Mathematical Biology

The increasing study of realistic and practically useful mathematical models in population biology, whether we are dealing with a human population with or without its age distribution, population of an endangered species, bacterial or viral growth and so on, is a reflection of their use in helping to understand the dynamic processes involved and in making practical predictions. The study of population change has a very long history: in 1202 an exercise in an arithmetic book written by Leonardo of Pisa involved building a mathematical model for a growing rabbit population; we discuss it later in Chapter 2. Ecology, basically the study of the interrelationship between species and their environment, in such areas as predator-prey and competition interactions, renewable resource management, evolution of pesticide resistant strains, ecological and genetically engineered control of pests, multi-species societies, plant-herbivore systems and so on is now an enormous field. The continually expanding list of applications is extensive as are the number of books on various aspects1 of the field. There are also highly practical applications of single-species models in the biomedical sciences; in Section 1.5 we discuss two examples of these which arise in physiology. Here, and in the following three chapters, we consider some deterministic models by way of an introduction to the field. The excellent books by Hastings (1997) and Kot (2001) are specifically on ecological modelling. Elementary introductions are also given in the textbooks by Edelstein-Keshet (1988) and Hoppensteadt and Peskin (1992).

Differential equation models, whether ordinary, delay, partial or stochastic, imply a continuous overlap of generations. Many species have no overlap whatsoever between successive generations and so population growth is in discrete steps. For primitive organisms these can be quite short in which case a continuous (in time) model may be a reasonable approximation. However, depending on the species the step lengths can vary widely. A year is common. With fruit fly emergence from pupae it is a day, for cells it can be a number of hours while for bacteria and viruses it can be considerably less. In the models we discuss in this chapter and later in Chapter 5 we have scaled the time-step to be 1. Models must thus relate the population at time t + 1, denoted by N_t+1, in terms of the population N_t at time t. This leads us to study difference equations, or discrete models, of the form N_t+1 = N_tF(N_t) = f(N_t), (2.1) where f(N_t ) is in general a nonlinear function of N_t. The first form is often used to emphasise the existence of a zero steady state. Such equations are usually impossible to solve analytically but again we can extract a considerable amount of information about the population dynamics without an analytical solution. The mathematics of difference equations is now being studied in depth and applied in diverse fields: it is a fascinating subject having given rise to some totally unexpected phenomena some of which we discuss later. Difference equation models are also proving of use in a surprisingly wide spectrum of biomedical areas such as cancer growth (see, for example, the article by Cross and Cotton 1994), aging (see, for example, the article by Lipsitz and Goldberger 1992), cell proliferation (see, for example, the article by Hall and Levinson 1990) and genetics (see, for example, the chapter on inheritance in the book by Hoppensteadtand Peskin 1992 and the book by Roughgarden 1996.)

When species interact the population dynamics of each species is affected. In general there is a whole web of interacting species, sometimes called a trophic web, which makes for structurally complex communities. We consider here systems involving 2 or more species, concentrating particularly on two-species systems. The book by Kot (2001) discusses such models (including age-structured interacting population systems) with numerous recent practical examples. There are three main types of interaction. (i) If the growth rate of one population is decreased and the other increased the populations are in a predator- prey situation. (ii) If the growth rate of each population is decreased then it is competition. (iii) If each population’s growth rate is enhanced then it is called mutualism or symbiosis.

This chapter introduces a new use of mathematical modelling and a new approach to the modelling of social interaction using difference equation models such as we discussed in Chapter 3. These equations express, in mathematical form, a proposed mechanism of change of marital interaction over time. The modelling is designed to suggest a precise mechanism of change. In much of this book the aim of the methodology is quantitative. That is, on the basis of our psychological understanding we write down, in mathematical form, the causes of change in the dependent variables. In the field of family psychology, however, statistical analysis is the usual analytical approach and, furthermore, generally based on linear models. In recent years it has become increasingly clear that most systems are highly nonlinear. The new approach to studying marital interaction with mathematical models was initiated by J. M. Gottman, based on his extensive studies of family interaction, and J.D. Murray (see the book by Gottman et al. 2002 for considerably more psychological detail and several case studies which have used the modelling technique and philosophy described in this chapter). The material we discuss here is based in large part on the paper by Cook et al. (1995).

Biochemical reactions are continually taking place in all living organisms and most of them involve proteins called enzymes, which act as remarkably efficient catalysts. Enzymes react selectively on definite compounds called substrates. For example, haemoglobin in red blood cells is an enzyme and oxygen, with which it combines, is a substrate. Enzymes are important in regulating biological processes, for example, as activators or inhibitors in a reaction. To understand their role we have to study enzyme kinetics which is mainly the study of rates of reactions, the temporal behaviour of the various reactants and the conditions which influence them. Introductions with a mathematical bent are given in the books by Rubinow (1975), Murray (1977) and the one edited by Segel (1980). Biochemically oriented books, such as Laidler and Bunting (1977) and Roberts (1977), go into the subject in more depth.

