Stability and Bifurcation Analysis of the Impact of Refuge on the Dissolved Oxygen-Plankton Interaction
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Abstract
The suggested mathematical model for studying the effect of refuge on the dissolved oxygen in the plankton ecosystem is based on measurements of dissolved oxygen, phytoplankton, and zooplankton populations. The aim of this work is to find out the potential equilibrium and to investigate their behaviour. The study shows that there are three points of equilibrium. The feasibility requirements and stability conditions for all steady states are determined. Using the consumption of oxygen by zooplankton as a bifurcation parameter, we test for the presence of Hopf- bifurcation for the interior equilibrium. It is shown what conditions must be met for stable limit cycles. Finally, a numerical simulation is conducted to back up the analytical findings. It shows when the stability criteria are met, the solution of the proposed system constantly oscillates around the positive stable state. In addition, the solution exhibits limit cycle behaviour for small changes in certain parameters.
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