Phase drift on networks of coupled crystal oscillators for precision timing.

Precise time dissemination and synchronization have been some of the most important technological tasks for several centuries. Since the early 1800s, it was realized that precise time-keeping devices having the same stable frequency and precisely synchronized can have important applications in navigation. In modern times, satellite-based global positioning and navigation systems such as the GPS use the same principle. However, even the most sophisticated satellite navigation equipment cannot operate in every environment. In response to this need, we present a computational and analytical study of a network-based model of a high-precision, inexpensive, coupled crystal oscillator system and timing (CCOST) device. A bifurcation analysis (carried out by the authors in a related publication) [Buono et al., SIAM J. Appl. Dyn. Syst. 17, 1310 (2018)1536-004010.1137/16M1066154] of the network dynamics shows a wide variety of collective patterns, mainly various forms of discrete rotating waves and synchronization patterns. Results from computer simulations seem to indicate that, among all patterns, the standard traveling wave pattern in which consecutive crystals oscillate out of phase by 2π/N, where N is the network size, leads to phase drift error that decreases as 1/N as opposed to 1/sqrt[N] for an uncoupled ensemble. The results should provide guidelines for future experiments, design, and fabrication tasks.

Pattern formation in a nonlocal mathematical model for the multiple roles of the TGF-ß pathway in tumour dynamics.

The growth and invasion of cancer cells are very complex processes, which can be regulated by the cross-talk between various signalling pathways, or by single signalling pathways that can control multiple aspects of cell behaviour. TGF-ß is one of the most investigated signalling pathways in oncology, since it can regulate multiple aspects of cell behaviour: cell proliferation and apoptosis, cell-cell adhesion and epithelial-to-mesenchimal transition via loss of cell adhesion. In this study, we use a mathematical modelling approach to investigate the complex roles of TGF-ß signalling pathways on the inhibition and growth of tumours, as well as on the epithelial-to-mesenchimal transition involved in the metastasis of tumour cells. We show that the nonlocal mathematical model derived here to describe repulsive and adhesive cell-cell interactions can explain the formation of new tumour cell aggregations at positions in space that are further away from the main aggregation. Moreover, we show that the increase in cell-cell adhesion leads to fewer but larger aggregations, and the increase in TGF-ß molecules - whose late-stage effect is to decrease cell adhesion - leads to many small cellular aggregations. Finally, we perform a sensitivity analysis on some parameters associated with TGF-ß dynamics, and use it to investigate the relation between the tumour size and its metastatic spread.

A spatial theory for characterizing predator-multiprey interactions in heterogeneous landscapes.

Trophic interactions in multiprey systems can be largely determined by prey distributions. Yet, classic predator-prey models assume spatially homogeneous interactions between predators and prey. We developed a spatially informed theory that predicts how habitat heterogeneity alters the landscape-scale distribution of mortality risk of prey from predation, and hence the nature of predator interactions in multiprey systems. The theoretical model is a spatially explicit, multiprey functional response in which species-specific advection-diffusion models account for the response of individual prey to habitat edges. The model demonstrates that distinct responses of alternative prey species can alter the consequences of conspecific aggregation, from increasing safety to increasing predation risk. Observations of threatened boreal caribou, moose and grey wolf interacting over 378 181 km(2) of human-managed boreal forest support this principle. This empirically supported theory demonstrates how distinct responses of apparent competitors to landscape heterogeneity, including to human disturbances, can reverse density dependence in fitness correlates.

Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation.

The study of self-organised collective animal behaviour, such as swarms of insects or schools of fish, has become over the last decade a very active research area in mathematical biology. Parabolic and hyperbolic models have been used intensively to describe the formation and movement of various aggregative behaviours. While both types of models can exhibit aggregation-type patterns, studies on hyperbolic models suggest that these models can display a larger variety of spatial and spatio-temporal patterns compared to their parabolic counterparts. Here we use stability, symmetry and bifurcation theory to investigate this observation more rigorously, an approach not attempted before to compare and contrast aggregation patterns in models for collective animal behaviors. To this end, we consider a class of nonlocal hyperbolic models for self-organised aggregations that incorporate various inter-individual communication mechanisms, and take the formal parabolic limit to transform them into nonlocal parabolic models. We then discuss the symmetry of these nonlocal hyperbolic and parabolic models, and the types of bifurcations present or lost when taking the parabolic limit. We show that the parabolic limit leads to a homogenisation of the inter-individual communication, and to a loss of bifurcation dynamics (in particular loss of Hopf bifurcations). This explains the less rich patterns exhibited by the nonlocal parabolic models. However, for multiple interacting populations, by breaking the population interchange symmetry of the model, one can preserve the Hopf bifurcations that lead to the formation of complex spatio-temporal patterns that describe moving aggregations.

Movement responses of caribou to human-induced habitat edges lead to their aggregation near anthropogenic features.

The assessment of disturbance effects on wildlife and resulting mitigation efforts are founded on edge-effect theory. According to the classical view, the abundance of animals affected by human disturbance should increase monotonically with distance from disturbed areas to reach a maximum at remote locations. Here we show that distance-dependent movement taxis can skew abundance distributions toward disturbed areas. We develop an advection-diffusion model based on basic movement behavior commonly observed in animal populations and parameterize the model from observations on radio-collared caribou in a boreal ecosystem. The model predicts maximum abundance at 3.7 km from cutovers and roads. Consistently, aerial surveys conducted over 161,920 km(2) showed that the relative probability of caribou occurrence displays nonmonotonic changes with the distance to anthropogenic features, with a peak occurring at 4.5 km away from these features. This aggregation near disturbed areas thus provides the predators of this top-down-controlled, threatened herbivore species with specific locations to concentrate their search. The edge-effect theory developed here thus predicts that human activities should alter animal distribution and food web properties differently than anticipated from the current paradigm. Consideration of such nonmonotonic response to habitat edges may become essential to successful wildlife conservation.