Abstract
A theoretical model developed by Stone describing a three-level trophic system in the Ocean is analysed. The system consists of two distinct predator-prey networks, linked by competition for nutrients at the lowest level. There is also an interaction at the level of the two preys, in the sense that the presence of one is advantageous to the other when nutrients are low. It is shown that spontaneous oscillations in population numbers are possible, and that they result from a Hopf bifurcation. The limit cycles are analysed using Floquet theory and are found to change from stable to unstable as a solution branch is traversed.
Highlights
In a recent paper, Stone 1 presented a mathematical model of a three-level trophic food web involving ocean-dwelling microorganisms
Once we were satisfied that apparently oscillatory solutions were present using a shooting algorithm based on Newton’s method, we were able to find if the resultant solutions were periodic limit cycles
Only one predicts long-term survival for all four species. It is around this equilibrium point that we centred our study
Summary
Stone 1 presented a mathematical model of a three-level trophic food web involving ocean-dwelling microorganisms. Van der Stap et al 9 showed that population stability of phytoplankton occurred when the phytoplankton had a defence mechanism that affected the uptake interaction of the Zooplankton From these studies, it appears that qualitative behaviour, such as population stability oscillation and chaos, is highly dependent on the number of species present and the conditions under which the results were obtained. We use methods from Dynamical Systems theory to analyse the three-level trophic food web described by Stone 1 for further discussion on Dynamical Systems theory, see Murray 15 and Edelstein-Keshet 18 This model displays the interaction between Phytoplankton, Bacteria, Protozoa, Zooplankton, and Nutrients.
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