Partial differential equations of chemotaxis and angiogenesis.

*(English)*Zbl 0990.35014In 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.

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.

Reviewer: Michael Grinfeld (Glasgow)

##### 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) |

##### Keywords:

no-flux boundary conditions; reinforced random walks; exact solutions; finite time collapse; blow-up; travelling wave; similarity solutions
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\textit{B. D. Sleeman} and \textit{H. A. Levine}, Math. Methods Appl. Sci. 24, No. 6, 405--426 (2001; Zbl 0990.35014)

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