#### Event Details

**Título:**

*Logistic growth described by birth-death and diffusion processes*

**Impartida por:**Paola Paraggio (Università degli Studi di Salerno)

**Resumen:**

Growth curves with sigmoidal behavior are widely used for data analysis among several
fields of application. In general terms, a sigmoidal function is a positive, bounded and
differentiable function with positive derivative. Usually, its graph has a characteristic Sshape: it thus shows a slow growth at the beginning and an exponential growth that then
slows gradually until it reaches an equilibrium value (usually called carrying capacity).

During the years, several sigmoidal curves have been introduced (such as the Gompertz
or the Korf curves), but in general with sigmoidal function we refer to the case of the logistic
growth.

Traditionally, most of the aforementioned curves derive from the solution of ordinary
differential equation and, in this sense, they are deterministic. In order to incorporate random
influences, the dynamic growth models appeared, among them especially diffusion processes.
They are constructed in such a way that their mean function is a certain sigmoidal growth
curve. On the other hand, several investigations propose a different stochastic counterpart
and they define particular birth-death processes whose mean functions correspond to the
proposed growth model. In this talk we will investigate both of these two strategies in the
case of the classical logistic growth model.

Since there are several real situations in which the maximum level of growth is reached
after successive stages, in each of which there is a deceleration followed by an explosion
of the exponential type, the sigmoidal curves with more than one inflection point (called
multisigmoidal) seem to be interesting in real applications, especially if we consider the
logistic curve.

The forecasting of oil production has been a problem of great current interest, given its
fundamental role in the world’s economy. Oil production is known to be cyclical: in any
given system, after it reaches its peak, a decline will begin. With this in mind, the geologist
M.K. Hubbert, correctly estimated that oil production in USA would peak around 1970. His
prediction was based on the study of a special bell-shaped curve (called Hubbert function)
which is the first derivative with respect to the time of the logistic function. Some aspects
of this theory has lead researchers to extend/modify the original model. One such aspect is related to the fact that Hubbert provides a forecast with only one peak in oil production, but in other works examples were presented showing that oil production in several countries
cannot be represented by a single Hubbert cycle and the peaks can be more than one.

Since the peaks of the Hubbert function correspond to the inflection points of the logistic
curve, we propose a multisigmoidal logistic growth model useful in application and we define
both a birth-death and diffusion process whose mean function is of multisigmoidal logistic
type.

17 de diciembre de 2019, 12:00,

*Seminario de la 1º planta*del IEMath-GR