A canonical model for production and distribution of root mass in space and time

A mathematical model is presented for the production and distribution of plant root mass in a substrate in space and time. This model is intended to be used for the interpretation of data on root density profiles. It is assumed that a pulse of root apices appears at the surface of the substrate at zero time, emerging from seeds or stems, and that these apices migrate downwards in the substrate, leaving trails of root mass as they go, giving birth to daughter apices, and dying by terminal differentiation. The model has at least 3 parameters, representing, respectively, initial root generating capacity of root axes at zero time, net rate of root apex reproduction (or extinction), and rate of downwards migration of root apices. Two versions of the model are explored, a nondiffusive version and a diffusive version in which there is an additional parameter representing diffusivity of root apices. It is shown that both versions of this theoretical mechanistic model predict that the profile of root density with depth tends to a steady-state exponential form, parameterised by a density of root mass at zero depth and a logarithmic rate of change of root mass with respect to depth. This form has previously appeared in the literature as an empirical model. Data analyses are performed, using this model, on a set of data in which cultures of plants were subjected to treatment with auxin or by genetic manipulation, and the effects of the various treatments on the parameters of the model are described. Interpretations of the experimental results are proposed in the light of the theoretical results derived in this paper.