A whole new approach is needed if we are to consider the complete route taken by all members of a developing population. For example, we may wish to study the space–time development of an invading species of ant as it reproduces and spreads across a region. The resulting map will resemble a ‘tree’, with current ant positions corresponding to branch buds, births to branch forks, deaths to branch ends, and the paths between such events to the branches themselves. Though such scenarios are not often encountered in population dynamics, tree-like structures abound in biology, and a way of describing and analyzing them is clearly needed. Obvious examples include lung-airways, neural and arterial networks, and plant rhizome systems; examples of more abstract networks include the concepts of food webs and dominance relations in animal society. MacDonald (1983) provides an excellent overview of this potentially vast subject area. His presentation is eminently readable by mathematician and biologist alike, and provides an ideal starting point for readers wishing to pursue this highly absorbing subject.

In order to find our way around a tree or network (the former implies the absence of closed paths, the latter does not) we need to define an ordering over the connected branches. Fortunately, geographers spent considerable effort in the 1950s and 1960s investigating various possibilities for stream and river networks, and this work has been of considerable benefit to subsequent biological research.

Gause's conclusion that a predator–prey system is inherently self-annihilating without some outside interference such as immigration (Section 6.1.2) was questioned by Huffaker (1958). He claimed that Gause had used too simple a microcosm, and so set out to learn whether an adequately large and complex experiment could be constructed in which the predator–prey relation would not be self-exterminating. We therefore now ask, ‘What will be the effect, if any, of accepting that individuals rarely mix homogeneously over the whole site but that they develop instead within separate sub-regions?’.

This question is an old one, for as early as 1927 A. J. Nicholson asked Bailey (1931) to investigate mathematically the abundance of two species which interact in the following manner. Members of the host species lay eggs and then die. These eggs are then searched for by members of the parasite species who traverse at random a specific area during their lifetime. Host eggs which survive this search develop into adults; those that are found are attacked and a parasite egg is deposited on them. New generations then repeat this process indefinitely.

Huffaker's experiments

Huffaker selected the six-spotted mite, Eotetranychus sexmaculatus, as the prey species and the predatory mite, Typhlodromus occidentalis, as the predator species because earlier observations had revealed this Typhlodromus as being a voracious enemy of the six-spotted mite. It was known to develop in great numbers on oranges infested with the prey species, to destroy essentially the entire infestation, and then to die en masse.

The geographic distribution of a species over its range of habitats, and the associated dynamics of population growth, are inseparably related, a fact which no complete study of population development can afford to ignore (see Levin, 1974). Thus whilst the assumption that populations develop at a single location is ideal for mathematical purposes, in real life we must accept that individuals seldom mix homogeneously over the whole region available to them but develop instead within separate sub-regions. Indeed, this is precisely the reason why we extended the non-spatial predator–prey process (Chapter 6) to allow individuals of either species to migrate between separate sites (Chapter 7). Having shown that spatial and non-spatial predator–prey behaviour can be very different, we clearly need to extend our spatial framework to cover more general population processes.

Most species attempt to migrate for a variety of both individual and population reasons, including: search for food; territorial extension for increasing population needs; widening the available gene pool; and minimizing the probability of extinction. Migration can range from being purely local, e.g. aquatic life in a small pond, to extensive migration patterns covering a fair part of the Earth's surface, e.g. birds, locusts, salmon, caribou, and viruses. Moreover, migration can occur either between distinct sites, such as neighbouring valleys or islands in an archipaelego, or else it can occur within continuous media such as the air or sea.

So far we have just considered single-species population dynamics. However, in nature organisms do not generally exist in isolated populations but they live alongside organisms from many other species. Whilst a large number of these species will be unaffected by the presence or absence of one another, in some cases two or more species will interact competitively. Such competition may either be for common resources that are in short supply, such as food or space, or it may be that organisms from different species attack each other directly.

