Mathematical Biosciences and Engineering, 2006, 3(1): 189-204. doi: 10.3934/mbe.2006.3.189.

70K20, 76E30, 34D20, 37B25, 70K15, 93D30, 92A15,35K57.

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A nonlinear $L^2$-stability analysis for two-species population dynamics with dispersal

1. University of Naples Federico II, Department of Mathematics and Applications ''R. Caccioppoli", Complesso Universitario Monte S. Angelo. Via Cinzia, 80126 Napoli

The nonlinear $L^2$-stability (instability) of the equilibrium states of two-species population dynamics with dispersal is studied. The obtained results are based on (i) the rigorous reduction of the $L^2$-nonlinear stability to the stability of the zero solution of a linear binary system of ODEs and (ii) the introduction of a particular Liapunov functional V such that the sign of $\frac{dV}{dt}$ along the solutions is linked directly to the eigenvalues of the linear problem.
  Article Metrics

Keywords Liapunov direct method; nonlinear stability; reaction diffusion equations.; two-species population dynamics

Citation: Salvatore Rionero. A nonlinear $L^2$-stability analysis for two-species population dynamics with dispersal. Mathematical Biosciences and Engineering, 2006, 3(1): 189-204. doi: 10.3934/mbe.2006.3.189


This article has been cited by

  • 1. A. A. Hill, Global stability for penetrative double-diffusive convection in a porous medium, Acta Mechanica, 2008, 200, 1-2, 1, 10.1007/s00707-007-0575-0
  • 2. Bruno Buonomo, Salvatore Rionero, On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate, Applied Mathematics and Computation, 2010, 217, 8, 4010, 10.1016/j.amc.2010.10.007
  • 3. Salvatore Rionero, $$L^2$$ L 2 -energy decay of convective nonlinear PDEs reaction–diffusion systems via auxiliary ODEs systems, Ricerche di Matematica, 2015, 64, 2, 251, 10.1007/s11587-015-0231-2
  • 4. Isabella Torcicollo, On the non-linear stability of a continuous duopoly model with constant conjectural variation, International Journal of Non-Linear Mechanics, 2016, 81, 268, 10.1016/j.ijnonlinmec.2016.01.018
  • 5. Bruno Buonomo, Salvatore Rionero, Linear and nonlinear stability thresholds for a diffusive model of pioneer and climax species interaction, Mathematical Methods in the Applied Sciences, 2009, 32, 7, 811, 10.1002/mma.1068
  • 6. Salvatore Rionero, Isabella Torcicollo, On the dynamics of a nonlinear reaction–diffusion duopoly model, International Journal of Non-Linear Mechanics, 2018, 99, 105, 10.1016/j.ijnonlinmec.2017.11.005
  • 7. G. Mulone, B. Straughan, An operative method to obtain necessary and sufficient stability conditions for double diffusive convection in porous media, ZAMM, 2006, 86, 7, 507, 10.1002/zamm.200510272
  • 8. Brian Straughan, A note on convection with nonlinear heat flux, Ricerche di Matematica, 2007, 56, 2, 229, 10.1007/s11587-007-0016-3
  • 9. Isabella Torcicollo, On the dynamics of a non-linear Duopoly game model, International Journal of Non-Linear Mechanics, 2013, 57, 31, 10.1016/j.ijnonlinmec.2013.06.011
  • 10. Salvatore Rionero, Isabella Torcicollo, Stability of a Continuous Reaction-Diffusion Cournot-Kopel Duopoly Game Model, Acta Applicandae Mathematicae, 2014, 132, 1, 505, 10.1007/s10440-014-9932-x
  • 11. Salvatore Rionero, On the nonlinear stability of nonautonomous binary systems, Nonlinear Analysis: Theory, Methods & Applications, 2012, 75, 4, 2338, 10.1016/
  • 12. Bruno Buonomo, Deborah Lacitignola, Modeling peer influence effects on the spread of high–risk alcohol consumption behavior, Ricerche di Matematica, 2014, 63, 1, 101, 10.1007/s11587-013-0167-3
  • 13. Eric Avila-Vales, Bruno Buonomo, Analysis of a mosquito-borne disease transmission model with vector stages and nonlinear forces of infection, Ricerche di Matematica, 2015, 64, 2, 377, 10.1007/s11587-015-0245-9
  • 14. G. Mulone, S. Rionero, W. Wang, The effect of density-dependent dispersal on the stability of populations, Nonlinear Analysis: Theory, Methods & Applications, 2011, 74, 14, 4831, 10.1016/
  • 15. Florinda Capone, On the dynamics of predator-prey models with the Beddington–De Angelis functional response, under Robin boundary conditions, Ricerche di Matematica, 2008, 57, 1, 137, 10.1007/s11587-008-0026-9
  • 16. Monica De Angelis, Pasquale Renno, Existence, uniqueness and a priori estimates for a nonlinear integro-differential equation, Ricerche di Matematica, 2008, 57, 1, 95, 10.1007/s11587-008-0028-7

Reader Comments

your name: *   your email: *  

Copyright Info: 2006, Salvatore Rionero, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved