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Partial differential equations of chemotaxis and angiogenesis. (English) Zbl 0990.35014
In this paper the authors motivate the study of the PDE system ${\partial P\over \partial t} = D \nabla \cdot (P \nabla (\ln (P/\Phi(\omega)))),$ ${\partial\omega \over \partial t} = {\mathbf F}(P,\omega),$ and examine some of its properties. Here $$x \in \Omega \subset \mathbb{R}^n$$, $$t > 0$$, $$D>0$$, and the system is supplemented with appropriate initial and no-flux boundary conditions. $$P$$ is a population density and $$\omega$$ is an $$m$$-dimensional vector of control parameters.
The main applications of the system above, described in Section 1 of the paper, are in the modelling of chemotaxis, in which case $$P(x,t)$$ is typically the density of myxobacteria cells and $$\omega$$ is the concentration of the chemotactic signal produced by and reacted to by the bacteria. Alternatively, in a model of tumour angiogenesis, $$P(x,t)$$ stands for the density of endothelial cells, in which case $$\omega$$ has at least two components, the concentration of tumour angiogenesis factor(s) and of fibronectin.
In Section 2 the authors review the theory of reinforced random walks and, by passing to a continuum limit, derive from the master equations the above PDEs. This section is best read in conjunction with the work of H. G. Othmer and A. Stevens [SIAM J. Appl. Math. 57, 1044-1081 (1997; Zbl 0990.35128)]. After a survey of numerical simulations in Section 3, the authors consider in Section 4 exact solutions of the system for the case of $$\Omega$$ being one-dimensional, $$\omega$$ scalar, and particular choices of $$\Phi(\omega)$$ and $${\mathbf F}$$. A wealth of phenomena including finite time collapse and blow-up is discovered. In Section 5 the authors survey results on aggregation following H. A. Levine and B. D. Sleeman [SIAM J. Appl. Math. 57, 683-730 (1997; Zbl 0874.35047)]. In Section 6 existence of travelling wave and other similarity solutions is discussed, while in Section 7 implications of inhomogeneity in the medium through which the cells diffuse for transition probability rates in the master equation is considered.

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 92C17 Cell movement (chemotaxis, etc.) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 92C50 Medical applications (general)
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