### Abstract

The problem of alignment of cells (or other objects) that interact in an angle-dependent way was described in Mogilner and Edelstein-Keshet (1995). In this sequel we consider in detail a special limiting case of nearly complete alignment. This occurs when the rotational diffusion of individual objects becomes very slow. In this case, the motion of the objects is essentially deterministic, and the individuals or objects tend to gather in clusters at various orientations. (Numerical solutions show that the angular distribution develops sharp peaks at various discrete orientations.) To understand the behaviour of the deterministic models with analytic tools, we represent the distribution as a number of δ-like peaks. This approximation of a true solution by a set of (infinitely sharp) peaks will be referred to as the peak ansatz. For weak but nonzero angular diffusion, the peaks are smoothed out. The analysis of this case leads to a singular perturbation problem which we investigate. We briefly discuss other applications of similar techniques.

Original language | English (US) |
---|---|

Pages (from-to) | 811-842 |

Number of pages | 32 |

Journal | Journal of Mathematical Biology |

Volume | 34 |

Issue number | 8 |

State | Published - 1996 |

### Fingerprint

### Keywords

- Orientation selection
- Parallel cells
- Peak ansatz
- Total alignment

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

### Cite this

*Journal of Mathematical Biology*,

*34*(8), 811-842.

**Selecting a common direction II. Peak-like solutions representing total alignment of cell clusters.** / Mogilner, Alexander; Edelstein-Keshet, Leah; Ermentrout, G. Bard.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 34, no. 8, pp. 811-842.

}

TY - JOUR

T1 - Selecting a common direction II. Peak-like solutions representing total alignment of cell clusters

AU - Mogilner, Alexander

AU - Edelstein-Keshet, Leah

AU - Ermentrout, G. Bard

PY - 1996

Y1 - 1996

N2 - The problem of alignment of cells (or other objects) that interact in an angle-dependent way was described in Mogilner and Edelstein-Keshet (1995). In this sequel we consider in detail a special limiting case of nearly complete alignment. This occurs when the rotational diffusion of individual objects becomes very slow. In this case, the motion of the objects is essentially deterministic, and the individuals or objects tend to gather in clusters at various orientations. (Numerical solutions show that the angular distribution develops sharp peaks at various discrete orientations.) To understand the behaviour of the deterministic models with analytic tools, we represent the distribution as a number of δ-like peaks. This approximation of a true solution by a set of (infinitely sharp) peaks will be referred to as the peak ansatz. For weak but nonzero angular diffusion, the peaks are smoothed out. The analysis of this case leads to a singular perturbation problem which we investigate. We briefly discuss other applications of similar techniques.

AB - The problem of alignment of cells (or other objects) that interact in an angle-dependent way was described in Mogilner and Edelstein-Keshet (1995). In this sequel we consider in detail a special limiting case of nearly complete alignment. This occurs when the rotational diffusion of individual objects becomes very slow. In this case, the motion of the objects is essentially deterministic, and the individuals or objects tend to gather in clusters at various orientations. (Numerical solutions show that the angular distribution develops sharp peaks at various discrete orientations.) To understand the behaviour of the deterministic models with analytic tools, we represent the distribution as a number of δ-like peaks. This approximation of a true solution by a set of (infinitely sharp) peaks will be referred to as the peak ansatz. For weak but nonzero angular diffusion, the peaks are smoothed out. The analysis of this case leads to a singular perturbation problem which we investigate. We briefly discuss other applications of similar techniques.

KW - Orientation selection

KW - Parallel cells

KW - Peak ansatz

KW - Total alignment

UR - http://www.scopus.com/inward/record.url?scp=0029681920&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029681920&partnerID=8YFLogxK

M3 - Article

C2 - 8858852

AN - SCOPUS:0029681920

VL - 34

SP - 811

EP - 842

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 8

ER -