We consider a bistable reaction-diffusion-advection system as a population model describing the growth of biological individuals which move by diffusion and chemotaxis. Assuming that the diffusion rate and the chemotactic rate are both very small with the growth rate, we introduce the limiting system by using the singular limit procedure and study the dynamics of growth patterns arising in this system. It is shown that in two dimensional domain planar traveling front solutions are transversally stable when the chemotaxis effect is weak and, when it becomes stronger, they are destabilized. But, these results depend on the sign of the second derivative of the sensitive function. Numerical simulations reveal that the destabilized solution evolves into complex patterns with dynamic spot structures and so on. To explain the appearance of these complex patterns, we obtain the estimate of the attractor dimension from below and it tends to infinity as the chemotaxis effect is increasing.

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The purpose of this article is to describe some properties of solutions to the Keller-Segel model and a simplified Keller-Segel model. The Keller-Segel model was introduced to describe chemotactic aggregation of cellular slime molds, and the simplified Keller-Segel model was also introduced to describe the same phenomenon. Moreover, the simplified Keller-Segel model describes the gravitational interaction of particles. The solutions to those models represent the density of cells or particles. Then, the solutions are functions of spatial and time variables. Those systems have blowup solutions. We say that a solution blows up at finite time, if maximum value of the solution in the spatial domain tends to infinity as time variable tends to the finite time. We think that the blowup corresponds to aggregation of the cells or the particles. In this article, we describe some results about blowup solutions and the relation between those results and the biological phenomenon.

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We consider the following reaction-diffusion equation (KS)_m. Throughout this article, we assume that N≥1, and that m, q, and γ are the constants satisfying m>1, q≥2, γ≥0, respectively. This equation is often called the Keller-Segel model describing the motion of the chemotaxis molds. Here u(x,t) and v(x,t) denote the density of amoebae and the concentration of the chemo-attractant, respectively. In the semi-linear case i.e., m=1, up to the present there have been many results for q=2. As for the quasi-linear case m>1, the balance between diffusion strength m and non-linearity effect q plays an important role for existence of global solutions to (KS)_m. In this article, we introduce recent results on global existence and blow-up in a finite time of solution of (KS)_m with the critical exponet q=m+2/N. Furthermore, we introduce a significant feature of degenerate one such as the property of finite speed of propagation for solutions, which is different phenomena from that of non-degenerate one. In addition, we characterize the interface curve as the solution of a certain ordinary differential equation associated with (KS)_m. We also state the so-called ε-regularity theorem of (KS)_m. Finally, we give several open problems.

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In the field of Life and Biological Science, the understanding of tumour growth has made enormous progress in recent years. In this paper we study a couple of mathematical models of tumour angiogenesis with rigorous background to gain a mathematical characterization of tumour angiogenesis. One is proposed by Othmer and Stevens and developed by Levine and Sleeman, which has been arised in the theory of reinforced random walk. Another is considered by Anderson and Chaplain, based on a number of pathological and physiological researches. Mathematical analysis and characterization of these models are done by showing existence of global in time solution of the models and the asymptotic behaviour of them. Then we attempt to clarify the gap and equivalency between these models mathematically and investigate basic and essential mathematical properties common to them and pull out the relevance of these properties to angiogenesis.

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We are concerned with numerical methods for the Keller-Segel system that describes the aggregation of slime molds resulting from their chemotactic features. Whereas the system has positivity and mass conservation properties, it is not certain that numerical schemes for the system preserve these conservation properties. In the present article, we review two conservative numerical schemes proposed by the author and discuss how to choose the time increments and the space meshes in order to realize those conservation properties. The first one is the finite-difference method that makes use of the semi-implicit time discretization with the time-increment control and the upwind difference approximation. The second is the finite-element method that is an application of Baba-Tabata's conservative upwind finite element. Conservative properties are proved via the theory of M-matrices. Error analysis of the finite-element scheme is also summarized. In particular, we have error estimates with explicit convergence rates by virtue of the analytical semigroup theory.

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