Last modified: 2014-03-28

#### Abstract

A species is called semelparous if its specimen reproduces only once in the lifetime, and usually dies afterwards. We consider a population such that individuals may give birth only at a given age. Discrete-time models of semelparous population have been intensively studied recently, because they have unexpected asymptotic properties, such as the extinction of all but one year classes. It is an interesting question if a continuous time age-structured model may exhibit similar properties.

We present a non-linear McKendrick-type age-structured model, given by a linear partial differential equation with a nonlinear boundary condition. Properties of measure-valued periodic solutions of this system are investigated. We observe that there exists a unique nonnegative stationary distribution which is often unstable. We investigate also the behavior of classical solutions that converge to periodic degenerated Dirac measure solutions, which means that the population asymptotically consists of individuals at the same age. Such a phenomenon is observed in nature in some insects populations. The model is compared to discrete-time nonlinear Leslie models.

References:

- J. M. Cushing,
*Three stage semelparous Leslie models*, J. Math. Biol.**59**(2009), 75-104. - R. Rudnicki and R. Wieczorek,
*Asymptotic analysis of a semelparous species model*, Fundamenta Informaticae**103**(2010), 219-233. - R. Rudnicki and R. Wieczorek,
*On a nonlinear age-structured model of semelparous species*,*submitted*.