The practical consequence of independent development is that individual members of a population must be able to live in virtually unrestricted environments in which no intraspecific competition can occur. Obvious potential situations are where species have been introduced into, or have invaded, isolated areas. Unfortunately, very few such cases have been extensively studied, though one that has is the invasion of Great Britain by the collared dove, Streptopelia decaocto (Hutchinson, 1978; Hengeveld, 1989). The bird spread westwards in Europe and started to breed in Britain in 1954. For the next ten years reasonably accurate censuses exist, but then the dove became sufficiently common to reduce its appeal to bird watchers and so data collection became inadequate. The rise in population growth between 1955 and 1963 (Figure 2.1) shows an almost linear relationship between the logarithm of population size and time. As we shall soon see, this is associated with independent growth during the early years; breeding birds were pioneers with no immediate neighbours. However, by the mid-1960s the dove had colonized most of Britain and competition for resource was beginning to take effect, shown by the crude 1970 estimate which is quite out of line with the values for earlier years.

If we ignore the invasive spatial element, the development of the dove population involves just two features, birth and death. Since each of these is easiest to understand in isolation, we shall first study them separately (Sections 2.1 and 2.2) before bringing them together as the simple birth–death process (Section 2.3).

The remarkable variety of dynamic behaviour exhibited by many species of plants, insects and animals has stimulated great interest in the development of both biological experiments and mathematical models. From a relatively slow start in the 1920s and 1930s, the pace of research has quickened dramatically over the past few years. Unfortunately, however, ideas have polarized at the same rate. Theoreticians often model purely in terms of manipulating mathematical equations, throwing in the occasional biological reference merely to gain practical respectability; whilst biologists may develop vaguely plausible deterministic models which reflect mathematical hope rather than biological reality.

Many researchers still use one approach to the total exclusion of the other. The reasons are two-fold. First, pioneering biological studies were greatly influenced by deterministic mathematics, and reluctance to accept stochastic ideas is still ingrained. Second, too many mathematicians are taught in a practical vacuum, with the result that instead of using mathematics to interpret and understand biological phenomena they become transfixed by the models themselves.

In this book we develop a unifying approach. First, we show that both deterministic and stochastic models have important roles to play and should therefore be considered together; popular deterministic ideas of logistic, chaotic and predator–prey relationships can change markedly when viewed in a stochastic light.

Second, in biology we are often asked to infer the nature of population development from a single data set, yet different realizations of the same process can vary enormously.

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.