Bulletin of Mathematical Biology Vol. 54, No. 1, pp. 149 161, 1992. Pergamon Press pie. Printed in Great Britain. Society for Mathematical Biology

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Matrix Population Models: Construction, Analysis and Interpretation, Hal Caswell. Sinauer Associates, Sunderland MA, 1989. $50.00 (cloth), $28.95 (paper), 328 pp.

Parameter Estimation in Ecology: The Link Between Data and Models, O. Richter and D. Srndgerath. VCH Publishers, New York, NY, 1990. $110 (cloth), 218 pp.

"I speak not this in estimation, As what I think might be, but what I know Is ruminated, plotted and set down. . . "

King Henry IV, Part I

The intricacies of life histories occupy a cherished position in mathematical biology. Some fairly elaborate mathematics has been applied to answering real questions in this area, without invoking too many fanciful assumptions about model form. This is rare in mathematical ecology. The key to these successes has been that the questions themselves could be expressed precis'ely enough that they naturally dictated the important features of the model, pre-empting much of the need for additional, and less natural, assumptions to be brought in, deus ex machina, during the course of the subsequent mathematical analysis. In particular, the very fact of age-structure poses a worthy mathematical challenge, but the mathematical representation of age-structure is fairly straightforward. As this discipline matures and evolves, it is well for us to keep a close eye on the questions, to be assured that the natural correspondence between question and model is maintained. This is not an easy task, for the same formulation may appear as a question or as an answer. For example, when the question is to predict population growth rate from the so-called life table of vital rates, Lotka's equation serves as an answer. Yet that same equation motivates a question, when we ask how a life history should be optimized. The matter may become more blurred ff we ask how life histories evolve, for we then begin to vacillate between pretending that we know what natural selection maximizes and discovering that we do not, depending on context. In the former case, we can devote our attention to the details of the

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ways in which the maximization is achieved, and in the latter, we simply construct selection models in order to see what, if anything, gets maximized.

Finally, the same model may serve in very applied problems, such as population prediction, or in abstruse exercises in explanation via hypothetical entertaining of counterfactual conditionals, where the criteria for adequacy of the model may be quite different. To an extent, this resolves itself into attitudes that are specific to the application rather than to the underlying mathematics. In human demography it is generally taken for granted that population predictions are always wrong, since there is a well documented history of this common outcome, and it is acknowledged that the economic and political consequences of very modest-sized errors can be significant. In most wildlife management applications we may be blissfully unconcerned with whether the predictions are right or wrong, since in the short-term we seldom have access to sufficient data that might tell us which was the case. In the new applied interest in population extinction rates (called viability analysis in the discipline of conservation biology) there is a reasonably subtle, but not intractable, issue as to what we might even mean by the model being right or wrong, since the prediction is a probability density of survival times, whereas the population in question will simply survive until it dies out, once.

Caswell has recently published a book on matrix models for population growth and life history analysis, which collects a number of advanced topics that are not treated, e n m a s s e , in any other text. In particular, the extension of the models to consideration of stage-stratified, rather than age-stratified, state descriptions has suffered a neglect which Caswell has almost single- handedly attempted to rectify in a substantial series of publications spanning more than 'a decade. As a consequence of these efforts, there now exists a parallel, applying to stage-structured models, for essentially every important theorem which has been developed for the age-structured models. At the same time, Caswell has adopted a curious, but effective, graph-theoretic notation for representing these models and for classifying their properties. It is good to have this important family of models clearly and carefully explained, in a consistent notation, to the accompaniment of a rich array of examples, in a single volume.

Other advanced topics treated here, and generally neglected in other book- length presentations, have to do with population dynamics in time-varying environments, drawing especially on literature originally authored by Cohen and by Tuljapurkar. Two further topics which ought to be considered elementary, rather than advanced, in any quantitative science with pretense to application, but which in fact are sorely neglected in ecology, are parameter estimation and statistical inference. I was heartened that Caswell devotes some attention to both, the former meriting a section of 13 pages, the other a full chapter of its own, with a chapter on a different topic intervening. I will expand

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later on the symptomatic oddity of this separation of parameter estimation and statistical inference.

In some ways, Caswell's exclusive focus on discrete-time age- or stage- structured models, is a handicap, since, as I will explain presently, certain phenomena are better explored in simpler models. (Yes, continuous-time, time- homogeneous models, are dynamically simpler than difference equations, even though their notation may appear more forbidding to the uninitiated.) In particular, when a phenomenon appears too complicated in the determinants of its behaviour for ready generalization in a simpler model, what is gained by embedding it in a more complicated model?

