Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

Oscillations in a delay-logistic equation


Author: K. Gopalsamy
Journal: Quart. Appl. Math. 44 (1986), 447-461
MSC: Primary 34K15; Secondary 92A15
DOI: https://doi.org/10.1090/qam/860898
MathSciNet review: 860898
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient conditions are derived for all nonconstant nonnegative solutions of the equations of the form

$\displaystyle \frac{{dx\left( t \right)}}{{dt}} = x\left( t \right)\left\{ {a - \sum\limits_{j = 1}^n {{b_j}x\left( {t - {\tau _j}} \right)} } \right\}$

and

$\displaystyle \frac{{dx\left( t \right)}}{{dt}} = x\left( t \right)\left\{ {a - b\int_{ - \infty }^t {k\left( {t - s} \right)x\left( s \right)ds} } \right\}$

to be oscillatory about their respective positive steady states. The results are complementary to those in [15].

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

  • [1] R. Bellman and K. L. Cooke, Differential-difference equations, Academic Press, New York, 1963 MR 0147745
  • [2] R. D. Braddock and P. Van den Driessche, On a two lag differential delay equation, J. Aust. Math. Soc. Ser. B 24, 292-317 (1983) MR 680841
  • [3] S. N. Chow, Existence of periodic solutions of autonomous functional differential equations, J. Differential Equations 15, 350-378 (1974) MR 0336003
  • [4] J. R. Claeyssen, Effect of delays on functional differential equations, J. Differential Equations 20, 404-440 (1976) MR 0435553
  • [5] W. J. Cunningham, A nonlinear differential-difference equation of growth, Proc. Natl. Acad. Sci. U.S.A. 40, 708-713 (1954) MR 0067343
  • [6] J. M. Cushing, Integrodifferential equations and delay models in population dynamics, Lecture notes in biomathematics, vol. 20, Springer-Verlag, Berlin, 1977 MR 0496838
  • [7] J. M. Cushing, Time delays in single species growth models, J. Math. Biol. 4, 257-264 (1977) MR 0682256
  • [8] B. F. Dibrov, M. A. Levshits, and M. A. Volkenstein, Mathematical model of immune processes, J. Theor. Biol. 65, 609-631 (1977) MR 0444099
  • [9] L. E. El'sgol'ts and S. B. Norkin, Introduction to the theory and application of differential equations with deviating arguments, Academic Press, New York, 1973 MR 0352647
  • [10] M. D. Fargue, Réductibilité des systemes hereditaires à des systemes dynamiques, C. R. Acad. Sci. Paris Ser. B. 277, 471-473 (1973) MR 0333630
  • [11] A. C. Fowler, Linear and nonlinear instability of heat exchangers, J. Inst. Math. Appl. 22, 361-382 (1978) MR 516555
  • [12] A. A. Francis, I. H. Herron, and C. McCalla, Speculative demand with supply response lag, in Nonlinear systems and applications (V. Lakshmikantham, ed.), pp. 603-610, Academic Press, New York, 1977 MR 0459530
  • [13] J. C. Friedly and V. S. Krishnan, Predictions of non-linear flow oscillations in boiling channels, AIChE Symp. Ser. 68, 127-135 (1974)
  • [14] K. Gopalsamy, Stability, instability, oscillation and nonoscillation in scalar integrodifferential systems, Bull. Aust. Math. Soc. 28, 233-246 (1983) MR 729010
  • [15] K. Gopalsamy, Nonoscillation in a delay-logistic equation, Quart. Appl. Math. In press MR 793526
  • [16] K. Gopalsamy, Global asymptotic stability in Volterra's population, J. Math. Biol. 19 157-168 (1984) MR 745849
  • [17] F. S. Grodins, J. Buell, and A. J. Bart, Mathematical analysis and digital simulation of the respiratory control system, J. Appl. Physiol. 22, 260-276 (1967)
  • [18] K. P. Hadeler and J. Tomiuk, Periodic solutions of functional differential equations, Arch. Rat. Mech. Anal. 65 87-95 (1977) MR 0442417
  • [19] J. K. Hale, Nonlinear oscillations in equations with delays, Lect. Appl. Math. 17 (1979) MR 564916
