Revue des Energies Renouvelales Vol. 6 N () Modeling a fertilisant dynamic transformation in soil-plant system Z. Aouachria * Energetics Applied Physics Laoratory, Department of Physics Faculty Hadj Lahdar University, Av. Chahid Bouhlouf Mohamed El Hadi, Batna, Algeria (reçu le Juillet accepté le Mars ) Astract - The application of fertilizers is one of the conditions for otaining the resistant cultures. Effectiveness of contriution fertilizers such as phosphorus, the nitrogen, etc depends on many factors to nowing: properties of the soil, the request of the plants out of fertilizers (phosphorescence for eample) and also on the form and the application method of fertilizers. Several results show that rational of organic and inorganic manure applications not only increase the production ut also can support a stale level of the production. The mathematical model of the transformation of fertilizer in the system soil plant is presented in this wor. We showed that there is a stale mode for the system soil plants. The strategy of the fertilizer application, permitted to provide this regime, has een determinate. Résumé L application des engrais est l une des conditions pour otenir des cultures résistantes. L efficacité de la contriution des engrais, tels que le phosphore, l azote, dépend de nomreu facteurs à savoir: les propriétés du sol, la demande des plantes en fertilisants (phosphorescence par eemple) et aussi sur la forme et la méthode d'application de ces engrais. Plusieurs résultats montrent que l application rationnelle des engrais organiques et inorganiques est non seulement d augmenter la production, mais prend également en charge un niveau stale de la production. Le modèle mathématique de la transformation d engrais dans le système sol-plante est présenté dans ce travail. Nous avons montré qu il eiste un mode stale pour le système solplante. La stratégie de l application des engrais, permettant à fournir ce régime, a été déterminée. Key words: Modeling - Dynamic Transformation - Fertilizer - System Soil-plant - Stale mode.. INTRODUCTION The development of information and technology in the field of the chemical products for agriculture relates to a numer of majorities of science directions of modern agriculture. The level of the quantitative description of the agro-ecosystem, lie an oject of research, determines the scientific value, the importance of the practice and the competitiveness of the agro-industrial one. An important element of advanced information on agro-technology is the mathematical model. Lie we let us now it that, for eample, phosphoric manures relate to the macro-element of the nutrition of the plants. The regular use of manures increases all shapes of the nitrogen s or phosphoric compounds. * aouachria@gmail.com
Z. Aouachria The group of the mineral components, such as the calcium, iron, and aluminum phosphate, is eposed to the attitude of the important changes quantitative and qualitative. The quantity of the ase forms of the mineral phosphate components in the various grounds is determined y the characteristics of the processes of these grounds. The proportion of aluminum phosphate increases consideraly in the various grounds ecause this is due to the complete soluility of phosphoric manures in water. In semi-arid agriculture for the caronated grounds, a long application of manures maes the influence appreciale, Guseynov (96) and Mamdov (96). The residual component of fertilization on the nutrition of the plant activates phosphoric salt. The share of a compound of fertilization of manures can e in contact with a structure of the organic sustances, roots of the plants and teture of ground. They ecome accessile to the plants in a process of mineralization. The application of the radioactive isotopes changes the design aout the character of the transformation of the minerals in the grounds and the availaility of their plants. For the first year of fertilization y manures, the plants use only a part of entering manure and the ground can e enriched for eample y digestile phosphates. The remainder is maintained in motionless form in the ground for an application or closer action. At the time of a contriution in ecess of the components in the grounds, the nutrition of the seedlings is carried out after action of the applications of old manures. It is possile to reduce, y such grounds, or to always correct the application of the quantity of components of fertilization. The strategy of fertilization is the principle of recommendation to otain the development of agriculture. It is ovious that to continue to use this practice of wor will have as consequence the construction of a fertile ground and increases its productivity, Wang De-ren (998).. GENERATING THE PROBLEM Taing into account these reasons, we want to develop a mathematical model of the dynamics of made-up fertilizing in the system soil plants. The mechanism of the transformation of this element, contained in manure, is given y the following diagram (Fig. ). Fig. : Diagram of the mechanism of fertilizer transformation
