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.

This paper continues the study of the initial value problem of a chemotaxisgrowth system. In the previous paper [13], we have handled the case when the sensitivity function χ(ρ) is regular. In this paper we are concerned with the case when the function has singularity at ρ = 0 like χ(ρ) = log ρ or − 1 ρ. We verify global existence of solutions and discuss some asymptotic behaviour of solutions.

Abstract. This paper is concerned with the initial value problem for some di¤usion system which describes the process of a pattern formation of biological individuals by chemotaixis and growth. In the paper Osaki et al. [13], exponential attractors have been constructed for the dynamical system determined by this problem. The exponential attractor is one of limit sets which is a positively invariant compact set with finite fractal dimension and which attracts every trajectory in an exponential rate. In this paper we study another feature of exponential attractors, that is we show that the approximate solution also gets close to the exponential attractor in an exponential rate and remains in its neighborhood forever. Our methods are available to any other exponential attractors determined by interaction-di¤usion systems.

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