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Bifurcation diagram of the Ricker model with carrying capacity of 1000. The Ricker model, named after Bill Ricker, is a classic discrete 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, [1]
The two great economic revolutions that marked human history up to 1900—the agricultural and industrial revolutions—greatly increased the Earth's human carrying capacity, allowing human population to grow from 5 to 10 million people in 10,000 BCE to 1.5 billion in 1900. [44]
The logistic model takes the shape of a sigmoid curve and describes the growth of a population as exponential, followed by a decrease in growth, and bound by a carrying capacity due to environmental pressures. [4]
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.
The model can also be written in the form of a differential equation: = with initial condition: P(0)= P 0. This model is often referred to as the exponential law. [5] It is widely regarded in the field of population ecology as the first principle of population dynamics, [6] with Malthus as the founder.
The 1972 book The Limits to Growth discussed the limits to growth of society as a whole. This book included a computer-based model which predicted that the Earth would reach a carrying capacity of ten to fourteen billion people after some two hundred years, after which the human population would collapse. [7]
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
The Gompertz curve or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes growth as being slowest at the start and end of a given time period.