On a property of a generalized Kolmogorov population model

Shair Ahmad; Alan C. Lazer

doi:10.3934/dcds.2013.33.1

Article Contents

2013, Volume 33, Issue 1: 1-6. Doi: 10.3934/dcds.2013.33.1

This issue Previous Article Tribute to our friend and colleague Jean Mawhin Next Article Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential

On a property of a generalized Kolmogorov population model

Shair Ahmad^1, and
Alan C. Lazer^2,

1.
Department of Mathematics, University of Texas at San Antonio, San Antonio, Texas 78249
2.
Department of Mathematics, University of Miami, Coral Gables, Miami, Florida 33124

Received: July 31, 2011

Revised: December 31, 2011

Published: December 2012

Abstract Related Papers Cited by

Abstract

We consider Kolmogorov-type systems which are not necessarily competitive or cooperative. Our main result shows that such systems cannot have nontrivial periodic solutions whose orbits are orbitally stable. We obtain our results under two assumptions that we consider to be natural assumptions.

Keywords:

Mathematics Subject Classification: 34C60, 92D25.

Citation:

Related Papers

Cited by

References

[1]	S. Ahmad and A. C. Lazer, Average growth and total permanence in a competitive Lotka-Volterra system, Ann. Mat. Pura Appl., 185 (2006), S47-S67.doi: 10.1007/s10231-004-0136-2.
[2]	C. Cosner and R. S. Cantrell, "Spatial Ecology via Reaction-Diffusion Equations," John Wiley, 2003.
[3]	K. P. Hadeler and D. Glas, Quasimonotone systems and convergence to equilibrium in a population genetic model, J. Math. Anal. Appl., 95 (1983), 297-303.doi: 10.1016/0022-247X(83)90108-7.
[4]	M. W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc., 11 (1984), 1-64.doi: 10.1090/S0273-0979-1984-15236-4.
[5]	J. Jiang, Attractors for strictly monotone flows, J. Math. Anal. Appl., 162 (1991), 210-223.doi: 10.1016/0022-247X(91)90188-6.
[6]	A. N. Kolmogorov, "Sulla Teoria Di Volterra Della Lotta Per L'esistenza," Giorn. Instituto Ital. Attuari, 1936.
[7]	H. L. Smith, "Monotone Dynamical Systems: An introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, 1995.
[8]	F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math., 21 (1992), 224-250.
[9]	M. L. Zeeman, Hopf bifurcation in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Syst., 8 (1993), 189-216.doi: 10.1080/02681119308806158.