A Distributed-Delay DE (DDDE) model for the nitrogen cycle in pasture
Contributing Author: G. C. Wake (Massey University Auckland, New Zealand)
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 the clover which is then subsequently used by rye-grass which is the larger biomass in the composite pasture. Soil nitrogen is difficult to measure in situ and so the model here uses a distributed differential-delay system, with just two components for the respective biomasses. This enables a smaller set of variables and parameters to be used and the important thresholds for sudden changes of outcomes with parameter value changes to be determined, and related to the underlying biological processes. The dynamical system behaviour is extremely diverse, largely because of the distributed delay. These sorts of non-local equations are rarely studied and there is little generic theory available. Monocultures, coexisting mixtures and limit cycle behaviour are possible long-term attracting outcomes. The match of this model to observed outcomes is encouraging, and this appears to be a good simple model for nitrogen-cycling in the soil. The basis of the model is easily generalised to multiple-species models with seasonal variation or other externally-driven inputs. The presentation will include generally discussion on DDDE models. The support of a Maclaurin Fellowship of NZIMA is gratefully acknowledged.