Correcting mean-field approximations for birth-death-movement processes

Abstract

On the microscale, migration, proliferation and death are crucial in the development, homeostasis and repair of an organism; on the macroscale, such effects are important in the sustainability of a population in its environment. Dependent on the relative rates of migration, proliferation and death, spatial heterogeneity may arise within an initially uniform field; this leads to the formation of spatial correlations and can have a negative impact upon population growth. Usually, such effects are neglected in modeling studies and simple phenomenological descriptions, such as the logistic model, are used to model population growth. In this work we outline some methods for analyzing exclusion processes which include agent proliferation, death and motility in two and three spatial dimensions with spatially homogeneous initial conditions. The mean-field description for these types of processes is of logistic form; we show that, under certain parameter conditions, such systems may display large deviations from the mean field, and suggest computationally tractable methods to correct the logistic-type description.

Distinguishing between mechanisms of cell aggregation using pair-correlation functions

“…Clearly, an extensive verification process is a considerable undertaking and has not been attempted here. There are several options for extending this work with an obvious extension being the consideration of other mechanisms, such as incorporating cell adhesion, cell death and cell processes within a developing tissue [3,7,21]. Similarly, this work has only considered cell populations composed of a single cell type; however, the formation of aggregates consisting of two or more cell types is also of interest [2,14,19,25].…”

Experimental and Modelling Investigation of Monolayer Development with Clustering

“…We will consider the case where the initial distribution of agents is spatially uniform and we denote the number of agents on the lattice at time t by Q(t). Discrete simulations of this process can be performed using the Gillespie algorithm (Baker and Simpson 2010;Gillespie 1977). Such simulations of this discrete model have been presented, at length, previously (Baker and Simpson 2010).…”

“…We consider a model of cell motility and proliferation which has been described, and analysed, previously (Baker and Simpson 2010). The model can be implemented using a discrete random-walk framework (Codling et al 2008) and the continuum limit of the model can be obtained by techniques outlined in (Baker and Simpson 2010).…”

“…The model can be implemented using a discrete random-walk framework (Codling et al 2008) and the continuum limit of the model can be obtained by techniques outlined in (Baker and Simpson 2010). In brief, the discrete model consists of a two-dimensional square lattice with lattice spacing Δ.…”

“…Our recent work has focused on adapting these techniques to discrete models of cell migration, proliferation and death processes which include finite size effects by allowing, at most, only one agent to occupy a particular location in space. This theoretical work implies that rapid cell proliferation (Baker and Simpson 2010; or strong cell-to-cell adhesion (Johnston et al 2012) can lead to clustering (Simpson et al 2010a). Our theoretical work has shown that alternative models, based on studying the dynamics of individuals, pairs of individuals, triplets of individuals, and so on (Bolker and Pacala 1997;Illian et al 2008;Law and Dieckmann 2000;Law et al 2003;Murrell et al 2004), together with an appropriate moment closure approximation (Hansen and McDonald 2006;Singer 2004;Raghib et al 2011), gives an improved prediction of the system behaviour (Baker and Simpson 2010;Johnston et al 2012).…”