Special Issue on Spatial Moment Techniques for Modelling Biological Processes

Abstract

Over the last two decades, there has been an increasing awareness of, and interest in, the use of spatial moment techniques to provide insight into a range of biological and ecological processes. Models that incorporate spatial moments can be viewed as extensions of mean-field models. These mean-field models often consist of systems of classical ordinary differential equations and partial differential equations, whose derivation, at some point, hinges on the simplifying assumption that individuals in the underlying stochastic process encounter each other at a rate that is proportional to the average abundance of individuals. This assumption has several implications, the most striking of which is that mean-field models essentially neglect any impact of the spatial structure of individuals in the system. Moment dynamics models extend traditional mean-field descriptions by accounting for the dynamics of pairs, triples and higher n-tuples of individuals. This means that moment dynamics models can, to some extent, account for how the spatial structure affects the dynamics of the system in question.Previous applications of moment dynamics models have focused on several application areas, such as: ecological dynamics

Spatially inhomogeneous population dynamics: beyond the mean field approximation

“…The standard methods cannot handle this singularity because they are built on regular finite-difference schemes. In the ISMD/KSA/DP approach, the above singularity is removed since this part of the dynamics is integrated analytically by the product (12) of exponential transformations (13) and (15), guarantying the positiveness of the spatial moments. For instance, the rhs of the equalities in Eq.…”

Spatial population dynamics: Beyond the Kirkwood superposition approximation by advancing to the Fisher–Kopeliovich ansatz

“…For overcoming drawbacks of existing PD models, over the last two decades there has been an increasing interest in developing spatial moment dynamics (SMD) [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In SMD the populations are described by time-dependent spatial moments (aka reduced distribution [13] or correlation [15,20,24] functions).…”

“…The SMD models were applied to ecological dynamics, surface chemistry reactions, spatial epidemics, herding behaviour, predator-prey metapopulations (see [2,17] and the references therein). These models are particularly useful for detecting patchiness and clustering [11,26] in the spatial distribution of different organisms, such as trees in a beech forest [27] or breast cancer cells at an in vitro growth-to-confluence assay [28].…”

The effect of domain growth on spatial correlations

“…IBMs allow an intuitive representation of cells (referred to as 'agents' in the IBM), and allow for complex behaviours, such as cell-cell interactions and volume exclusion, to be easily assigned to agents in the model [9][10][11][12]. Importantly, IBMs can capture the effects of spatial correlations and heterogeneity in agent populations, and the ramifications spatial correlations can have on density-dependent processes such as cell migration and proliferation [13][14][15][16][17][18][19][20][21][22][23].…”

“…However, in certain scenarios standard mean-field partial differential equation (PDE) descriptions of IBMs, such as those describing the expected evolution of the population density, suffer from the limitation that they neglect to incorporate the impact of spatial correlations and clustering. Therefore, in order to derive accurate continuum approximations of IBMs it is often necessary to include the effects of spatial correlations in continuum models [14][15][16][17][18][19][20][21][22][24][25][26][27][28][29]. Furthermore, having the mathematical tools to directly compute spatial correlations allows them to be analysed, which can give important insights into the biological process being studied.…”