Stochastic foundations in movement ecology. Anomalous diffusion, front propagation and random searches
This book, written in an interdisciplinary spirit and for a broad (but mathematically minded) audience, attempts to provide an overview of the essential mathematical, physical and modelling aspects which describe the motion of living entities at various scales. Since the movement of living entities involves a certain level of randomness, the mathematical scaffolding of the book lies in the theory of stochastic processes, rather than being deterministic. A distinguishing feature is the emphasis put on the multilevel description of stochastic processes and stochastic dispersal models (called microscopic, mesoscopic and macroscopic levels therein). Clear distinctions between these levels of description are made, their characteristic features, fundamental assumptions, advantages and limitations being highlighted. This book is divided into three parts. The first part, consisting in two chapters, provides a survey of the essential results in probability theory (first chapter) and an introduction to the concept of stochastic processes and of their micro, meso and macroscopic descriptions (second chapter). The reader is also generously provided with many concrete examples from population ecology. The second part of the book, consisting in four chapters, provides a comprehensive review of the stochastic modelling approach towards describing ecological movements. Chapter 3 refers to micro, meso and macroscopic definitions of dispersal and their ability (or lack thereof) to incorporate nontrivial effects such as memory (correlations), finite jump speeds, long-distance dispersal and velocity drifts. It also describes the opposite of dispersal, aggregation, in the form of animal grouping. This chapter does not only state and describe, but also discuss, explain and clarify, passing through multiple levels of description (micro, meso and macro) whenever necessary. Chapter 4 shows how differentiation on multiple levels of description helps when introducing more advanced concepts such as memory in the transport process. More complex situations which arise under non-Markovian conditions are considered, with the aim of introducing the concept of anomalous diffusion, through the process of taking the macroscopic limit of continuous-time random walks. Chapter 5 discusses reaction-dispersion processes, that is, dispersal processes coupled with birth and death phenomena. The joint effects of reaction and dispersal are shown to give rise to nontrivial dynamical behavior. If this dynamics occurs through an initially unoccupied territory, this leads to the idea of waves of propagation, or fronts, which are essential to describe the phenomena of biological invasion. Chapter 6 introduces the essential ideas of random search theory for animals, addressing the concepts of search and uncertainty in behavioural ecology within the framework of animal foraging. The essential behavioural mechanisms underlying efficient random searching are deduced by interpreting the equations of movement from a biological viewpoint. The third part of the book, consisting in three chapters, is concerned with specific applications of the previously introduced approaches and concepts, further augmenting the clarity of a presentation which already had a predominantly applied nuance. Chapter 7 is concerned with cell motility, with a focus on active, rather than passive, cell motion. Individual cell trajectories are analyzed, emphasizing common statistical features found in experiments concerning basic parameters such as velocity distribution, correlation and mean square displacement. The purpose is to illustrate the variety of phenomena that one may possibly encounter and the physical and mathematical frameworks which are available to explain them. The applicability of anomalous diffusion as a reasonable concept to describe cell motion is also discussed. Chapter 8 reviews relevant models for biological invasions, providing examples of how dispersal kernels and diffusion coefficients can be estimated from available data, and how the advance front speed can be predicted. Chapter 9 is concerned with field and lab experiments that approach the idea of a random search in different ecological contexts, discussing typical problems and limitations which occur when attempting to provide experimental evidence. The occurrence of stochastic adaptive mechanisms in lab experiments is also discussed. Finally, an appendix containing particulars on certain mathematical and numerical tools is also provided. This book is interdisciplinary, intuitive and versatile, bridging a gap between mathematics and ecology in a pedagogically sound and scientifically relevant manner, from which a broad audience may benefit.
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