American Institute of Mathematical Sciences

June  2015, 4(2): 193-203. doi: 10.3934/eect.2015.4.193

Modeling plant nutrient uptake: Mathematical analysis and optimal control

 1 UMR Espace-Dev, Université de la Guyane, UR, UM2, IRD, 2091 Route de Baduel, 97306 Cayenne (Guyane), France 2 UMR Espace-Dev, Université de la Guyane, UR, UM2, IRD, Campus de TrouBiran, Route de Baduel, 97337 Cayenne (Guyane), France 3 INRA, UR1321, ASTRO AgroSystèmes TROpicaux, 97170 Petit-Bourg (Guadeloupe), France 4 INRA, UMR 1391 ISPA, F-33140 Villenave d’Ornon, France

Received  April 2014 Revised  September 2014 Published  May 2015

The article studies the nutrient transfer mechanism and its control for mixed cropping systems. It presents a mathematical analysis and optimal control of the absorbed nutrient concentration, governed by a transport-diffusion equation in a bounded domain near the root system, satisfying to the Michaelis-Menten uptake law.
The existence, uniqueness and positivity of a solution (the absorbed concentration) is proved. We also show that for a given plant we can determine the optimal amount of required nutrients for its growth. The characterization of the optimal control leading to the desired concentration at the root surface is obtained. Finally, some numerical simulations to evaluate the theoretical results are proposed.
Citation: Loïc Louison, Abdennebi Omrane, Harry Ozier-Lafontaine, Delphine Picart. Modeling plant nutrient uptake: Mathematical analysis and optimal control. Evolution Equations and Control Theory, 2015, 4 (2) : 193-203. doi: 10.3934/eect.2015.4.193
References:
 [1] G. Allaire, Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation, Oxford Science Publications, 2007. [2] J. H. Cushman, Nutrient transport inside and outside the root rhizosphere: Generalized model, Soil Science, 138 (1984), 164-171. [3] D. Daudin and J. Sierra, Spatial and temporal variation of below-ground N transfer from a leguminous tree to an associated grass in an agroforestery system, Agriculturen Ecosystems and Environment, 126 (2008), 275-280. doi: 10.1016/j.agee.2008.02.009. [4] A. El Jai, A. J. Pritchard, M. C. Simon and E. Zerrik, Regional controllability of distributed systems, International Journal of Control, 62 (1995), 1351-1365. doi: 10.1080/00207179508921603. [5] A. El Jai, Analyse régionale des systèmes distribués, Control Optimisation and Calculus of Variation (COCV), 8 (2002), 663-692. doi: 10.1051/cocv:2002054. [6] M. Griffon, Nourrir la Planète, Odile Jacob Ed., 2006. [7] 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 (1983), 403-413. doi: 10.1007/BF02374895. [8] R. Jalonen, P. Nygren and J. Sierra, Root exudates of a legume tree as a nitrogen source for a tropical fodder grass, Cycling in Agroecosystems, 85 (2009), 203-213. doi: 10.1007/s10705-009-9259-6. [9] R. Jalonen, P. Nygren and J. Sierra, Transfer of nitrogen from a tropical legume tree to an associated fodder grass via root exudation and common mycelial networks, Plant, Cell & Environment, 32 (2009), 1366-1376. doi: 10.1111/j.1365-3040.2009.02004.x. [10] S. Lenhart and J. T. Workman, Control Applied to Biological Models,, Chapman & Hall, (). [11] J.-L. Lions, Optimal Control for Partial Differential Equations, Dunod, 1968. [12] J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Dunod, Paris, 1970. [13] L. Louison, Analysis and Optimal Control of Transport-Diffusion Problems of Incomplete Data: Agroecology Application to Nutrient Uptake in Polluted Soils,, PhD Thesis, (). [14] P. H. Nye, The effect of the nutrient intensity and buffering power of a soil, and the absorbing power, size and root hairs of a root, on nutrient absorption by diffusion, Plant and Soil, 25 (1966), 81-105. doi: 10.1007/BF01347964. [15] P. H. Nye and F. H. C. Marriott, A theoretical study of the distribution of substances around roots resulting from simultaneous diffusion and mass flow, Plant and Soil, 3 (1969), 459-472. doi: 10.1007/BF01881971. [16] D. Picart and B.-E. Ainseba, Parameter identification in multistage population dynamics model, Nonlinear Ananlysis: Real world Applications, 12 (2011), 3315-3328. doi: 10.1016/j.nonrwa.2011.05.030. [17] M. Ptashnyk, Derivation of a macroscopic model for nutrient uptake by hairy-roots, Nonlinear Analysis: Real World Applications, 11 (2010), 4586-4596. doi: 10.1016/j.nonrwa.2008.10.063. [18] J. F. Reynolds and J. Chen, Modelling whole-plant allocation in relation to carbon and nitrogen supply: Coordination versus optimization, Plant and Soil, 185 (1996), 65-74. doi: 10.1007/BF02257565. [19] T. Roose, Mathematical Model of Plant Nutrient Uptake, College, University of Oxford, 2000. [20] T. Roose, A. C. Fowler and P. R. Darrah, A mathematical model of plant nutrient uptake, J. Math. Biology, 42 (2001), 347-360. doi: 10.1007/s002850000075. [21] A. Schnepf, T. Roose and P. Schweiger, Impact of growth and foraging strategies of arbuscular mycorrhizal fungi on plant phosphorus uptake, Plant and Soil, (2008), 85-99. [22] P. B. Tinker and P. H. Nye, Solute Movement in the Rhizosphere, Oxford University, (2000). [23] H. A. Van den Berg, Y. N. Kiseley and M. V. Orlov, Optimal allocation of building blocks between nutrient uptake systems in a microbe, J. Math Biology, 44 (2002), 276-296. doi: 10.1007/s002850100123.

