Random walk models in the life sciences: including births, deaths and local interactions

Abstract

Random walks and related spatial stochastic models have been used in a range of application areas including animal and plant ecology, infectious disease epidemiology, developmental biology, wound healing, and oncology. Classical random walk models assume that all individuals in a population behave independently, ignoring local physical and biological interactions. This assumption simplifies the mathematical description of the population considerably, enabling continuum-limit descriptions to be derived and used in model analysis and fitting. However, interactions between individuals can have a crucial impact on population-level behaviour. In recent decades, research has increasingly been directed towards models that include interactions, including physical crowding effects and local biological processes such as adhesion, competition, dispersal, predation and adaptive directional bias. In this article, we review the progress that has been made with models of interacting individuals. We aim to provide an overview that is accessible to researchers in application areas, as well as to specialist modellers. We focus particularly on derivation of asymptotically exact or approximate continuum-limit descriptions and simplified deterministic models of mean-field behaviour and resulting spatial patterns. We provide worked examples and illustrative results of selected models. We conclude with a discussion of current areas of focus and future challenges.

References in corpus (20)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems
• Physical Models of Collective Cell Migration
• Modeling tumor cell migration: from microscopic to macroscopic
• Simulation of stochastic reaction-diffusion processes on unstructured meshes
• Excluded-volume effects in the diffusion of hard spheres
• Diffusion of multiple species with excluded-volume effects
• Learning differential equation models from stochastic agent-based model simulations
• Exclusion process for particles of arbitrary extension: Hydrodynamic limit and algebraic properties
• Capture of particles undergoing discrete random walks
• Learning Mean-Field Equations from Particle Data Using WSINDy
• Coexistence in locally regulated competing populations and survival of branching annihilating random walk
• Diffusion of finite-size particles in confined geometries
• Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population
• Empirical quantification of predictive uncertainty due to model discrepancy by training with an ensemble of experimental designs: an application to ion channel kinetics
• Spatially inhomogeneous population dynamics: beyond the mean field approximation
• Random walk models in the life sciences: including births, deaths and local interactions
• Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media
• Tracer diffusion beyond Gaussian behavior: explicit results for general single-file systems
• Establishment and Fecundity in Spatial Ecological Models: Statistical Approach and Kinetic Equations

Cited by in corpus (2)

• Random walk models in the life sciences: including births, deaths and local interactions
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