Nonlinear Dynamics of Interacting Populations

By Alexander D. Bazykin
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Cover of Nonlinear Dynamics of Interacting Populations Nonlinear Dynamics of Interacting Populations

Alexander D. Bazykin, translation editors Alexander Khibnik and Bernd Krauskopf, World Scientific Series on Nonlinear Science, Series A, Volume 11, (1998), 215 pp. price £ 52.-;
Reviewer: Odo Diekmann, Mathematisch Instituut, University of Utrecht, The Netherlands.

Level: introductory.
When trying to uncover the cause-effect relationship between mechanisms and phenomena, it makes sense to isolate a small part of the world (which we then usually call a "system"). In physics one does this both experimentally and theoretically. In population biology, where experiments are much more cumbersome (if not impossible) theoretical thought experiments prevail. Frequently, the gained understanding of an isolated small part serves as a building block for later more encompassing analyses.

This elegant and useful book expounds the methodology sketched above in a systematic manner. It explains how various mechanisms are represented by nonlinear terms in ordinary differential equations for population sizes. Labeling models according to the mechanisms that are incorporated, one obtains a suite of low dimensional dynamical systems that can be studied, starting at the simple end and progressing towards the complex "end", in search for qualitative understanding of the phenomena that they generate. Bifurcation theory is the guiding principle : one tries to partition the parameter space into regions of equivalent behaviour, with due attention for the boundaries of such regions and the associated changes in behaviour. Pursueing this last aspect, one encounters organising centres at parameter points where the highest order degeneracy occurs.

After the untimely death of the author in 1994, the production of the present book was coordinated by translation editors Alexander Khibnik and Bernd Krauskopf. In the foreword they say : "This text can be used as a guided tour to bifurcation theory from the applied point of view". I fully agree and highly recommend the book to both theoretical population biologists and applied mathematicians with an interest in dynamical systems.

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