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 & 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).   Google Scholar

[2]

J. H. Cushman, Nutrient transport inside and outside the root rhizosphere: Generalized model,, Soil Science, 138 (1984), 164.   Google Scholar

[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.  doi: 10.1016/j.agee.2008.02.009.  Google Scholar

[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.  doi: 10.1080/00207179508921603.  Google Scholar

[5]

A. El Jai, Analyse régionale des systèmes distribués,, Control Optimisation and Calculus of Variation (COCV), 8 (2002), 663.  doi: 10.1051/cocv:2002054.  Google Scholar

[6]

M. Griffon, Nourrir la Planète,, Odile Jacob Ed., (2006).   Google Scholar

[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.  doi: 10.1007/BF02374895.  Google Scholar

[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.  doi: 10.1007/s10705-009-9259-6.  Google Scholar

[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, 32 (2009), 1366.  doi: 10.1111/j.1365-3040.2009.02004.x.  Google Scholar

[10]

S. Lenhart and J. T. Workman, Control Applied to Biological Models,, Chapman & Hall, ().   Google Scholar

[11]

J.-L. Lions, Optimal Control for Partial Differential Equations,, Dunod, (1968).   Google Scholar

[12]

J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications,, Dunod, (1970).   Google Scholar

[13]

L. Louison, Analysis and Optimal Control of Transport-Diffusion Problems of Incomplete Data: Agroecology Application to Nutrient Uptake in Polluted Soils,, PhD Thesis, ().   Google Scholar

[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.  doi: 10.1007/BF01347964.  Google Scholar

[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.  doi: 10.1007/BF01881971.  Google Scholar

[16]

D. Picart and B.-E. Ainseba, Parameter identification in multistage population dynamics model,, Nonlinear Ananlysis: Real world Applications, 12 (2011), 3315.  doi: 10.1016/j.nonrwa.2011.05.030.  Google Scholar

[17]

M. Ptashnyk, Derivation of a macroscopic model for nutrient uptake by hairy-roots,, Nonlinear Analysis: Real World Applications, 11 (2010), 4586.  doi: 10.1016/j.nonrwa.2008.10.063.  Google Scholar

[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.  doi: 10.1007/BF02257565.  Google Scholar

[19]

T. Roose, Mathematical Model of Plant Nutrient Uptake,, College, (2000).   Google Scholar

[20]

T. Roose, A. C. Fowler and P. R. Darrah, A mathematical model of plant nutrient uptake,, J. Math. Biology, 42 (2001), 347.  doi: 10.1007/s002850000075.  Google Scholar

[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.   Google Scholar

[22]

P. B. Tinker and P. H. Nye, Solute Movement in the Rhizosphere,, Oxford University, (2000).   Google Scholar

[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.  doi: 10.1007/s002850100123.  Google Scholar

show all references

References:
[1]

G. Allaire, Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation,, Oxford Science Publications, (2007).   Google Scholar

[2]

J. H. Cushman, Nutrient transport inside and outside the root rhizosphere: Generalized model,, Soil Science, 138 (1984), 164.   Google Scholar

[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.  doi: 10.1016/j.agee.2008.02.009.  Google Scholar

[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.  doi: 10.1080/00207179508921603.  Google Scholar

[5]

A. El Jai, Analyse régionale des systèmes distribués,, Control Optimisation and Calculus of Variation (COCV), 8 (2002), 663.  doi: 10.1051/cocv:2002054.  Google Scholar

[6]

M. Griffon, Nourrir la Planète,, Odile Jacob Ed., (2006).   Google Scholar

[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.  doi: 10.1007/BF02374895.  Google Scholar

[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.  doi: 10.1007/s10705-009-9259-6.  Google Scholar

[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, 32 (2009), 1366.  doi: 10.1111/j.1365-3040.2009.02004.x.  Google Scholar

[10]

S. Lenhart and J. T. Workman, Control Applied to Biological Models,, Chapman & Hall, ().   Google Scholar

[11]

J.-L. Lions, Optimal Control for Partial Differential Equations,, Dunod, (1968).   Google Scholar

[12]

J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications,, Dunod, (1970).   Google Scholar

[13]

L. Louison, Analysis and Optimal Control of Transport-Diffusion Problems of Incomplete Data: Agroecology Application to Nutrient Uptake in Polluted Soils,, PhD Thesis, ().   Google Scholar

[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.  doi: 10.1007/BF01347964.  Google Scholar

[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.  doi: 10.1007/BF01881971.  Google Scholar

[16]

D. Picart and B.-E. Ainseba, Parameter identification in multistage population dynamics model,, Nonlinear Ananlysis: Real world Applications, 12 (2011), 3315.  doi: 10.1016/j.nonrwa.2011.05.030.  Google Scholar

[17]

M. Ptashnyk, Derivation of a macroscopic model for nutrient uptake by hairy-roots,, Nonlinear Analysis: Real World Applications, 11 (2010), 4586.  doi: 10.1016/j.nonrwa.2008.10.063.  Google Scholar

[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.  doi: 10.1007/BF02257565.  Google Scholar

[19]

T. Roose, Mathematical Model of Plant Nutrient Uptake,, College, (2000).   Google Scholar

[20]

T. Roose, A. C. Fowler and P. R. Darrah, A mathematical model of plant nutrient uptake,, J. Math. Biology, 42 (2001), 347.  doi: 10.1007/s002850000075.  Google Scholar

[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.   Google Scholar

[22]

P. B. Tinker and P. H. Nye, Solute Movement in the Rhizosphere,, Oxford University, (2000).   Google Scholar

[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.  doi: 10.1007/s002850100123.  Google Scholar

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