Barber-Cushman Model

The Barber- Cushman model is a model that was developed in the early 1980s (see several references below). It simulates nutrient uptake by roots. It does so by assuming that roots are evenly distributed in the soil, and nutrient flow in the soil to the roots can be described by a one dimensional radial flow. The model includes a diffusion (dispersion) and a mass flow component which both may cause nutrients to move towards the roots. Once nutrients have reached the root surface, they are taken up actively by the root. Active uptake is described by michealis menten kinetics in the model.

     Barbara Cushman Model 

Figure 1: schematic representation of Barber-Cushman model.

The Governing Formulas:


Some Arbitrary Results of the Model as an example:BCM results

Example sensitivity analysis

read me

You can download a copy of the script that I used below. This script does not contain the full barber-cushman model. It calculates uptake for a 1 cm root segment. It does not inlcude root growth, which would require additional integration (which is done quite easily). The script, like in the paper, has a constant delta r. This makes it hard to find a good combination of delta r and delta t, which can especially be problematic for mobile nutrients and fine roots. Note that, because the model assumes an arbitrary compartment inside the root, the delta r must be smaller than the radius of the root. Othewise you may get strange, inverted results.


As far as I know the model is only available as closed software under the name  NST 3 (which includes improvements such as an increasing delta r). I made my own version, using the papers listed below. I wrote it in R , and open source scripting language. The code is distributed under the GPL 3 license, which means you can use and change the code for free, as long as you keep it free. So no commercial use, no closed software distribution of this code. I would appreciate it if you send me suggestions or changes to the code. You can post them on the sourceforge webpage.

You can download the code from sourceforge . No warrenties, what so ever.


  1. S A Barber, Soil nutrient bioavailability: a mechanistic approach (John Wiley and Sons, New York, USA, 1995).  
  2. S. Itoh and S A Barber, “A numerical solution of whole plant nutrient uptake for soil-root systems with root hairs,” Plant and Soil 70, no. 3 (1983): 403-413.  
  3. S.A. Barber and J.H. Cushman, “Nitrogen uptake model for agronomic crops,” in Modeling Waste Water Renovation- Land Treatment, ed. I.K. Iskandar (Wiley Interscience, New York, 1981), 382-409.  
  4. M. Silberbush and S. Barber, “Sensitivity of simulated phosphorus uptake to parameters used by a mechanistic-mathematical model,” Plant and Soil 74, no. 1 (February 28, 1983): 93-100.  
  5. M. Silberbush and S. A. Barber, “Phosphorus and Potassium Uptake of Field-Grown Soybean Cultivars Predicted by a Simulation Model,” Soil Sci Soc Am J 48, no. 3 (May 1, 1984): 592-596.