## Carrying capacities

Figure 5.7a-c reiterates the fact that as density increases, the per capita birth rate eventually falls and the per capita death rate eventually rises. There must, therefore, be a density at which these curves cross. At densities below this point, the birth rate exceeds

Figure 5.7 Density-dependent birth and mortality rates lead to the regulation of population size. When both are density dependent (a), or when either of them is (b, c), their two curves cross. The density at which they do so is called the carrying capacity (K). Below this the population increases, above it the population decreases: K is a stable equilibrium. However, these figures are the grossest of caricatures. The situation is closer to that shown in (d), where mortality rate broadly increases, and birth rate broadly decreases, with density. It is possible, therefore, for the two rates to balance not at just one density, but over a broad range of densities, and it is towards this broad range that other densities tend to move.

Figure 5.8 Some general aspects of intraspecific competition. (a) Density-dependent effects on the numbers dying and the number of births in a population: net recruitment is 'births minus deaths'. Hence, as shown in (b), the density-dependent effect of intraspecific competition on net recruitment is a domed or 'n'-shaped curve. (c) A population increasing in size under the influence of the relationships in (a) and (b). Each arrow represents the change in size of the population over one interval of time. Change (i.e. net recruitment) is small when density is low (i.e. at small population sizes: A to B, B to C) and is small close to the carrying capacity (I to J, J to K), but is large at intermediate densities (E to F). The result is an 'S'-shaped or sigmoidal pattern of population increase, approaching the carrying capacity.

Figure 5.8 Some general aspects of intraspecific competition. (a) Density-dependent effects on the numbers dying and the number of births in a population: net recruitment is 'births minus deaths'. Hence, as shown in (b), the density-dependent effect of intraspecific competition on net recruitment is a domed or 'n'-shaped curve. (c) A population increasing in size under the influence of the relationships in (a) and (b). Each arrow represents the change in size of the population over one interval of time. Change (i.e. net recruitment) is small when density is low (i.e. at small population sizes: A to B, B to C) and is small close to the carrying capacity (I to J, J to K), but is large at intermediate densities (E to F). The result is an 'S'-shaped or sigmoidal pattern of population increase, approaching the carrying capacity.

the death rate and the population increases in size. At densities above the crossover point, the death rate exceeds the birth rate and the population declines. At the crossover density itself, the two rates are equal and there is no net change in population size. This density therefore represents a stable equilibrium, in that all other densities will tend to approach it. In other words, intraspecific competition, by acting on birth rates and death rates, can regulate populations at a stable density at which the birth rate equals the death rate. This density is known as the carrying capacity of the population and is usually denoted by K (Figure 5.7). It is called a carrying capacity because it represents the population size that the resources of the environment can just maintain ('carry') without a tendency to either increase or decrease.

However, whilst hypothetical populations caricatured by line drawings like Figures 5.7a-c can be characterized by a simple carrying capacity, this is not true of any natural population. There are unpredictable environmental fluctuations; individuals are affected by a whole wealth of factors of which intraspecific competition is only one; and resources not only affect density but respond to density as well. Hence, the situation is likely to be closer to that depicted in Figure 5.7d. Intraspecific competition does not hold natural populations to a predictable and unchanging level (the carrying capacity), but it may act upon a very wide range of starting densities and bring them to a much narrower range of final densities, and it therefore tends to keep density within certain limits. It is in this sense that intraspecific competition may be said typically to be capable of regulating population size. For instance, Figure 5.9 shows the fluctuations within and between years in populations of the brown trout (Salmo trutta) and the grasshopper, Chorthippus brunneus. There are no simple carrying capacities in these examples, but there are clear tendencies for the 'final' density each year ('late summer numbers' in the first case, 'adults' in the second) to be relatively constant, despite the large fluctuations in density within each year and the obvious potential for increase that both populations possess.

In fact, the concept of a population settling at a stable carrying capacity, even in caricatured populations, is relevant only to situations in which density dependence is not strongly overcom-pensating. Where there is overcompensation, cycles or even

Figure 5.9 Population regulation in practice. (a) Brown trout (Salmo trutta) in an English Lake District stream. A, numbers in early summer, including those newly hatched from eggs; o, numbers in late summer. Note the difference in vertical scales. (After Elliott, 1984.) (b) The grasshopper, Chorthippus brunneus, in southern England. •, eggs; +, nymphs; o, adults. Note the logarithmic scale. (After Richards & Waloff, 1954.) There are no definitive carrying capacities, but the 'final' densities each year ('late summer' and 'adults') are relatively constant despite large fluctuations within years.

real populations lack simple carrying capacities

Figure 5.9 Population regulation in practice. (a) Brown trout (Salmo trutta) in an English Lake District stream. A, numbers in early summer, including those newly hatched from eggs; o, numbers in late summer. Note the difference in vertical scales. (After Elliott, 1984.) (b) The grasshopper, Chorthippus brunneus, in southern England. •, eggs; +, nymphs; o, adults. Note the logarithmic scale. (After Richards & Waloff, 1954.) There are no definitive carrying capacities, but the 'final' densities each year ('late summer' and 'adults') are relatively constant despite large fluctuations within years.

chaotic changes in population size may be the result. We return to this point later (see Section 5.8).

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