Introduction

The corn and bean intercropping has several objectives including the efficient use of important resources such as water, light and nutrients, in economic and agronomic terms. It has been reported complementarity between maize and beans because they make better use of resources when they are interleaved than simple crops (^{Willey, 1990}; ^{Johanne and Lynch, 2012}), which is reflected in higher performance (^{Li et al., 2001}; ^{Malezieux et al., 2009}), the same level of inputs, and is attributed to topological arrangement of the crop.

The milpa system intercropped with fruit trees has been designed to promote food security, income, employment and family environment. In this system, corn, some legumes and fruit are interleaved in stripes. The system responds to the needs of small production units (usually 2 to 2.5 hectares divided into several properties), high degree of economic and social marginalization. Under these conditions and with a traditional management, staples maize and beans are produced with insufficient income to meet the needs of food and household income.

In crop production, seed, variety and fertilizers are considered among the most important inputs, as they constitute a significant proportion of production costs (^{Volke, 1982}). The classic econometric method to establish the optimum economic dose soil fertility studies can be summarized as follows: 1) design and establish factorial experiments with various forms of fertilization and treatment design according to the investigator; 2) it is assumed that nutritional deficiencies are reflected in the observed yields and performance approaches in terms of a function of input levels, showing as much as possible the law of diminishing returns and 3) the dose of nutrients is obtained that produce the farmer the maximum net income per hectare (^{Martínez and Martínez, 1996}; ^{Barrios-Ayala et al., 2003}; ^{Volke et al., 2005}; ^{Barrios et al., 2008}).

The relative efficiency of land (ERT) is the most commonly used measure to compare the productivity of intercropping (^{Vandermeer, 1989}; ^{Rezaei-Chianeh et al., 2011}) under arable land constraints. The ERT monoculture represents the area required for the performance of polyculture at the same level of inputs (^{Vandermeer, 1989}). In a variation of the formula of ERT may include net income to integrate economic and biological factor in the equation relative efficiency of the ERG gain.

The objective of this research was to determine the best combination of N, P, K and planting density for growing corn ´H-155´ and black bean shrub '8025' in different topological arrangements, defined in terms of profit or net income, the relative efficiency of land and the relative efficiency of total income. The assumption was that the effect of the topological arrangement without significantly increasing doses of N, P and K, planting density and cost, corn and shrubs interspersed bean increase performance and profits compared with those of monocultures at the same level of inputs.

Materials and methods

In the spring-summer (april 10-november 17, 2012) an experiment was conducted in the Valley Experimental Field of Mexico (CEVAMEX) "El Horno", the National Institute of Forestry, Agriculture and Livestock (INIFAP). The site, with fluvisol haplic floor (19° 29’ north latitude, 98° 53’ west longitude) has an elevation of 2 280 msnm. The average temperature during the growing season was 19.5 °C and precipitation of 315 mm.

An experimental design with three treatments divided large plot and 25 small plot treatments, with a repeat parcels handled. The large plot corresponded to the topological arrangement of white ´H-155´ and black bean '8025' in: A) simple culture of corn and bean (CSMF); B) two rows of corn alternating with two bean (MMFF); and C) a groove corn alternating with one bean (MFMF) (Figure 1).

Figure 1 Topological arrangements large plot evaluated as: A) simple culture maize and beans (CSMF); B) two rows of corn alternating with two bean (MMFF); and C) a row of maize alternating with one bean (MFMF). 

In each topological arrangement were evaluated the nitrogen factors (N), phosphorus (P), potassium (K) and plant density (D). Space exploration ranged from 30 to 150 and 0 to 50 kg N; 0 to 60 and 0 to 50 kg P₂O₅; 0 to 40 and 0 to 30 kg K₂O; and 31.5 to 43.5 and 60 to 120 plants in 0.5 ha of maize and beans, respectively. The combination of these factors resulted in 25 small plot treatments according to a central composite design rotatory (Table 1). This design was based on a factorial design 2⁴ (two levels of factors: N, P, K and D in which the levels of each variable were coded as -1 and +1 increased by eight axial points coded as -2 and +2, and a center point coded as 0 (^{Barrios-Ayala et al., 2003}; ^{Volke et al., 2005}). Thus, the number of treatments small plot by the central composite design rotatory corresponded to the formula 2^k+2k+n; i.e, 2⁴+24+1= 25 treatments, randomly distributed the three topological arrangements (75 total treatments). The treatments were perfectly aligned small plot field. Each treatment of small plot consisted of four rows of 0.8 m wide by 2.12 m, with the two center rows as useful plot.