A second restriction of focus is in the treatment of genetic mechanisms. Except for passing mention in the presentation of the secondary theorem of natural selection, there is no explicit representation of alleles; and recombina- tion is not mentioned at all (compare the book by Charlesworth, 1980, of one decade ago). The genetic models are exclusively those of quantitative genetics, which does not mention genes, but deals simply in heritable variation of traits subject to selection, with the mechanism of inheritance unspecified. The treatment draws primarily on some important papers by Lande. Vital rates are, of course, quantitative traits, but it does seem a little strange, at least pedagogically, to omit all genetic mechanism in a book which discusses evolutionary demography at length.

A third restriction of focus is the omission of the literature on harvested populations. This represents a missed opportunity in terms of some interesting strategies for parameter estimation, such as the fitting of parametrized survivorship curves or size-at-age curves and the estimation of vital rates from catch-at-age records. It also represents a missed opportunity in terms of the richest and most extensive source of data on non-human populations.

The book's strengths, in bringing together much important, advanced material which is not compiled elsewhere, combined with the above-mentioned limitations of scope, argues that this volume will best serve as an advanced graduate text for mathematically sophisticated students who have already been exposed to a more conventional course in population biology. Alternatively, the book may serve an applied modeller who is concerned only with the nuts and bolts of constructing a stage-structured representation of a population, and selects a few chapters for that purpose; in this respect the book is quite self- contained.

Caswell's enthusiasm for matrix models in general, and stage-stratified models in particular, motivates reflection on whether, and when, that preference is justified by convenience or correctness. Matrix projection models for populations always embody a choice of time scale. This choice is not innocuous. In the representation of age- or stage-class specific rates there is a Markovian assumption that finer structure (i.e. the details of age- or stage-

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structure within a class) is not relevant to the resulting per capita rates for the class. We note immediately that in general this can be mathematically true for, at most, one choice of time step size--if it is true for that time scale, then it obviously cannot be true for coarser time scales (unless, trivially, the rates are constant for fortuitous segments of the life history) and by inversion, it cannot be true for finer time scales (subject to the same exception). Fiddling the time scale has ramifications that reach beyond quantitative performance of the model-- i t can affect qualitative properties, resulting for example, in represen- tations of the same population which, respectively, do or do not exhibit convergence to stable age distribution, depending on time scale.

We may also wonder whether the Markovian assumption can be literally true for a biological population. It makes most sense in the case of a population where seasonality imposes a synchronization of reproduction, so that only certain families of age structure are realized. Then the appropriate choice of time scale is the one that corresponds to the period of the factor driving the synchronization.

Similar considerations apply to representation of the population structure by stages rather than age class. The projection matrix still has an implicit time step, but when stages no longer correspond to age classes of unit time step duration, non-zero elements of the transition matrix can appear in rather peculiar places. For example, a non-zero main diagonal element (which is inevitable if a stage has a duration of more than one time step but the model does not recognize age-within-stage) implies mathematically that some positive fraction of the individuals will stay in this stage for any arbitrarily long t ime--but this almost certainly is not what we intend in the model. At best, there are methods which will guarantee that the dominant eigenvalue comes 'out exactly correct in restructuring a true age-stratified model to some arbitrary form, with different structure, but there is no way to accomplish this and also make the other eigenvalues, which govern transient behaviour, exactly right.

Often enough, model structure is influenced by contingencies. We may lack age data, but have data on size classes; we may have age specific data for young age classes, but it may be too sparse for older age classes; we may choose a time step for computational convenience, etc. When this is the case, the form of the model can itself be inexact, in ways that will be of greater or lesser import, depending on the application and depending on the degree of failure of the Markovian assumption. There is at least one notorious example where the discrepancy was embarrasingly large. Let the user beware.

On the other hand, it may be that an age classification is insufficient to satisfy the Markovian assumption, because there are state descriptors other than age which are important to determining per capita birth rates or death rates. In this situation, use of a stage-stratified description is a concession to realism and

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accuracy, and not a mere dodge. But, to avoid such problems as non-zero elements on the main diagonal, I suspect that these models must be stage- and age-classified, at the cost of some clutter. Caswell's graph notation does help in dealing with the clutter.

Caswell's ambition in collecting very advanced methods in his book prompts another kind of assessment. Where are we, at this moment, relative to the grand questions in population dynamics? The central theoretical question, I believe, is: what difference does age- or stage-structure make? There are, I believe, three levels of answer to this question. These address: (1) what are the intrinsic dynamics of age- or stage-structure? (2) How do life history patterns, as summarized in age- or stage-specific vital rates evolve? (3) How can we carry out a population prediction for a population with a particular age- or stage- structure?

