Stochastic and continuum descriptions of population dynamics with spatial structure

Spatial structures are ubiquitous in populations of plants, animals and cells, typically occurring as clustering or segregation. These spatial structures influence how individuals interact and the overall population dynamics. Yet, these details are rarely accounted for in classical population dynamics models. Through Individual-based and continuum models, I show that spatial structures can dramatically alter population dynamics. The thesis specifically explores the role of spatial structure in biologically and ecologically relevant scenarios, such as the movement of cells in the presence of biological obstacles, directional movement of animals in response to interaction with others (chase-escape dynamics), predator-prey dynamics, and Allee kinetics

Spatial moment description of birth-death-movement processes incorporating the effects of crowding and obstacles

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

Spatial structure arising from chase-escape interactions with crowding

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.</p

Small-scale spatial structure affects predator-prey dynamics and coexistence

Small-scale spatial variability can affect community dynamics in many ecological and biological processes, such as predator-prey dynamics and immune responses. Spatial variability includes short-range neighbour-dependent interactions and small-scale spatial structure, such as clustering where individuals aggregate together, and segregation where individuals are spaced apart from one another. Yet, a large class of mathematical models aimed at representing these processes ignores these factors by making a classical mean-field approximation, where interactions between individuals are assumed to occur in proportion to their average density. Such mean-field approximations amount to ignoring spatial structure. In this work, we consider an individual-based model of a two-species community that is composed of consumers and resources. The model describes migration, predation, competition and dispersal of offspring, and explicitly gives rise to varying degrees of spatial structure. We compare simulation results from the individual-based model with the solution of a classical mean-field approximation, and this comparison provides insight into how spatial structure can drive the system away from mean-field dynamics. Our analysis reveals that mechanisms leading to intraspecific clustering and interspecific segregation, such as short-range predation and short-range dispersal, tend to increase the size of the resource species relative to the mean-field prediction. We show that under certain parameter regimes these mechanisms lead to the extinction of consumers whereas the classical mean-field model predicts the coexistence of both species.</p

Population dynamics with spatial structure and an Allee effect

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.</p

Agent-based modelling reveals the role of the tumor microenvironment on the short-term success of combination temozolomide/immune checkpoint blockade to treat glioblastoma

Glioblastoma is the most common and deadly primary brain tumor in adults. All glioblastoma patients receiving standardof- care surgery-radiotherapy-chemotherapy (i.e., temozolomide (TMZ)) recur, with an average survival time of only 15 months. New approaches to the treatment of glioblastoma, including immune checkpoint blockade and oncolytic viruses, offer the possibility of improving glioblastoma outcomes and have as such been under intense study. Unfortunately, these treatment modalities have thus far failed to achieve approval. Recently, in an attempt to bolster efficacy and improve patient outcomes, regimens combining chemotherapy and immune checkpoint inhibitors have been tested in trials. Unfortunately, these efforts have not resulted in significant increases to patient survival. To better understand the various factors impacting treatment outcomes of combined TMZ and immune checkpoint blockade, we developed a systems-level, computational model that describes the interplay between glioblastoma, immune, and stromal cells with this combination treatment. Initializing our model to spatial resection patient samples labeled using imaging mass cytometry, our model's predictions show how the localization of glioblastoma cells, influence therapeutic success. We further validated these predictions in samples of brain metastases from patients given they generally respond better to checkpoint blockade compared with primary glioblastoma. Ultimately, our model provides novel insights into the mechanisms of therapeutic success of immune checkpoint inhibitors in brain tumors and delineates strategies to translate combination immunotherapy regimens more effectively into the clinic.</p