Numerical Approaches for non-local Transport-Dominated PDE Models with Applications to Biology

TL;DR

This paper surveys numerical strategies for non-local, transport-dominated PDEs used to model collective migration in biology and ecology, with an emphasis on finite element methods. It highlights the challenges posed by non-local advective fluxes and introduces a semi-implicit FE scheme to stabilize oscillations, validated through 1D and 2D simulations for single- and multi-population models. The authors compare finite difference, finite volume, and FE approaches, detail a variational FE formulation and FFT-based non-local term evaluation, and demonstrate stable, physically plausible aggregation patterns. The work identifies open problems in stabilization, continuation, and complex geometries, arguing that FE methods offer a powerful framework to advance numerical studies of non-local biological and ecological systems.

Abstract

Transport-dominated partial differential equation models have been used extensively over the past two decades to describe various collective migration phenomena in cell biology and ecology. To understand the behaviour of these models (and the biological systems they describe) different analytical and numerical approaches have been used. While the analytical approaches have been discussed by different recent review studies, the numerical approaches are still facing different open problems, and thus are being employed on a rather ad-hoc basis for each developed non-local model. The goal of this review is to summarise the basic ideas behind these transport-dominated non-local models, to discuss the current numerical approaches used to simulate these models, and finally to discuss some open problems related to the applications of these numerical methods, in particular the finite element method. This allows us to emphasize the opportunities offered by this numerical method to advance the research in this field. In addition, we present in detail some numerical schemes that we used to discretize these non-local equations; in particular a new semi-implicit scheme we introduced to stabilize the oscillations obtained with classical schemes.

Figures (9)

• Figure 1: Difference between local and non-local interactions at the cellular level. In both cases, there are two types of interaction: bio-mechanical (represented by the solid and striped arrows) and bio-chemical (represented by bright pink molecules). On the left, we say that these interactions are local since they appear between directly neighboring cells. Whereas on the right, these interactions are non-local since the mechanical traction of cell 1 towards the left leads to the movement towards the left of cells 2 and 4 which are neighbors but also of cells 3 and 5 which are not directly neighboring. In addition, cell 1 can chemically interact with distant cells 3 and 5 via protrusions carrying a chemical messenger.
• Figure 2: Numerical resolution of equation \ref{['non_local model 1d']} using numerical schemes \ref{['scheme FD']} (red dashed curve) and \ref{['scheme VF']} (blue continuous curve). We show here the solution curves at times $t=0$, $t=0.01$, $t=0.5$, $t=0.75$, $t=1$ and $t=10$. The parameter values are as in Table \ref{['Tab:Param1d']}.
• Figure 3: Numerical solution of the equation \ref{['non_local model 1d']} using FEM implemented in FEniCS, at different time. Parameter values are as in Table \ref{['Tab:Param1d']}.
• Figure 4: Numerical simulations of model \ref{['non_local model 1d']}- \ref{['K 1D']}, using the parameter values in Table \ref{['Tab:Param1d']} (except $\alpha=15$) obtained with : (left column) the explicit scheme \ref{['explicit']}, (right column) the mix explicit-implicit scheme.
• Figure 5: Numerical solution of the bi-dimensional equation \ref{['nonlocal_model']} using FEM implemented in FEniCS. We show the solution at six different times: $t=0$, $t=1$, $t=1.5$, $t=2$, $t=10$, $t=100$. The parameter values used for these simulations are listed in Table \ref{['Tab:Param2d']}.

Theorems & Definitions (3)

• Remark 2
• Remark 3
• Remark 4