Existencia de soluciones periódicas para una extensión del modelo de Palomba

Abstract

In 1939 Italian economist G. Palomba [1] proposes the following model Lotka-Volterra predatorprey type with i; i, i = 1; 2, positive constants, to study an economic system where there are only two types of goods: consumer goods whose quantities are given by C1 C1( ) and capital goods whose quantities are given by the variable C2 C2( ). Following the mathematical and graphical treatment developed in [2], G. Palomba proved the oscillatory behavior (economic cycles) of the quantities of each good in his model. With the purpose of developing a little more the ideas found in [1], this document proposes a variation of the model when considering an exogenous and periodic increase in the production rates of capital goods and a negative e ect due to the increase of the quantities of consumer goods in the rate of production of capital goods. More precisely, we consider the system where b ; k nonnegative constants and h(! ) is a periodic function that simulates the net and exogenous e ect of seasons of higher and / or smaller production rate of capital goods. The Lower and Upper-solutions Method allow us to prove the existence and uniqueness of periodic solutions for the system (2), which, as in the Palomba model, represent an oscillatory behavior of consumer and capital goods.