UK Nonlinear News, February 1999

Nonlinear Dynamics of Interacting Populations

By Alexander D. Bazykin

Edited by Alexander I. Khibnik and Bernd Krauskopf.

Reviewed by Jonathan Swinton

World Scientific Series on Nonlinear Science, Volume A11
Publisher: World Scientific, Singapore
ISBN 9810216858
193 pages

Gilbert White wrote in 1778 of his observation that, year after year, Selborne contained eight breeding pairs of swifts (Micropus apus): an observation repeated by Lawton and May in 1983 (Nature, 306:732-733, 1984) who found twelve pairs in an otherwise much changed environment. This is perhaps the longest known ecological time series, and illustrates two facts relevant to this review. The first is that ecologists have long been interested in ecosystem stability, and the second is that such anecdotal data as a basis for theory is, while atypical, not unknown.

Given these two facts, the Lotka-Volterra equations have enjoyed sustained attention from mathematical ecologists. In particular, the mathematical neutral stability of the fixed point solution of the basic Lotka-Volterra equations for a predator-prey system is one natural starting point for the investigation of ecosystem stability. Bazykin lists seven possible ecological factors that could perturb this structurally unstable system, such as nonlinear reproduction or predation rates, and models each as a simple modification to the Lotka-Volterra system. He demonstrates that these factors can be divided into those that stabilise the equilibrium fixed point to become a global attractor, and those that destabilise it so that most trajectories go to infinity. Stabilisers include competition among prey or predators and nonlinearity in predation rates making rare prey disproportionately hard to find. Destabilisers include predator saturation and nonlinearity in reproduction rates making rare prey disproportionately hard to breed with (an Allee effect). This much has been analysed many times, and as Bazykin recognises, can all be contained within the analyses of Rosenzweig and MacArthur in 1963 (American Naturalist, 97:209-223). Moreover it is not hard to see that combining two stabilising factors remains stabilising.

At the heart of this book, though, is a systematic study of what happens when a stabilising factor is combined with a destabilising factor. There are four stabilisers and three destabilisers, and so twelve possible two-factor modifications of the Lotka-Volterra system to consider. The system is two-dimensional, and so the stage is set for a systematic display of bifurcation theory. We get transcritical and Hopf bifurcations, Takens-Bogdanov bifurcations, and ultimately a codimension-three bifurcation when a Takens-Bogdanov point degenerates (the unfolding of this last bifurcation is only conjecturally generic). Eventually we see, for a half dozen of the cases, complete descriptions of the bifurcation diagrams, with the pleasing air of inevitability that makes bifurcation so satisfying (Aha! The periodic orbit terminates in a homoclinic bifurcation!). Finally, a three factor model is analysed similarly. After a nod to competition models, the book finishes with extensions to three-population systems, and so of course much of the analytic completeness is lost. But there is a nice combination of numerical and analytical work and an introduction to the delightful area of Shil'nikov theory.

As the book's blurb itself says, this is a good example of bifurcation theory from the viewpoint of application. But the blurb, and the authors own words inside, spend a great deal more space extolling the virtue of this book as a contribution to ecology and aspires to present a 'much-needed tool, given the increasing anthropogenic load on the biosphere'. Does it succeed in these terms? No, I think. There is a great deal more to ecology than the Lotka-Volterra equations. The author cites the Odum book of 1971 as essentially his only reference for theoretical ecology, yet there are other key texts - May's Stability and Complexity of Model Ecosystems (1974) springs to mind - which have sparked a continuing debate about ecosystem stability (for example in Nature just as this review was written), which this book completely fails to engage in. One reason for this is clear: Bazykin was a mathematician; ultimately he was more interested in the bifurcation theory than the ecology. I described above how he constructs twelve combinations of ecological factors: he only analyses six because the rest don't lead to systems with qualitatively different dynamics and are thus uninteresting. But ecologists might think differently: it is interesting that interactions of different factors can lead to the same dynamic result. It is noticeable that when Bazykin got the chance to influence policy on the `anthropogenic load on the biosphere' as a Deputy Minister to Gorbachev, his tool of choice was the rather less mathematically elegant Geographical Information System. Nevertheless, there is much of value here for the ecologist. It is important that within such simple models it is possible to demonstrate such a wide range of dynamic behaviour: coexistence, exclusion, hysteresis, global bifurcations to extinction or large-scale oscillation. It is this variety of transitions, and the 'dangerous boundaries' (as Bazykin calls them) at which they occur, of which the ecologist needs to be aware.

This book has had a long struggle into English, and it shows. The author wrote the Russian edition in 1985, and died in 1994 before he could complete the translation. The reference list has only a handful of references post 1990, and those to a specific model; there is no reference to the explosion of work on Shil'nikov theory, for example. In essence this book remains the record of work of the 1970s and early 1980s. His editors have completed the book tidily, although their own interventions are a little obtrusive: the numerical phase plane diagrams they add are crudely drawn and rather pointless, while their rendering of `simpatrick' for `sympatric' is unlikely to win the confidence of ecologists. There is no index.

In summary, a fine guided tour, through a sustained application, of bifurcation theory as at the 1980s and a good basis for studying subsequent developments. But mathematical ecologists would be unwise to use this as an introduction to the theory of interacting populations.

Jonathan Swinton, 11/10/98. Department of Zoology and King's College Research Centre, University of Cambridge, Cambridge CB2 1ST
Thanks to Rebecca Hoyle and Pej Rohani for useful comments.

UK Nonlinear News thanks World Scientific for providing a copy of this book for review.

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