Stability and bifurcation of aquatic food chain model

Abstract

The objective of this paper is to study the dynamical behavior of an aquatic food chain system. A mathematical model that includes nutrients and harmful phytoplankton and zooplankton is proposed and analyzed. The phytoplankton produces a toxic substance as a strategy of defense against predation by zooplankton. Its assumed that all the feeding processes in this food chain are depending on the Holling type-II functional response. Further a portion of dead phytoplankton and zooplankton are return to the nutrient due to the decomposition process. This model is represented mathematically by the set of nonlinear differential equations. The existence, uniqueness and boundedness of the solution of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. The occurrence of local bifurcation (such as saddle-node, transcritical and pitchfork) a long with Hopf bifurcation near each of the equilibrium points are discussed. Finally, numerical simulation is used to study the global dynamics of this model.

Keywords: nutrient , phytoplankton, zooplankton, stability analysis, stability bifurcation.