Modeling a fertilisant dynamic transformation in soil-plant system The dependence of the dynamics of the asorption of made up fertilizing y the plants, G f, with its quantity in the soil, f, is represented in the following diagram, (Fig. ). Fig. : Variation of the asorption of made up fertilizing y the plant The relationship etween G f and f depends mainly on the plant stage of growth. Part OM corresponds to a case where the quantity of made-up fertilizing does not satisfy the minimum need for the plant. Part MN corresponds to the case where the quantity is not needed for the plant and the asoring quantity is proportional to the quantity of fertilizer, f, and Part NP corresponds to the supersaturating case, when asorption of the plant decreases and accurate superaundance in the plant (poisoning effect). From Dzhafarov (986) we show the result of the mechanism of transition of many forms of fertilizing component for the vineyard, (Fig. ). We will simulate the variation of the quantity of various forms of fertilizer during one year. It is nown, Dzhafarov (986), that each year roughly % of phosphorus in moile form passes in the motionless form and % are required y the plant. In the same way, not less than % of phosphorus reserves in the soil are accessile to the plants. Another part of phosphorus in the soil is sujected to the difficulties of access or inaccessiility in the made up ones for the plants. Fig. : the mechanism of transition from the various forms components fertilizing for the vineyard
To moile To immoile To manure To manure To plant Losses 6 Z. Aouachria It is considered that the quantity of the phosphorus which was drawn up in the soil lie manure remains for the years a small quantity. The Tale elow reflects the quantitative variations of the various shapes of fertilizer. Tale : Quantity variations of the various forms of fertilizing compound Moile in soil In manure of st year In manure of rd year In manure of rd year Immoile in soil a a µ µ a a. MODELING OF THE DYNAMIC TRANSFORMATION OF COMPONENT IN A SYSTEM SOIL PLANT In order to determine the equations controlling the dynamic change of the products of manures in the system soil plants, noted y; and quantities of a component in the soil, in the moile and motionless form respectively, at the eginning of the period of the seed (either = ). In the same way we indicate y, and the quantities of this component, which were put in the soil in the current year, that in the previous year and the year which respectively precedes the last year y fertilization. Suitale parameters of the eginning of the net year indicated y h with j =,,, and. For the calculation of the assessment of the various forms we can write, from Tale, thus: µ µ The total assumed compound flu which passes in the plant is calculated y () () We notice that all the coefficients of the equations () and () are positive and are connected y group relations epressing the conservation laws of the total fertilizing component quantity, so we have;
Modeling a fertilisant dynamic transformation in soil-plant system 7 µ µ It comes to us the idea to wonder aout certain questions. Is it possile each year, to use an identical quantity of manure to ensure the stailization of the quantity of the various forms from made up fertilizing? It is possile thus to satisfy the needs for the plant in certain fertilizing compound?. Study of stailization We say that there is a stale mode, if with the annual urial of a constant quantity of manure in the soil, quantity of composed in moile form and motionless echange in wea and unimportant affinity. For a given quantity of the component put in the soil in the current year, equations: u, there is a single stale mode. This translates to the system of () µ u µ () We replace and in equation () y ( µ u ) and ( µ µ u ). Then the system () is reduced to the equivalent matri equation: ( E A) B () with B, A and u µ µ. E is the unit matri and For simplicity writing we omit the superscript. By taing into account the relations of conservation () one otains. det (E A ) ( ) a (6) u is a fied point of the system of equations (), for a given value of u with: / and ( a ) / a. Consequently the vector u u u µ u. Process of stailization Let us choose the mode u. For the rd year the dynamics of variation of made up fertilizer will e descried y the system of equations according to:
8 Z. Aouachria a a µ u µ a a u u (7) By replacing follows: and in the second equation of this system, we can write it as A B (8) where,,,, B (,) and µ µ µ.. Proposal That is to say u u u for a given value, u, if the condition a a a (9) is checed, then lim( ) u To justify it, let us consider ((A()) ) From this one has A where A ma[a a,a a ] According to the condition () a a a a a a a a Consequently, the matri A is a contracted form and { u } (=,,, ) is lie a fundamental sequence: Then u Au u u with (, ) (A u) ( Au u) A( u ) Then the proposal is justified. In addition, the soil-plant system satisfying the condition (9) is termed favourale.
