Bifurcations and periodic orbits in variable population interactions

Jorge Rebaza

doi:10.3934/cpaa.2013.12.2997

Article Contents

2013, Volume 12, Issue 6: 2997-3012. Doi: 10.3934/cpaa.2013.12.2997

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Bifurcations and periodic orbits in variable population interactions

Jorge Rebaza^1,

1.
Department of Mathematics, Missouri State University, Springfield, MO 65897

Received: October 31, 2012

Revised: January 31, 2013

Published: October 2013

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Abstract

Variable population interactions with harvesting on one of the species are studied. Existence and stability of equilibria and existence of periodic solutions are established, existence of some bifurcation phenomena are analytically and numerically studied, explicit threshold values are computed to determine the kind of interaction (mutualism, competition, host-parasite) between the species, and several numerical examples are provided to illustrate the main results in this work. A brief discussion on the influence of the harvesting function on the dynamics of the model is also included. Hopf bifurcations and periodic solutions are found for the first time in this kind of models.

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Mathematics Subject Classification: Primary: 37N25, 58F17; Secondary: 92D25.

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