Mathematical analysis of a time-delayed model for cocoa yield

Abstract

Abstract Cocoa is an important crop that is predominantly grown in the western part of Africa. However, there have been fluctuations and declining trends in production and several factors have been identified to be responsible for this. A significant factor is the effect of climate variation, which could result in a low farm-level yield. Therefore, to understand the contribution of climate variability on the farm-level yield, we construct and analyse a time-delayed model to capture the effect of rainfall on cocoa production. This work uses a system of differential equations to model the crop transition from the flowering stage to pod formation, pod ripening and then to harvesting. We introduce a periodic forcing function into the model of flowering to account for the impact of seasonal rainfall variations. This leads to a novel nonlinear parametrically forced ODE for the flowering with periodically varying coefficients, which is coupled to a time-delayed model for the ripened pod formation and then harvesting. We perform an analysis of all parts of the system proving that it has a periodic solution when (parametrically) forced periodically, and we then conduct an asymptotic analysis on this periodic solution to show how its rich behaviour depends on the parameters of the climatic forcing in the model.