## Modelling complex life cycles

The population models in the previous chapters have assumed that all the individuals are the same age or at the same stage in their life cycle. Here we will introduce models which can take account of individuals of different age, stage, or size. In particular we will use matrices to summarize the structure and parameters of a population composed of organisms with complex life histories. There is only space here for a short treatment of what is a rich and fascinating area of ecological modelling (Caswell 2000a).

Individuals of long-lived species may have widely varying patterns of pre-reproductive and reproductive life (Fig. 4.1). It will be assumed that, although generations overlap, reproduction occurs at certain times of year and therefore discrete time models are appropriate. From the perspective of population dynamics there are two important differences between long-lived organisms with overlapping generations and annual or short-lived organisms with separate generations. First, long-lived organisms may delay reproduction for 1 or more years and, second, they may survive after reproduction to reproduce again. In all of these cases the life history of an individual may be categorized according to its age (e.g. time of first reproduction), stage (e.g. adult or juvenile) or size (e.g. only plants over a certain size can reproduce).

Imagine a species, the individuals of which breed once a year, starting at age 3 years and which live to a maximum of 5 years. The reproduction and survival of these organisms can be described by a set of first-order difference equations. These give either the survival of individuals of different age or the reproductive output of individuals aged 3-5 years. Assume that the age-specific fecundity and survival parameter values are density-independent and are constant from year to year. For example, survival from birth to age 1 is described as:

Number of individuals aged 1 (in year t + 1) = number born (aged 0 in year t)

x fraction surviving from age 0 to 1

This can be represented algebraically:

In equation 4.1 the double subscript for the number of individuals (N) indicates the age class and the time (year). For the survival parameter (s) the double subscript describes the ages over which survival is considered. We can write similar equations describing the survival for the other age classes:

The fraction of individuals surviving from birth (age 0) to age 5 is therefore the multiple of the separate survival values from ages 0 to 1, 1 to 2 and so on; that is, s0,1s1,2s2,3s3,4s4,5. We will assume that any individuals surviving to reproduce at age 5 then die. Therefore for any given value of N0, N5 could be predicted.

An equation is also required for the production of offspring (age 0 individuals in year t) by individuals aged 3-5 in the same year (t):

f3, f4 and f5 are age-specific fecundity parameters representing the average number of offspring per individual of that age in year t. Multiplying by s0,1 gives an equation determining the number of offspring surviving to age 1 in year t + 1 (see equation 4.1):

Equations 4.1-4.6 provide a complete description of the density-independent survival and fecundity of individuals in this age-structured population. We could explore by simulation the dynamics of this population, using these equations. Alternatively we can employ analytical techniques, in which case it is helpful to rewrite the equations in a different form, employing a matrix structure. As we do so, you might wish to consider whether you expect any fundamental differences in the dynamics of this population to the one described by equation 2.2.

Equations 4.2 to 4.6 can be represented as three matrices:

 f N1 f 0 0 S0,1 f3 S0,1 f4 S0,1 f5 ^ f N1 ^ N2 0 0 0 0 N 2 N3 = 0 S2,3 0 0 0 N3 N 4 0 0 S3,4 0 0 N4 1N5 ; V 0 0 0 S4,5 0 y 1 n5 J

Three matrices are required to summarize the five difference equations. There are two column matrices representing the number of individuals at ages 1-5 at times t + 1 and t (vt+1 and vt respectively). These column matrices are referred to as the population-structure vectors or age-distribution vectors. There is also one square matrix, M, which gives all of the fecundity and survival values and is known as the population projection matrix. To check that equation 4.7 is equivalent to equations 4.2-4.6 you can multiply out the matrix and population-structure vector on the right-hand side of the equation. For readers unfamiliar with matrix multiplication, you begin by multiplying the five coefficients in the top row of the square matrix M by the corresponding population sizes in the column matrix vt (0 x N1, 0 x N2, s0f x N3 and so on) and add the resulting five multiplied pairs of values to give Ni in vt+:. This process is then repeated with the next row, again multiplying by the corresponding values of N1-N5 in vt and summing the five multiples. This process is repeated for all five rows of the matrix M. Representation of age-structured populations in this manner was first described by Bernardelli (1941), Lewis (1942) and Leslie (1945, 1948).

Matrix equations such as equation 4.7, representing a set of difference equations, can be written in a general form to describe any age- or stage-structured population:

where vt and vt+1 are population vectors of the numbers of individuals at different ages (or sizes or stages) at t and t + 1 respectively, and M is a square matrix in which the number of columns and rows is equal to the number of age classes. You will see the similarity of this to equation 2.2, Nt+1 = %Nt. This similarity is considered in the next section as we proceed with an analytical study of the dynamics of equation 4.8.