Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs Modeling the interactions among phythopatogens and phyllosphere microorganisms for the biological disease control of Olea europaea L. Paula Baptista a , Iulia Martina Bulai ⁎ ,1,b , Teresa Gomes a , Ezio Venturino 1,c a CIMO, School of Agriculture, Polytechnic Institute of Bragança, Campus de Santa Apolónia, Bragança 5300-253, Portugal b Department of Information Engineering, University of Padova. Via Gradenigo, 6/B, 35131, Padova, Italy c Dipartimento di Matematica “Giuseppe Peano” Università di Torino, via Carlo Alberto 10, Torino 10123, Italia ARTICLEINFO Keywords: Olive tree disease Mathematical model Bistability Oscillating behavior ABSTRACT 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. 1. Introduction The olive tree (Olea europaea L.) has a great economic importance in the Mediterranean region [1]. This has been relevant all throughout history since very ancient times, dating back up to at least the classical Greek civilisation [2]. This plant however may be subject to several diseases that can hinder its growth, reducing the olives production and even cause its death [3]. These facts entail considerable economic losses for the farmers. The main diseases that affect olive trees are mostly caused by fungi and bacteria, which can infect several parts of the plant (roots, stem, fruits and leaves) [3]. Nowadays, olive diseases control programs rely mostly on chemical control by application of copper-based fungi- cides [3]. Besides having limited efficacy, this control measure is not compatible with sustainable production systems (organic and in- tegrated production) which are the pillars of the European Model for Agriculture, according the Directive 2009/128/EC. In olives produc- tion, the plant protection strategy must follow the Guidelines for in- tegrated production of olives [4]. Thus, a need to develop novel and environment-friendly control strategies for the biological management of olive diseases has recently become an important issue, involving both biological issues as well as applied mathematics concepts and is leading to an interesting research topic. The aerial parts of the olive plants (phyllosphere) are colonized by a diverse microbial community (mostly bacteria and filamentous fungi), which can grow both epiphytically on the surface of plant tissues and endophytically within the tissues [5]. These microorganisms interact with each other and with the host plant, mediating several ecosystem processes by altering plant traits [6], including disease resistance traits [7]. Phyllosphere microorganisms can reduce the infection of plant tissues by pathogens either directly, through the production of antag- onistic molecules and competition for resources, or indirectly by in- duction of plant resistance response [8]. Because of the importance of the microbiome for host fitness and function, there is a growing desire to model and manage host-microbiome interactions [9], to improve crop yields [10,11]. In this way, the development of the so-called “microbiome-driven cropping systems” might result in the next re- volution in agriculture, resulting in a more sustainable system for plant production [10,11]. Phyllosphere-associated microorganisms may be explored, in an integrative perspective, for designing new strategies for the biological control of olive diseases. Indeed, in the laboratory the salient compo- nents of the antagonistic molecules of these microorganisms can be artificially produced and then sprayed on the infected trees, to control the infection. However, the assessment of the impact that such possible strategies have on the cultures cannot be evaluated easily in the field. https://doi.org/10.1016/j.mbs.2018.12.002 Received 29 May 2018; Received in revised form 4 September 2018; Accepted 3 December 2018 ⁎ Corresponding author. E-mail addresses: pbaptista@ipb.pt (P. Baptista), iuliam@live.it (I.M. Bulai), ezio.venturino@unito.it (E. Venturino). 1 Member of the research group GNCS of INdAM. Mathematical Biosciences 308 (2019) 42–58 Available online 04 December 2018 0025-5564/ © 2018 Elsevier Inc. All rights reserved. T