Yihong Du est Professeure à l'University of New England, Australie.
Titre : Trichotomy dynamics of a free boundary model for biological invasion
Résumé : It is well known that the reaction-diffusion equation (u_t=du_{xx}+f(u)) with compactly supported nonnegative initial functions exhibits trichotomy dynamics for bistable and combustion type (f(u)) ([Zlatos 2006], [Du-Matano 2010]). The same is true for the corresponding Stefan type free boundary problems [Du-Lou 2015]. In this talk, I’ll report some recent work that reveals a rather different type of trichotomy for this reaction-diffusion equation under a new set of free boundary conditions, arising as a model for the invading dynamics of a species with density (u(t,x)) and evolving range ([0,h(t)]).
The evolution of the free boundary (x=h(t)) is governed by (h'(t)=-(d/\delta)u_x(t,h(t))) and (u(t,h(t))=\delta\in(\theta_f,1)), with (\theta_f\in[0,1)) uniquely determined by (f), which allow (h(t)) to advance as well as to retreat when time increases. At the fixed boundary (x=0), (u(t,0)=\delta_0\geq 0) represents an intervention. Here (f(u)) may be of monostable, bistable, or combustion type.
In the biologically interesting case (\delta_0<\delta), the long-time dynamics has exactly three scenarios: successful spreading, finite-time vanishing, and a transition state characterized by convergence to the unique stationary solution of the free boundary problem. A notable feature of the model is that it does not enjoy the usual order-preserving property, which is intrinsically linked to several new phenomena.
The talk is based on joint work with Wenjie Ni, Hongkai Cao and Xiaoyan Zhang.