Population growth cannot continue to infinitely large sizes because resources within any environment are finite. The most common population growth model representing growth in a limited environment is a logistic equation (commonly called the Verhulst-Pearl logistic equation) that has a maximum limit at the carrying capacity (K) of the environment:
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Figure 3 Population growth forms for species with continuous generations. A: Exponential (geometric) population growth in an environment with unlimited resources (eqn , r = 0.085, N0 = 10). B: Logistic (sigmoid) population growth in an environment with limited resources (eqn , r = 0.085, K=500, N0 = 10).
where r is the intrinsic rate of increase and the (1 — N/K) term represents the effect of individuals in the same population competing for resources (i.e., the effect of intraspecific competition). The growth curve is sigmoid shaped (B, Figure 3). The rate of change in population size is highest at low population sizes and has a value of zero once the population reaches carrying capacity. The Verhulst-Pearl model assumes the population size at time t only depends on the conditions at t, K is a constant, and every new individual decreases the rate of population increase by the fraction 1/K, causing the decrease in r to be a linear function of N. At any initial value of N, the model will approach K monotonically.
The Verhulst-Pearl model can be modified by adding an exponent (0) to the (1 — N/K) term allowing a nonlinear relationship between r and N. The 0-logistic model is as follows:
where 0 controls the shape of the relationship between r and N. The value of 0 depends on how individuals of a given population interact at different values of N. The growth form of this model is sigmoid for all values of 0, yet different values of 0 have different ecological interpretations. If 0 >1 there is a convex relationship between r and N, inferring that a population grows rapidly until it approaches Kthen growth slows rapidly. If 0 = 1, then the model behaves exactly like eqn  (i.e., the negative effect of each individual is the same, regardless of population size). If 0 <1 then there is a concave relationship between r and N, implying that the per capita reduction in population growth is greater at lower values of N.
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