Sowing was manual and on it all the P, K and a third of the N corn and all three elements applied to beans; the remaining two thirds of N were applied to the first hoeing corn.

The experiment was maintained without restriction moisture by furrow irrigation and weeding was manual. Bean harvest was 125 days after planting; plants of each experimental unit were removed entire plot, dried in the shade, is peel and weight of the clean grain stubble was recorded. From this data was estimated the grain yield in 0.5 ha (Y_f).

The corn was harvested at 221 days after sowing. All ears of each useful plot were harvested, weighed and shelled. Grain moisture at harvest time was determined with a John Deere portable meter (SW 16060^®, Ill. EEUU). The grain yield Y_m with 14% humidity, was estimated to 0.5 ha, occupied by corn on topological surface arrangements. The corn stover was cut and weighed the day after the corn harvest. Stubble moisture was determined by gravimetric method and was considered to calculate the yield of stover (Y_r) in 0.5 ha at 14% moisture.

With dataY_m, Y_r and Y_f observed each treatment was calculated the relative efficiency of land (ERT) (^{Vandermeer, 1989}) and the relative efficiency of the gain (ERG), with the following equations:

Where: profit or net income (IN); corresponded to total income (IT); minus total cost (CT), according to relationship IN= IT-CT. The IT of each species was obtained by multiplying the yields of corn stover and grain for their respective prices at the point of sale, ditto for the grain of beans. The CT was the sum of the variable cost (CV) plus the fixed costs (CF). The CV of each treatment was the sum of the quantities of N, P, K and D multiplied by its respective price (^{Volke, 1982}). The price list considered in the calculation is presented in Table 2.

Using SAS program (^{SAS Institute, 2005}), the regression models for the dependent variables were adjusted: IN, Y_m, Y_r and Y_f. The general model equations was:

Dependent variable= µ + a₁ + a₂ + N + P + K + D + NP + NK + ND + PK +PD +KD+ N² + P² + K² + D² + a₁N + a₁P + a₁K + a₁D + a₁NP + a₁NK + a₁ND + a₁PK + a₁PD + a₁KD + a₁N² +a₁P² + a₁K² + a₁D² + a₂N + a₂P +a₂K + a₂D + a₂NP + a₂NK + a₂ND + a₂PK + a₂PD + a₂KD + a₂N² + a₂P² + a₂K² +a₂D² + a₂D² + e

Topological arrangements were included as auxiliary variables (dummy) that in the case of three topological arrangements took values of: a₁= 0 and a₂= 0 for settlement CSMF, a₁= 1 and a₂= 0 for MMFF and a₁= 0 and a₂= 1 for MFMF. The response of topological arrangements was the algebraic sum of terms without "a" plus counterparts terms corresponding to the effect of management factors; that is, that all terms with "a" in the equations are differences in response between topological arrangements (^{Barrios et al., 2003}; ^{Barrios et al., 2008}). The factors were coded into five levels, equally spaced, as:

The regression procedure (backward sls= 0.2) included significant factors (p< 0.05) for each dependent variable. With IN equation resulting from an optimization program in SAS^® which showed the maximum net income (NI) or gain with its combination of N, P, K and D of each topological arrangement it was made.

Once known the level of factors that maximized the gain in the three topological arrangements optimum economic dose (DOE) simple crops of corn and beans to compare with other arrangements at the same level of input was considered. For this input levels of the DOE they were replaced in the regression equations to determine the effect of the topological arrangement Ŷ_m, Ŷ_r, Ŷ_f.

Results and discussion

With the equation of estimated net income (ÎN) (Table 3) the DOE that maximized the gain of each topological arrangement was obtained. The biggest gain was $33 942 and was achieved with the topological arrangement interleaved with the combination MMFF per hectare of 34-0-40 kg of NPK and 29 500 maize plants and 11-030 kg of NPK and 80 000 plants beans (Table 4), since this arrangement is needed less N, which according to ^{Lithourgidis et al. (2011)} is an environmentally important aspect intercropping systems because it involves less use of inputs and production costs. This "DOE" in the MMFF system is suitable for the case of a producer who has satisfied the requirements of maize and beans from the family diet and is interested in the maximum net income.

