Biological Modelling with Nonlocal Advection Diffusion Equations
arXiv:2307.14396 · doi:10.1142/S0218202524400025
Abstract
The employment of nonlocal PDE models to describe biological aggregation and other phenomena has gained considerable traction in recent years. For cell populations, these methods grant a means of accommodating essential elements such as cell adhesion, critical to the development and structure of tissues. For animals, they can be used to describe how the nearby presence of conspecifics and/or heterospecifics influence movement behaviour. In this review, we will focus on classes of biological movement models in which the advective (or directed) component to motion is governed by an integral term that accounts for how the surrounding distribution(s) of the population(s) impact on a member's movement. We recount the fundamental motivation for these models: the intrinsic capacity of cell populations to self-organise and spatially sort within tissues; the wide-ranging tendency of animals towards spatial structuring, from the formations of herds and swarms to territorial segregation. We examine the derivation of these models from an individual level, illustrating in the process methods that allow models to be connected to data. We explore a growing analytical literature, including methods of stability and bifurcation analysis, and existence results. We conclude with a short section that lays out some future challenges and connections to the modelling of sociological phenomena including opinion dynamics.
References in corpus (20)
- Novel type of phase transition in a system of self-driven particles
- Collective motion
- Physical Models of Collective Cell Migration
- A new model for self-organized dynamics and its flocking behavior
- A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
- The derivation of Swarming models: Mean-Field Limit and Wasserstein distances
- Pattern formation with a conservation law
- Locust Dynamics: Behavioral Phase Change and Swarming
- Nonparametric inference of interaction laws in systems of agents from trajectory data
- Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus
- Predicting local and non-local effects of resources on animal space use using a mechanistic step selection model
- Bridging from single to collective cell migration: A review of models and links to experiments
- A unifying framework for quantifying the nature of animal interactions
- Spatial memory and taxis-driven pattern formation in model ecosystems
- The undecided have the key: Interaction-driven opinion dynamics in a three state model
- Noisy bounded confidence models for opinion dynamics: the effect of boundary conditions on phase transitions
- Territorial pattern formation in the absence of an attractive potential
- Territorial Developments Based on Graffiti: a Statistical Mechanics Approach
- Modeling Opinion Dynamics: Theoretical analysis and continuous approximation
- A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis