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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[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].Google Scholar

[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. Google Scholar

[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].Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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].Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[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].Google Scholar

[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. Google Scholar

[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].Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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].Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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
[1]

Xiongxiong Bao, Wan-Tong Li, Zhi-Cheng Wang. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Communications on Pure & Applied Analysis, 2020, 19 (1) : 253-277. doi: 10.3934/cpaa.2020014

[2]

Daozhou Gao, Xing Liang. A competition-diffusion system with a refuge. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 435-454. doi: 10.3934/dcdsb.2007.8.435

[3]

Yukio Kan-On. Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system. Conference Publications, 2013, 2013 (special) : 427-436. doi: 10.3934/proc.2013.2013.427

[4]

Wei-Ming Ni, Masaharu Taniguchi. Traveling fronts of pyramidal shapes in competition-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 379-395. doi: 10.3934/nhm.2013.8.379

[5]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083

[6]

Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055

[7]

E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39

[8]

Zhen-Hui Bu, Zhi-Cheng Wang. Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Communications on Pure & Applied Analysis, 2016, 15 (1) : 139-160. doi: 10.3934/cpaa.2016.15.139

[9]

Chiun-Chuan Chen, Yin-Liang Huang, Li-Chang Hung, Chang-Hong Wu. Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats. Communications on Pure & Applied Analysis, 2020, 19 (1) : 1-18. doi: 10.3934/cpaa.2020001

[10]

M. Guedda, R. Kersner, M. Klincsik, E. Logak. Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1589-1600. doi: 10.3934/dcdsb.2014.19.1589

[11]

Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817

[12]

Shi-Liang Wu, Cheng-Hsiung Hsu. Propagation of monostable traveling fronts in discrete periodic media with delay. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2987-3022. doi: 10.3934/dcds.2018128

[13]

Lia Bronsard, Seong-A Shim. Long-time behavior for competition-diffusion systems via viscosity comparison. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 561-581. doi: 10.3934/dcds.2005.13.561

[14]

Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059

[15]

Danielle Hilhorst, Masato Iida, Masayasu Mimura, Hirokazu Ninomiya. Relative compactness in $L^p$ of solutions of some 2m components competition-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 233-244. doi: 10.3934/dcds.2008.21.233

[16]

Li-Jun Du, Wan-Tong Li, Jia-Bing Wang. Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1187-1213. doi: 10.3934/mbe.2017061

[17]

Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116

[18]

Martin Gugat, Falk M. Hante, Markus Hirsch-Dick, Günter Leugering. Stationary states in gas networks. Networks & Heterogeneous Media, 2015, 10 (2) : 295-320. doi: 10.3934/nhm.2015.10.295

[19]

Shi-Liang Wu, Cheng-Hsiung Hsu. Entire solutions with merging fronts to a bistable periodic lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2329-2346. doi: 10.3934/dcds.2016.36.2329

[20]

Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks & Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (20)
  • HTML views (1)
  • Cited by (1)

Other articles
by authors

[Back to Top]