Analytical Solutions Of Nutrient Uptake Model By Plant Roots

The root system is vegetative organ of plants that provide essential functions for survival and growth,such as anchoring plants,absorbing nutrients,storing water,synthesizing transport organic matter.Plants require 16-18 essential nutrient elements to sustain the lifecycles.Plants directly obtain the carbon,hydrogen and oxygen from the air,and the other elements(e.g.nitrogen,phosphorus,potassium)are extracted from the soil by roots.Nutrients generally migrate to the root surface by three ways:interception,diffusion and mass flow.Due to the limited contact area between soil particles and roots,the amount of intercepted nutrients are very small(only 5% of the total plant requirement),therefore the diffusion and the mass flow are predominant.At present,the process of nutrient absorption is described by convective-diffusion equations.The main contents of this thesis is to build mathematical methods to obtain the analytical solutions,approximate analytical solutions and numerical solutions and thesis consists of three parts:In the first part,Nye,Tinker and Barber considered the conservation of solid and liquid nutrients in the soil,and proposed a Nye-Tinker-Barber model for nutrient uptake by single roots.According to the fast-slow process in the Nye-Tinker-Barber model,the rhizosphere is divided into the out field and the inner field.The outer solution is obtained by the similarity method,and the inner solution is obtained by the rescaling and the singular perturbation methods.We expand the outer solution as series from root surface and matching it with the inner solution and obtain the analytical solutions of the root surface nutrient uptake flux and rhizosphere solute concentrations.Numerical simulations are take on the uptake fluxes of six elements(N,K,P,Mg,S and Ca)and the solute concentrations of two elements(N and K).Comparisons are made among the numerical solutions,analytical solutions and previous results.In the second part,the citrate secreted by the rice roots will promote the absorption of phosphate,and this process is described by the Kirk model.Since the Kirk model is semi-coupled,we can divide the Kirk model into a citrate sub-model and a phosphate sub-model.In the citrate sub-model,we obtain the analytical solution of citrate with the methods of Laplace transform,inverse Laplace transform and convolution theorem.The citrate solution is substituted into the phosphate sub-model,and the analytical solution of phosphate is obtained by the separation variable method.The existence of the solutions can be proved by the comparison test,the Abel discriminating method.We make a bounded estimation of solutions,and the maximum value is confirmed by numerical simulation.In the third part,when the free metals concentration at the root surface decreases,the dissociation of the complexes also increase the uptake,in cadmium-contaminated soils.The state of complexes is described by different models.When the complexes are fully inert complex,the process of cadmium absorption by the root can be described by the Barber model.When the complexes are fully labile complex,the absorption of cadmium ions by the root system can be described by the Lin model.Both models consist of two convection-diffusion equations with functional response terms,and the right boundary is the free boundary.Firstly,determining the right boundary condition,and Solve the steady-state solution of the differential equation,the parameters are determined by the boundary and initial conditions.The cumulative flux is determined by the ion concentration of the root surface.