Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Oscillations and global attractivity in a discrete delay logistic model


Authors: S. A. Kuruklis and G. Ladas
Journal: Quart. Appl. Math. 50 (1992), 227-233
MSC: Primary 92D25; Secondary 34K15, 92B05
DOI: https://doi.org/10.1090/qam/1162273
MathSciNet review: MR1162273
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Consider the discrete delay logistic model

$\displaystyle {N_{t + 1}} = \frac{{\alpha {N_t}}}{{1 + \beta {N_{t - k}}}}, \qquad \left( 1 \right)$

where $ \alpha \in \left( {1, \infty } \right), \beta \in \left( {0, \infty } \right)$, and $ k \in \mathbb{N} = \left\{{0, 1, 2,...} \right\}$. We obtain conditions for the oscillation and asymptotic stability of all positive solutions of Eq. (1) about its positive equilibrium $ \left( {\alpha - 1} \right)/\beta $. We prove that all positive solutions of Eq. (1) are bounded and that for $ k = 0$ and $ k = 1$ the positive equilibrium $ \left( {\alpha - 1} \right)/\beta $ is a global attractor.

References [Enhancements On Off] (What's this?)

  • [1] L. Brand, A sequence defined by a difference equation, Amer. Math. Monthly 62, 489-492 (1955) MR 1529078
  • [2] I. Györi and G. Ladas, Oscillation Theory of Delay Difference Equations with Applications, Oxford Univ. Press, 1991 MR 1168471
  • [3] I. Györi and G. Ladas, Linearized oscillations for equations with piecewise constant arguments, Differential Integral Equations J. 2, 123-131 (1989) MR 984181
  • [4] G. Ladas, Recent developments in the oscillation of delay difference equations, Differential Equations: Stability and Control, Marcel Dekker, 1990 MR 1096768
  • [5] S. Levin and R. May, A note on difference-delay equations, Theoret. Population Biol. 9, 178-187 (1976) MR 0504043
  • [6] R. M. May, Biological populations obeying difference equations: stable points, stable cycles, and chaos, J. Theoret. Biol. 51, 511-524 (1955)
  • [7] A. J. Nicholson, Compensatory reactions of populations to stresses and their evolutionary significance, Austral. J. Zool. 2, 9-65 (1954)
  • [8] E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, 1969 MR 0252051
  • [9] E. C. Pielou, Population and Community Ecology, Gordon and Breach, New York, 1974

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 92D25, 34K15, 92B05

Retrieve articles in all journals with MSC: 92D25, 34K15, 92B05


Additional Information

DOI: https://doi.org/10.1090/qam/1162273
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society