Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling

Understanding interactions between populations: Individual based modelling and quantification using pair correlation functions

Understanding the underlying mechanisms that produce the huge variety of swarming and aggregation patterns in animals and cells is fundamental in ecology, developmental biology, and regenerative medicine, to name but a few examples. Depending upon the nature of the interactions between individuals (cells or animals), a variety of different large-scale spatial patterns can be observed in their distribution; examples include cell aggregates, stripes of different coloured skin cells, etc. For the case where all individuals are of the same type (i.e., all interactions are alike), a considerable literature already exists on how the collective organisation depends on the inter-individual interactions. Here, we focus on the less studied case where there are two different types of individuals present. Whilst a number of continuum models of this scenario exist, it can be difficult to compare these models to experimental data, since real cells and animals are discrete. In order to overcome this problem, we develop an agent-based model to simulate some archetypal mechanisms involving attraction and repulsion. However, with this approach (as with experiments), each realisation of the model is different, due to stochastic effects. In order to make useful comparisons between simulations and experimental data, we need to identify the robust features of the spatial distributions of the two species which persist over many realisations of the model (for example, the size of aggregates, degree of segregation or intermixing of the two species). In some cases, it is possible to do this by simple visual inspection. In others, the features of the pattern are not so clear to the unaided eye. In this paper, we introduce a pair correlation function (PCF), which allows us to analyse multi-species spatial distributions quantitatively. We show how the differing strengths of inter-individual attraction and repulsion between species give rise to different spatial patterns, and how the PCF can be used to quantify these differences, even when it might be impossible to recognise them visually.

Filling the gaps: A robust description of adhesive birth-death-movement processes

Existing continuum descriptions of discrete adhesive birth-death-movement processes provide accurate predictions of the average discrete behavior for limited parameter regimes. Here we present an alternative continuum description in terms of the dynamics of groups of contiguous occupied and vacant lattice sites. Our method provides more accurate predictions, is valid in parameter regimes that could not be described by previous continuum descriptions, and provides information about the spatial clustering of occupied sites. Furthermore, we present a simple analytic approximation of the spatial clustering of occupied sites at late time, when the system reaches its steady-state configuration.

The Confinement-Escape Problem of a Defender Against an Evader Escaping from a Circular Region

In this paper, we first formulate the confinement-escape problem of a defender and an evader who attempts escaping from a circular region, which differs from the traditional pursuit-evasion problems. In our setting of the confinement-escape problem, the defender is restricted to move and patrol on the circle, trying to prevent possible escape of the evader who is initially located inside the circle. We describe and characterize some general properties of the problem, and then design two bio-inspired control strategies for the evader and the defender, respectively. In addition, we illustrate some possible motion patterns of the system, investigate the escaping time as a function of the relative-initial-positions of the agents, as well as the winning sets of the two players, respectively, under different system-parameters. To that end, we characterize the contour lines of the winning sets with their gradient properties. Finally, we indicate the abrupt phase transitions between successful confinement and escaping, revealing the strong sensitivity and nonlinearities of the system under critical conditions.

Collective Motion of Swarming Agents Evolving on a Sphere Manifold: A Fundamental Framework and Characterization

Collective motion of self-propelled agents has attracted much attention in vast disciplines. However, almost all investigations focus on such agents evolving in the Euclidean space, with rare concern of swarms on non-Euclidean manifolds. Here we present a novel and fundamental framework for agents evolving on a sphere manifold, with which a variety of concrete cooperative-rules of agents can be designed separately and integrated easily into the framework, which may perhaps pave a way for considering general spherical collective motion (SCM) of a swarm. As an example, one concrete cooperative-rule, i.e., the spherical direction-alignment (SDA), is provided, which corresponds to the usual and popular direction-alignment rule in the Euclidean space. The SCM of the agents with the SDA has many unique statistical properties and phase-transitions that are unexpected in the counterpart models evolving in the Euclidean space, which unveils that the topology of the sphere has an important impact on swarming emergence.