Birth-death-movement processes, modulated by interactions between individuals, are fundamental to many cell biology processes. A key feature of the movement of cells within in vivo environments is the interactions between motile cells and stationary obstacles. Here we propose a multi-species model of individual-level motility, proliferation and death. This model is a spatial birth-death-movement stochastic process, a class of individual-based model (IBM) that is amenable to mathematical analysis. We present the IBM in a general multi-species framework and then focus on the case of a population of motile, proliferative agents in an environment populated by stationary, non-proliferative obstacles. To analyse the IBM, we derive a system of spatial moment equations governing the evolution of the density of agents and the density of pairs of agents. This approach avoids making the usual mean-field assumption so that our models can be used to study the formation of spatial structure, such as clustering and aggregation, and to understand how spatial structure influences population-level outcomes. Overall the spatial moment model provides a reasonably accurate prediction of the system dynamics, including important effects such as how varying the properties of the obstacles leads to different spatial patterns in the population of agents.

Population dynamics including a strong Allee effect describe the situation where long-term population survival or extinction depends on the initial population density. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to extinction, whereas initial densities above the threshold lead to survival. Mean-field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as competition and dispersal. The influence of non-mean-field effects has not been studied in the presence of an Allee effect. To address this, we develop an individual-based model that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure neglected by mean-field models. For example, there are cases where the mean-field model predicts extinction but the population actually survives. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.

Gene expression is an intrinsically noisy process, typically, producing mRNAs and proteins in bursts. An important description of such stochastic processes can be done in terms of the mean first passage time (MFPT), i.e., the time taken by mRNAs/proteins to reach a particular threshold. We study the role of burstiness on MFPT and obtain an analytical expression for different models of transcriptional and translational bursts. Our analytical results and numerical simulations confirm that MFPT monotonically decreases with burstiness.

Movement of individuals, mediated by localised interactions, plays a key role in numerous processes including cell biology and ecology. In this work, we investigate an individual-based model accounting for various intraspecies and interspecies interactions in a community consisting of two distinct species. In this framework we consider one species to be chasers and the other species to be escapees, and we focus on chase-escape dynamics where the chasers are biased to move towards the escapees, and the escapees are biased to move away from the chasers. This framework allows us to explore how individual-level directional interactions scale up to influence spatial structure at the macroscale. To focus exclusively on the role of motility and directional bias in determining spatial structure, we consider conservative communities where the number of individuals in each species remains constant. To provide additional information about the individual-based model, we also present a mathematically tractable deterministic approximation based on describing the evolution of the spatial moments. We explore how different features of interactions including interaction strength, spatial extent of interaction, and relative density of species influence the formation of the macroscale spatial patterns.

Birth-death-movement processes, modulated by interactions between individuals, are fundamental to many biological processes such as development, repair and disease. Similar interactions are also relevant in ecology. A key feature of the movement of cells within in vivo environments are the interactions between motile cells and stationary obstacles, such as the extracellular matrix and stationary macromolecules. Here we propose a multi-species individual-based model (IBM) of individual-level motility, proliferation and death. In particular, we focus on examining the case where we consider a population of motile, proliferative agents within an environment that is populated by stationary, non-proliferative obstacles. To provide a mathematical foundation for the analysis of the IBM, we derive a system of spatial moment equations that approximately governs the evolution of the density of agents and the density of pairs of agents. By using a spatial moment approach we avoid making the usual mean field assumption so that our IBM and continuous model are able to predict the formation of spatial structure, such as clustering and aggregation. We explore several properties of the obstacle field, such as systematically varying the obstacle density, obstacle size, and the nature and strength of the interactions between the motile, proliferative agents and the stationary, non-proliferative obstacles. Overall we find that the spatial moment model provides a reasonably accurate prediction of the dynamics of the system, including subtle but important effects, such as how varying the properties of the obstacles leads to different patterns of clustering and segregation in the population.