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
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Abstract: Consider the discrete delay logistic model \[ {N_{t + 1}} = \frac {{\alpha {N_t}}}{{1 + \beta {N_{t - k}}}}, \qquad \left ( 1 \right )\] 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.
- Louis Brand, Classroom Notes: A Sequence Defined by a Difference Equation, Amer. Math. Monthly 62 (1955), no. 7, 489–492. MR 1529078, DOI https://doi.org/10.2307/2307362
- I. Győri and G. Ladas, Oscillation theory of delay differential equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. With applications; Oxford Science Publications. MR 1168471
- I. Győri and G. Ladas, Linearized oscillations for equations with piecewise constant arguments, Differential Integral Equations 2 (1989), no. 2, 123–131. MR 984181
- G. Ladas, Recent developments in the oscillation of delay difference equations, Differential equations (Colorado Springs, CO, 1989) Lecture Notes in Pure and Appl. Math., vol. 127, Dekker, New York, 1991, pp. 321–332. MR 1096768
- Simon A. Levin and Robert M. May, A note on difference-delay equations, Theoret. Population Biol. 9 (1976), no. 2, 178–187. MR 504043, DOI https://doi.org/10.1016/0040-5809%2876%2990043-5
R. M. May, Biological populations obeying difference equations: stable points, stable cycles, and chaos, J. Theoret. Biol. 51, 511–524 (1955)
A. J. Nicholson, Compensatory reactions of populations to stresses and their evolutionary significance, Austral. J. Zool. 2, 9–65 (1954)
- E. C. Pielou, An introduction to mathematical ecology, Wiley-Interscience A Division of John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0252051
E. C. Pielou, Population and Community Ecology, Gordon and Breach, New York, 1974
L. Brand, A sequence defined by a difference equation, Amer. Math. Monthly 62, 489–492 (1955)
I. Györi and G. Ladas, Oscillation Theory of Delay Difference Equations with Applications, Oxford Univ. Press, 1991
I. Györi and G. Ladas, Linearized oscillations for equations with piecewise constant arguments, Differential Integral Equations J. 2, 123–131 (1989)
G. Ladas, Recent developments in the oscillation of delay difference equations, Differential Equations: Stability and Control, Marcel Dekker, 1990
S. Levin and R. May, A note on difference-delay equations, Theoret. Population Biol. 9, 178–187 (1976)
R. M. May, Biological populations obeying difference equations: stable points, stable cycles, and chaos, J. Theoret. Biol. 51, 511–524 (1955)
A. J. Nicholson, Compensatory reactions of populations to stresses and their evolutionary significance, Austral. J. Zool. 2, 9–65 (1954)
E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, 1969
E. C. Pielou, Population and Community Ecology, Gordon and Breach, New York, 1974
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Article copyright:
© Copyright 1992
American Mathematical Society