Extreme stability and almost periodicity in a discrete logistic equation. (English) Zbl 0954.39005

Tohoku Math. J., II. Ser. 52, No. 1, 107-125 (2000); correction ibid. 53, No. 4, 629-631 (2001).
In this interesting paper, the authors obtain a sufficient condition for the existence of a globally asymptotically stable positive almost periodic solution to the discrete logistic equation \[ x(n+1)= x(n) \exp \left[r(n) \left(1-{x(n) \over K(n)}\right) \right],\quad n\in Z= \{0,1,2, \dots\};\;x(0)>0,\tag{1} \] where \(\{r(n)\}\), \(\{K(n)\}(n\in Z)\) are strictly positive sequences of real numbers. The following main results are established in the paper:
Theorem 3.2. Suppose that \[ 0<\inf_{n\in Z}r(n)\leq r(n)\leq \sup_{n\in Z}r(n)< \infty, \quad 0<\inf_{n\in Z}K(n)\leq K(n)\leq\sup_{n\in Z}K(n) <\infty \] hold. Then equation (1) is extremely stable in the sense of T. Yoshizawa, i.e., for any two positive solutions \(\{x(n)\}\), \(\{y(n)\}\) to (1) we have \(\lim_{n\to\infty}|x(n)-y(n)|=0\).
Theorem 4.1. Suppose that the strictly positive sequences \(\{r(n)\}\), \(\{K(n)\}(n\in Z)\) in Theorem 3.2 are almost periodic and the additional condition \(\sup_{n\in Z}r(n)<2\) holds. Then equation (1) has a unique globally asymptotically stable almost periodic solution.
The last existence condition is also necessary for the corresponding autonomous equation of equation (1). The authors conjectured in the paper that the condition on \(\{r(n)\}\) can be replaced by \[ 0<\lim_{n \to\infty} \left[{1\over n}\sum^{n-1}_{j=0} r(j)\right]<2. \]


39A11 Stability of difference equations (MSC2000)
92D25 Population dynamics (general)
92D40 Ecology
Full Text: DOI


[1] M S BOYCE AND D J. DALEY, Population tracking of fluctuating environments and natural selection for tracking ability, Amer Natur 115 (1980), 480-491
[2] B D COLEMAN, Nonautonomous logistic equations as models of the adjustment of populations to environ mental change, Math. Biosci 45 (1979), 159-173 · Zbl 0425.92013
[3] B D COLEMAN, Y H HSIEH AND G P KNOWLES, On the optimal choice of r for a population in periodic environment, Math Biosci 46 (1979), 71-85 · Zbl 0429.92022
[4] K L COOKE AND W HUANG, A theorem of George Seifert and an equation with state-dependent delay, Delay and Differential Equations, 65-77, World Scientific, Singapore, 1992 · Zbl 0820.34042
[5] K L COOKE AND J WIENER, Retarded differential equations with piecewise constant delays, J Math Ana Appl 99(1984), 265-297. · Zbl 0557.34059
[6] C CORDUNEANU, Almost periodic discrete processes, Libertas Math 2 (1982), 159-16 · Zbl 0548.39002
[7] C CORDUNEANU, Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics 22, Interscience, New York, 1968 · Zbl 0175.09101
[8] A M FINK, Almost Periodic Differential Equations, Lecture Notes in Math 377, Springer Verlag, Berlin New York, 1974. · Zbl 0325.34039
[9] M E FISHER AND B S GOH, Stability results for delayed-recruitment models in population dynamics, Math Biol 19 (1984), 147-156 · Zbl 0533.92017
[10] K GOPALSAMY, Stability and oscillations in delay differential equations of population dynamics, Mathemat ics and its Applications 74, Kluwmer Acad, Dordrecht, 1992 · Zbl 0752.34039
[11] K GOPALSAMY, I GYORI AND G LADAS, Oscillations of a class of delay equations with continuous an piecewise constants arguments, Funkcial Ekvac 32 (1989), 395-406 · Zbl 0697.34059
[12] K GOPALSAMY AND X Z HE, Dynamics of an almost periodic logistic integrodifferential equation, Meth ods Appl Anal 2(1995), 38-66 · Zbl 0835.45004
[13] K GOPALSAMY, M R S KULENOVIC AND G LADAS, On a logistic equation with piecewise constant arguments, Differential Integral Equations 4 (1990), 215-223 · Zbl 0727.34061
[14] I GYORI AND G LADAS, Oscillation Theory of Delay Differential Equations with Applications, Oxfor Math Monogr, Clarendon Press, Oxford, 1991 · Zbl 0780.34048
[15] A HALANAY, Solutions periodiques et presque-periodiques des systemes d’equations aux differences finies, Arch Rational Mech Anal 12(1963), 134-149 · Zbl 0112.06106
[16] M H MoADAB, Discrete dynamical systems and applications, Ph D Thesis, University of Texas, Arlington, 1988
[17] R M MAY, Biological populations obeying difference equations: stable points, stable cycles and chaos, Theoret Biol 51 (1975), 511-524
[18] R M MAY AND G F OSTER, Bifurcation and dynamic complexity in simple ecological models, Ame Natural 110 (1976), 573-599
[19] A M SAMOILENKO AND N A PERESTYUK, Impulsive Differential Equations, World Sci Ser Nonlinea Sci Ser A Monogr Treatises 14, World Sci Publishing, River Edge, NJ, 1995 · Zbl 0837.34003
[20] J WIENER, Differential equations with piecewise constant delays, Trends in theory and practice of nonlinea differential equations, Lecture Notes in Pure and Appl Math 90, Dekker, New York, 1984 · Zbl 0531.34059
[21] T YOSHIZAWA, Extreme stability and almost periodic solutions of functional-differential equations, Arc Rational Mech Anal 17 (1964), 148-170 · Zbl 0132.06601
[22] T YOSHIZAWA, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Appl Math Sci 14, Springer Verlag, New York-Heidelberg, 1975 · Zbl 0304.34051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.