The quintessential feature of age- or stage-structured models is that the population structure itself can vary under its own dynamics, with attendant variation in the total per capita birth rate and death rate of the population. By this we mean that even in an idealized representation, where the age- or stage- specific vital rates are constant, so that the projection matrix applies linearly, the population composition will, in general, vary under the projection process. The linearity property is preserved where the vital rates are allowed to be time- varying, but are not functions of the population vector. The fact of change in population composition, even when the vital rates are independent of the population state description, motivates us to wonder how wildly the population structure might vary, under the influence of linear projection processes. The answers constitute the various ergodicity theorems of population dynamics.

The first answer, applying for time-constant vital rates, was proven by Sharpe and Lotka, for the continuous-time model, 80 years ago: the population structure converges to a characteristic form which is determined by the vital rates, and at a rate determined by the vital rates. This result applies conditionally for matrix models- -a sufficient condition is that births occur in at least two age classes whose age designations are relatively prime. A deeper answer, proven 30 years ago for the discrete-time case in a remarkable monograph by Lopez, applies with time-varying vital rates: under conditions essentially the same as for convergence with constant vital rates, the population structures of populations subject to a common sequence of time-varying vital rates will converge, and the form to which they converge is determined by the recent history of vital rates. A recent extension, proven by Cohen, is that if the sequence of vital rates can be represented as a process of sampling a distribution of projection matrices, then the distributions of population structures resulting from independent realizations of the sampling process will converge, with essentially the same conditions applying.

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The net result of these ergodicity properties, when they apply, is that age- and stage-structure do not vary so erratically as to produce dynamics that are terribly different from those of non-structured population models. In this sense, the lesson of the ergodicity theorems whose proofs represent such a major accomplishment, is that population structure does not matter, at least qualitatively. This is not to say that the behaviour of structured population models is necessarily simple or transparent; but it does assert that when the behaviour is not simple or transparent, the cause lies in features of the model other than the mere fact of population structure.

It is now well known, for example, that the incorporation of density- or frequency-dependence in the vital rates can open up a daunting range of dynamical complexity, which, for all practical purposes defies sweeping generalization. But this same complexity can be induced, by density feedback, in extremely simple, unstructured, difference equation models of population growth. In other words, it is the intensity of the discrete-time (or time-lagged) feedback process that is responsible for the dynamic complexity. Furthermore, since transitions from one dynamic regime to the next are known to be the abrupt responses to continuous shift in the values of parameters affecting the feedback intensity, this aspect of the dynamics of actual populations cannot reasonably be portrayed as the exclusive domain of theoretical inquiry-- elucidation of real dynamics will have to rest on secure measurement of real parameter values. For this reason, I do not know what is gained strategically, in terms of intuitive insight or technical tractability, by attempting to study these processes in the abstract in the more complicated, structured population models.

Where age- or stage-structure clearly does matter is in the evolution of the vital rates which determine the life history pattern. This operates via the quantitative effects of the vital rates on the population growth rate or genotypic growth rate. Caswell's formulae for sensitivities of the growth rate provide a powerful and convenient tool for exploring these effects. Related theory provides algorithms which elucidate the age- or stage-specific criteria for maximizing the growth rate. What emerges, is that, as expected from the above comments about ergodicity, there are no surprises caused by considerations of age- or stage-structure in evolution. Analogues of Fisher's Fundamental Theorem appear to apply for structured population models in much the same circumstances under which the theorem applies for unstructured models. With no density- or frequency-dependence, and with simple (e.g. clonal or intermediate heterozygote fitness and random mating) genetics, the population growth rate increases under selection in a way that is elegantly describable. With density-dependence and a constant environment, and with simple genetics, the population size increases under selection. With density- dependence and a fluctuating environment, a balance emerges between

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selection for population growth and for population size (so-called r and K selection). With frequency-dependent selection we cannot generalize, except tautologically, as to the property (or properties) that increase under selection. This limitation of our theoretical understanding of the process with frequency- dependence is, again, due to a feature which operates in unstructured models just as intractably as in structured population models.

Caswell seems not to be sure whether he really wants to deal with the third issue--prediction. In Chapter 2, there is a somewhat laboured page distinguishing between projection and forecasting. Caswell asserts that the former constitutes an exploration of hypothetical scenarios subject to whatever assumptions were made, whereas the latter is to be interpreted literally as a prediction about the future. At least with regard to the methods of Chapter 2, my impression is that the author's sympathies are for projection rather than forecasting. But of course, population models do get used for forecasting, to help make decisions about pest control, harvest quotas, environmental impacts, park management, etc. And so, inevitably, we come to be concerned with the relationship between reality and model output. More to the point, we would like to say something quantitative about the expected match, based on examination of the data that are in hand for any particular forecast. This brings us to the topics of parameter estimation, which is the process of obtaining parameter values from data, and statistical inference, which includes the process of attaching a measure of certainty to parameter estimates.

