Masayasu Mimura

Reaction–diffusion modelling of bacterial colony patterns

It is well known from experiments that bacterial species Bacillus subtilis exhibit various colony patterns. These are essentially classified into five types in the morphological diagram, depending on the substrate softness and nutrient concentration. (A) diffusion-limited aggregation-like; (B) Eden-like; (C) concentric ring-like; (D) disk-like; and (E) dense branching morphology-like. There arises the naive question of whether the diversity of colony patterns observed in experiments is caused by different effects or governed by the same underlying principles. Our research has led us to propose reaction–diffusion models to describe the morphological diversity of colony patterns except for Eden-like ones.

Spatial segregation in competitive interaction-diffusion equations

SummaryThe effect of cross-population pressure on the Volterra type dynamics for two competing species is investigated. On the basis of cross-diffusion induced instability, spatial segregation is studied. Spatially discrete models are also discussed. It is shown that this effect has a tendency to enhance the stability assuring coexistence of species. In continuous and discrete cases, time-dependent segregation processes are studied numerically.

On a diffusive prey-predator model which exhibits patchiness

Abstract Spatial heterogeneity (patchiness) in certain predator-prey situations has been observed even though their environment appears homogeneous. As a model mechanism to explain this patchiness phenomenon we propose a predator-prey interaction system with diffusive effects. We show that when the diffusion of the prey is small compared with that of the predator the non-linearity which we call a hump effect in the prey interaction, is a key mechanism for the system to exhibit, asymptotically in time, stable heterogeneity in a bounded domain with zero flux boundary conditions. The model can reasonably be applied to certain terrestrial plant-herbivore systems.

Aggregating pattern dynamics in a chemotaxis model including growth

A population model including diffusion, chemotaxis and growth is studied. Assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate, we derive a new equation to describe the time-evolution of the aggregating region of biological individuals and show the conditions for the existence and stability of radially symmetric equilibrium solutions of the equation, which indicate the aggregation of individuals.

Higher-dimensional localized patterns in excitable media

Abstract The excitable reaction-diffusion equation model of the form eτu t =e 2 ▿ 2 u+ƒ(u)–v , v t =▿ 2 v+u−γv is considered. When ƒ(u) is assumed to be of McKeans piecewise linear type, the interfacial approach can be applied to the stability of various localized patterns in higher-dimensional spaces. It is shown that a band-shaped localized pattern is destabilized into a zig-zag mode or a varicose mode and that a disk-shaped localized pattern is destabilized into static modes and, when τ is small, into an oscillatory mode like a “breather motion”. Numerical simulations are performed to confirm such destabilizations for more general nonlinear functions ƒ(u) .

Layer oscillations in reaction-diffusion systems

This paper considers a two-component system of reaction-diffusion equations involving two parameters

Multiple Solutions of Two-Point Boundary Value Problems of Neumann Type with a Small Parameter

\varepsilon

Interface growth and pattern formation in bacterial colonies

and

A picture of the global bifurcation diagram in ecological interacting and diffusing systems

\tau

Formation of colony patterns by a bacterial cell population

: \[ \varepsilon \tau u_1 = \varepsilon ^2 u_{xx} + f( u,v ),\qquad v_...