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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.