## Exponential and Geometric Models

The simplest model of population growth describes change in numbers as the initial population size times the per capita rate of increase (see Fig. 6.7) (Berryman 1997, Price 1997). This model integrates per capita natality, mortality, immi-

Time

| Exponential and logistic models of population growth. The exponential model describes an indefinitely increasing population, whereas the logistic model describes a population reaching an asymptote at the carrying capacity of the environment (K).

Time

| Exponential and logistic models of population growth. The exponential model describes an indefinitely increasing population, whereas the logistic model describes a population reaching an asymptote at the carrying capacity of the environment (K).

gration, and emigration per unit time as the instantaneous or intrinsic rate of increase, designated r:

where N = natality, I = immigration, M = mortality, and E = emigration, all instantaneous rates.

Where cohort life table data, rather than time-specific natality, mortality, and dispersal, have been collected, r can be estimated as follows:

where R0 is replacement rate, and T is generation time.

The rate of change for populations with overlapping generations is a function of the intrinsic (per capita) rate of increase and the current population size. The resulting model for exponential population growth is as follows:

where Nt is the population size at time t, and N0 is the initial population size. This equation also can be written as follows:

For insect species with nonoverlapping cohorts (generations), the replacement rate, R0, represents the per capita rate of increase from one generation to the next. This parameter can be used in place of r for such insects. The resulting expression for geometric population growth is as follows:

where Nt is the population size after t generations.

Equations 6.3-6.5 describe density-independent population growth (Fig. 6.7). However, as discussed earlier in this chapter, density-dependent competition, predation, and other factors interact to limit population growth.

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