Mass-Selection in Alignment Models with Non-Deterministic Effects

Abstract

In this paper we consider a kinetic model for alignment of cells or filaments with probabilistic turning. For this equation existence of solutions is known, see [6 Kang , K. , Perthame , B. , Stevens , A. , Velazquez , J. J. L. ( 2009 ). An integro- differential equation model for alignment and orientational aggregation . J. Diff Equations 246 : 1387 – 1421 .[Crossref] , [Google Scholar]]. To understand its qualitative behavior, especially with respect to the selection of orientations and mass distributions for long times, the model is approximated by a diffusion equation in the limit of small deviations of the interactions between the cell bundles. For this new equation existence of steady states is shown. In contrast to the kinetic equation discussed in [6 Kang , K. , Perthame , B. , Stevens , A. , Velazquez , J. J. L. ( 2009 ). An integro- differential equation model for alignment and orientational aggregation . J. Diff Equations 246 : 1387 – 1421 .[Crossref] , [Google Scholar]] with deterministic turning, where local stability of two opposite orientations was shown but no selection of mass could be observed, for the new approximating problem with probabilistic turning additionally mass selection takes place. In the limit of small diffusion, steady states can only be constructed, if the aligning masses are either equal or the total mass is concentrated in one direction. By numerical simulations we tested stability of these steady states and for situations with 4 symmetrically placed smooth distributions of alignment. Convergence of the numerical code was proved. The simulations suggest, that only the 2- and the 1-peak steady states can be stable, whereas the 4 peak steady state is always unstable. We conjecture that the noise in the system is responsible for this final selection of masses. There exist other steady states with an arbitrary number of aligned bundles of cells or filaments, but we suspect that, as numerically shown for the 4 peak case, these multi-peak states are all unstable.