Abstract : Reaction-diffusion equations have been a powerful tool in studying population dynamics since the seminal works of Fisher, Kolmogorov-Petrovsky-Piskunov, Turing, and many others. In recent years many important questions from ecology, such as habitat fragmentation and shifting environment change, and life sciences, such as tumor growth, required new mathematical models and gave rise to challenging problems in analysis. This mini-symposium aims to showcase some recent development in the theory of reaction-diffusion equations and its applications to some emerging ecological and evolutionary questions.
[00123] Propagation direction of the traveling wave in the Lotka-Volterra competition-diffusion system
Format : Talk at Waseda University
Author(s) :
Chiun-Chuan Chen (National Taiwan University)
Abstract : We consider the two species Lotka-Volterra competition-diffusion system and assume that the species are in strong competition. It is well-known that up to a translation, there exists a unique monotone traveling wave solution. The sign of the wave speed provides significant information about which species can wipe out the other. However, it is a difficult problem to determine the sign of the speed. In this talk, we will present some estimates for the relation between the wave speed and the growth rates of the species. With these estimates, we are able to determine the speed sign under some suitable conditions.
[00124] Principal eigenvalue problem with large advection in 2 dimensional case
Format : Talk at Waseda University
Author(s) :
Maolin Zhou (Nankai University)
Abstract : In this talk, we will discuss the recent progress on the limit of the principal eigenvalue of some second order operators with large advection: 1. degenerate case; 2. parabolic case (1-d); 3. elliptic case (2-d).
[00099] Some game theoretical models in population dynamics
Format : Talk at Waseda University
Author(s) :
Idriss Mazari-Fouquer (CEREMADE, Paris Dauphine Université, PSL)
Abstract : We review several recent contributions aimed at providing a better understanding of optimal fishing strategies in spatial ecology: how should one fish in order to maximise the fishing output? If several players are fishing, can they reach an equilibrium situation? Formulated in terms of optimal control problems and Nash equilibria properties, these problems are amenable to mathematical analysis, and we present a variety of related results. Joint work with D. Ruiz-Balet.
[00125] Free boundary problem for the curve shortening flow with driving force in undulating cylindrical domains
Format : Talk at Waseda University
Author(s) :
Ryunosuke Mori (Meiji University)
Abstract : We study a free boundary problem for the curve shortening flow with driving force in a two-dimensional cylindrical domain with periodically undulating boundary. We consider the large time behavior of the graph-like solution. The classical solution generally does not exist in global time, since the solution may touch the boundary of the domain in finite time. To overcome this difficulty, we prove that the set of singularities of the solution is discrete under some assumptions.
[00120] Front Propagation in the Shadow Wave-Pinning Model
Format : Talk at Waseda University
Author(s) :
Daniel Gomez (University of Pennsylvania)
King-Yeung Lam (The Ohio State University)
Yoichiro Mori (University of Pennsylvania)
Abstract : In this paper we consider a non-local bistable reaction-diffusion equation arising from cell polarization. A typical solution of this model exhibits an interface with velocity regulated by the total mass. The feedback between mass-conservation and bistablity causes the interface to approach a fixed limit. In the limit of a small diffusivity $\varepsilon^2\ll 1$, we prove that for any $0<\gamma<1/2$ the interface can be estimated within $O(\varepsilon^\gamma)$ of the location as predicted using formal asymptotics.
[00109] Propagation speeds in a shifting environment
Format : Talk at Waseda University
Author(s) :
Thomas Giletti (University of Lorraine)
Abstract : I will discuss the asymptotic behavior of solutions of reaction-diffusion equations with shifting heterogeneity. Such a situation arises in the modeling of population dynamics under an environmental change, due to global warming or the invasion by competing species. Two situations will be considered, depending on whether the reaction or the diffusion is heterogeneous. We will see that the heterogeneity may modify the nature of the propagation by inducing some unexpected threshold or acceleration phenomena.
[00067] Lotka-Volterra competition-diffusion system: the critical competition case
Format : Talk at Waseda University
Author(s) :
Dongyuan XIAO (Meiji University)
Matthieu Alfaro (Universite de Rouen Normandie)
Abstract : We consider the competition system in the so-called critical competition case. The associated ODE system then admits infinitely many equilibria. We first show the non-existence of ultimately monotone traveling waves. Next, we study the large-time behavior of the solution of the Cauchy problem with a compactly supported initial datum and provide a sharp description of the profile of the solution.
[00080] Coexistence of strains in some reaction-diffusion systems for infectious disease
Format : Talk at Waseda University
Author(s) :
Lou Yuan (Shanghai Jiao Tong University)
Rachidi Salako (University of Nevada at Las vegas)
Abstract : We study the global dynamics of some reaction-diffusion systems for multiple strains and investigate how the coexistence of strains is impacted by the movement of populations and spatial heterogeneity of the environment. For the case of two strains, general conditions for the existence, uniqueness and stability of coexistence steady states are found. Surprisingly, when there is no coexistence of strains, it is possible for the weak strain to be dominant for intermediate diffusion rates, in strong contrast to small and large diffusion cases where the weak strain goes extinct.
