# American Institute of Mathematical Sciences

July  2007, 8(1): 115-125. doi: 10.3934/dcdsb.2007.8.115

## Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems

 1 Department of Information and Communication Sciences, Kyoto Sangyo University, Kyoto 603-8555

Received  November 2005 Revised  February 2006 Published  April 2007

This paper investigates the existence of travelling waves for the two component higher order autocatalytic reaction-diffusion systems by the phase plane analysis. We prove the existence of travelling waves for the system without decay for two extreme cases: the non-diffusive reactant case and the equal diffusive case. We further discuss the existence problem for the system with higher order decay when the reactant does not diffuse. Our analysis also gives the estimate of the minimal propagation speeds in terms of the order of autocatalysis.
Citation: Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 115-125. doi: 10.3934/dcdsb.2007.8.115
 [1] Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707 [2] Hao Wen, Jianhua Huang, Yuhong Li. Propagation of stochastic travelling waves of cooperative systems with noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (10) : 5779-5803. doi: 10.3934/dcdsb.2021295 [3] Yaping Wu, Niannian Yan. Stability of traveling waves for autocatalytic reaction systems with strong decay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1601-1633. doi: 10.3934/dcdsb.2017033 [4] Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775 [5] Khaled Mohammed Saad, Eman Hussain Faissal AL-Sharif. Comparative study of a cubic autocatalytic reaction via different analysis methods. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 665-684. doi: 10.3934/dcdss.2019042 [6] Yunfeng Jia, Yi Li, Jianhua Wu. Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4785-4813. doi: 10.3934/dcds.2017206 [7] Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769 [8] Da-Peng Li. Phase transition of oscillators and travelling waves in a class of relaxation systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2601-2614. doi: 10.3934/dcdsb.2016063 [9] C. van der Mee, Stella Vernier Piro. Travelling waves for solid-gas reaction-diffusion systems. Conference Publications, 2003, 2003 (Special) : 872-879. doi: 10.3934/proc.2003.2003.872 [10] Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591 [11] Sheng-Chen Fu. Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 189-196. doi: 10.3934/dcdsb.2011.16.189 [12] Sheng-Chen Fu, Je-Chiang Tsai. Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4041-4069. doi: 10.3934/dcds.2013.33.4041 [13] K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591 [14] Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1827-1842. doi: 10.3934/dcdsb.2020364 [15] Dmitry Treschev. Travelling waves in FPU lattices. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867 [16] Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126 [17] Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245 [18] Yong Zhou, Zhengguang Guo. Blow up and propagation speed of solutions to the DGH equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 657-670. doi: 10.3934/dcdsb.2009.12.657 [19] Antoine Benoit. Finite speed of propagation for mixed problems in the $WR$ class. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2351-2358. doi: 10.3934/cpaa.2014.13.2351 [20] Matthew H. Chan, Peter S. Kim, Robert Marangell. Stability of travelling waves in a Wolbachia invasion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 609-628. doi: 10.3934/dcdsb.2018036

2021 Impact Factor: 1.497