Analysis of a splitting-differentiation population model leading to cross-diffusion

TL;DR

Simulation experiments indicate that the numerical scheme arising from the approximation introduced in this article outperforms those of the existent models from the stability point of view.

Abstract

Starting from the dynamical system model capturing the splitting-differentiation process of populations, we extend this notion to show how the speciation mechanism from a single species leads to the consideration of several well known evolution cross-diffusion partial differential equations. Among the different alternatives for the diffusion terms, we study the model introduced by Busenberg and Travis, for which we prove the existence of solutions in the one-dimensional spatial case. Using a direct parabolic regularization technique, we show that the problem is well posed in the space of bounded variation functions, and demonstrate with a simple example that this is the best regularity expected for solutions. We numerically compare our approach to other alternative regularizations previously introduced in the literature, for the particular case of the contact inhibition problem. Simulation experiments indicate that the numerical scheme arising from the approximation introduced in this article outperforms those of the existent models from the stability point of view.

Figures (4)

• Figure 1: Experiment 1. Odd rows correspond to the approximated solution of problem (P)$_\delta$ for several time slices, $t$, while even rows correspond to the solution of problem (P)$_B$. Rows correspond to different mesh sizes, captured by parameter $h$. Mind the different vertical scales among time slices.
• Figure 2: Experiment 1. Each row corresponds to the oscillation measure $\textrm{osc}(u)(t)$ of the solutions approximated by (P)$_\delta$ and (P)$_B$, respectively. Columns correspond to different mesh sizes, captured by parameter $h$.
• Figure 3: Experiment 2. Exact solution (dots) against approximated solutions (solid lines). Odd rows correspond to the approximated solution of problem (P)$_\delta$ for several time slices, $t$, while even rows correspond to the solution of problem (P)$_B$. Rows correspond to different mesh sizes, captured by parameter $h$. Mind the different vertical scales among time slices.
• Figure 4: Experiment 2. Two first rows correspond to the oscillation measure $\textrm{osc}(u)(t)$ of the solutions approximated by (P)$_\delta$ and (P)$_B$, respectively. Two last rows correspond to the relative error between the exact solution and the solution approximated by (P)$_\delta$ and (P)$_B$, respectively. Columns correspond to different mesh sizes, captured by parameter $h$.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Remark 1