## Section: New Results

### New results in dynamical systems theory applied to population dynamics

Participants : Arnaud Ducrot, Pierre Magal.

#### Size structured populations and integrated semigroup theory

A joint work of P. Magal, J. Chu, S. Ruan and A. Ducrot studied the possibility to have Hopf bifurcations for some size structured model with random noise by using the theory of integrated semigroup. After reformulating the problem in term of abstract Cauchy problem, the authors studied the characteristic equation associated to the problem to study bifurcations in the dynamical system. This work provides a fine description of the resulting dynamics. This study leads to the appearance of new bifurcation phenomena. It has been published by Journal of Differential Equations [27] .

In a joint work of P. Magal and S. Ruan, the existence and smoothness of the center manifold for abstract non-densely defined Cauchy problems is investigated. This is allows in particular to study the existence the bifurcation properties of the various class of infinite dimensional dynamical system, such as delay differential equations, age-structured models, and some class of parabolic problems with non-linear and non-local boundary conditions. In this work an application to the existence of Hopf bifurcation to a class of age structured model has been studied. The work has been published in 2009 the Memoirs of American Mathematical Society.

In a joint work of A. Ducrot, P. Magal and K. Prevost we investigate some mathematical properties of a class of linear abstract Cauchy problem involving almost sectorial operators by using the theory of integrated semigroups. Some results of well posedness as well as some perturbation result of linear operators are obtained. This work will appear in Journal of Evolution Equations [30] .