Stable age-distribution

Suppose that in a constant environment, the survival probability of a female individual with age , is given by , and that its reproduction rate is . Let denote the number of females at time $t$ having an age somewhere in the interval . Then the total number of individuals, , is given by . How will the population develop if we specify the composition of a population at time $t=0$, say by ? To evaluate the population, we need to know the birth rate at time $t$, . If the reproduction rate was a constant , we could easily state . However, because reproduction rate is age dependent, we must calculate birth rates in every 'age class' and 'add them up', or in mathematical terms

So if we know we can calculate . On the other hand we can see that is determined by in the following way: individuals having age at time $t$ are the individuals which were born at time and have survived up to age . Again in mathematical terms

Notice that the right-hand side of this expression contains in fact a probabilistic term, . On the population level it can be interpreted as 'the fraction surviving age '. Substitution of (4.2) in (4.1) gives the following integral equation for

In this equation our assumed knowledge of the population at time $t=0$ has not yet been used. This can be achieved by making a census in time in (4.3). If we can rewrite (4.2) as Substitution in (4.3) results in

This equation is known as the renewal equation. It can be read as follows: births at time $t$ originate from individuals born since $t=0$ and from individuals already present in the population at $t=0$. The latter term is usually called . We assume there is some value such that and for ( can be interpreted as the maximal age). Then for . Now suppose that the solution of (4.4) is of the form , for some value for , then (4.4) reduces (for ) to

Dividing by we get the following equation for

This equation is known as the characteristic equation. It is possible to show that, under some smoothness restrictions on the reproduction function , this equation has exactly one real root . The other roots are complex and have a real part smaller than . The general solution appears to be a linear combination . For large values of $t$ the exponential will be dominant, so the the asymptotic solution will be . (The smoothness restrictions on are violated if for instance reproduction is only possible at certain particular ages. The information about the composition of the initial population then never gets lost.) From (4.2) it follows that and that . This implies that represents the (eventual) population growth rate and that not only the total number of individuals is eventually growing exponentially, but also the number in each age class. Now we define the so-called stable age distribution as

It can be conceived as the probability density of the age of a randomly selected individual in a really large population, exponentially growing in a constant environment. As an example, consider a micro-organism growing exponentially from volume to volume in interval , after which it divides into two parts of volume again. Here for . If there are no actual deaths, we can fictively put and . From (4.5) we obtain . In principle, a population of such micro-organisms has no stable age distribution because of the mentioned phenomenon that the initial age distribution continues to exercise its influence. In practice however, there will be enough scatter among the division intervals of the daughter cells to destroy synchronicity. It seems safe therefore to neglect this problem and obtain from (4.6) the semi-stable age distribution

Alternatively, we can recast this into the semi-stable size distribution by substituting and into (4.7). We obtain . See Fig.4.1. Thus we see that the size distribution is independent from the population growth rate. In practice it will be dependent nonetheless, because at high population growth rates cells start a new DNA duplication cycle, before the former one is fully completed. This results in increasing cell sizes at high population growth rates. The notion of stable age distributions is important in the theory of population dynamics. For practical purposes it is essential to realize that they eventually occur only in populations living in constant environments. This condition will not occur for a long period, outdoors. Apart from seasonal changes in the environment, an exponentially growing population will soon exhaust its resources, depending on the population growth rate.

Figure 4.1: The stable age (left) and size (right) distribution of exponentially growing micro-organisms that divide in two parts.