Notwithstanding his apparent ambivalence, Caswell has far more to say on these topics than do other book-length treatments in the ecological literature on population dynamics. But it seems that his interest in statistical inference derives more from the tradition of hypothesis testing than from a concern with the predictive power of the models. The section on parameter estimation does not mention the sampling variances of the estimators, but there is an illustration of possible specification error resulting from choice of model. The discussion of the distinction between birth-pulse and birth-flow clarifies a matter that is often confused in applications. The chapter on statistical inference presents two approaches for putting a "confidence interval" on an estimate of the population growth rate obtained from a projection matrix. One is an approximate error-propagation formula, based on a linearization, assuming independence of the error in the various matrix elements. The example illustrating this approach does present the idea of sampling variation as a source of error in the matrix elements. The second approach is the resampling strategy of either a jack-knife estimate of variance, or a bootstrap estimate of the distribution of population growth rate. In both resampling strategies, Caswell proposes that the individual oroanism is the fundamental sampling unit, even though, in his examples, one individual may contribute to the estimate of several elements of the matrix as that individual's reproductive

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contribution is recorded for every year that it lived. Both approaches raise many obvious questions that are worth pursuing.

Overall, the contrast is striking between the primitive statistical methods and the sophistication of the mathematics in the dynamical models. But Caswell has faithfully represented the state of the science. There is a gap between the voluminous literature on the statistics of the elementary observations on abundance, survival, and reproduction (Seber's book, 1982, provides a good introduction to that literature) and the voluminous literature on the mathematics of ecological population dynamics. There is very little literature bridging the gap by considering the implications of the statistics of the observations for a dynamical model with parameter estimates based upon them. I hope that a generation of ambitious graduate students will recognize in this a promising research agenda.

Richter and S6ndgerath have recently published a book which bears on an important item in this proposed research agenda: the parameter estimation problem. The main focus is on the use of numerical methods that are implemented in readily available computer packages, for estimating para- meters by non-linear least squares or by maximum likelihood. The models that are considered include censusing, body growth, population dynamics, population spatial distribution, population interactions and ecosystem models. Among the population dynamics models that are treated is an age- within-stage representation, for which smoothing strategies are presented to obviate the evident impracticality, with sparse data, of separately estimating each element in the projection matrix. The basic idea is that there is an underlying function describing the age- or stage-dependence of the birth, death, or stage-transition coefficients, so that a few parameters describing this function can take the place of the set of (probably more numerous) age- or stage-subscripted values. This idea seems most attractive when the function can be biologically motivated, but the fact of reasonable smoothness in the age- or stage-dependencies can be treated simply as an empirical matter. The approaches presented by Richter and S6ndgerath can readily encompass parametrization of effects of additional factors, such as temperature or density or abundance of other species, on the dynamical coefficients.

Richter and S6ndgerath routinely report standard deviations of parameter estimates in their worked-out examples, and briefly describe the mechanics of obtaining these error estimates by application of asymptotic normal theory with maximum likelihood or non-linear least squares methods. But, curiously, the book has little to say on what these error estimates are used for. In particular, the natural extension of considering the propagation problem-- how the parameter error affects model behaviour, and the relation to predictive power -is not taken up. Nor is there discussion of the expected consequences of the approximations and assumptions which went into the error estimates.

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There is a useful discussion on the relationship between design of the data coverage, the resulting parameter covariance matrix, and estimability.

The authors describe simulation methods in connection with the discussions of spatial distribution and of sampling for abundance estimation, but it seems that their main intention in so doing was didactic. There is no intimation in this section that simulation can be used for serious work, such as variance estimation or as a solution to the propagation problem (contrast Morgan's book-length treatment, 1984, of simulation methods and their applications).

This book would serve well as the subject of a graduate seminar course for students who face the threat of data. At the price, which is roughly in order of magnitude more per page than the cost of xerography, students may want to share copies. The book is clearly written (despite more than the usual number of typographic errors) and it treats an important topic. Eventually it would be nice to have a thorough compendium which marries a concern for the statistical properties of what goes into an ecological model with a concern for the statistical properties (and relation to reality) of what comes out of the model. But we have to start somewhere.

LITERATURE

Charlesworth, B. 1980. Evolution in Age-structured Populations. Cambridge: Cambridge University Press.

Morgan, B. J. T. 1984. Elements of Simulation. London: Chapman and Hall. Seber, G. A. F. 1982. The Estimation of Animal Abundance and Related Parameters, 2nd edn. New

York: Macmillan.

DANIEL GOODMAN Department of Biology

Montana State University Bozeman, MT 59717, U.S.A.

Evolutionary Instability, Gebhard Geiger. Springer, Berlin, 1990. DM79.00 (paper), 167 pp.

The volume under review is about certain aspects of "sociobiology". As I understand the issue, the basic question is the extent to which social behaviors