fronts for bistable and ignition reactions, ArXiv e-prints, 2016, arXiv: 1503.07599 [math. AP]. doi: 10.1016/j.anihpc.2016.11.004.

show all references

References:
[1]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.  doi: 10.1002/cpa.3022.

[2]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. i. species persistence, J. Math. Biol., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.

[3]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. ii. biological invasions and pulsating travelling fronts, J. Math. Pures Appl. (9), 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.

[4]

H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math., 68 (2015), 1014-1065.  doi: 10.1002/cpa.21536.

[5]

M. ContiS. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560.  doi: 10.1016/j.aim.2004.08.006.

[6]

E. C. M. CrooksE. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36. 

[7]

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

[8]

E. N. Dancer and Y. H. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations, 114 (1994), 434-475.  doi: 10.1006/jdeq.1994.1156.

[9]

E. N. Dancer and Z. M. Guo, Some remarks on the stability of sign changing solutions, Tohoku Math. J. (2), 47 (1995), 199-225.  doi: 10.2748/tmj/1178225592.

[10]

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.

[11]

E. N. DancerK. Wang and Z. Zhang, Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005.  doi: 10.1090/S0002-9947-2011-05488-7.

[12]

E. N. Dancer and Z. Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, J. Differential Equations, 182 (2002), 470-489.  doi: 10.1006/jdeq.2001.4102.

[13]

D. G. de Figueiredo and E. Mitidieri, Maximum principles for linear elliptic systems, Rend. Istit. Mat. Univ. Trieste, 22 (1990), 36-66. 

[14]

W. Ding, F. Hamel and X. -Q. Zhao, Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat, ArXiv e-prints, arXiv: 1408.0723 [math. AP].

[15]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.

[16]

J. Fang, X. Yu and X. -Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, ArXiv e-prints, arXiv: 1504.03788 [math. AP].

[17]

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.

[18]

J. E. Furter and J. López-Gómez, On the existence and uniqueness of coexistence states for the lotka-volterra competition model with diffusion and spatially dependent coefficients, Nonlinear Anal., 25 (1995), 363-398.  doi: 10.1016/0362-546X(94)00139-9.

[19]

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.

[20]

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.

[21]

L. Girardin and G. Nadin, Competition in periodic media: Ⅱ-Segregative limit of pulsating fronts and "Unity is not Strength"-type result, ArXiv e-prints, arXiv: 1611.03237 [math. AP].

[22]

J.-S. Guo and C.-H. Wu, Recent developments on wave propagation in 2-species competition systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2713-2724.  doi: 10.3934/dcdsb.2012.17.2713.

[23]

V. HutsonY. Lou and K. Mischaikow, Spatial heterogeneity of resources versus lotka-volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.

[24]

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.

[25]

X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces, Trans. Amer. Math. Soc., 278 (1983), 21-55.  doi: 10.2307/1999300.

[26]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl. (9), 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.

[27]

G. Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation, European J. Appl. Math., 22 (2011), 169-185.  doi: 10.1017/S0956792511000027.

[28]

C.-V. Pao, Coexistence and stability of a competition-diffusion system in population dynamics, J. Math. Anal. Appl., 83 (1981), 54-76.  doi: 10.1016/0022-247X(81)90246-8.

[29]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[30]

X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, Journal of Dynamics and Differential Equations, (2015), 1-26.  doi: 10.1007/s10884-015-9426-1.

[31]

A. Zlatos, Existence and non-existence of transition fronts for bistable and ignition reactions, ArXiv e-prints, 2016, arXiv: 1503.07599 [math. AP]. doi: 10.1016/j.anihpc.2016.11.004.

Table  .  Contents
Introduction 1341
1. Preliminaries and main results 1343
1.1. Preliminaries 1343
1.2. Two main results and a conjecture 1345
1.3. A few more preliminaries 1346
2. Existence of pulsating fronts 1348
2.1. Aim: Fang-Zhao's theorem 1348
2.2. Stability of all extinction states 1348
2.3. Instability of all periodic coexistence states 1349
2.4. Counter-propagation 1357
2.5. Existence of pulsating fronts connecting both extinction states 1358
Acknowledgments 1359
REFERENCES 1359
Introduction 1341
1. Preliminaries and main results 1343
1.1. Preliminaries 1343
1.2. Two main results and a conjecture 1345
1.3. A few more preliminaries 1346
2. Existence of pulsating fronts 1348
2.1. Aim: Fang-Zhao's theorem 1348
2.2. Stability of all extinction states 1348
2.3. Instability of all periodic coexistence states 1349
2.4. Counter-propagation 1357
2.5. Existence of pulsating fronts connecting both extinction states 1358
Acknowledgments 1359
REFERENCES 1359
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