One-day workshop on pattern dynamics appearing in reaction-diffusion system

The sign of the traveling wave speed in the two species Lotka-Volterra competition-diffusion system under strong competition provides 

important information about the competitive advantage of the species. To study the sign problem of the speed, we propose a mini-max characterization 

for the coefficients in the reaction terms when the wave speed is zero. 

We consider bounded and unbounded traveling wave solutions for one-dimensional reaction-diffusion equation. We show a Hadeler-Rothe type variational characterization of the traveling wave speed for several types of nonlinearity by phase plane method.

Using the variational characterization, we give one example of the approximation value of traveling wave speed for bistable case. This talk is based on a joint work with Hirokazu Ninomiya (Meiji University).

Bacterial cell populations exhibit diverse growth morphologies and collectively form a robust system that can withstand environmental fluctuations. The diversity of macroscopic spatiotemporal patterns and flexible environmental responses in morphogenesis are supported by a variety of cellular states. Therefore, to understand the morphogenesis of bacterial populations, it is necessary to construct and analyze multilevel mathematical models that connect the cellular and tissue levels. Here we propose two multilevel modeling methods.

Some analytical and numerical results are presented for pattern formation properties associated with novel types of reaction-diffusion (RD) systems that involve the coupling of bulk diffusion in the interior of a multi-dimensional spatial domain to nonlinear processes that occur either on the domain boundary (bulk-membrane models) or within localized compartments or ``cells'' (bulk-cell models) that are confined within the domain. For a class of bulk-membrane system in the unit disk, a weakly nonlinear analysis is used to characterize Turing and Hopf bifurcations that can arise from the linearization around a radially symmetric, but spatially non-uniform, steady-state of the bulk-membrane system. For bulk-cell models, with dynamically active intracellular kinetics, we show that

synchronized intracellular oscillations can occur for a collection of spatially segregrated cells that are coupled via a scalar bulk diffusion field. Applications of bulk-cell systems to some specific biological problems, including Turing pattern formation and quorum sensing, are outlined and some open problems in this area are discussed.

In this talk, we revisit the speed selection problem for minimal traveling wave speed in scalar reaction-diffusion equations with monostable nonlinearity. We will discuss sufficient and necessary conditions that determine the linear or nonlinear selection mechanisms. 

The presentation is based on joint work with Dongyuan Xiao and Maolin Zhou.

We consider bistable reaction-diffusion equations in funnel-shaped domains of R^N made up of straight parts and conical parts with positive opening angles. We study 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, by proving especially some new Liouville type results on stable solutions of semilinear elliptic equations in the whole space R^N . 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 openingangle α 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; α) for which the propagation is complete. This is a joint work with Francois Hamel.