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**Dynamical systems analysis of a five-dimensional trophic food web model in the southern oceans.**
*(English)*
Zbl 1196.37125

The paper investigates the solutions of a dynamical system representing a theoretical model for a three-level trophic system in the ocean. The system consists of two distinct prey-predator networks, linked by competition for nutrients at the lowest level. There is also an interaction at the level of the two preys; the presence of one of the preys gives an advantage to the other one when nutrients are low. The model displays the interaction between phytoplankton, bacteria, zooplankton and nutrients. A particular emphasis in the analysis is on the stability of the steady-state populations and self-sustained solutions. Previous work on this model considered that the nutrient concentration was invariable, showing that the system becomes degenerate, in the sense that the unforced equations gave rise to equilibrium points that are centres. In this paper, the nutrient concentration is considered to be variable and, instead of degeneracy, Hopf bifurcation was observed for a range of parameter values. Using the Floquet theory, it is shown that the limit cycles change the stability as one moves along the solution branches. Some numerical results in MATLAB are presented at the end of the paper.

Reviewer: Iulian Stoleriu (Iaşi)

### Software:

Matlab### References:

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