American Institute of Mathematical Sciences

Advanced
July  2003, 9(4): 1049-1061. doi: 10.3934/dcds.2003.9.1049

Positive steady--states for two interacting species models with linear self-cross diffusions

Kimun Ryu 1, and  Inkyung Ahn 1,

1. 

Department of Mathematics, Korea University, Jochiwon, Chung-nam 339-700, South Korea, South Korea

Received  April 2002 Revised  December 2002 Published  April 2003

In this paper, we discuss the positive steady-state existence for predator-prey and competing interaction systems between two species with linear self-cross diffusions. The methods employed are the decomposing operators and the theory of fixed point index on cones in a Banach space. We give sufficient conditions for the existence of positive solutions. The conditions are given in terms of the signs of the principal eigenvalues of certain differential operators.
Keywords: fixed point index., linear self-cross diffusion, Elliptic system, positive coexistence, decomposing operator.
Mathematics Subject Classification: 35J60, 92D2.
Citation: Kimun Ryu, Inkyung Ahn. Positive steady--states for two interacting species models with linear self-cross diffusions. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1049-1061. doi: 10.3934/dcds.2003.9.1049
[1]

Kimun Ryu, Inkyung Ahn. On certain elliptic systems with nonlinear self-cross diffusions. Conference Publications, 2003, 2003 (Special) : 752-759. doi: 10.3934/proc.2003.2003.752
[2]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381
[3]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536
[4]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[5]

Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial & Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487
[6]

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
[7]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[8]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193
[9]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[10]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035
[11]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435
[12]

Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185
[13]

Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193
[14]

Yuxia Guo, Jianjun Nie. Classification for positive solutions of degenerate elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1457-1475. doi: 10.3934/dcds.2018130
[15]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[16]

Anotida Madzvamuse, Raquel Barreira. Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2775-2801. doi: 10.3934/dcdsb.2018163
[17]

Jing Liu, Xiaodong Liu, Sining Zheng, Yanping Lin. Positive steady state of a food chain system with diffusion. Conference Publications, 2007, 2007 (Special) : 667-676. doi: 10.3934/proc.2007.2007.667
[18]

Benlong Xu, Hongyan Jiang. Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4255-4266. doi: 10.3934/dcdsb.2018136
[19]

Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control & Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217
[20]

Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (29)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences