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Abstract: The paper presents a detailed study of a four component population dynamical model consisting of the biomasses of carbon of phytoplankton , nitrogen of phytoplankton , carbon of zooplankton and total free nutrient . In this model, both the influence of the internal (carbon) and external (nitrogen) nutrient on the growth of phytoplankton is included. This is usually termed a variable yield model in contrast to the constant yield model analyzed by J. Monod [Recherches sur la croissance des cultures bacteriennes, Herman, Paris, 1942. ]. The existence and stability of various steady states are discussed analytically and numerically using a computer simulation (AUTO) designed by Doedel [E.J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer. 30 (1981) 265–284; E. Doedel, X. Wang, T. Fairgrieve, AUTO: Software for continuation and bifurcation problems in ordinary differential equations, in: Applied Mathematics Reports, California Institute of Technology, 1994]. Various bifurcation diagrams so obtained support the analytical results and identified complex behavior which cannot be obtained analytically. The purpose of this paper is as a preliminary step to determine the change in the richness of behavior as complexity, such as spatial compartments, is added, which in limiting form would be a continuous spatial dependence (as a partial differential equation model). [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study a non-Kolmogrov type prey-predator system. First, we investigate the linear stability of the model by analyzing the associated characteristic equation of the linearized system. Second, we show that the system exhibits the Hopf bifurcation. The stability and direction of the Hopf bifurcation are determined by applying the norm form theory and center manifold theorem. Finally, numerical simulations are performed to illustrate the obtained results. [Copyright &y& Elsevier]