Resource budget model with Duffing oscillator for dynamics of synchronized biennial-bearing olives in the Levant

Abstract

We develop and analyze a temporally continuous spatially lumped resource budget model to explain the dynamics of synchronized biennial-bearing olives in the Levant, specifically focusing on Syria, the region’s foremost olive-producing country. The model is a time-continuous counterpart of the celebrated resource budget model. It consists of a Duffing oscillator coupled with a dynamical model of pollination with an external force propelling olive growth by photosynthesis. The reference data are obtained from statistical databases of international organizations and our own observation systems in Jordan, a country neighboring Syria, providing a wealth of information to refine the model structure. An intensive review of Syria’s modern history involving significant shifts in agricultural policy and social stability leads to a conclusion that the model should comprehend the anomaly of olive yield interacting with socio-political factors as an autonomous behavior. The conventional mathematical methodology analyzes the model’s characteristics, such as solutions’ non-negativity, boundedness, and stability. The system is stable during pollination off-season but may become unstable and unbounded during pollination on-season, which is a property that the time-discrete resource budget model cannot reproduce. A significant finding is that coupling individual fruit trees by anemophily is not essential in synchronization, overturning earlier studies in the literature. The values of model parameters that best fit the historical data of olive yield in Syria result in bounded chaos. With alternative parameter values, the model could exhibit periodic oscillation, instability, or blowing up, as clearly shown in bifurcation diagrams.