Continuous Population Models for Single Species

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).