## Determination of the eigenvalue and eigenvector

To proceed with the analytical investigation we will take a much simpler age-structured population and then discuss more complicated examples in the light of results from the simpler version. Consider a population of biennial plants (Fig. 4.2). The plant population has two age classes, which correspond to particular developmental stages. In the first year the plant forms rosettes following the germination of over-wintering seed. In the second year the surviving rosettes flower, set seed and then die. We will assume that the plant is a strict biennial: it always flowers in the second year (assuming it survives) and always dies after flowering. This model could also be described as a stage-structured population (Lefkovitch 1965; see Manly 1990 for an overview of matrix models of stage-structured populations) composed of rosettes and flowering plants. It is a coincidence in this case that each stage survives for one unit of time: in most cases this would not be true; for example, a tree species may spend many years at one defined stage. The dynamics of the population can be summarized with two first-order equations:

where R is the number or density of rosette plants, F is the number of flowering plants, f is the average number of viable seed per flowering plant, s01 represents the fraction of seed surviving between dispersal from the mother plant to rosette formation and s1,2 describes the fraction of rosettes surviving until flowering.

In constructing such models it is often the case that stages such as seed are omitted. This will depend on the units of time chosen for the model and the census time. For example, we could have examined changes from spring to autumn and autumn to spring in which case seed may need to be included as a specified stage, or at least a seed/small rosette stage.

Fig. 4.2 Representation of the life cycle of a biennial plant species with fecundity f) and survival at two different stages (s01 and s12). Seed as a separate stage is not included in this model.

As before, it is possible to write equations 4.9 and 4.10 in matrix notation (the algebraic shorthand for the matrices is indicated below them):