  The paper was suggested by a brief note of the second author about the
application of the Hubbert curve to predict decay of resource exploitation. A
further suggestion came from the interpretation of the Hubbert curve in terms
of a specific Lotka Volterra (LV) equation. The link with population dynamics
was obvious as logistic function and LV equation were proposed within the
demography science field. Mathematical population dynamics has a history of
about two centuries. The first principle and model of population dynamics can
be regarded the exponential law of Malthus. In the XIX century, the Malthusian
demographic model was first refined to include mortality rate by Gompertz. In
the early XIX century the model was further refined by Verhulst by introducing
the standard logistic function. The previous models only concern the population
of a single species. In the early XX century, the American demographer Lotka
and the Italian mathematician Volterra proposed a pair of state equations which
describe the population dynamics of two competing species, the predator and the
prey. This paper is concerned with the single and two-species fundamental
equations: the logistic and LV equation. The paper starts with the generalized
logistic equation whose free response is derived together with equilibrium
points and stability properties. The parameter estimation of the logistic
function is applied to the raw data of the US crude oil production. The paper
proceeds with the Lotka Volterra equation of the competition between two
species, with the goal of applying it to resource exploitation. At the end, a
limiting version of the LV equation is studied since it describes a competition
model between the production rate of exploited resources and the relevant
capital stock employed in the exploitation.
