A system of models for the life cycle of a biological organism

SUMMARY This paper is concerned with stochastic models for the representation of the development of a biological organism through recognizable distinct stages. The models are based upon the assumption that the periods spent in each stage of development, excluding the possibility of death, are independent random variables with a characteristic form of probability distribution. Within each stage the organism is liable to be taken by predators or to die from other causes, and incidents of this type are assumed to occur as events in a Poisson process. The development of the theory for various forms of distribution of the period spent in a given stage is considered, including the negative exponential and second and third order special Erlangian distributions. An example is given of the application of the proposed models to the analysis of sampling data from a study of the life cycle of the grasshopper, Corthippus parallelus. The main features of the life cycle of a biological organism exhibit a similar pattern over a wide variety of different types and species. The birth of the organism occurs at a clearly defined point in time and the organism then passes through a period of growth and development until it reaches maturity. In certain types of organism this process is characterized by transition through a number of distinct and easily recognizable states in turn. An insect, for example, passes through a succession of larval instars. In other types of organism the process of development is less well defined, although it is usually possible to divide the life cycle into discrete states by reference to the presence or absence or to the size of characteristic features of the organism. At every moment of its life the organism is liable to suffer death, either as a result of the action of predators, of accidents or for other reasons. If the organism does reach maturity, its life will eventually be terminated, either by natural or other causes. In order to carry out quantitative studies of a population of a given type of organism it is often helpful to set up a mathematical model to represent the process of birth, development and death. Since there will generally be variations from one organism to another within the same population, such a model must preferably embody a stochastic or random element, as the assessment of individual variability will form an essential part of the description of the life cycle. The main features which must be taken into account are the distributions of the times of birth, of the periods spent in each stage of development and of mortality in the various stages. The process of growth, as revealed by the size of the organism at any given stage in its development, may also be of interest. This paper is concerned with a model which was originally developed to describe the life cycle of the grasshopper, Corthippus parallelus. The model is, however, of more general application, not only to other biological organisms, but also to studies of the structure of human populations. For example, the 'population' may correspond to a large organization, 'birth' may correspond to the recruit