# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The effects of state-dependent time delay on a stage-structured population growth model. (English) Zbl 0777.92014

This 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 $H$.

Reviewer: O.Arino (Pau)
##### MSC:
 92D25 Population dynamics (general) 92D40 Ecology 34K20 Stability theory of functional-differential equations