The auxiliary region method: A hybrid method for coupling PDE- and Brownian-based dynamics for reaction-diffusion systems

Reaction-diffusion systems are used to represent many biological and physical phenomena. They model the random motion of particles (diffusion) and interactions between them (reactions). Such systems can be modelled at multiple scales with varying degrees of accuracy and computational efficiency. When representing genuinely multiscale phenomena, fine-scale models can be prohibitively expensive, whereas coarser models, although cheaper, often lack sufficient detail to accurately represent the phenomenon at hand. Spatial hybrid methods couple two or more of these representations in order to improve efficiency without compromising accuracy. In this paper, we present a novel spatial hybrid method, which we call the auxiliary region method (ARM), which couples PDE and Brownian-based representations of reaction-diffusion systems. Numerical PDE solutions on one side of an interface are coupled to Brownian-based dynamics on the other side using compartment-based "auxiliary regions". We demonstrate that the hybrid method is able to simulate reaction-diffusion dynamics for a number of different test problems with high accuracy. Further, we undertake error analysis on the ARM which demonstrates that it is robust to changes in the free parameters in the model, where previous coupling algorithms are not. In particular, we envisage that the method will be applicable for a wide range of spatial multi-scales problems including, filopodial dynamics, intracellular signalling, embryogenesis and travelling wave phenomena.Comment: 29 pages, 14 figures, 2 table

The pseudo-compartment method for coupling PDE and compartment-based models of diffusion

Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, due to recent advances in computational power, the simulation, and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction-diffusion systems in which regions of low copy numbers exist. The stochastic models with which we shall be concerned in this manuscript are referred to as `compartment-based'. These models are characterised by a discretisation of the computational domain into a grid/lattice of `compartments'. Within each compartment particles are assumed to be well-mixed and are permitted to react with other particles within their compartment or to transfer between neighbouring compartments. We develop two hybrid algorithms in which a PDE is coupled to a compartment-based model. Rather than attempting to balance average fluxes, our algorithms answer a more fundamental question: `how are individual particles transported between the vastly different model descriptions?' First, we present an algorithm derived by carefully re-defining the continuous PDE concentration as a probability distribution. Whilst this first algorithm shows strong convergence to analytic solutions of test problems, it can be cumbersome to simulate. Our second algorithm is a simplified and more efficient implementation of the first, it is derived in the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by comparing the averaged simulations to analytic solutions of PDEs for mean concentrations.Comment: MAIN - 24 pages, 10 figures, 1 supplementary file - 3 pages, 2 figure

Mathematical Modelling of Turning Delays in Swarm Robotics

We investigate the effect of turning delays on the behaviour of groups of differential wheeled robots and show that the group-level behaviour can be described by a transport equation with a suitably incorporated delay. The results of our mathematical analysis are supported by numerical simulations and experiments with e-puck robots. The experimental quantity we compare to our revised model is the mean time for robots to find the target area in an unknown environment. The transport equation with delay better predicts the mean time to find the target than the standard transport equation without delay.Comment: Submitted to the IMA Journal of Applied Mathematic

Refining self-propelled particle models for collective behaviour

Swarming, schooling, flocking and herding are all names given to the wide variety of collective behaviours exhibited by groups of animals, bacteria and even individual cells. More generally, the term swarming describes the behaviour of an aggregate of agents (not necessarily biological) of similar size and shape which exhibit some emergent property such as directed migration or group cohesion. In this paper we review various individual-based models of collective behaviour and discuss their merits and drawbacks. We further analyse some one-dimensional models in the context of locust swarming. In specific models, in both one and two dimensions, we demonstrate how varying the parameters relating to how much attention individuals pay to their neighbours can dramatically change the behaviour of the group. We also introduce leader individuals to these models with the ability to guide the swarm to a greater or lesser degree as we vary the parameters of the model. We consider evolutionary scenarios for models with leaders in which individuals are allowed to evolve the degree of influence neighbouring individuals have on their subsequent motion

The NHS is facing the bleakest midwinter

With the NHS caught in a vicious cycle of connected pressures, we are heading for a very bleak midwinter, say Christina Pagel and Christian A Yates