SciELO - Scientific Electronic Library Online

 
vol.109 número5-6An assessment of the atmospheric nitrogen budget on the South African HighveldA tool for modernisation? The Boer concentration camps of the South African War, 1900-1902 índice de autoresíndice de assuntospesquisa de artigos
Home Pagelista alfabética de periódicos  

Serviços Personalizados

Artigo

Indicadores

Links relacionados

  • Em processo de indexaçãoCitado por Google
  • Em processo de indexaçãoSimilares em Google

Compartilhar


South African Journal of Science

versão On-line ISSN 1996-7489
versão impressa ISSN 0038-2353

Resumo

VAN DER HOFF, Quay; GREEFF, Johanna C.  e  KLOPPERS, P Hendrik. Numerical investigation into the existence of limit cycles in two-dimensional predator-prey systems. S. Afr. j. sci. [online]. 2013, vol.109, n.5-6, pp.01-06. ISSN 1996-7489.

There has been a surge of interest in developing and analysing models of interacting species in ecosystems, with specific interest in investigating the existence of limit cycles in systems describing the dynamics of these species. The original Lotka-Volterra model does not possess any limit cycles. In recent years this model has been modified to take disturbances into consideration and allow populations to return to their original numbers. By introducing logistic growth and a Holling Type II functional response to the traditional Lotka-Volterra-type models, it has been proven analytically that a unique, stable limit cycle exists. These proofs make use of Dulac functions, Liénard equations and invariant regions, relying on theory developed by Poincaré, Poincaré-Bendixson, Dulac and Liénard, and are generally perceived as difficult. Computer algebra systems are ideally suited to apply numerical methods to confirm or refute the analytical findings with respect to the existence of limit cycles in non-linear systems. In this paper a class of predator-prey models of a Gause type is used as the vehicle to illustrate the use of a simple, yet novel numerical algorithm. This algorithm confirms graphically the existence of at least one limit cycle that has analytically been proven to exist. Furthermore, adapted versions of the proposed algorithm may be applied to dynamic systems where it is difficult, if not impossible, to prove analytically the existence of limit cycles.

Palavras-chave : Lotka-Volterra models; predator-prey systems; stable limit cycle; Poincaré mapping; numerical method.

        · texto em Inglês     · Inglês ( pdf )

 

Creative Commons License Todo o conteúdo deste periódico, exceto onde está identificado, está licenciado sob uma Licença Creative Commons