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2018 | Buch

Hyperbolic and Kinetic Models for Self-organised Biological Aggregations

A Modelling and Pattern Formation Approach

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Über dieses Buch

This book focuses on the spatio-temporal patterns generated by two classes of mathematical models (of hyperbolic and kinetic types) that have been increasingly used in the past several years to describe various biological and ecological communities. Here we combine an overview of various modelling approaches for collective behaviours displayed by individuals/cells/bacteria that interact locally and non-locally, with analytical and numerical mathematical techniques that can be used to investigate the spatio-temporal patterns produced by said individuals/cells/bacteria. Richly illustrated, the book offers a valuable guide for researchers new to the field, and is also suitable as a textbook for senior undergraduate or graduate students in mathematics or related disciplines.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
The modelling and investigation of self-organised biological aggregations is a research area that has undergone rapid expansion over the past years. Self-organised aggregations are found in swarms of insects, schools of fish, flocks of birds, mammal herds, bacteria and even human crowds. The complex spatial and spatial-temporal patterns exhibited by these aggregations, from milling schools of fish and zigzagging flocks of birds, to rippling waves in Myxobacteria colonies, have been the starting point of the research in this area. Various types of mathematical models have been derived to reproduce the observed patterns, and to propose hypotheses about the mathematical and biological mechanisms behind these patterns. Some of these models explicitly include various local and nonlocal communication mechanisms. In this Chapter, we present an overview the research area (while briefly mentioning a few individual-based models that are related to the models which will be discussed throughout the rest of the book), emphasise the reason for our approach on focusing only on two specific classes of models (hyperbolic and kinetic), and summarise the patterns that we will discuss throughout the rest of the chapters.
Raluca Eftimie
Chapter 2. A Short Introduction to One-Dimensional Conservation Laws
Abstract
The one-equation advection models that are being used to describe the movement of various animal populations have been extensively investigated over the last decades. Since the theory behind these equations is well known (and can be found in any textbook on hyperbolic conservation laws), our goal here is to give the reader a brief review of this theory (while leaving behind most technical details). This approach will help the reader understand the analytical results presented in the upcoming chapters regarding the existence and the types of patterns displayed by various hyperbolic models for populations dynamics that exist in the literature.
Raluca Eftimie
Chapter 3. One-Equation Local Hyperbolic Models
Abstract
The first step in the investigation of transport models for aggregation and movement, is represented by the study of one-equation models. To emphasise the complexity of these models, we start with a variety of hyperbolic models for car traffic and pedestrian traffic (since the models for collective movement of pedestrians are a natural extension of the car traffic models, and moreover traffic-like aspects can be found in many biological systems). Next, we discuss models for animal movement that incorporate constant or linear velocity functions. We review also models with reaction terms describing the inflow/outflow of cars and populations. In the context of animal movement, we present in more detail an analytical investigation of the speed of travelling waves. We conclude with a very brief discussion of numerical approaches for advection equations.
Raluca Eftimie
Chapter 4. Local Hyperbolic/Kinetic Systems in 1D
Abstract
Local hyperbolic systems have been first introduced to describe the movement of a population formed of left-moving and right-moving individuals, in response to the local density of their neighbours. These types of models (also called discrete-speed kinetic models, since they incorporate individual-level information regarding the movement direction of cell/bacteria/individuals into macroscopic models for population dynamics) are applied to describe biological phenomena characterised by sharp turning behaviours (as observed, for example, in bacteria or cells). In this Chapter we discuss these hyperbolic systems in a step-by-step manner: we start with conservative systems with density-dependent turning rates, then we discuss systems with density-dependent speeds, and we conclude by discussing systems that include population dynamics (described by death and birth terms). We also present in more detail an analytical investigation of the stability of spatially-homogeneous steady states and spatially-heterogeneous travelling waves.
Raluca Eftimie
Chapter 5. Nonlocal Hyperbolic Models in 1D
Abstract
