4.8 Population projection models

      4.8.1 Population projection matrices

A more general, more powerful, and therefore more useful method of analysing and interpreting the fecundity and survival schedules of a population with overlapping generations makes use of the population projection matrix (see Caswell, 2001 , for a full exposition). The word ‘projection’ in its title is important. Just like the simpler methods above, the idea is not to take the current state of a population and forecast what will happen to the population in the future, but to project forward to what would happen if the schedules remained the same. Caswell uses the analogy of the speedometer in a car: it provides us with an invaluable piece of information about the car’s current state, but a reading of, say, 80 km h⁻¹ is simply a projection, not a serious forecast that we will actually have travelled 80 km in one hour’s time.

      life cycle graphs

The population projection matrix acknowledges that most life cycles comprise a sequence of distinct classes with different rates of fecundity and survival: life cycle stages, perhaps, or size classes, rather than simply different ages. The resultant patterns can be summarised in a ‘life cycle graph’, though this is not a graph in the everyday sense but a flow diagram depicting the transitions from class to class over each step in time. Two examples are shown in Figure 4.15 (see also Caswell, 2001). The first (Figure 4.15a) indicates a straightforward sequence of classes where, over each time step, individuals in class i may: (i) survive and remain in that class (with probability p_i ); (ii) survive and grow and/or develop into the next class (with probability g_i ); and (iii) give birth to m_i newborn individuals into the youngest/smallest class. Figure 4.15b then goes on to show that a life cycle graph can also depict a more complex life cycle, for example with both sexual reproduction (here, from reproductive class 4 into ‘seed’ class 1) and vegetative growth of new modules (here, from ‘mature module’ class 3 to ‘new module’ class 2). Note that the notation here is slightly different from that in life tables like Table 4.2. There the focus was on age classes, and the passage of time inevitably meant the passing of individuals from one age class to the next: p values therefore referred to survival from one age class to the next. Here, by contrast (as in Table 4.1), an individual need not pass from one class to the next over a time step, and it is therefore necessary to distinguish survival within a class (p values here) from passage and survival into the next class (g values).