Now there is considerable evidence to suggest that species population stability is typically greater in communities with many interacting species than in simple ones. For example, it has been noted that simple laboratory predator–prey populations characteristically undergo violent oscillations; cultivated land and orchards have shown themselves to be fairly unstable; whilst the rain forest, a highly complex structure, appears to be very stable. On closer examination, however, the issue clouds over since species integration in a complex community is a highly non-linear affair, and quite remarkable instabilities can ensue from the introduction or removal of a single species (May, 1971b). We shall therefore ignore the difficult world of three or more interacting populations and concentrate on just two (an extremely important field of study in its own right).

Before we begin it is worthwhile repeating Park's (1954) warning that the functional existence of inter-species competition may be inferred from a body of data even when no such inter-species dependence exists.

Of all areas of ecology, population biology is perhaps the most mathematically developed, and has involved a long history of mathematicians fascinated by problems associated with the dynamics of population development. Interest was induced by early studies of small mammals and laboratory controlled organisms, since these easily lent themselves to a mathematical formulation. A great deal of more recent research is concerned with modelling multi-species and spatial population growth, though it is not clear just how effective these models are for predicting behaviour outside the laboratory. There is general uncertainty regarding whether populations in the natural environment are mostly regulated from within by density-dependent factors, or whether the main influence is due to external density-independent factors. Theoretical developments have generally followed the former route, primarily because there is much less information on external factors due to their complexity and variability (see Gross, 1986).

Throughout most of this text we shall therefore disregard the (generally unknown) external influences on population growth, and develop the ideas of density-dependence. Moreover, since even apparently minor modifications to simple biological models can lead to difficult, if not intractable, mathematics, we shall begin by investigating the simplest possible forms of model structure (Chapter 2). In these, members of a population are assumed to develop independently from each other, for then the resulting mathematical analyses are sufficiently transparent to enable useful biological conclusions to be drawn.

The fascination of natural communities of plants and animals lies in their endless variety. Not only do no two places share identical histories, climates or topography, but also climate and other environmental factors are constantly fluctuating. Such systems will therefore not exhibit the crisp determinacy which characterizes so much of the physical sciences (May, 1974a). Now in the preceding chapters we have implicitly assumed that the environment is unvarying; birth and death rates, carrying capacities, etc., have all been held constant through time and space. Thus our stochastic models have involved variation in the sense that random events occur with probabilities which depend only on population size.

However, the most striking features of life on this planet are directly attributable to the diurnal rotation of the Earth and its annual journey around the Sun. Indeed, the behaviour and reproductive cycles of living organisms are closely adapted to the regular alternation of summer and winter, or of wet season and dry season (Skellam, 1967). So as real environments are themselves uncertain, all parameters which characterize populations must exhibit random or periodic fluctuations to at least some degree. Thus even deterministic equilibrium is not an absolute fixed state, but is instead a ‘fuzzy’ value around which the biological system fluctuates.

We have already seen in Section 4.8 that static environment blowfly models (for example) do not produce sufficient variability, and so by admitting the reality of environmental variation we have a second, powerful source of variability at our disposal.

Some ten years ago, when completing with J.-B. Zuber a previous text on Quantum Field Theory, the senior author was painfully aware that little mention was made that methods in statistical physics and Euclidean field theory were coming closer and closer, with common tools based on the use of path integrals and the renormalization group giving insights on global structures. It was partly to fill this gap that the present book was undertaken. Alas, over the five years that it took to come to life, both subjects have undergone a new evolution. Disordered media, growth patterns, complex dynamical systems or spin glasses are among the new important topics in statistical mechanics, while superstring theory has turned to the study of extended systems, Kaluza–Klein theories in higher dimensions, anticommuting coordinates … in an attempt to formulate a unified model including all known interactions. New and sophisticated techniques have invaded statistical physics, ranging from algebraic methods in integrable systems to fractal sets or random surfaces. Powerful computers or special devices provide “experimental” means for a new brand of theoretical physicists. In quantum field theory, applications of differential topology, geometry, Riemannian manifolds, operator theory … require a deeper background in mathematics and a knowledge of some of its most recent developments. As a result, when surveying what has been included in the present volume in an attempt to uncover the basic unity of these subjects, the authors have the same unsatisfactory feeling of not being able to bring the reader really up to date.

