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Ecological Modelling 148 (2002) 307–310
Book review
www.elsevier.com/locate/ecolmodel
Matrix Population Models: Construction, Analysis,
and Interpretation
Author Hal Caswell, 2nd ed., Sinauer Associates,
Sunderland, Masssachusetts, 2001, xxii+722 pp.
ISBN 0-87893-096-5; $64.95
If writing books is an art, then writing books
on the science of ecological modelling is certainly
a ‘modern art’. But should we, then, consider
writing books on ‘Population Models’ as a pop-
art? Certainly not. It is, rather, the highest art to
conjugate the three basic ‘dimensions’ of mod-
elling, i.e. ‘Construction, Analysis, and Interpreta-
tion’, into a single, readable and self-sufficient,
hence workable, creation. The new book by Hal
Caswell, published a decade after its first edition,
is an excellent example of such a creation, where
all these ‘dimensions’, though mandatory to any
model, are so strong and in such a harmony that
the author’s announcing them explicitly in the
title seems no longer extraneous.
Conceptions of age or size classes and of life-cy-
cle stages are among the simplest and most natu-
ral ones in population biology. The reputation of
a mathematical object such as matrices is, on the
contrary, esoteric outside mathematics, appalling
even by the fact A×B"B×A, unlike with usual
numbers. But it is however matrices which for-
malise such a natural idea that the age (respec-
tively size or stage) structure of a population
causes the population dynamics, thus giving rise
to the population projection matrix— perhaps, the
most urgent and efficient tool of population
modelling.
Nineteen chapters of the second edition vs. 10
chapters of the former one are just a quantitative
indication of the enormous progress achieved in
the theory and application of matrix population
models since the publication of the first edition.
The new chapters expand the subject in essentially
those directions, which lead us from simplifica-
tions of the classic Leslie matrix (the simplest
linear model with constant survival and fertility
rates) to more realistic situations in problems of
0304-3800/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.
PII: S0304-3800(02)00002-9
Book re6iew308
conservation biology, pest control or optimal har-
vesting. Those directions include parameter esti-
mation and sensitivity analysis, statistical
inference, seasonal and stochastic variations of
the environment, demographic stochasticity, sex-
ual dimorphism, and density dependence.
The author has taken much care to make the
book an efficient guide for readers down these far
from simple ‘routes’. Chapter 1, called ‘Introduc-
tion’, while stating just a general idea of intimate
links between the individual life cycle and popula-
tion dynamics, introduces the reader into how
well the monograph on this fascinating subject is
organised, rather than into the subject itself. In
particular, it is already here that MatLab-users
will be pleased to find the author in their, now
expanding, ranks and non-users will encounter
one more argument against their abstinence.
The real introduction to the subject is given in
Chapter 2 and it begins, of course, with the Leslie
formalism for a population subdivided into dis-
crete age classes of the same duration which coin-
cides with the time step of the model. The Leslie
matrix arises here in a natural and logical way,
together with the basic dynamic equation, as a
particular case of the projection matrix. Equally
logically the directions for further model develop-
ment are revealed, whereby the postulate of con-
stant vital rates might be relaxed in one way or
another. The fundamental link between those
rates and the life tables of traditional demography
gives way to correct modes of statistical estima-
tion for the model parameters.
Little known details of Patrick H. Leslie’s biog-
raphy clarify his role in the development of the
well-known formalism, and a short digression into
the history of demography and ecology in the
60–70s explains why the formalism failed to find
an immediate response in contemporary minds.
To have the person of Leonard P. Lefkovitch
introduced here too is as relevant and logical as
Lefkovitch’s extension of the Leslie formalism to
‘ecological studies where the age of an individual
is rarely known’was logical and pragmatic: classi-
fying individuals by developmental stage rather
than by chronological age. This idea is compre-
hensively presented in Chapter 4, while Chapter 3
poses a question that generalises Lefkovitch in
quite a mathematical way: ‘What, in general,
should be considered state 6ariables in population
models?’The author suggests a reasonable way to
answer originating from statistical analysis of the
dataset(s) available, for instance, on age, body
size, and weight measurements.
The central place in the book is logically occu-
pied by Chapter 4 on ‘Stage-classified Matrix
Models’, with the ‘stages’understood in a gener-
alised sense: either as the life-cycle stages, or
multiple habitats of the population, or stages of
ecological succession which the habitat of a given
species may have reached, etc., the generality aris-
ing in the matrix formalism, while the biological
meaning of model parameters being always case-
specific. Concepts and notions are introduced and
illustrated here which are fundamental to analyse
any kind of matrix model. These are: the life cycle
graph (‘isomorphic’to the projection matrix), the
projection equation and its general solution, the
eigen6alues and eigen6ectors as the major quanti-
tative characteristics of population dynamics in
the long-term and the stable stage distribution,
irreducible and primiti6ematrices responsible for
con6ergence and ergodicity in model trajectories.
