A cell-cell repulsion model on a hyperbolic Keller-Segel equation

Bifurcation analysis of critical values for wound closure outcomes in wound healing experiments

A nonlinear partial differential equation containing a nonlocal advection term and a diffusion term is analyzed to study wound closure outcomes in wound healing experiments. There is an extensive literature of similar models for wound healing experiments. In this paper we study the character of wound closure in these experiments in terms of the sensing radius of cells and the force of cell-cell adhesion. We prove a bifurcation result which differentiates uniform closure of the wound from nonuniform closure of the wound, based on a critical value [Formula: see text] of the force of cell-cell adhesion parameter [Formula: see text]. For [Formula: see text] the steady state solution [Formula: see text] of the model is stable and the wound closes uniformly. For [Formula: see text] the steady state solution [Formula: see text] of the model is unstable and the wound closes nonuniformly. We provide numerical simulations of the model to illustrate our results.

Waves in a Hyperbolic Predator–Prey System

We address a hyperbolic predator–prey model, which we formulate with the use of the Cattaneo model for chemosensitive movement. We put a special focus on the case when the Cattaneo equation for the flux of species takes the form of conservation law—that is, we assume a special relation between the diffusivity and sensitivity coefficients. Regarding this relation, there are pieces arguing for its relevance to some real-life populations, e.g., the copepods (Harpacticoida), in the biological literature (see the reference list). Thanks to the mentioned conservatism, we get exact solutions describing the travelling shock waves in some limited cases. Next, we employ the numerical analysis for continuing these waves to a wider parametric domain. As a result, we discover smooth solitary waves, which turn out to be quite sustainable with small and moderate initial perturbations. Nevertheless, the perturbations cause shedding of the predators from the main core of the wave, which can be treated as a settling mechanism. Besides, the localized perturbations make waves, colliding with the main core and demonstrating peculiar quasi-soliton phenomena sometimes resembling the leapfrog playing. An interesting side result is the onset of the migration waves due to the explosion of overpopulated cores.