A delay model for the growth of ryegrass–clover mixtures: formulation and preliminary simulations

A model which describes the interaction of ryegrass and clover when grown together is proposed and analysed. The main biological feature of interest is the fixation of nitrogen by clover which is then subsequently used by ryegrass. In view of the difficulty of obtaining reliable data for the nitrogen level, this is modelled by a distributed differential-delay system. This enables a smaller set of variables and parameters to be used and the thresholds for sudden changes of outcomes with parameter values to be determined and related to the underlying biological processes. The dynamical system behaviour is extremely diverse, largely because of the effect of the distributed delay. Monocultures, coexisting mixtures, and limit cycle behaviour are possible long-term attracting outcomes. Numerical simulations of the model reveal that often two of these outcomes are possible for a given set of parameter values and which one is preferred depends on the initial conditions. The biological basis of the paper is proposed and then the model is analysed. Simulations and response outcomes are given for various values of the distributed delay. These indicate that this is a realistic model with an interpretable link to the underlying processes. The basis of the model is easily generalised to multiple-species models with seasonal variation or other externally-driven inputs.