More and more experimental studies show that nonlocal interactions play a role in the majority of biological aggregations. In this chapter we describe a few classes of hyperbolic models that include nonlocal interactions among cells/bacteria/animals, which can influence (1) their turning behaviour, (2) their speeding behaviour, or (3) both turning and speeding behaviours. In addition to emphasising the complexity of the numerical patterns that can be exhibited by these nonlocal models, we also present in more detail some analytical approaches, such as bifurcation theory or parabolic limits, that are being used to classify the various patterns exhibited by these nonlocal models. We conclude by discussing the patterns exhibited by a class of nonlocal models that incorporate explicitly stochastic environmental effects that could impact animal movement.
Raluca Eftimie
Chapter 6. Multi-Dimensional Transport Equations
Abstract
One-dimensional (1D) models are simple enough to be investigated analytically and numerically. However, they are not very biological realistic since the majority of behaviours observed in nature occur in 2D or 3D. In this chapter we review different types of 2D kinetic models derived to investigate the movement and local/nonlocal interactions of individuals inside one or multiple populations of cells/bacteria/animals/pedestrians. In this context, we also discuss a few explicit multi-scale models which assume that the dynamics of the (cell) populations depend on their molecular-level states. While the majority of the models considered here are deterministic, we draw attention to a few models that incorporate explicit stochastic events. In addition to reviewing all these different kinetic models, we present in more detail a few analytical approaches used to investigate these models: from the derivation of mean-field models, to hyperbolic and parabolic scaling, and grazing collision limits.
Raluca Eftimie
Chapter 7. Numerical Approaches for Kinetic and Hyperbolic Models
Abstract
Due to the complexity of hyperbolic and kinetic models discussed in the previous chapters, it is difficult to gain much understanding of the behaviour of the models only from analytical results. As we have already seen throughout this study, numerical approaches are critical when trying to unravel the patterns exhibited by these models. There are a large variety of approaches to discretise and simulate numerically the kinetic and hyperbolic models described in the previous sections. However, due to the intense activity of this field, it is impossible to do a detailed review of all numerical schemes developed over the past 50–60 years. Therefore, in this chapter we briefly discuss some of these approaches, to give the reader a glimpse of the large variety of numerical schemes existent in the literature. We start by discussing a few numerical methods for macroscopic hyperbolic models, followed by a discussion on the numerical methods for more complex (and higher dimension) kinetic equations. We conclude this chapter with a brief overview of different boundary conditions.
Raluca Eftimie
Chapter 8. A Few Notions of Stability and Bifurcation Theory
Abstract
While numerical approaches are a very important step in investigating the patterns exhibited by the hyperbolic and kinetic models discussed in the previous chapters, they could be slow and might not offer a full understanding of the models’ dynamics due to the very large parameter space associated with some models. In contrast, stability theory could identify the parameter conditions under which a pattern could form, and eventually could become unstable giving rise to a different pattern. A deeper understanding of the formation of various spatial and spatio-temporal patterns is offered by bifurcation theory, which can distil the mathematical and biological mechanisms not only behind the formation of patterns, but also behind the transitions between different spatial and spatio-temporal patterns. In this Chapter, we review some basic notions of linear stability analysis for pattern formation in ordinary differential equations and partial differential equations, as well as basic notions of symmetry theory and bifurcation theory. We also present in more details the weakly nonlinear analysis approach for pattern investigation and classification. Finally, we discuss some drawbacks of bifurcation theory (e.g. centre manifold reduction) for infinite-dimensional dynamical systems.
Raluca Eftimie
Chapter 9. Discussion and Further Open Problems
Abstract
The aim of this study was to review some of the local and nonlocal kinetic and hyperbolic models derived over the last few years to investigate movement and pattern formation across different biological communities. The emphasis was on the modelling of various self-organised behaviours for cell/animal aggregations, and the analytical and numerical methods used to investigate these behaviours. In this final chapter we discuss the biological relevance of some of the classical assumptions incorporated into the mathematical models, as well as the relevance of some spatial and spatio-temporal patterns presented in this review. We also discuss some of the open problems in the area, and suggest possible directions of future research.
Raluca Eftimie
Backmatter
Metadaten
Titel
Hyperbolic and Kinetic Models for Self-organised Biological Aggregations
verfasst von
Raluca Eftimie
Copyright-Jahr
2018
Electronic ISBN
978-3-030-02586-1
Print ISBN
978-3-030-02585-4
DOI
https://doi.org/10.1007/978-3-030-02586-1