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January  2013, 33(1): 1-6. doi: 10.3934/dcds.2013.33.1

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, United States
2. 

Department of Mathematics, University of Miami, Coral Gables, Miami, Florida 33124, United States

Received  August 2011 Revised  January 2012 Published  September 2012

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: cooperative, Kolmogorov, Lotka--Volterra, dissipative, Competitive, attracting orbit, positive cone..
Mathematics Subject Classification: 34C60, 92D2.
Citation: Shair Ahmad, Alan C. Lazer. On a property of a generalized Kolmogorov population model. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 1-6. doi: 10.3934/dcds.2013.33.1
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.  Google Scholar

[2]

C. Cosner and R. S. Cantrell, "Spatial Ecology via Reaction-Diffusion Equations," John Wiley, 2003.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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J. Jiang, Attractors for strictly monotone flows, J. Math. Anal. Appl., 162 (1991), 210-223. doi: 10.1016/0022-247X(91)90188-6.  Google Scholar

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A. N. Kolmogorov, "Sulla Teoria Di Volterra Della Lotta Per L'esistenza," Giorn. Instituto Ital. Attuari, 1936. Google Scholar

[7]

H. L. Smith, "Monotone Dynamical Systems: An introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, 1995.  Google Scholar

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F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math., 21 (1992), 224-250.  Google Scholar

[9]

M. L. Zeeman, Hopf bifurcation in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Syst., 8 (1993), 189-216. doi: 10.1080/02681119308806158.  Google Scholar

show all references

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.  Google Scholar

[2]

C. Cosner and R. S. Cantrell, "Spatial Ecology via Reaction-Diffusion Equations," John Wiley, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Jiang, Attractors for strictly monotone flows, J. Math. Anal. Appl., 162 (1991), 210-223. doi: 10.1016/0022-247X(91)90188-6.  Google Scholar

[6]

A. N. Kolmogorov, "Sulla Teoria Di Volterra Della Lotta Per L'esistenza," Giorn. Instituto Ital. Attuari, 1936. Google Scholar

[7]

H. L. Smith, "Monotone Dynamical Systems: An introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, 1995.  Google Scholar

[8]

F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math., 21 (1992), 224-250.  Google Scholar

[9]

M. L. Zeeman, Hopf bifurcation in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Syst., 8 (1993), 189-216. doi: 10.1080/02681119308806158.  Google Scholar

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