‘Matrix Models’as a special kind of mathemat-
ical object have certain relationships with other
known kinds of model, and a Fundamental
Monograph could not cast a veil over this aspect.
Indeed, it is the depth of treatment of this aspect,
which distinguishes H. Caswell’s book from all
others on the subject. Formal links are high-
lighted throughout the Monograph, and whole
chapters have been written wherever interesting
model applications arise from those links. Thus,
the life cycle graph is considered, in Chapter 5, as
the graph of a finite Markov chain, and canonical
results of the absorbing-chain theory ‘provide
ways to calculate age-specific parameters [indices]
from stage-classified models’—for example, how
the survival and fertility rates depend on age,
what is the mean age at first reproduction, etc. In
Chapter 7, the life cycle graph is studied by means
of ‘z-transformation’of a discrete-variable func-
tion, and this enables one ‘to derive simple alge-
braic formulas for the characteristic equation and
the …eigenvectors of a stage-classified projection
matrix’immediately from the graph.
Book re6iew 309
‘Structured Population Models’are frequently
understood, in the literature, as continuously
structured ones, and these are what Chapter 8 is
especially concerned with. The general pattern of
choice between discrete and continuous descrip-
tion of the time and population structure is
reflected in a 2×2 matrix (Table 8.1) —a trivial
one for an expert, but quite informative for a
beginner—where each particular choice is ac-
corded its own mathematical apparatus and where
the ‘Matrix Models’occupy their legal place. The
problem of choice—the most difficult and eternal
one—has both objective aspects and subjective
ones, and the author shares openly with us the
conjectures, which had determined the choice he
apparently made.
That choice was made many years ago, and
these past years have seen convincing evidence of
its practical implications. The author’s experience,
truly invaluable in the practice of population
modelling, has been embodied in one more valu-
able feature of the book, namely, its vivid ‘vector’
to applications, which threads together and con-
solidates all the manifold contents of the book,
much like the ‘Passing Line’in the famous paint-
ing by W. Kandinsky threads together and unites
the entire multi-image composition. The point is
not that each idea, formal conception or theoreti-
cal issue has unfailingly been illustrated by rele-
vant examples. Rather, all the theoretic contents
of the book are predetermined by applications of
‘Matrix Models’and, furthermore, many chapters
are of vividly applied character, e.g. Chapters 6,
Parameter Estimation; 9, Sensitivity Analysis; 10,
Life Table Response Experiments; 12, Statistical
Inference; 18, Conservation and Management.
‘Matrix’is a synonym for ‘linear operator’in
standard courses of linear algebra, and the ‘Ma-
trix Population Models’are therefore perceived as
primarily linear models. But this is ‘Nonlinear
Ecology’which has become the most popular
slogan of the 20th century and which will be so
perhaps during the next millennium too. How
does the author of a Monograph on ‘Population
Models’respond to this challenge of the century?
With aplomb. His capacious Chapter 16 on ‘Den-
sity-dependent Models’begins with a trivial ob-
servation that the vital rates may well be
functions of the total population size or of partic-
ular stage abundances, the matrix form of the (no
longer linear!) population equations being still
conserved. The stage-structured models really har-
bour much more opportunities to account for
density effects than the non-structured ones, to
which the notorious bifurcations (qualitative
changes of dynamic regimes) and chaos (a special
regime never possible in linear models) historically
owe their appearance in population contexts.
These opportunities are conclusively demon-
strated both in theory and interesting practice of
case studies on population dynamics of Tribolium
beetles in the laboratory and spotted owls Strix
occidentalis caurina in the field.
The author himself states his goal ‘to make
matrix population models accessible to the biolo-
gist who wants to use them’. Many sections have
therefore been written as an exciting introduction
into the subject, while the mathematics needed
have been provided for in an Appendix (whose
logical structure has been markedly improved vs.
the first edition). The didactic aspect is highly
strong in the book, although it is completely
deprived of any explicit and boring didactics. Pure
‘biological’passion for tables and diagrams really
improves understanding and ‘digestion’of the ma-
terial including that of mathematics (for example,
graphic diagrams 4.7 and 4.11 of formal proper-
ties in non-negative matrices are worth including
into any text on matrix theory that presents the
Perron–Frobenius Theorem). Even a mistake
made elsewhere by the author himself appears to
help understanding a methodically hard point
(Example 4.1 of constructing the life cycle graph
for teasel Dipsacus syl6estris).
But, in my view, the book is no less interesting
or useful for a more advanced reader, although a
rigorous mathematician would easily find certain
points to criticise. The monograph gives certainly
an impressive picture of the current state-of-the-
art in a rapidly developing field of ecological
modelling. Both the scope of issues and the means
by which they are exposed show the author’s
desire to create a book that might be interesting
and useful not only for the beginner, but for the
expert too. Hal Caswell has succeeded remarkably
in doing both.
Book re6iew310
Dmitrii O. Logofet
Laboratory of Mathematical Ecology,
RAS,
3
Pyzhe6sky Lane,
109017
Moscow,
Russia
E-mail: danilal@postman.ru