  • [20] U. an der Heiden, Analysis of neural networks, Lect. Notes Biomath. 35, Springer-Verlag, Berlin-Heidelberg-New York (1980) MR 617008
  • [21] E. Hewitt and K. Stromberg, Real and abstract analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1965 MR 0367121
  • [22] G. E. Hutchinson, Circular causal systems in ecology, Ann. N. Y. Acad. Sci. 50, 221-240 (1948)
  • [23] G. S. Jones, On the nonlinear differential-difference equation $ f'\left( x \right) = - \alpha f\left( {x - 1} \right)\left[ {1 + f\left( x \right)} \right]$, J. Math. Anal. Appl. 4, 440-469 (1962) MR 0151690
  • [24] G. S. Jones, The existence of periodic solutions of $ f'\left( x \right) = - \alpha f\left( {x - 1} \right)\left[ {1 + f\left( x \right)} \right]$, J. Math. Anal. Appl. 5, 435-450 (1962) MR 0141837
  • [25] S. Kakutani and L. Markus, On the nonlinear difference-differential equation $ y'\left( t \right) = \left[ {A - By\left( {t - \tau } \right)} \right]y\left( t \right)$, in Contributions to the theory of nonlinear oscillations, IV, Annals of Mathematics Study 41, Princeton Univ. Press, Princeton, N. J. (1958) MR 0101953
  • [26] R. L. Kitching, Time, resources and population dynamics in insects, Aust. J. Ecol. 2, 31-42 (1977)
  • [27] J. J. Levin, Boundedness and oscillation of some Volterra and delay equations, J. Differential Equations 5, 369-398 (1969) MR 0236642
  • [28] N. MacDonald, Time lags in biological models, Lect. Notes Bio-math. 27, Springer-Verlag, Berlin (1978) MR 521439
  • [29] M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science 197, 287-289 (1977)
  • [30] R. M. May, Stability and complexity in model ecosystems, Princeton Univ. Press, Princeton, N. J., 1973
  • [31] J. Maynard Smith, Models in ecology, Cambridge Univ. Press, 1974
  • [32] R. K. Miller, Nonlinear Volterra integral equations, Benjamin, Menlo Park, 1971 MR 0511193
  • [33] A. D. Myschkis, Lineare differentialgleichungen mit nacheilendem argument, Deutscher Verlag der Wissenschaften, Berlin, 1955 MR 0073844
  • [34] R. D. Nussbaum, Differential delay equations with two time lags, Mem. Amer. Math. Soc. No. 205, 16, 1-62 (1975) MR 0492729
  • [35] E. C. Pielou, An introduction to mathematical ecology, Wiley, New York, 1969 MR 0252051
  • [36] J. F. Perez, C. P. Malta, and F. A. B. Coutinho, Qualitative analysis of oscillations in isolated populations of flies, J. Theor. Biol. 71, 505-514 (1978) MR 0472128
  • [37] H. Stech, The effect of time lags on the stability of the equilibrium state of a population growth equation, J. Math. Biol. 5 115-130 (1978) MR 645053
  • [38] H. Stech, The Hopf-bifurcation: a stability result and application, J. Math. Anal. Appl. 71, 525-546 (1979) MR 548781
  • [39] C. E. Taylor and R. R. Sokol, Oscillations in housefly population sizes due to time lags, Ecology 57, 1060-1067 (1976)
  • [40] G. C. Walter, Delay differential equation models for fisheries, J. Fish. Res. Bd, Can. 30, 930-945 (1973)
  • [41] P. Waltman, Deterministic threshold models in the theory of epidemics, Lect. Notes Biomath. 1, Springer-Verlag, Berlin-Heidelberg-New York (1974) MR 0359874
  • [42] P. J. Wangersky and W. J. Cunningham, Time lag in population models, Cold Spring Harbor Symp. Quant. Biol. 22, 329-338 (1957)
  • [43] E. M. Wright, A nonlinear difference-differential equation, J Reine Angew. Math. 194, 66-87 (1955) MR 0072363
  • [44] V. I. Zubov, Mathematical methods for the study of automatic control, Pergamon, Oxford (1962)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 34K15, 92A15

Retrieve articles in all journals with MSC: 34K15, 92A15


Additional Information

DOI: https://doi.org/10.1090/qam/860898
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society