Age-Structured and Stage-Structured
|1.||Age-structured models: Life tables and the Leslie matrix|
|2.||Stage-structured matrix models|
|3.||Integral projection models|
|4.||Continuous-time models with age structure|
|5.||Applications and extensions|
When all individuals in a population are identical, we can characterize the population just by counting the number of individuals. However, the individuals within many animal and plant populations differ in important ways that influence their current and future prospects of survival and reproduction. For example, larger individuals typically have greater chances of survival, produce more and sometimes larger offspring, and often have slower growth rates. In such cases, characterizing the population structure—the numbers of individuals of each different type—is critical for understanding how the population will change through time. In this chapter, we examine some of the main types of models used for describing and forecasting the dynamics of structured populations. Age-structured models in discrete time, appropriate for populations in seasonal environments, were developed centuries ago by the great mathematician Leonhard Euler (1707–1783). These are considered first, before moving to models where individuals are characterized by their stage in the life cycle (e.g., seed versus flowering plant, larva versus adult). Next we look at how to incorporate differences among individuals that vary continuously, such as size. Having explored discrete-time models, we briefly turn to continuoustime models and then present some applications and extensions.
age structure. Distribution of ages in a population
matrix. A rectangular array of symbols, which could represent numbers, variables, or functions
AND THE LESLIE MATRIX
The simplest age-structured models assume that each individual’s chance of survival and reproduction depends only on its age; there are no effects of population density. The standard model counts only females (assuming no shortage of mates) and assumes that all births occur in a single birth pulse immediately before the population is censused (a so-called postbreeding census). The population dynamics is then summarized by the following equations:
where na(t) is the number of individuals of age a at time t, fa, and pa are, respectively, the average fecundity and the probability of survival to age a + 1 of age a individuals, and A is the maximum possible age (or the maximum age at which reproduction occurs, if postreproductives are omitted from the population count). Because births occur just before the next census, fa = pama + 1, where ma + 1 is the number of offspring produced by an age a + 1 female.