We study a non-linear age-structured population models for semalparous species. Semelparous species are those whose individuals reproduce only once in their lives and die afterwards. Using rigorous computer assisted methods based on Conley-Morse graphs and covering relations we showed that for given range of parameters almost all initial populations converge to a population consisted only of individuals at one age. Therefore long-time behavoiur of population depends only on one dimensional dynamics of a system on the boundary of the domain.
Over the past few decades, the spontaneous formation of patterns such as spatially periodic rolls, hexagonal cell structures, and spiral waves in spatially extended systems has attracted much attention. In the context of the modified Swift-Hohenberg PDE, some of these interesting interfaces can be modelled as modulated fronts, i.e. as waves that are time-periodic in an appropriate co-moving coordinate frame. Via the appropriate change of coordinates introduced in [Doelman, Sandstede, Scheel and Schneider. European J. of Appl. Math. 14 (1), 2003], it is known that computing the modulated fronts reduces to compute heteroclinic orbits between equilibria of a given system of second order ODEs. In this talk, we introduce a computational method to prove existence of some of these connecting orbits, hence leading to rigorous statements about co-existence of different type of non trivial patterns for the original PDE. The rigorous method combines linear splines, the parameterization method of invariant manifold, fixed point theory and interval arithmetics. This is joint work with J.B. van den Berg, J.-P. Lessard and J.D. Mireles James.
A method based on Chebyshev series to rigorously compute solutions of the Ginzburg-Landau equation is proposed. The idea is to recast solutions as fixed points of an operator defined on a Banach space of rapidly decaying Chebyshev coefficients and to combine analytic estimates and the contraction mapping theorem to show the existence of a unique genuine solution nearby an approximate solution. With this approach, it is possible to answer some open questions regarding the co-existence of non trivial solutions of the Ginzburg-Landau model of superconductivity. This is joint work with J.-P. Lessard (U. Laval).
In this talk, we introduce a rigorous computational method for periodic orbits of dissipative PDEs. The idea is to consider a space-time Fourier expansion of a periodic solution and to solve for its Fourier coefficients in a space of algebraically decaying sequences. The rigorous computation is based on the radii polynomials approach, which provide an efficient means of constructing a ball centered at a numerical approximation which contains a genuine periodic solution. We apply this method to show the existence of several periodic solutions in the Kuramoto-Sivashinsky equation. This is joint work with Marcio Gameiro (Sao Carlos, Brazil).