Spatial Dynamics of Some Reaction-Diffusion Models in Heterogeneous Media

Title:  Spatial Dynamics of Some Reaction-Diffusion Models in Heterogeneous Media

Supervisor: Prof. François Hamel

Co-supervisor: Prof. Xing Liang

Guillemette Chapuisat	Aix-Marseille Université
Arnaud Ducrot   (rapporteur)	Université Le Havre Normandie
Francois Hamel (supervisor)	Aix-Marseille Université
Xing Liang    (co-supervisor)	University of Science and Technology of China
Yuan Lou	Ohio State University & Shanghai Jiao Tong University
Philippe Souplet	Université Sorbonne Paris Nord
Xuefeng Wang   (rapporteur)	The Chinese University of Hong Kong-Shenzhen
Yaping Wu	Capital Normal University, China

Abstract: We first study the effect of the geometry of the underlying domain on propagation phenomena of bistable equations. We consider bistable equations in funnel-shaped domains of \mathbb{R}^N made up of straight parts and conical parts with positive opening angles. We investigate the large time dynamics of entire solutions emanating from a planar front in the straight part of such a domain and moving into the conical part. We show a dichotomy between blocking and spreading. We also show that any spreading solution is a transition front having a global mean speed, which is the unique speed of planar fronts, and that it converges at large time in the conical part of the domain to a well-formed front whose position is approximated by expanding spheres. Moreover, we provide sufficient conditions on the size R of the straight part of the domain and on the opening angle $\alpha$ of the conical part, under which the solution emanating from a planar front is blocked or spreads completely in the conical part. We finally show the openness of the set of parameters $(R,\alpha)$ for which the propagation is complete. Then, we consider a one-dimensional patchy model made up of a succession of reaction-diffusion equations in homogeneous media, where novel interface matching conditions are introduced to reflect the movement behavior of individuals when they come to the edge of a patch. Firstly, we consider this model in a spatially periodic environment. We  establish the well-posedness rigorously for the Cauchy problem. We then investigate the spreading properties  and the existence of pulsating traveling waves in the positive and negative directions. Secondly, we study a simplified two patchy model in \mathbb{R} which consists of two homogeneous habitats. Our interest is to investigate propagation dynamics  of solutions to the Cauchy problem with compactly supported initial data in different reaction combinations. We first derive the spreading properties of solutions in the KPP-KPP case. Then, in the KPP-bistable framework we investigate different conditions under which  the solutions of the Cauchy problem may show different dynamics in the bistable patch, that is, blocking, virtual blocking or propagation. In particular, when propagation occurs, a global stability result is proved. The results in the KPP-bistable frame can also be extended to the bistable-bistable setting with certain hypotheses.