Pair formation models with maturation period.

*(English)*Zbl 0808.92024This paper deals with a model of demography, taking into account the formation of pairs, and a maturation period from the time a pair has been formed and the time when it gives birth to new individuals. The model is a set of three equations describing the evolution of single males, single females and couples. No crowding effect is considered, so that birth and death effects are counted linearly. Nonlinearity occurs from the pair formation term which is supposed to be homogeneous of degree one.

An age-dependent version of the equation is presented in section 2 of the paper which, under some simplifying assumptions on the coefficients, leads to a system of equations of the above type satisfied by the respective total single male or female populations and couples. Analysis of the directional stability of the nontrivial steady-state (the asynchronous exponential stability) is then performed in sections 3 to 5 of the paper. This extends previous work by the author on homogeneous ordinary differential equations [see notably J. Differ. Equations 95, No. 1, 183-202 (1992; Zbl 0747.34030)].

There are two major problems faced here: 1) in contrast to the O.D.E. case, normalizing the solutions (in the vector case) by dividing them by a kind of norm does not lead to the same equation. The equation obtained is not even a delay equation. But, the normalized equation allows the author to show that the original equation has a simple dominant eigenvalue. 2) As in the O.D.E. case, the linear equation to be analyzed does not yield a positive semigroup with respect to the usual order. But, the author shows that it may be turned into a positive semigroup with respect to a suitable order and concludes stability from this.

An age-dependent version of the equation is presented in section 2 of the paper which, under some simplifying assumptions on the coefficients, leads to a system of equations of the above type satisfied by the respective total single male or female populations and couples. Analysis of the directional stability of the nontrivial steady-state (the asynchronous exponential stability) is then performed in sections 3 to 5 of the paper. This extends previous work by the author on homogeneous ordinary differential equations [see notably J. Differ. Equations 95, No. 1, 183-202 (1992; Zbl 0747.34030)].

There are two major problems faced here: 1) in contrast to the O.D.E. case, normalizing the solutions (in the vector case) by dividing them by a kind of norm does not lead to the same equation. The equation obtained is not even a delay equation. But, the normalized equation allows the author to show that the original equation has a simple dominant eigenvalue. 2) As in the O.D.E. case, the linear equation to be analyzed does not yield a positive semigroup with respect to the usual order. But, the author shows that it may be turned into a positive semigroup with respect to a suitable order and concludes stability from this.

Reviewer: O.Arino (Pau)

##### MSC:

92D25 | Population dynamics (general) |

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

34K20 | Stability theory of functional-differential equations |

##### Keywords:

two-sex problem; age-structure; delay equation; homogeneous equation; spectral bound; irreducible matrix; model of demography; formation of pairs; maturation period; directional stability; nontrivial steady-state; asynchronous exponential stability; dominant eigenvalue; positive semigroup##### Citations:

Zbl 0747.34030
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DOI

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##### References:

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