| Fig. 4. a) The type of population growth to equilibrium (as shown in Figure 3) is obtained when rates of birth and death change linearly with an increase in population size. Specifically, the birth rate declines and the death rate increases until they are balanced at the population's carrying capacity; that is, the change in the rates depends on the size, or density, of the population. Another scenario b) is when the birth and death rates do not depend on density, but instead remain constant over a range of population sizes. In this instance, populations theoretically should continue to grow quickly and not show any deceleration of growth even at high numbers. More realistically, c) unbounded growth is not the rule in nature because environments are not limitless, although small populations may grow at rates not influenced by their density until the populations grow quite large, at which point effects of density on growth may suddenly appear. Real data from the kinds of large-bodied, long-lived species that humans frequently harvest (such as large mammals) often show this kind of relation between birth and death rates as population size changes. Harvesting can lower population sizes of such a mammal species to where the population might respond according to the approximations of maximum sustained yield and into a region where the assumptions of maximum sustained yield are violated. The relationship between the size of a population at one time (Nt) and its size at some later time after a harvest (Nt+1) is very different when birth and death rates are d) density-dependent or e) density-independent. The diagonal line connects values of Nt and Nt+1 that are equal. The population's compensation for density reduction can cause an increase in population size at a later time, but the response is very different for a population at a size where the birth and death rates are not influenced by density. If birth and death rates are influenced by density, the population trajectory over time may f) fluctuate around an equilibrium size. If birth and death rates are density-independent, the fluctuations may g) be of increasing amplitude, suggestive of chaotic behavior. The fluctuations of increasing amplitude seen in g) increase the probability that the population could, by chance, become reduced to zero, which means, of course, extinction. Such cycles are observed in long-term data sets from marine fisheries and white-tailed deer. | |
