You are here

Integrodifference Equations in Spatial Ecology

Frithjof Lutscher
Publication Date: 
Number of Pages: 
Interdisciplinary Applied Mathematics
[Reviewed by
Andrew Krause
, on
Integrodifference Equations in Spatial Ecology is a mathematical overview of these models and their applications in population dynamics. As the forward by Mark Lewis indicates, this book provides a valuable summary for a developing field, essentially covering the entire literature in the area and explaining all of the key results and approaches in a friendly textbook approach. The book is primarily aimed at graduate students or researchers and does not contain exercises or other pedagogical formatting common to some undergraduate textbooks (though an accompanying website includes some course materials as well as computational tools and other resources). Most of the presentation is self-contained, though a number of technical details would be hard to follow if the reader is not somewhat familiar with related models such as difference equations or reaction-diffusion models. Overall the book does an excellent job in surveying these models, highlighting the successes and remaining challenges of research in this field. The book is far more specialized, but of a similar presentation and comprehensiveness as J. D. Murray's volumes on Mathematical Biology.
The book is organized into three parts. It starts with eight chapters on foundations covering what the models are, how they can be derived and related to empirical data, and the basics of analytical and numerical methods used to analyze them. This foundational material also covers an introduction to the typical questions asked of such models including population persistence and spreading. The following four chapters cover applications, including empirically-relevant measurements, as well as approximate analytical techniques used to analyze these models. The vast majority of these first twelve chapters consider relatively simple settings of single populations with idealized life histories, though Allee effects and other nonlinear aspects are discussed. Finally, the last five chapters cover more detailed mathematical extensions into structured populations, multi-species models, and spatially heterogeneous environments as well as time-dependent (nonautonomous) variants. 
Likely mirroring the literature, the presentation discusses both heuristic and ecological arguments, as well as rigorous functional-analytic approaches, though it does not use a formal theorem-proof style in the majority of the text (and only contains a handful of what might be called rigorous proofs). This balance, along with the extensive bibliography, is executed exceptionally well. Someone more interested in either the rigorous development of the theory or its application to real populations, can easily follow the ideas as presented in the text and find a plethora of further reading material. Overall I would highly recommend this book both as an interesting setting for studying dynamical systems in their own right, and as a less-well-known, but plausibly very powerful, way to model population dynamics.
Dr. Andrew Krause is a Departmental Lecturer in Applied Mathematics at the University of Oxford. His research is primarily in mathematical biology and nonlinear dynamical systems. More information about him can be found at