Journal of Mathematics Research; Vol. 8, No. 3; June 2016 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education Hopf-bifurcation Limit Cycles of an Extended Rosenzweig-MacArthur Model Enobong E. Joshua 1 , Ekemini T. Akpan 2 , Chinwendu E. Madubueze 3 1 Department of Mathematics and Statistics, University of Uyo, Uyo, Nigeria. 2 Department of Science Education, University of Uyo, Uyo, Nigeria. 3 Department of Mathematics/Statistics/Computer Science, University of Agriculture, makurdi, Nigeria. Correspondence: Ekemini T. Akpan, Department of Science Education, Mathematics Unit. University of Uyo, P.M.B 1017, Uyo, Nigeira. Tel: 234-813-037-6772. E-mail: ekeminitakpan@uniuyo.edu.ng Received: March 18, 2016 Accepted: March 29, 2016 Online Published: May 9, 2016 doi:10.5539/jmr.v8n3p22 URL: http://dx.doi.org/10.5539/jmr.v8n3p22 Abstract In this paper, we formulated a new topologically equivalence dynamics of an Extended Rosenzweig-MacArthur Model. Also, we investigated the local stability criteria, and determine the existence of co-dimension-1 Hopf-bifurcation limit cycles as the bifurcation-parameter changes. We discussed the dynamical complexities of this model using numerical responses, solution curves and phase-space diagrams. Keywords: Hopf-bifurcation, stability criteria, limit cycles 1. Introduction In general, any nonlinear dynamical system contains certain parameters called bifurcation parameters or controlled-free parameters, and thus it’s an imperative to study the qualitative behaviors of such robust systems as the parameters are varied. This study of bifurcation analysis includes the post-critical behaviors of the nonlinear system in the neighborhood of the critical points called Hopf-bifurcation limit cycles or periodic solutions (Liao &Yu, 2007). The complexities of such nonlinear dynamics include heteroclinic orbits, homoclinic orbits and chaos which envisaged essential global behaviors (Kutnzetsov, 1995; Sun & Luo 2005; Wang & Zhao, 2011). Mathematical models of multiple interacting species that exhibit such rich local and global dynamical behaviors (i.e., stability of equilibra, local and global bifurcations, limit cycles, peak-to-peak dynamics) includes Rosenzweig-MacArthur tri-trophic food chain models; it predicts and depicts real ecological system (Rosenweig-MacArthur, 1963; Keshet-Edelstien, 2005; Brauer, & Chavez-Castillo, 2012; Kar, Ghorai, Batabyal, 2012; Sanjaya, Sunaryo, Salleh, & Mamat 2013; Haque, Ali, & Chakravarty, 2013). In this paper, we obtained a topologically equivalence dynamics of an Extended Rosenweig-MarArthur Model, and studied codimension-1 Hopf-bifurcation limit cycles (periodic behaviors). These limit cycles occur naturally in the ecosystem, and using CASS (maple) we obtained their solution space and phase space diagrams, see (Naji, Upadhyay, & Rai, 2010; Candalen, & Rinaldi, 1999; Wiggin, 2010; Seydel, 2010, & Shavin, 2015). 2. Model Formulation and Boundedness The Extended Rosenzweig-MacArthur (ERM) model formulated and studied by Feng, Freeze, Lu, and Rocco (2014) is given as: 1 = 1 �1 − 1 �− 2 1 1 + 1 2 − 3 1 1 + 1 3 2 = 2 2 2 1 + 1 1 − 2 2 − 3 2 2 + 2 3 3 = 3 3 2 2 + 2 3 − 3 3 + 3 3 1 1 + 1 3 1.1 where, 1 (), 2 (), 3 () are population biomass densities of prey, predator and super-predator respectively, and constant parameters with significance ecological implications. We obtained a topologically equivalence dynamical model through the non-dimensionalization of the state variables as 22