Water Management under Deficit Irrigation

Depending on the production function, irrigation water cost function and the sale price of yield, previously, lengthy mathematical expressions have been developed to estimate the optimal levels of water used in deficit irrigation, that would maximize yield (Wm), under water limiting (Ww), and under land limiting (Wl) and the amount of water that leads to income equal to income of Wm when land is limited (Wel), and that leads to farm income equal to farm income of Wm when water is limited Wew . Therefore, in this paper, firstly the previous lengthy expressions for (Wel) and (Wew) were simplified. Secondly, simple expressions for both (Wel) and (Wew) were derived under presence of rainfall. Thirdly, very simple mathematical relations between (Wm & Wl) and (Wm & Ww) were derived in order to determine much easier expressions than those previously derived for Wel, and Wew respectively, with and without rainfall.


INTRODUCTION
The increasing use of irrigation water has led to a deterioration in water in many areas. In the race to boost agricultural productivity, irrigation will become more dependent on poorly developed and almost uncontrolled water sources [1]. Irrigated agriculture needs to contribute an increasing share of food production to meet the growing demands of the growing population. In the face of climate change impacts, many arid and semi-arid areas suffer from water shortages. Drought and increased water demand have led to the need for irrigation water management. Water stress negatively affects plants. Depending on the degree and extent of water deficit stress, plants growth and the physiological functions can be limited and affected [2], [3].
Water shortage is the main impediment of agricultural production, so agricultural research in arid and semi-arid areas is mainly focused on the relationship between crop production and water use. The challenge in agriculture is to increase the economic return from the lowest amount of water, which is either reducing water consumption with increasing production, socalled irrigation water management, or increase production with a fixed amount of water (agricultural management) that will eventually lead to more water production. Irrigation water management to increase production will not be the same in all areas. In areas where sufficient water is available, full irrigation can be an appropriate option while in areas where water is limited; irrigation is considered an appropriate strategy for more efficient use of resources available water.
The use of management strategies to achieve sustainable production as the economic dividend is an appropriate management. However, in all cases it is necessary to know the relationship between water and production (the productivity equation) in order to achieve sustainable production in optimum quantities of irrigation water. Crop production functions is a convenient tool for irrigation management and yield response estimation in Water scarcities [4].
The objective of this research is firstly, to simplify the lengthy equations that derived by (English, 1990 andCapra et al., 2008) for the optimum water level at which net income equals that at full irrigation when land is limited (W el ), and when water is limited (W ew ). Secondly, to derive expressions for both (W el ) and (W ew ) under presence of rainfall. Thirdly, the main objective is to derive easy to apply equations for both (W el ) and (W ew ) depending on irrigation water level that, maximizes yield per unit land (W m ), and that maximizes net income per unit land (W l ), and that maximizes net income per unit of water (W w ) which can be found with simplified equations and much easier than the previously derived equations.

THE PROPOSED METHODOLOGY 2.1. Conceptual model
By adopting the production function ( ) (kg or tonha -1 ) and irrigation water cost function of ( ) ($ha -1 ) and the sale price of yield P, W is the amount of irrigation water, W m , which represents the amount of irrigation water that will maximize yields, can be found by setting the derivative of production function to zero, ∂Y(w)/∂w=0. Where y(w) = yield/unit land kg or tonha -1 ). C(w) = cost/unit land ($ha -1 ). a, b, c = production function constant. A, B = irrigation water cost function constant. P= sale price of yield ($kg -1 or $ton -1 ). W = the amount of irrigation water (m 3 ha -1 ).
The net income function of applied water ( ) ( ) ( )($ha -1 ), the net farm income from all irrigated land ($) If(w)= ( ) Area. When land is limited, the amount of irrigation water that gives maximum profit or income (w l ), can be found by setting the derivative of the income function to zero ∂iL(w)/∂w= 0. The amount of irrigation water that gives maximum farm income when water is limited (w w ) can be found by taking the derivative of the farm income function and set it to zero ∂If(w)/∂w=0. w el represents the amount of irrigation water that leads to profit or net income equal to the net income of W m when land is limited il(w m )= il(w el ), when irrigation water is limited W ew , represents the amount of irrigation water that leads to profit or net farm income, equal to the net income of the farm when using Wm If(w m )= If(w ew ). Figure (1) shows the quadratic production function and irrigation water cost function and W m , W l and W el .

General mathematical composition 2.2
Using quadratic production function and linear cost function, [9] presented general expression of W m , W l and W w and lengthy equations with two factors Z 1 and Z 2 to be compensated in equations 6A and 7A in order to determine W el and W ew respectively Also [10] presented lengthy equations for W el and W ew . [11] gave full expressions of W m , W l and W w with the quadratic production function and quadratic cost function and also presented lengthy equations for W el and W ew with two factors Z 1 and Z 2 as shown in the equations 1B-7B.
In presence of rainfall, using quadratic production function and linear cost function [12] modified full expression for W l , W w [7]also presented full expressions for W m , W l , W w but, without expressions for W el and W ew , as shown in the equations 1C-5C.
Where a′, b', c' = production function constants in presence of rainfall. R = rain. w′ = w+ R

Current research derivation
As mentioned above the optimum water levels w el and w ew were derived previously but their derivation were complicated and lengthy. To achieve the main objective of our study, simple expressions for w el and w ew with different production and cost functions were very essential to derive simple mathematical relations between w m , w l , ww in order to determine w el and w ew . Therefore, firstly the lengthy equations that derived by [9] were simplified as shown in equations 8A, 9A respectively. As their derivations were complicated and lengthy, they were given in Annex A.
Using both quadratic production and cost functions, simple expressions were derived in compare to the lengthy equations that previously derived by [11] with two factors Z 1 and Z 2 . As their full derivations were complicated and lengthy, they were given in Annex B.

Derivation of rained condition
Under rained condition full expressions for W el and W ew were derived using quadratic production function and linear cost function as in equations 6C and 7C respectively and their full derivation are given in Annex C.

The relationship between W m , W l and W el
The current study presented simple mathematical relations to determine w el and w ew From equations 10 and 11 results simple relation to determine w el as in equation (12).
( ) The comprehensiveness of this relationship can be verified by using the values of equations 3B and 4B in equation (12) to find the amount of equation 8B or to verify its application by using the values of equations 3C and 4C in equation (12) to find the amount of equation 6C, and figure (2) shows the relation between ( ⁄ & ) ⁄

The relationship between W m , W w and W ew
Simple expression was derived for W ew depending on W m , W w . From equation 5A results

CONCLUSION
It is easy to fully express the amounts of irrigation water referred to above W m , W l , W w , regardless to the formula of both the productivity function or the cost function, but there is a great difficulty in fully expressing each of the water quantities W el and W ew . So the research presented equations (12) and (15) or figures (2) and (3) where through them and amounts of water W el and W ew can be easily found depending on the amount of water W m , W l , W w .