19The readily available global rock phosphate (P) reserves may run out within the next 50-130 20 years, causing soils to have a reduced P concentration which will affect plant P uptake. Using 21 a combination of mathematical modelling and experimental data we investigated potential 22 plant-based options for optimising crop P uptake in reduced soil P environments. 23By varying the P concentration within a well-mixed agricultural soil, for high and low P (35.5 24 to 12.5 mg l -1 respectively, using Olsen's P index), we investigated branching distributions 25 within a wheat root system that maximise P uptake. 26Changing the root branching distribution from linear (evenly spaced branches) to strongly 27 exponential (a greater number of branches at the top of the soil), improves P uptake by 142%

We develop a simple mathematical model for forced flow of culture medium through a porous scaffold in a tissue engineering bioreactor. Porous-walled hollow fibres penetrate the scaffold and act as additional sources of culture medium. The model, based on Darcy's law, is used to examine the nutrient and shear stress distributions throughout the scaffold. We consider several configurations of fibres and inlet and outlet pipes. Compared with a numerical solution of the full Navier-Stokes equations within the complex scaffold geometry, the modelling approach is cheap, and does not require knowledge of the detailed microstructure of the particular scaffold being used. The potential of this approach is demonstrated through quantification of the effect the additional flow from the fibres has on the nutrient and shear-stress distribution.

Plants rely on the conducting vessels of the phloem to transport the products of photosynthesis from the leaves to the roots, or to any other organs, for growth, metabolism and storage. Transport within the phloem is due to an osmotically-generated pressure gradient and is hence inherently nonlinear. Since convection dominates over di↵usion in the main bulk flow, the e↵ects of di↵usive transport have generally been neglected by previous authors. However, di↵usion is important due to boundary layers that form at the ends of the phloem, and at the leaf-stem and stem-root boundaries. We present a mathematical model of transport which includes the e↵ects of di↵usion. We solve the system analytically in the limit of high Munch number which corresponds to osmotic equilibrium, and numerically for all parameter values. We find that the bulk solution is dependent on the di↵usion-dominated boundary layers. Hence, even for large Péclet number, it is not always correct to neglect di↵usion. We consider the cases of passive and active sugar loading and unloading. We show that for active unloading the solutions diverge with increasing Péclet. For passive unloading the convergence of the solutions is dependent on the magnitude of loading. Di↵usion also permits the modelling of an axial e✏ux of sugar in the root zone which may be important for the growing root tip and for promoting symbiotic biological interactions in the soil. Therefore, di↵usion is an essential mechanism for transport in the phloem and must be included to accurately predict flow.

At a time of increasing global demand for food, dwindling land and resources, and escalating pressures from climate change, the farming industry is undergoing financial strain, with a need to improve e ciency and crop yields. In order to improve e ciencies in farming, and in fertiliser usage in particular, understanding must be gained of the fertiliser-to-crop-yield pathway. We model one aspect of this pathway; the transport of nutrients within the vascular tissues of a crop plant from roots to leaves. We present a mathematical model of the transport of nutrients within the xylem vessels in response to the evapotranspiration of water. We determine 7 di↵erent classes of flow, including positive unidirectional flow, which is optimal for nutrient transport from the roots to the leaves; and root multidirectional flow, which is similar to the hydraulic lift process observed in plants. We also investigate the e↵ect of di↵usion on nutrient transport and find that di↵usion can be significant at the vessel termini especially if there is an axial e✏ux of nutrient, and at night when transpiration is minimal. Models such as these can then be coupled to whole-plant models to be used for optimisation of nutrient delivery scenarios.

Management and irrigation of plants increasingly relies on accurate mathematical models for the movement of water within unsaturated soils. Current models often use values for water content and soil parameters that are averaged over the soil profile. However, many applications require models to more accurately represent the soil-plant-atmosphere continuum, in particular, water movement and saturation within specific parts of the soil profile. In this paper a mathematical model for water uptake by a plant root system from unsaturated soil is presented. The model provides an estimate of the water content level within the soil at different depths, and the uptake of water by the root system. The model was validated using field data, which includes hourly water content values at five different soil depths under a grass/herb cover over one year, to obtain a fully calibrated system for plant water uptake with respect to climate conditions. When compared quantitatively to a simple water balance model, the proposed model achieves a better fit to the experimental data due to its 2 ability to vary water content with depth. To accurately model the water content in the soil profile, the soil water retention curve and saturated hydraulic conductivity needed to vary with depth.