#### Introduction to population dynamics and resource exploitation

##### Enrico Canuto, Daniele Mazza

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.

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