Modeling a fertilisant dynamic transformation in soil-plant system 9.. Proposal For a certain reserve of compound fertilizer in a favourale soil, it is possile to mae such a stailizing element, which for the third year, would meet the needs of plants in phosphorus. To proof this, note, the total need of the plant in this compound, is nown. Let s write the equation of the assessment u u uµ uµ () From this, we can write: u () ( µ µ ) Then such choice (preceding proposal) will result in stailization and simultaneously will satisfy the need for the plant. Let us suppose that at the eginning of the -th year the quantities of the various forms of phosphorus (,,, ) are nown. Whether are possile operating y the quantity of phosphorus, apply in to soil, u, u, u, u, during sequence years to achieve a stale mode, i.e., that on ending years, quantity of phosphorus, accordingly ecame: u } =,,, will e called an application strategy of u, u, µ u, µ u, for the eforehand chosen value u. In the case, { fertilizing compound... Proposal For a any initial set () and final epectations (,,, ) eist the strategy to apply a fertilizing compound, if the following inequality is true. a (a a (a µ ( a ) (a a a a ( a (a a Let us proof this proposal. For doing this, we note that ) (a a µ )) µ ) ) () B a a a a µ Then the assessment of composed for the fourth year end must e descried y the following sequence: B Bu B Bu u B B u Bu u
Z. Aouachria B B u B u Bu u The last equation can e written in the equivalent form B B u B u Bu u And so on. The last equation can e written as u u B () u u The system () is solved if its matri is not degenerated. So let us compute the determinant of its matri; det det µ µ det By the form, the columns of determinant are lined y the evident relations A Consequently det And if V is not an appropriate eigenvector of the matri A to one of its eigenvalues, which is equal to, aa (a a ) (( a a ( )a )/) i.e if the previous inequality is true, we have: a a (a a µ µ, a ) a (a ( a a Taing into account an eplicit mode eigenvalues, ().. CONCLUSION a (a µ ) µ )), we otain the condition We have estalished that the annual regulation dose of applied fertilizer is most effective approach for a given soil plant system, which is characterized y a whole
Modeling a fertilisant dynamic transformation in soil-plant system comple of parameters when calculated. Also we have shown that it is a stale regime for a system soil plant. A strategy has een determinate to otain this stale regime. Then, according to the initial parameters, it is possile to wor directly under the regime of the application of a constant quantity of the component in the soil or to estalish this regime along the first four years. REFERENCES [] R.K. Guseynov, Conditions of Increasing the Effectiveness of Phosphoric Fertilizers, Bau s, Ana s Pulishing House Azera, 96. [] G.Sh. Mamdov, Ecologecal Models of the Fertility of the Basic Types of Soils of Azerajan, A series in Agriculture, AzNIINTI. Bau 6s, 99. [] Wang De-ren and Lu Wan-fang, Nutrient Balance of Nitrogen, Phosphorus and Potassium under Triple Cropping Systems Based on Rice, Better Crops International, Vol., N, 998. [] M.I. Dzhafarov, Prolem of Phosphates in the Soils in Azerajan Agriculture, Bau 88, Ana s Pulishing House Azera, 986 [] A.B. Pashayev, E.N. Saziev, G.M. Mammadov, S.M Eyuova and V.F. Guliyev, Mathematical Modeling of the Transformation Dynamics of Phosphorus in Soil Plant, Selçu Journal of Applied Mathematical, Vol. 7, N, pp. -, 6.