An economical model with Allee effect

• Autor(a) principal: Luís, Rafael
• Data de Publicação: 2009
• Outros Autores: Elaydi, Saber, Oliveira, Henrique
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.13/3776
• topic: Profit rate; Organic composition of the capital; Exploitation rate; Allee effect; Non-autonomous periodic systems; Bust and boom in the profit rate; .; Faculdade de Ciências Exatas e da Engenharia

We formulate a mathematical model based on Marx theory of economics. The profit rate r is considered as a function of both the exploitation rate e and the organic composition of the capital k. This model possesses a new property, commonly used in biology, called the Allee effect, in which the profit rate declines to zero if it falls below a certain threshold. It is represented by the difference equation rnþ1 ¼ fa(rn), which is a family of unimodal maps depending on the parameter a, where a measures the relative growth of the exploitation rate when the profit rate is zero. Moreover, the model predicts a period-doubling bifurcation scenario as the parameter a increases. Finally, we allow the parameter a fluctuate periodically which leads to a periodic non-autonomous difference equations.