Biased Random Walks, Partial Differential Equations and Update Schemes

Abstract

There is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a collection of noninteracting biased random walkers on a one-dimensional lattice is considered. The usual master equation approach requires that two continuum limits, involving three parameters, namely step length, time step and the random walk bias, approach zero in a specific way. We are interested in the case where the two limits are not consistent. New results are obtained using a Fokker–Planck equation and the results are highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to an accurate macroscopic description of stochastic local movement rules in agent-based models when the lattice spacing represents a physical object such as cell diameter.

Survival probability for a diffusive process on a growing domain

“…In these types of problems the population density profile is thought to evolve according to an advection-diffusion mechanism [18][19][20][21][22][23][24], The diffusion mechanism is associated with individual cells or molecules in the population undergoing an undirected random walk, and the advection mechanism describes the directed motion of individuals driven by the underlying tissue growth. Since the rate of advection is spatially dependent [18][19][20][21][22][23][24], the advection process also gives rise to a dilution effect.…”

“…Since the rate of advection is spatially dependent [18][19][20][21][22][23][24], the advection process also gives rise to a dilution effect. Here, we consider a diffusion process on a one-dimensional growing domain, 0 < x < L{t), where L(t) is the increasing length of the domain.…”

In-Silico Modeling of Tumor Spheroid Formation and Growth

“…Development of biological patterns, such as animal coat markings and evolution of the enteric nervous system (ENS), is another type of process that often involves cellular proliferation and reaction/diffusion of materials [ 40 , 41 , 42 ]. To describe such processes, the evolution of a domain boundary is incorporated into the mathematical modeling [ 43 , 44 , 45 , 46 ]. This boundary evolution of patterns is very similar to that of tumors, i.e., growth, from a mathematical point of view.…”

Exact Solutions of Linear Reaction-Diffusion Processes on a Uniformly Growing Domain: Criteria for Successful Colonization

“…In addition to considering particular biological applications, other studies have focused on examining more theoretical questions associated with reactive transport processes on growing domains. Most notably, several previous studies have examined the relationship between discrete random walk models and associated continuum partial differential equation (PDE) descriptions [ 10 – 14 ].…”