American Institute of Mathematical Sciences

Advanced
August  2006, 15(3): 725-746. doi: 10.3934/dcds.2006.15.725

Periodic solutions for a 3x 3 competitive system with cross-diffusion

Salomé Martínez 1, and  Wei-Ming Ni 2,

1. 

Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, silla 170 Correo 3, Santiago, Chile
2. 

School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Received  September 2005 Revised  January 2006 Published  April 2006

In this paper we study the role of cross-diffusion in the existence of spatially non-constant periodic solutions for the Lotka-Volterra competition system for three species. By properly choosing cross-diffusion coefficients, we show that Hopf bifurcation occurs at a constant steady state. Furthermore, these spatially nonhomogeneous periodic solutions are stable if diffusion rates are in appropriate ranges.
Keywords: 3 x 3 competitive system, Cross-diffusion, periodic solutions..
Mathematics Subject Classification: Primary: 35K55, 35K5.
Citation: Salomé Martínez, Wei-Ming Ni. Periodic solutions for a 3x 3 competitive system with cross-diffusion. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 725-746. doi: 10.3934/dcds.2006.15.725
[1]

Yuan Lou, Salomé Martínez, Wei-Ming Ni. On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 175-190. doi: 10.3934/dcds.2000.6.175
[2]

Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185
[3]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193
[4]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[5]

Pietro Baldi. Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 883-903. doi: 10.3934/dcds.2006.15.883
[6]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
[7]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435
[8]

F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239
[9]

Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145
[10]

Michael Winkler, Dariusz Wrzosek. Preface: Analysis of cross-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : i-i. doi: 10.3934/dcdss.20202i
[11]

Maxime Breden, Christian Kuehn, Cinzia Soresina. On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics, 2021, 8 (2) : 213-240. doi: 10.3934/jcd.2021010
[12]

Yaru Hu, Jinfeng Wang. Dynamics of an SIRS epidemic model with cross-diffusion. Communications on Pure and Applied Analysis, 2022, 21 (1) : 315-336. doi: 10.3934/cpaa.2021179
[13]

Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161
[14]

Willian Cintra, Carlos Alberto dos Santos, Jiazheng Zhou. Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3913-3931. doi: 10.3934/dcdsb.2021211
[15]

Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2813-2829. doi: 10.3934/dcdsb.2017152
[16]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 719-730. doi: 10.3934/dcds.2004.10.719
[17]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1193-1200. doi: 10.3934/dcds.2003.9.1193
[18]

Peng Feng, Zhengfang Zhou. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1145-1165. doi: 10.3934/cpaa.2007.6.1145
[19]

Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147
[20]

Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (152)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy