The effects of state-dependent time delay on a stage-structured population growth model.

*(English)*Zbl 0777.92014This paper continues the study of a model initiated by W. G. Aiello and H. I. Freedman [Math. Biosci. 101, No. 2, 139-153 (1990; Zbl 0719.92017)], of a two-stage population with density-dependent time delay. W. G. Aiello, H. I. Freedman and J. Wu [SIAM J. Appl. Math. 52, No. 3, 855-869 (1992; Zbl 0760.92018)] showed that, under certain conditions, the problem is well-posed in the sense that solutions with nonnegative initial values stay nonnegative. Here, the main issue is on stability properties, possible changes of stability and bifurcation, and attractivity of equilibria.

Let us give a brief outline of the paper: Section 2 shows that no Hopf bifurcation can arise, in the sense of that the characteristic equation near any strictly positive equilibrium never has imaginary roots. Section 3 characterizes the onset of linearized instability in terms of the coefficients. It is shown that instability goes together with the creation of multiple equilibria. Section 4 considers the attractivity region of each equilibrium. Most of the analysis is based on the properties of a scalar function \(H\), whose fixed points yield the equilibria. In particular, Lemma 6 states that the attractivity region of any equilibrium can be determined in terms of the attractivity region of the corresponding fixed point of \(H\).

Let us give a brief outline of the paper: Section 2 shows that no Hopf bifurcation can arise, in the sense of that the characteristic equation near any strictly positive equilibrium never has imaginary roots. Section 3 characterizes the onset of linearized instability in terms of the coefficients. It is shown that instability goes together with the creation of multiple equilibria. Section 4 considers the attractivity region of each equilibrium. Most of the analysis is based on the properties of a scalar function \(H\), whose fixed points yield the equilibria. In particular, Lemma 6 states that the attractivity region of any equilibrium can be determined in terms of the attractivity region of the corresponding fixed point of \(H\).

Reviewer: O.Arino (Pau)

##### MSC:

92D25 | Population dynamics (general) |

92D40 | Ecology |

34K20 | Stability theory of functional-differential equations |

##### Keywords:

state-dependent time delay; stage-structured population growth; attractivity of equilibria; Hopf bifurcation; characteristic equation; imaginary roots; linearized instability; multiple equilibria; fixed points
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\textit{Y. Cao} et al., Nonlinear Anal., Theory Methods Appl. 19, No. 2, 95--105 (1992; Zbl 0777.92014)

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##### References:

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