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A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with the equation. where. is the carrying capacity, the supremum of the values of the function; is the logistic growth rate, the steepness of the curve; and. is the value of the function's midpoint.
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.
A Malthusian growth model, sometimes called a simple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert Malthus, who wrote An Essay on the Principle of Population (1798), one of the earliest and most ...
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves.
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 ). Under the logistic model, population growth rate between these two limits is most often assumed to be sigmoidal (Figure 1).
Logistic map. The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations.
The logistic growth curve is initially very similar to the exponential growth curve. When population density is low, individuals are free from competition and can grow rapidly.
Beverton–Holt model. The Beverton–Holt model is a classic discrete-time population model which gives the expected number n t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation, Here R0 is interpreted as the proliferation rate per generation and K = ( R0 − 1) M is the ...