A new field opened when modern computers offered the possibility of performing extensive simulations of large systems. This allows known behaviours to be checked and provides an exploratory guide in circumstances where theoretical tools are absent. Measurements of observables can be compared both to existing theoretical expectations – providing a crucial test for their validity – and to experiments – checking the modelling of a physical system –. This chapter presents the background material needed to design simulations on a (relatively) large scale. Some numerical examples have already been presented in previous chapters, and we only give a few further illustrations, pertaining mainly to lattice gauge theory, the usefulness of which relies extensively on this method. We also describe a practical implementation of real space renormalization, known as the Monte Carlo Renormalization Group method (Ma, Swendsen, Wilson). Finally, we discuss specific issues relevant to the extension of the simulations to fermionic systems.

Algorithms

Generalities

Systems with up to 106 to 107 variables can be handled by computers, and these numbers may soon be significantly increased. The measurements can be sufficiently numerous to allow statistical accuracy. Although simulated systems still have a modest size as compared to macroscopic systems, collective effects already clearly appear and accurate results about critical phenomena emerge from the numerical simulations. It turns out, when investigating more closely the available numerical methods, that one gets a better insight into the foundations of equilibrium statistical physics, the ergodicity problems, the meaning of probabilities and, last but not least, ways to approach equilibrium.

This chapter is devoted to technicalities related to various expansions already encountered in volume 1, mostly those that derive from the original lattice formulation of the models, be it high or low temperature, strong coupling expansions and to some extent those arising in the guise of Feynman diagrams in the continuous framework. We shall not try to be exhaustive, but rather illustrative, relying on the reader's interest to investigate in greater depth some aspects inadequately treated. Nor shall we try to explore with great sophistication the vast domain of graph theory. There are, however, a number of common features, mostly of topological nature, which we would like to present as examples of the diversity of what looks at first sight like straightforward procedures.

General Techniques

Definitions and notations

Let a labelled graph G be a collection of v elements from a set of indices, and l pairs of such elements with possible duplications (i.e. multiple links). We shall also interchangeably use the word diagram instead of graph. This abstract object is represented by v points (vertices) and l links. Each vertex is labelled by its index.

The problem under consideration will define a set of admissible graphs, with a corresponding weight ω(G) (a real or complex number) according to a well-defined set of rules. We wish to find the sum of weights over all admissible graphs.

Up to now, continuous field theory has appeared as a tool in the study of critical phenomena. Conversely, techniques from statistical mechanics can be useful in field theory. In 1973, Wilson proposed a lattice analog of the Yang–Mills gauge model. Its major aim was to explain the confinement of quarks in quantum chromodynamics. The lattice implementation of a local symmetry yields a transparent geometric interpretation of the gauge potential degrees of freedom, the latter being replaced by group elements assigned to links. Strong coupling expansions predict a linearly rising potential energy between static sources. Complex phase diagrams emerge when gauge fields are coupled to matter fields, and new phenomena appear, such as the absence of local order parameters. The discretization of fermions leads also to interesting relations with topology. This chapter is devoted to the theoretical developments of these ideas.

Generalities

Presentation

Schematic as they are, statistical models have directly a physical background at any temperature. Lattices may represent the crystalline structure of solids. They play an important role at short distance, but become irrelevant in the critical region, except as a regulator for the field theory describing the approach to critical points. The opposite point of view can also be considered. A lattice is artificially introduced as a regulator for a continuous field theory. The lattice system has no physical meaning, but can be studied at any “temperature”, so that one can get information about its critical region, hopefully described by the initial continuous theory.