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Logistic growth regressions carry significant uncertainty when data is available only up to around the inflection point of the growth process. Under these conditions, estimating the height at which the inflection point will occur may have uncertainties comparable to the carrying capacity (K) of the system.
For the competition equations, the logistic equation is the basis. The logistic population model, when used by ecologists often takes the following form: = (). Here x is the size of the population at a given time, r is inherent per-capita growth rate, and K is the carrying capacity.
The logistic growth curve depicts how population growth rate and carrying capacity are inter-connected. As illustrated in the logistic growth curve model, when the population size is small, the population increases exponentially. However, as population size nears carrying capacity, the growth decreases and reaches zero at K. [20]
The logistic model (or logistic function) is a function that is used to describe bounded population growth under the previous two assumptions. The logistic function is bounded at both extremes: when there are not individuals to reproduce, and when there is an equilibrium number of individuals (i.e., at carrying capacity ).
The logistic map is a polynomial ... where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity ...
The Pearl-Reed logistic equation can be integrated exactly, and has solution = + where C = 1/N(0) − 1/K is determined by the initial condition N(0). The solution can also be written as a weighted harmonic mean of the initial condition and the carrying capacity,
This is a departure from the logistic growth equation = where N = population size; r = intrinsic rate of increase; K = carrying capacity; A = critical point; and dN/dt = rate of increase of the population.
Using these techniques, Malthus' population principle of growth was later transformed into a mathematical model known as the logistic equation: = (), where N is the population size, r is the intrinsic rate of natural increase, and K is the carrying capacity of the population.