REVIEWS MAY, R. M. (1973). Stability and complexity in model ecosystems. (Monographs in population biology 6. Ed. by R. H. MAcART~UR)--Princeton N. J., Princeton Univ. Press., ix + 235 pp., cloth $11.5o; paper $ 4.95. This book is a valuable addition to the literature of theoretical ecology. It surveys a large number of mathematical results of the author published in the preceding three years. The empirical contents of these results are stressed. Special models of various complexities are introduced and their interrelations are discussed in order to show how these models m a y be considered as metaphors for real ecological systems. Mathematical details are delegated to five appendices. As a result the book may well be read by ecologists who are mathematically orientated but lack a firm background in formal mathematics. The book begins with an introduction in which the outlines are sketched. In the second chapter various stability-concepts are discussed with special emphasis on neighbourhood stability, and the relations between deterministic models and models with inherent (demographic) and environmental stochasticity are outlined. Incidentally, the terms "analogous (homologous) differential and difference equations" as used by MAY (meaning that the right hand sides are identical) bears the false suggestion that the models are also conceptually related, whereas the difference equation analogues of the differential equation models usually are not even valid population models as the population densities may become negative. For example the discrete time model which is conceptually most akin to the logistic growth model is probably the hyperbolic reproduction curve model (see e.g.C.T. DE WIT (I960), On competition. Agric. Res. Rep. 66.8, Wageningen). However, for the discussion of neighbourhood stability the concept will do, but only if it is introduced after the linearisation has been performed. In Chapter three diverse ramifications of the Lotka-Volterra predation and competition models with m a n y species are discussed, whereas in Chapter four somewhat more complicated models with at most two species are considered. In this last chapter a great deal of space is devoted to the existence of limit cycles. The main conclusion that emerges from these two chapters is that trophic complexity does not entail stability as a logical necessity (trophic complexity being measured as the number of species or food links involved). On the contrary in these simple models Acta BiotheoreticaXXIV (1975)

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increased trophic complexity usually makes for lowered stability. If we want to salvage at least some part of the appealing stability and complexity thesis we shall have to look for special behavioural mechanisms or special life histories etc. These special features, however, will be the outcome of a long game of trial and error, only the more stable interaction being now observed as such. Chapter five illustrates how some of the results of the previous chapters should be modified for models with randomly varying environments, and in Chapter six MAY goes into limiting niche overlap for simple models with randomly fluctuating environments. It turns out that the limit to species packing on a one dimensional resource continuum, e.g. food size, depends only very weakly on the degree of environmental variability, as long as the fluctuations are not so severe that even one species alone verges on extinction, or virtually absent, in which case species may be packed arbitrarily close. In between, the food sizes of species adjacent on the resource continuum have to differ at least by an amount roughly equal to the standard deviation of the food sizes taken by either individual species. Some analogous results are derived for more resource dimensions. From these results a reverse conclusion concerning the relation between stability and complexity may be drawn: in an evolutionary sense environmental stability will promote trophic complexity. This idea is taken up again in the final chapter where some speculations concerning the conservation of the number of potential ecological niches in the geol%4cal past are given. In conclusion, this is a book which any theoretically orientated ecologist should read, or else he should refrain from any comment concerning the relation between complexity and stability.

J. A. J.

METZ

Institute for Theoretical Biology of the State University Leiden LIEBERSTEIN, f-I, M. (I973). Mathematical physiology; blood flow and electrically active ceils. (Modern analytic and computational methods in science and mathematics 4 o. Ed. by R. BELLMAN)--New York, London, Amsterdam, Amer. Elsevier Publ. Comp., xiv + 377 PP., Dfl. 61,5o.

The present reviewer began his task as a biologist highly interested in mathematical model building but trained in other fields of biology, hoping to find a survey of the subject matter indicated in the title. Acta B,iotheoretica XXIV (1975)

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However, he met with a deception. LIEBERSTEIN'S book is certainly not a textbook, but a highly personal account of the work of the author and his collaborators, published during the late I96O'S in Mathematical Biosciences and this journal. The first I23 pages, dealing with blood flow in the aorta, are in this reviewer's opinion the best part of the book. The basic physiological assumptions are clearly expounded and a great deal of attention is devoted to the approximations made. First the movement of the wall of the aorta is assumed to be known. It is left undecided whether the resulting partial differential equations have unique solutions given appropriate initial and boundary conditions. Instead, the author proves that (I) for some boundary conditions at least one solution exists, and (2) that all solutions belonging to a particular class, which is defined b y a condition of a different kind, will converge to each other in mean square as time progresses {this proof seems to contain an error, however). Statements (I) and (2) together are exactly what is needed for practical purposes only. An iteration procedure is given which will converge to a representative solution, starting from the solution for rigid walls for which an appropriate expression is derived first. The first iterate is thereupon modified (deleting terms that we m a y hope to be small) in such a way that (I) the unknown pattern of wall movements does not enter into the expressions except for the local radius, and (2) time enters into the expressions only in the local pressure gradient, together with its first derivate in the length direction, and their time derivatives up to some convenient order, which is all the information that we m a y obtain by measurement devices that exist or m a y become available with the present technology. The resulting formulae allow us to compute approximately the velocity field for a particular cross section as a function of its radius. Next a differential equation for the tension v e r s u s radius relation of the aorta wall is derived in which some functionals of the velocity field enter. A guess for an initial condition may be made from an estimate for the diameter at peak pressure. In this way we may try to determine the i1r v i v o tension v e r s u s stretch relation from pressure measurements. Altogether, the first part of the book presents an imaginative application of modern mathematical techniques to a biophysical problem. Applying the results in experimental practice will probably still require a considerable cooperative effort of both applied mathematicians and bioengineers. It is to be hoped that this line of research and related ones will be pursued in the near future. The second part of the book (225 pages) dealing with nervous conduc-

