Necessary and sufficient conditions for the oscillations of a multiplicative delay logistic equation
Authors:
S. R. Grace, I. Győri and B. S. Lalli
Journal:
Quart. Appl. Math. 53 (1995), 69-79
MSC:
Primary 34K15
DOI:
https://doi.org/10.1090/qam/1315448
MathSciNet review:
MR1315448
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Abstract: Necessary and sufficient conditions are presented for the oscillation of all positive solutions of the multiplicative delay logistic equation \[ \frac {{dN\left ( t \right )}}{{dt}} = r\left ( t \right )N\left ( t \right ) \left ( {1 - \prod \limits _{j = 1}^m { \left ( {\frac {{N\left ( {g_j}\left ( t \right ) \right .}}{K}} \right ) } } \right )\] about the positive equilibrium $K$. The cases when $r\left ( t \right ) = r$ and ${g_j}\left ( t \right ) = t - {\tau _j}$ or ${g_j}\left ( t \right ) = \left [ {t - {k_j}} \right ]$, [•] denoting the greatest integer function, where $r, {\tau _j}$, and ${k_j}$ are positive constants, $j = 1, 2,...,m$, are also included.
J. Caperon, Time lag in population growth response of Isochrysis Galbana to a variable nitrate environment, Ecology 5, 188–192 (1969)
K. Gopalsamy, Global stability in the delay logistic equation with discrete delays, Houston J. Math (to appear)
K. Gopalsamy and B. S. Lalli, Oscillatory and asymptotic behavior of a multiplicative delay logistic equation, Dynamics and Stability of Systems 7, 35–42 (1992)
I. Györi and G. Ladas, Oscillation theory of Delay Differential Equations, Oxford Science Publications, Oxford, 1991
I. Györi and G. Ladas, Linearized oscillations for equations with piecewise constant arguments, Differential and Integral Equations 2, 123–131 (1989)
G. E. Hutchinson, Circular causal systems in ecology, Ann. New York Acad. Sci. 50, 221–246 (1948)
G. S. Jones, On the nonlinear differential difference equation $f’\left ( x \right ) = f\left ( x - 1 \right )\left [ 1 + f\left ( x \right ) \right ]$, J. Math. Anal. Appl. 4, 440–469 (1962)
S. Kakutani and L. Markus, On the nonlinear difference differential equation $\dot y\left ( t \right ) = \\ \left [ A - By\left ( t - \tau \right ) \right ]y\left ( t \right )$, Contributions to the Theory of Nonlinear Oscillations 4, 1–18 (1958)
S. M. Lenhart and C. C. Travis, Global stability of a biological model with time delay, Proc. Amer. Math. Soc. 96, 75–78 (1986)
J. Maynard Smith, Models in Ecology, Cambridge Univ. Press, Cambridge, 1974
A. J. Nicholson, An outline of the dynamics of animal populations, Austral. Zool. 2, 9–65 (1954)
A. J. Nicholson, The self adjustment of populations to change, Cold Spring Harbor, Symposium on Qualitative Biology 22, 153–173 (1957)
E. E. Smith, Quantitative aspects of population growth in the dynamic of growth processes (E. J. Boel, ed.), Princeton Univ. Press, Princeton, N.J., 1954
K. L. Cooke and J. Weiner, A survey of differential equations with piecewise continuous arguments, Lecture Notes in Mathematics, vol. 1475 (S. Busenberg and M. Martelli, eds.), Springer-Verlag, Berlin-Heidelberg-New York, 1991
E. M. Wright, A nonlinear difference differential equation, J. Reine Angew. Math. 194, 66–87 (1955)
J. Caperon, Time lag in population growth response of Isochrysis Galbana to a variable nitrate environment, Ecology 5, 188–192 (1969)
K. Gopalsamy, Global stability in the delay logistic equation with discrete delays, Houston J. Math (to appear)
K. Gopalsamy and B. S. Lalli, Oscillatory and asymptotic behavior of a multiplicative delay logistic equation, Dynamics and Stability of Systems 7, 35–42 (1992)
I. Györi and G. Ladas, Oscillation theory of Delay Differential Equations, Oxford Science Publications, Oxford, 1991
I. Györi and G. Ladas, Linearized oscillations for equations with piecewise constant arguments, Differential and Integral Equations 2, 123–131 (1989)
G. E. Hutchinson, Circular causal systems in ecology, Ann. New York Acad. Sci. 50, 221–246 (1948)
G. S. Jones, On the nonlinear differential difference equation $f’\left ( x \right ) = f\left ( x - 1 \right )\left [ 1 + f\left ( x \right ) \right ]$, J. Math. Anal. Appl. 4, 440–469 (1962)
S. Kakutani and L. Markus, On the nonlinear difference differential equation $\dot y\left ( t \right ) = \\ \left [ A - By\left ( t - \tau \right ) \right ]y\left ( t \right )$, Contributions to the Theory of Nonlinear Oscillations 4, 1–18 (1958)
S. M. Lenhart and C. C. Travis, Global stability of a biological model with time delay, Proc. Amer. Math. Soc. 96, 75–78 (1986)
J. Maynard Smith, Models in Ecology, Cambridge Univ. Press, Cambridge, 1974
A. J. Nicholson, An outline of the dynamics of animal populations, Austral. Zool. 2, 9–65 (1954)
A. J. Nicholson, The self adjustment of populations to change, Cold Spring Harbor, Symposium on Qualitative Biology 22, 153–173 (1957)
E. E. Smith, Quantitative aspects of population growth in the dynamic of growth processes (E. J. Boel, ed.), Princeton Univ. Press, Princeton, N.J., 1954
K. L. Cooke and J. Weiner, A survey of differential equations with piecewise continuous arguments, Lecture Notes in Mathematics, vol. 1475 (S. Busenberg and M. Martelli, eds.), Springer-Verlag, Berlin-Heidelberg-New York, 1991
E. M. Wright, A nonlinear difference differential equation, J. Reine Angew. Math. 194, 66–87 (1955)
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© Copyright 1995
American Mathematical Society