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References:
 [1] G. Allaire, Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation, Oxford Science Publications, 2007. [2] J. H. Cushman, Nutrient transport inside and outside the root rhizosphere: Generalized model, Soil Science, 138 (1984), 164-171. [3] D. Daudin and J. Sierra, Spatial and temporal variation of below-ground N transfer from a leguminous tree to an associated grass in an agroforestery system, Agriculturen Ecosystems and Environment, 126 (2008), 275-280. doi: 10.1016/j.agee.2008.02.009. [4] A. El Jai, A. J. Pritchard, M. C. Simon and E. Zerrik, Regional controllability of distributed systems, International Journal of Control, 62 (1995), 1351-1365. doi: 10.1080/00207179508921603. [5] A. El Jai, Analyse régionale des systèmes distribués, Control Optimisation and Calculus of Variation (COCV), 8 (2002), 663-692. doi: 10.1051/cocv:2002054. [6] M. Griffon, Nourrir la Planète, Odile Jacob Ed., 2006. [7] 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 (1983), 403-413. doi: 10.1007/BF02374895. [8] R. Jalonen, P. Nygren and J. Sierra, Root exudates of a legume tree as a nitrogen source for a tropical fodder grass, Cycling in Agroecosystems, 85 (2009), 203-213. doi: 10.1007/s10705-009-9259-6. [9] R. Jalonen, P. Nygren and J. Sierra, Transfer of nitrogen from a tropical legume tree to an associated fodder grass via root exudation and common mycelial networks, Plant, Cell & Environment, 32 (2009), 1366-1376. doi: 10.1111/j.1365-3040.2009.02004.x. [10] S. Lenhart and J. T. Workman, Control Applied to Biological Models,, Chapman & Hall, (). [11] J.-L. Lions, Optimal Control for Partial Differential Equations, Dunod, 1968. [12] J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Dunod, Paris, 1970. [13] L. Louison, Analysis and Optimal Control of Transport-Diffusion Problems of Incomplete Data: Agroecology Application to Nutrient Uptake in Polluted Soils,, PhD Thesis, (). [14] P. H. Nye, The effect of the nutrient intensity and buffering power of a soil, and the absorbing power, size and root hairs of a root, on nutrient absorption by diffusion, Plant and Soil, 25 (1966), 81-105. doi: 10.1007/BF01347964. [15] P. H. Nye and F. H. C. Marriott, A theoretical study of the distribution of substances around roots resulting from simultaneous diffusion and mass flow, Plant and Soil, 3 (1969), 459-472. doi: 10.1007/BF01881971. [16] D. Picart and B.-E. Ainseba, Parameter identification in multistage population dynamics model, Nonlinear Ananlysis: Real world Applications, 12 (2011), 3315-3328. doi: 10.1016/j.nonrwa.2011.05.030. [17] M. Ptashnyk, Derivation of a macroscopic model for nutrient uptake by hairy-roots, Nonlinear Analysis: Real World Applications, 11 (2010), 4586-4596. doi: 10.1016/j.nonrwa.2008.10.063. [18] J. F. Reynolds and J. Chen, Modelling whole-plant allocation in relation to carbon and nitrogen supply: Coordination versus optimization, Plant and Soil, 185 (1996), 65-74. doi: 10.1007/BF02257565. [19] T. Roose, Mathematical Model of Plant Nutrient Uptake, College, University of Oxford, 2000. [20] T. Roose, A. C. Fowler and P. R. Darrah, A mathematical model of plant nutrient uptake, J. Math. Biology, 42 (2001), 347-360. doi: 10.1007/s002850000075. [21] A. Schnepf, T. Roose and P. Schweiger, Impact of growth and foraging strategies of arbuscular mycorrhizal fungi on plant phosphorus uptake, Plant and Soil, (2008), 85-99. [22] P. B. Tinker and P. H. Nye, Solute Movement in the Rhizosphere, Oxford University, (2000). [23] H. A. Van den Berg, Y. N. Kiseley and M. V. Orlov, Optimal allocation of building blocks between nutrient uptake systems in a microbe, J. Math Biology, 44 (2002), 276-296. doi: 10.1007/s002850100123.
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