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tion is less satisfying. After reviewing some basic electrophysiology in Chapter 6 a short review is given in Chapter 7 of HODGKIN'S and HUXLEY'S model formulation for the space clamped squid axon together with a somewhat more elaborate review of the history of the transatlantic cable, in order to introduce the essential role induction (and the resulting hyperbolic differential equation) played in the design of a reliable communication line. When, for a propagating impulse, HODGKIN and HUXLEY solved the parabolic differential equation, which resulted from a combination of their system of first order local membrane equations with the equation for passive electrodiffusive spread, they found that the existence of a solution of the resulting two point boundary value problem for a system of ordinary differential equations (second order in voltage, first order in the other variables) depended critically on the propagation rate 0. In this way 0 may be determined from first principles. In their shooting method this critical dependence on 0 was expressed as a gross instability when an incorrect guess for 0 was made. LIEBERSTEIN interprets this as evidence that their basic equation is incorrect--the propagation of a pure wave should in his opinion be described by a hyperbolic equation-and he introduces terms corresponding to an induction effect into HODGKIN'S and HUXLEY'S equations. The line inductance is determined from the observed velocity of a travelling pulse, assuming that the wave operator part of the equation has to be identically satisfied for such a pulse. A system of first order ordinary differential equations remains which allows us to calculate the form of the impulse. In Chapter 8 LIEBERSTEIN describes previously unpublished results concerning the existence and uniqueness of solutions of his equations, using classical results. Moreover, an equivalent integral equation is formulated, in such a way that iteration of the integral operator will also lead to a solution of the initial value problem. From this iteration procedure it is argued that the first order differential equation for the travelling pulse mentioned above may be considered either as a singular perturbation of the original partial differential equation or as an equation for a time asymptotic steady state. The reasoning by which these results are reached is highly heuristic; constructing a more rigorous version will certainly take a good deal of work still. The same remarks apply to some of the steps in the existence proofs. The first part of Chapter 8 is moreover relatively inaccessible, due to a cumbersome logical notation and an exceptionally large number of small errors. In the last paragraphs of Chapter 8 LIEBERSTEIN treats a mechanism which might give rise to a line inductance, inversely proportional to the

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8x

square of the radius of the axon; a relation which has to be assumed in order to get a correspondence with the observed proportionality of propagation rate and the square root of the radius (whereas the form of the impulse is independent of the radius). Incidentally, it should be remarked that this proportionality has been demonstrated to hold true for all models of nervous conduction which do not incorporate inductive effects. LIEBEt~STEIN does not mention this. In the last two chapters LIEBERSTEIN reports numerical studies corroborating his conclusions concerning the singular perturbation interpretation of the ordinary differential equation for a travelling pulse, parameter studies concerning the onset of repetitive firing and numerical studies on diverse modifications of the reformulated HODGtClN-HUXLEY equations pertaining to other electrophysiological phenomena. It is remarkable that the form of the travelling pulse resulting from LIEBE~STEIN'S equations seems to give a somewhat better fit to empirical data than the pulse that resulted from the original equations, whereas the equations themselves seem to differ quite a lot at first sight. Concluding remarks: this book is certainly not a textbook on mathematical physiology. Its mathematical level is much too high for a physiologist, whereas for a mathematician a much too narrow view of the subject matter is given. This is especially true for the second part in which the author elaborately expounds his own research together with much anecdotal information, without mentioning much about the results concerning the original parabolic equations. This omission is rather heavy, moreover, since, contrary to the opinion LIEBERSTEIN expresses, the research concerning travelling wave-like solutions of nonlinear parabolic equations seems to hold great promises. In fact this is one of the areas where biology has given impetus to interesting mathematical developments, starting with the discovery of the "wave of advantageous genes" by FISI~R in 1937. On the other hand the book is highly interesting and certainly deserves to be read in conjunction with other books and papers, which give more in the way of a survey of one of the fields indicated in its title. An advantage of the book is moreover, that it spans the whole range from experiment to sophisticated mathematics. This applies especially to the first part, since it presents an attitude towards biological problems which in this reviewer's opinion deserves to spread more widely.

J. A. J. METZ Institute for Theoretical Biology of the State University Leiden~ Netherlands