# American Institute of Mathematical Sciences

• Previous Article
Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems
• PROC Home
• This Issue
• Next Article
Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations
2009, 2009(Special): 24-33. doi: 10.3934/proc.2009.2009.24

## Characterizing the existence of coexistence states in a class of cooperative systems

 1 Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom 2 Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040-Madrid

Received  July 2008 Revised  July 2009 Published  September 2009

This paper characterizes the existence of coexistence states for a class of sublinear elliptic cooperative systems with a linear equation and the other non-linear, but yet linear on a subdomain of the underlying domain. The analysis of this problem is imperative for ascertaining the dynamics of wider general classes of cooperative systems with spatially heterogeneous nonlinearities, like those introduced in López-Gómez & Molina-Meyer [10] for weakly coupled systems. Our characterization relies upon a spectral bound associated to a certain non-local second order differential operator, which has not been previously documented in the literature.
Citation: Pablo Álvarez-Caudevilla, Julián López-Gómez. Characterizing the existence of coexistence states in a class of cooperative systems. Conference Publications, 2009, 2009 (Special) : 24-33. doi: 10.3934/proc.2009.2009.24
 [1] Pablo Álvarez-Caudevilla, Julián López-Gómez. The dynamics of a class of cooperative systems. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 397-415. doi: 10.3934/dcds.2010.26.397 [2] 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 [3] Tetsu Mizumachi. Instability of bound states for 2D nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 413-428. doi: 10.3934/dcds.2005.13.413 [4] Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1819-1835. doi: 10.3934/dcdss.2021038 [5] Mikko Kaasalainen. Dynamical tomography of gravitationally bound systems. Inverse Problems and Imaging, 2008, 2 (4) : 527-546. doi: 10.3934/ipi.2008.2.527 [6] Luís Simão Ferreira. A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. Kinetic and Related Models, 2022, 15 (1) : 91-117. doi: 10.3934/krm.2021045 [7] Isabel Coelho, Carlota Rebelo, Elisa Sovrano. Extinction or coexistence in periodic Kolmogorov systems of competitive type. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5743-5764. doi: 10.3934/dcds.2021094 [8] Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035 [9] J. Földes, Peter Poláčik. On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 133-157. doi: 10.3934/dcds.2009.25.133 [10] Hao Wen, Jianhua Huang, Yuhong Li. Propagation of stochastic travelling waves of cooperative systems with noise. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021295 [11] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 [12] Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013 [13] 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 [14] Lirong Huang, Jianqing Chen. Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system. Electronic Research Archive, 2020, 28 (1) : 383-404. doi: 10.3934/era.2020022 [15] Xia Sun, Kaimin Teng. Positive bound states for fractional Schrödinger-Poisson system with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3735-3768. doi: 10.3934/cpaa.2020165 [16] Kaimin Teng, Xian Wu. Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1157-1187. doi: 10.3934/cpaa.2022014 [17] M. R. S. Kulenović, Orlando Merino. Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1141-1156. doi: 10.3934/dcdsb.2006.6.1141 [18] Dan Li, Hui Wan. Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4145-4183. doi: 10.3934/dcds.2021032 [19] Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97 [20] Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems and Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239

Impact Factor: