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Oscillation and stability in a genotype selection model with several delays. (English) Zbl 0862.39006

The difference equation \(y_{n+1}= y_nE_{kn}/(1+y_n(E_{kn}-1))\) with the abbreviation \[ E_{kn}= \exp\Biggl(\beta\Biggl(1-\sum^k_{i=0} \alpha_iy_{n-i}\Biggr)\Biggr) \] is studied as a model for frequency dependent natural selection, where \(\beta\), \(\alpha_k\) are positive and \(\alpha_0,\dots,\alpha_{k-1}\) nonnegative numbers. Under certain conditions, every solution oscillates about the equilibrium \(1/\alpha\) with \(\alpha=\sum^k_{i=0}\alpha_i\) and \(1/\alpha\) is stable, respectively. In the special case \(k=0\), conditions are given such that \(1/\alpha\) is a global attractor of all positive solutions, and chaotic behavior appears, respectively.
Reviewer: L.Berg (Rostock)

MSC:

39A11 Stability of difference equations (MSC2000)
92D25 Population dynamics (general)
39A12 Discrete version of topics in analysis
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References:

[1] Cooke, K.L. 1990. ”Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments”. Vol. 633, IMA Preprint series.
[2] Fisher M.E., Bull. Math. Biol. 41 pp 861– (1979)
[3] Grove E.A., Quart. Appl. Math. 52 pp 499– (1994)
[4] Grove E.A., Differential Equations and Dynamical Systems 1 pp 35– (1993)
[5] Grove E.A., Dyn. Syst. Appl. 2 pp 243– (1993)
[6] Kocic V.L., Hiroshima Math. J. 22 pp 95– (1992)
[7] Kocic V.L., In Proceedings of the first world Congress of Nonlinear Analysts (1992)
[8] Kocic, V.L. and Ladas, G. 1993. ”Global Asymptotic Behavior of Nonlinear Difference Equations of Higher order with Applications”. Dordrecht: Kluwer Academic Publishers. · Zbl 0787.39001
[9] May R.M., Chaotic Behaviour of Determintistic Systems
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