Modeling the interactions among phythopatogens and phyllosphere microorganisms for the biological disease control of Olea europaea L

In this paper we formulate a model for assessing the interaction between the phytopathogen Spilocaea oleaginea and the phyllosphere microorganisms that are present in the olive tree leaves. The model describes the evolution in time of the foliage of the olive tree and the two different microorganisms, the phytopathogen fungi, that negatively affect the plant causing spots in the leaves, and the beneficial phyllosphere microorganisms, that help in keeping in check the invasion of the former. The system possesses five equilibria that are suitably analysed for feasibility and stability. The model shows interesting features: a bistable behavior, exhibited by three different pairs of equilibria. The separatrix surface of the basins of attraction of one such pair is computed. This allows the possible assessment of human intervention for control of the disease. Persistent oscillations via Hopf bifurcation are also discovered.