Lecture Room B, 4th Floor, The 3rd General Building, NTHU

(清華大學綜合三館四樓 B演講室)

Bifurcation Analysis for Structured Population Dynamics Models

Pierre Magal (University of Bordeaux)

Abstract:

The goal of this presentation is to present a bifurcation theory for structured population dynamics models corresponding to the recent book [8].

We will start by introducing the abstract problem for an example of age structured models and delay differential equation as an non-densely Cauchy problem. Next we will present some result on the existence and smoothness of the center manifold for a class of abstract non densely defined Cauchy problem [1]. We will turn to the existence of Hopf bifurcation [2] and the stability of the bifurcating periodic orbits [3]. The center manifold being defined only implicitly, in order to compute its Taylor expansion locally around an equilibrium, we will present a normal form theory [4]. The main result is to prove that it is effectively possible to simplify the hyperbolic part of the system by making a change of variable.

We will conclude our presentation with some example of application. We will first reconsider the first example age structured model for which the normal form has been studied in [6]. Then we will turn to an example of age structured with space [5]. We will conclude the presentation with an example of Bogdanov-Takens bifurcation [7].

Reference:

[1] P. Magal and S. Ruan (2009), Center Manifolds for Semilinear Equations with Non-dense Domain and Applications to Hopf Bifurcation in Age Structured Models, Memoirs of the American Mathematical Society 202, no. 951.

[2] Z. Liu, P. Magal, and S. Ruan (2011), Hopf Bifurcation for non-densely defined Cauchy problems, Zeitschrift fur Angewandte Mathematik und Physik, 62, 191-222.

[3] Z. Liu, P. Magal and S. Ruan (2012), Center-unstable manifold theorem for non-densely defined Cauchy problems, and the stability of bifurcation periodic orbits by Hopf bifurcation, Canadian Applied Mathematics Quarterly (20)2, 135-178.

[4] Z. Liu, P. Magal and S. Ruan (2014), Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differential Equations 257, 921-1011.

[5] Z. Liu, P. Magal and H. Tang (2015), Hopf bifurcation for a spatially and age struc-tured population dynamics model, DCDS B, 20 (6), 1735-1757.

[6] J. Chu, Z. Liu, P. Magal and S. Ruan (2016), Normal Forms for an Age Structured Model, Journal of Dynamics and Differential Equations 28, 733-761.

[7] Z. Liu, P. Magal and D. Xiao (2016), Bogdanov-Takens bifurcation in a predator prey model with age structure, Zeitschrift fuer Angewandte Mathematik und Physik 67:137.

[8] P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer-Verlag New-York, to appear.