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Linearized oscillations in population dynamics. (English) Zbl 0634.92013
A linearized oscillation theorem due to the authors and A. Meimaridou [Q. Appl. Math. 45, 155-164 (1987; Zbl 0627.34076)] and an extension of it are applied to obtain the oscillation of solutions of several equations which have appeared in population dynamics. They include the logistic equation with several delays, Nicholson’s blowflies model as described by W. S. C. Gurney, S. P. Blythe and R. M. Nisbet [Nature, Lond. 287, 17-21 (1980)] and the Lasota-Wazewska model of red blood cell supply in an animal. We also developed a linearized oscillation result for difference equations and applied it to several equations taken from the biological literature.
MSC:
92D25Population dynamics (general)
39A10Additive difference equations
34K99Functional-differential equations
34C15Nonlinear oscillations, coupled oscillators (ODE)
34K20Stability theory of functional-differential equations
References:
[1]Gopalsamy, K. 1986. ”Oscillations in a Delay-logistic Equation.”Quart. appl. Math. XLIV, 447–461.
[2]Gurney, W. S. C., S. P. Blythe and R. M. Nisbet. 1980. ”Nicholson’s Blowflies Revisited.”Nature, Lond. 287, 17–21. · doi:10.1038/287017a0
[3]Hunt, B. R. and J. A. Yorke. 1984. ”When all Solutions of x ' =- i=1 n q i (t)x(t-T i (t)) Oscillate.”J. diff. Eqns 53, 139–145. · Zbl 0571.34057 · doi:10.1016/0022-0396(84)90036-6
[4]Kakutani, S. and L. Markus. 1958. ”On the Non-linear Difference-differentiual Equation Y ˙(t)=A-By(t-τ)y(t) .”Contribution to the Theory of Nonlinear. Oscillations, Vol. 4, pp. 1–18. Princeton University Press.
[5]Kaplan, J. L. and J. A. Yorke. 1977. ”On the Nonlinear Differential Delay Equation x ˙(t)=-f(x(t),x(t-1)) .”J. diff. Eqns. 23, 293–314. · Zbl 0337.34067 · doi:10.1016/0022-0396(77)90132-2
[6]Kulenović, M. R., G. Ladas and A. Meimaridou. 1987. ”On Oscillation of Nonlinear Delay Differential Equations.”Quart. appl. Math. XLV, 155–164.
[7]Ladas, G. and I. P. Stavroulakis. 1982. Oscillations Caused by Several Retarded and Advanced Arguments.”J. diff. Eqns 44, 134–152. · Zbl 0477.34050 · doi:10.1016/0022-0396(82)90029-8
[8]May, R. M. 1975. ”Biological Populations Obeying Difference Equations: Stable Points, Stable Cycles and Chaos.”J. theor. Biol. 51, 511–524. · doi:10.1016/0022-5193(75)90078-8
[9]– and G. F. Oster. 1976. ”Bifurcations and Dynamic Complexity in Simple Ecological Models.”Am. Nat. 110, 537–599.
[10]Wright, E. M. 1955. ”A Nonlinear Difference-differential Equation.”J. Reine Angew. Math. 194, 66–87. · Zbl 0064.34203 · doi:10.1515/crll.1955.194.66