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On a two lag differential delay equation

Published online by Cambridge University Press:  17 February 2009

R. D. Braddock  and
P. van den Driessche
R. D. Braddock
Affiliation:
School of Australian Environmental Studies, Griffith University, Nathan, Queensland 4111.
P. van den Driessche
Affiliation:
Department of Mathematics, University of Victoria, Victoria, British Columbia, Canada, V8W 2Y2.
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Abstract

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The non-linear differential difference equation of the form

is investigated. This equation, with constant coefficients, is used to model the population level, N, of a single species, and incorporates two constant time lags T2 > T1 > 0; for example, regeneration and reproductive lags. The linear equation is investigated analytically, and some linear stability regions are described. The special case in which the two delay terms are equally important in self damping, B = C, is investigated in detail. Numerical solutions for this case show stable limit cycles, with multiple loops appearing when T2/T1 is large. These may correspond to splitting of major peaks in population density observations.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 3 , January 1983 , pp. 292 - 317
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Bailey, H. R. and Williams, M. Z., “Some results on the differential-difference equation ” J. Math. Anal. Appl. 15 (1966), 569–587.CrossRefGoogle Scholar
[2]Braddock, R. D. and van den Driessche, P., “A population model with time delay”, Math. Sci. 5 (1980), 55–65.Google Scholar
[3]Claeyssen, J. R., “Effect of delays on functional differential equations”, J. Differential Equations 20 (1976), 404–440.CrossRefGoogle Scholar
[4]Cooke, K. L. and Yorke, J. A., “Some equations modelling growth processes and gonorrhea epidemics”, Math. Biosci. 16 (1973), 75–101.CrossRefGoogle Scholar
[5]Dunkel, G., “Single species model for population growth depending on past history”, in Seminar on Differential Equations and Dynamical Systems, Lecture Notes in Math. 60 (Springer- Verlag, New York, 1968).Google Scholar
[6]Hale, J. K., Theory of functional differential equations, Appl. Math. Sci. 3 (Springer-Verlag, New York, 1977).CrossRefGoogle Scholar
[7]Hale, J. K., “Behaviour near constant solutions of functional differential equations”, J. Differential Equations 15 (1974), 278–294.CrossRefGoogle Scholar
[8]Hale, J. K., “Nonlinear oscillations in equations with delays”, in Lectures in Applied Math. 17, (Amer. Math. Soc. 1979).Google Scholar
[9]Hutchinson, G. E., “Circular causal systems in ecology”, Ann. N.Y. Acad. Sci. 50 (1948), 221–246.CrossRefGoogle ScholarPubMed
[10]Jones, G. S., “On the nonlinear differential difference equations f′(x) = −αf(x − 1)(1 + f(x))”, J. Math. Anal. Appl. 4 (1962), 440–469.CrossRefGoogle Scholar
[11]Kaplan, J. L. and Yorke, J. A., “Ordinary differential equations which yield periodic solutions of differential delay equations”, J. Math. Anal. Appl. 48 (1974), 317–324.CrossRefGoogle Scholar
[12]Kaplan, J. L. and Yorke, J. A., “On the stability of a periodic solution of a differential delay equation”, SIAMJ. Math. Anal. 6 (1975), 268–282.CrossRefGoogle Scholar
[13]Kitching, R. L., “Time, resources and population dynamics in insects”, Austral. J. Ecol. 2 (1977), 31–42.CrossRefGoogle Scholar
[14]May, R. M., Stability and complexity in model ecosystems (Princeton Univ. Press, Princeton, 1973).Google ScholarPubMed
[15]Smith, J. Maynard, Models in Ecology (CUP., Cambridge, 1974).Google Scholar
[16]Nussbaum, R. D., “Periodic solutions of some nonlinear autonomous functional differential equations”, Ann. Mat. Pura Appl. 101 (1974), pp. 263–306.CrossRefGoogle Scholar
[17]Nussbaum, R. D., “Differential delay equations with two time lags”, Mem. Amer. Math. Soc. No. 205, Vol. 16(1) (1975), 1–62.Google Scholar
[18]Nussbaum, R. D., “Periodic solutions of non-linear autonomous functional differential equations”, Lecture Notes in Math. 730 (Springer-Verlag, New York, 1979), 283–325.Google Scholar
[19]Pielou, E. C., An introduction to mathematical ecology (Wiley, New York, 1969).Google Scholar
[20]Pielou, E. C., Population and community ecology (Gordon and Breach, New York, 1974).Google Scholar
[21]Silkowski, R., “A star-shaped condition for stability of linear retarded functional differential equations”, Proc. Roy. Soc. Edinburgh Sect. A 83 (1979), 189–198.CrossRefGoogle Scholar
[22]Stech, H., “The effect of time lags on the stability of the equilibrium state of a population growth equation”, J. Math. Biol. 5 (1978), 115–130.CrossRefGoogle Scholar
[23]Stech, H., “The Hopf bifurcation: a stability result and application”, J. Math. Anal. Appl. 71 (1979), 525–546.CrossRefGoogle Scholar
[24]Walther, H.-O., “Existence of a non-constant periodic solution of a non-linear autonomous functional differential equation representing the growth of a single species population”, J. Math. Biol. 1 (1975), 227–240.CrossRefGoogle ScholarPubMed
[25]Wright, E. M., “A non-linear difference-differential equation”, J. Reine Angew. Math. 194 (1955), 66–87.CrossRefGoogle Scholar