Tohru Tsujikawa

Lower Estimate of the Attractor Dimension for a Chemotaxis Growth System

This paper estimates from below the attractor dimension of the dynamical system determined from a chemotaxis growth model which was presented by Mimura and Tsujikawa. It is already known that the dynamical system has exponential attractors and it is also known by numerical computations that the model contains various pattern solutions. This paper is then devoted to estimating the attractor dimension from below and in fact to showing that, as the parameter of chemotaxis increases and tends to infinity, so does the attractor dimension. Such a result is in a good correlation with the numerical results.

Travelling front solutions arising in the chemotaxis-growth model

We consider a bistable reaction-diffusion-advection system describing the growth of biological individuals which move by diffusion and chemotaxis. We use the singular limit procedure to study the dynamics of growth patterns arising in this system. It is shown that travelling front solutions are transversally stable when the chemotactic effect is weak and, when it becomes stronger, they are destabilized. Numerical simulations reveal that the destabilized solution evolves into complex patterns with dynamic network-like structures.

STATIONARY SOLUTIONS TO FOREST KINEMATIC MODEL

We continue the study of a mathematical model for a forest ecosystem which has been presented by Y. A. Kuznetsov, M. Y. Antonovsky, V. N. Biktashev and A. Aponina (A cross-diffusion model of forest boundary dynamics, J. Math. Biol. 32 (1994), 219–232). In the preceding two papers (L. H. Chuan and A. Yagi, Dynamical systemfor forest kinematic model, Adv. Math. Sci. Appl. 16 (2006), 393– 409; L. H. Chuan, T. Tsujikawa and A. Yagi, Aysmptotic behavior of solutions for forest kinematic model, Funkcial. Ekvac. 49 (2006), 427–449), the present authors already constructed a dynamical system and investigated asymptotic behaviour of trajectories of the dynamical system. This paper is then devoted to studying not only the structure (including stability and instability) of homogeneous stationary solutions but also the existence of inhomogeneous stationary solutions. Especially it shall be shown that in some cases, one can construct an infinite number of discontinuous stationary solutions. 2000 Mathematics Subject Classification. 35J60, 37L15, 37N25.

Exponential attractor for an adsorbate-induced phase transition model in non smooth domain

We improve our preceding result obtained in Tsujikawa and Ya gi [10]. We construct the similar exponential attractors to the same adsor bate-induced phase transition model as in [10] but in a convex domain by using the compac t smoothing property of corresponding nonlinear semigroup. In [10], the dom ain has been assumed to haveC3 regularity to ensure the squeezing property of semigroup.

Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system

This paper is concerned with stationary solutions of a reaction–diffusion-advection system arising in surface chemistry. Hildebrand et al (2003 New J. Phys. 5 61) have constructed stationary stripe (or spot) solutions of the system in the singular perturbation case and shown a numerical result that the set of stripe (or spot) solutions forms a saddle-node bifurcation curve with respect to a diffusion coefficient. In this paper, we introduce a shadow system in the limiting case that another diffusion and an advection coefficient tend to infinity. Furthermore we obtain the bifurcation structure of stationary solutions of the shadow systems in the one-dimensional case. This structure involves saddle-node bifurcation curves which support the above numerical result in Hildebrand et al (2003 New J. Phys. 5 61, figure 9). Our proof is based on the combination of the bifurcation, the singular perturbation and a level set analysis.

Propagation and Aggregation of Motile Cells of Escherichia coli Pattern

A concentric pulse by motile cells of Escherichia coli (E. coli) propagates and the cells aggregate to form self-organized patterns. We summarize experimental and numerical results on the self-organized pattern formation of E. coli to elucidate some aspects of its mechanism. Our presentation includes experiments on E. coli patterns, as well as numerical simulations on the basis of a reaction-diffusion-chemotaxis model. We find good agreement for one-dimensional propagating fronts in observation and simulation. However, corresponding results for two-dimensional circular bacterial clusters have still not been obtained.