Although living biological systems are immensely complex, they are at the same time highly ordered and compactly put together in a remarkably efficient way. Such systems concisely store the information and means of generating the mechanisms required for repetitive cellular reproduction, organisation, control and so on. To see how efficient they can be you need only compare the information storage efficiency per weight of the most advanced computer chip with, say, the ribonucleic acid molecule (mRNA) or a host of others: we are talking here of factors of the order of billions. This chapter, and the next two, are mainly concerned with oscillatory processes. In the biomedical sciences these are common, appear in widely varying contexts and can have periods from a few seconds to hours to days and even weeks. We consider some in detail in this chapter, but mention here a few others from the large number of areas of current research involving biological oscillators.

The reaction known as the Belousov-Zhabotinskii reaction is an important oscillating reaction discovered by the Russian Boris Belousov (1951), a biochemist, and is described in an unpublished paper, which was contemptuously rejected by a journal editor; at the time the accepted dogma was that oscillating reactions were simply not possible. A translation of the original article is given in the book edited by Field and Burger (1985). Eventually Belousov (1959) published a brief note in the obscure proceedings of a Russian medical meeting. Basically he found oscillations in the ratio of concentrations of the catalyst; in Belousov’s reaction it was cerium in the oxidation of citric acid by bromate. The oscillation manifested itself via a colour change as the cerium changed from Ce³⁺ to Ce⁴⁺ although it is more dramatic with an iron ion, ferroin where the colour is brick red when in the Fe^2e state and bright blue in the Fe^3e state. The study of this reaction was continued by Zhabotinskii (1964) and is now known as the Belousov-Zhabotinskii reaction or simply the BZ reaction. When the details of this important reaction and some of its dramatic oscillatory and wavelike properties finally reached the West in the 1970’s it provoked widespread interest and research. Belousov’s seminal work was finally, but posthumously, recognised in 1980 by his being awarded the Lenin Prize. Winfree (1984) gives a brief interesting description of the history of the Belousov-Zhabotinskii reaction. When the reactants can also diffuse a diverse menagerie of complex patterns can be formed and it is the latter which has sustained the continuing widespread interest among both biological and physical scientists.

The history of epidemics is an ever fascinating area; the 14th century Black Death is just the most famous epidemic historically (see Chapter 13, Volume II, which deals with the spatial spread of epidemics, for a brief history of it). In Europe, which had a population of around 85 million at the time, about a third of the population died.

In Chapter 11 we saw how diffusion, chemotaxis and convection mechanisms could generate spatial patterns; in Volume II we discuss mechanisms of biological pattern formation extensively. In Chapter 13, and Chapter 1 and Chapter 13, Volume II we show how diffusion effects, for example, can also generate travelling waves, which have been used to model the spread of pest outbreaks, travelling waves of chemical concentration, colonization of space by a population, spatial spread of epidemics and so on. The existence of such travelling waves is usually a consequence of the coupling of various effects such as diffusion or chemotaxis or convection. There are, however, other wave phenomena of a quite different kind, called kinematic waves, which exhibit wavelike spatial patterns, which depend on the coupling of biological oscillators whose properties relating to phase or period vary spatially. The two phenomena described in this chapter are striking, and the models we discuss are based on the experiments or biological phenomena which so dramatically exhibit them. The first involves the Belousov-Zhabotinskii reaction and the second, which is specifically associated with the swimming of, for example, lamprey and dogfish, illustrates the very important concept of a central pattern generator. The results we derive here apply to spatially distributed oscillators in general.

There is a vast number of phenomena in biology in which a key element or precursor to a developmental process seems to be the appearance of a travelling wave of chemical concentration, mechanical deformation, electrical signal and so on. Looking at almost any film of a developing embryo it is hard not to be struck by the number of wavelike events that appear after fertilisation. Mechanical waves are perhaps the most obvious. There are, for example, both chemical and mechanical waves which propagate on the surface of many vertebrate eggs. In the case of the egg of the fish Medaka a calcium (Ca⁺⁺) wave sweeps over the surface; it emanates from the point of sperm entry: we briefly discuss this problem in Section 13.6 below. Chemical concentration waves such as those found with the Belousov-Zhabotinskii reaction are visually dramatic examples (see Chapter 1, Volume II). From the analysis on insect dispersal in Section 11.3 in Chapter 11 we can also expect wave phenomena in that area, and in interacting population models where spatial effects are important. Another example, related to interacting populations, is the progressing wave of an epidemic, of which the rabies epizootic currently spreading across Europe is a dramatic and disturbing example; we study a model for this in some detail in Chapter 13. The movement of microorganisms moving into a food source, chemotactically directed, is another. The slime mould Dictyostelium discoideum is a particularly widely studied example of chemotaxis; we discuss this phenomenon later (see the photograph in Figure 1.1, Volume II which shows associated waves).

The problem with a good name for a new (or resurrected) field, particularly one such as fractal theory which can be visually dramatic and practised without much background and sophistication, is that uninformed proselytising and inappropriate use can raise unrealistic expectations as to its relevance and applicability. Catastrophe theory is another example: its overzealous mathematical practitioners did considerable harm to the cause of interdisciplinary science. Although chaos and fractal theory have been proposed by some as biological panaceas fortunately there are enough realists to counter this view and generally keep them in perspective.