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Bubbles for a Class of Delay Differential Equations

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Abstract

We analyze the effect of increasing mortality in usual delayed recruitment models of the form x′(t) = −δ x(t) + f (x(t − τ)). We consider constant effort harvesting, and discuss the phenomenon of bubbling, which appears when for parameters δ in an interval I there exists a unique positive stationary point K(δ) in the phase space \({C([-\tau,0],\mathbb{R})}\), and there exist δ 1 < δ 2 in I such that K(δ) is locally stable for \({\delta\in I{\setminus}[\delta_1,\delta_2]}\), and K(δ) is unstable for \({\delta\in(\delta_1,\delta_2)}\). We give a definition of bubbling in a more abstract setting, and show that it naturally appears in a variety of equations. For some nonlinearities rigorous proofs are available for the description of bubbles, for other nonlinearities we give numerical results to indicate the phenomenon of bubbling.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720, Szeged, Hungary

    Tibor Krisztin

  2. Departamento de Matemática Aplicada II, E.T.S.I. Telecomunicación, Universidad de Vigo, 36310, Vigo, Spain

    Eduardo Liz

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  1. Tibor Krisztin
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  2. Eduardo Liz
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Correspondence to Eduardo Liz.

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Krisztin, T., Liz, E. Bubbles for a Class of Delay Differential Equations. Qual. Theory Dyn. Syst. 10, 169–196 (2011). https://doi.org/10.1007/s12346-011-0055-8

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