"Each column represents one generation; each of rows 2 through 6 represents an age group.

"Each column represents one generation; each of rows 2 through 6 represents an age group.

Models based on differential equations such as this one are more appropriate to populations that breed continuously, with overlapping generations. On the other hand, equation (14.16) is better suited to species that breed at fixed intervals, once per season. The solution to equation (14.17), given an initial total population of N0, is the exponential relationship

and by comparing (14.16) and (14.18), we obtain r = ln l (14.19)

The values of l and r computed by equations (14.15) and (14.19) do not require age distribution information to predict growthâ€”only data on total population and the assumption that the population distribution is stable. Examples of these calculations are given in Table 14.7. Notice that they vary widely until stability emerges.

Generation

Figure 14.14 Predicted total population changes based on life table in Table 14.6.

An interesting parameter that can be computed from r is the doubling time, t2:

ln 2 0-693

The values of r and l could also be estimated directly from mortality and natality without having to simulate the population changes. Two intermediate values must first be computed: the net reproductive rate, R0, which is the average number of offspring produced per individual over a generation; and the mean length of a generation, G, which is the average time between the birth of an offspring and that of its parents:

where li = ni/n0 is the age distribution normalized to the first age group:

R0 ln Rp G

For the data from Table 14.6, we have R0 = 1.044, G = 2.47, and r = 0.0175 per unit time, for a doubling time t2 = 39.6 time units.

All populations are capable of exponential or geometric growth if favorable environmental conditions are maintained. However, this kind of growth is truly "explosive." Darwin gave an example of the elephant, which he predicted could grow from two individuals to 19 million over 750 years. This corresponds to a mere r = 0.021 per year, or a doubling time of 33 years. He also pointed out that although many parasites have a high intrinsic rate of increase, they are nevertheless rare. Clearly, the growth models described earlier need to take into account factors that limit growth, such as limiting factors, competition, and predation. These are the subjects of the next sections.

Single Species: The Logistic Equation A simple strategy to place a limit on the exponential growth of equation (14.17) is to have r decrease as N increases. A simple way to do this is to have it decrease linearly from a maximum specific growth rate, r0, when N is zero, down to zero when N increases to a particular level, K:

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