Diffusive and Chemotactic Cellular Migration: Smooth and Discontinuous Traveling Wave Solutions

• Published in: SIAM journal on applied mathematics 2005-01, Vol.65 (4), p.1420-1442
• Main Authors: Landman, K. A., M. J. Simpson, J. L. Slater, D. F. Newgreen
• Format: Article
• Language: English
• Subjects: Animal migration behavior; Applied mathematics; Cell adhesion & migration; Chemotactic factors; Chemotaxis; Developmental biology; Growth factors; Kinetics; Mathematical models; Phase plane; Population; Speed; Trajectories; Traveling waves
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/040604066

A mathematical model describing cell migration by diffusion and chemotaxis is considered. The model is examined using phase plane, numerical, and perturbation techniques. For a proliferative cell population, traveling wave solutions are observed regardless of whether the migration is driven by diffusion, chemotaxis, or a combination of the two mechanisms. For pure chemotactic migration, both smooth and discontinuous solutions with shocks are shown to exist using phase plane analysis involving a curve of singularities, and identical results are obtained numerically. Alternatively, pure diffusive migration and combinations of diffusive and chemotactic migration yield smooth solutions only. For all cases the wave speed depends on the exponential decay rate of the initial cell density, and it is bounded by a minimum value which is numerically observed whenever the initial cell distribution has compact support. The minimum wave speed$c_{min}$is proportional to$\sqrt{\chi}$or$\sqrt{D}$for pure chemotaxis and pure diffusion cases, respectively. The value of$c_{min}$for combined diffusion and chemotactic migration is examined numerically. The rate at which the mixed migration system approaches either a diffusion-dominated or chemotaxis-dominated system is investigated as a function of a dimensionless parameter involving$D/\chi$. Finally, a perturbation analysis provides details of the steep critical layer when$D/\chi \ll 1$, and these are confirmed with numerical solutions. This analysis provides a deeper qualitative and quantitative understanding of the interplay between diffusion and chemotaxis for invading cell populations.