Coding for the five levels of the study factors:

a₁= coding MMFF and a₂= MFMF as dummy variables (dumb) that can take the value 0 or 1 for its effect on the response variable. ÎN= net income estimate. Ŷ_m, Ŷ_r and Ŷ_f= estimated corn grain, corn stover and grain bean yields respectively.

The DOE estimated equation IN showed that is not required of P, in any topological arrangement, to achieve maximum income, this may be because the soil of the experimental site has been getting high and constant doses of fertilizers in the last 20 years, leaving a possible nutrient reservoir at least P. The MMFF arrangement provided $5 046.00 more profit and arrangement MFMF $2 595.00 more than the CSMF. These DOE for MMFF and MFMF arrangements are useful for producers who decide to adopt technology based on revenues generated (^{Seran and Brintha, 2010}).

^{Vandermeer (1989)} indicates that comparisons between topological arrangements must be made at the same level of inputs; thus, in the present study by contrasting earnings intercalated in these conditions system were obtained Ŷ_m, Ŷ_r and Ŷ_f, with their respective regression equations (Tables 3 and 5). Under these conditions it was also observed that the greatest gain was obtained with the arrangement MMFF ($29 293.00 ha^-1) compared to the CSMF ($28 895.00 ha^-1). At this level of input variable and fixed costs were the same in the topological arrangements so the difference between them was due to the performance of the two species. The MFMF arrangement in accordance with the treatment 20, the greater was obtained Y_m observed at 7.09 t in 0.5, which equals 14.18 t in 1 ha dispersed system. However, it was not treating or topological arrangement with which the highest gain was obtained ERT or greater, since there was a depression in bean yield compared to that obtained in CSMF. The decrease of 12-55% in performance or soya bean intercropped with maize has been reported by other authors (^{O'Callaghan et al., 1994}; ^{Lesoing and Francis et al., 1999}; ^{Matusso et al., 2014}). The revised literature suggests that this is due to interspecific competition for light, water and nutrient advantage seen in corn. Corn has more leaf area and root distribution (^{Hai-Yong et al., 2013}) that beans; thus, interleaving corn intercepts more photosynthetically active radiation that transforms grain biomass (^{Tsubo and Walker, 2002}).

The ERT older were 1.29 observed in the MMFF treatment system 13; while the maximum ERT MFMF was 1.27 in the treatment 20 (Tables 5 and 6). A value of ERT in MMFF 1.29 means that 1.29 is required is simple to achieve crop yields single crop, compared to the same level of inputs. On average, the ERT of the MMFF and MFMF systems was greater than one. Under production conditions where arable land is a limiting factor ERT is a good indicator to choose the best combination of inputs that maximizes.

In MFMF the ERT of 1.27 was accompanied by a high ERG (1.58) with a yield of grain like corn to 14.18 t ha^-1 dispersed system level inputs of treatment 20 (Tables 1 and 6).

In the topological arrangement MFMF, treatment 6 (60-45-15 kg N-P-K and 40 500 maize plants more 12.537.5-7.5 kg N-P-K and 105 000 bean plants per hectare) was obtained the biggest gain ($30 472.00 ha^-1) with a corn grain yield of 6.71 t ha^-1 in 0.5 occupied by corn. In addition, unpublished data showed that the protein maize grain in the topological arrangement MFMF (8.6%) is higher than in MMFF (8.3%) and CSM (7.8%) compared to the same dose of N, P, K and D. Thus, treatment 6 MFMF would be choice for producers as to obtain a good yield of corn grain quality, which is important because corn obtain 39% of the protein in the daily diet (^{Bourges, 2013}; ^{Turrent et al., 2013}). In terms of the higher net income ($33 942) it was obtained with the topological arrangement MMFF, treatment 4 (Tables 1 and 6).

Conclusions

In the maize crop ´H-155´ and black bean shrub '8025' the highest net income was presented with the interleaved topological arrangements (two rows of maize alternated with two bean and an alternating groove of each species) compared to simple crops.

In the topological arrangement of two rows of corn alternating with two bean was the biggest gain of $33 942.00 with the combination of inputs per hectare of 34-0-40 kg N-P-K in 29 500 plants of maize and was obtained 11-0 -30 kg N-P-K in 80 000 bean plants; i.e. $5 047.00 rather than single crop species. The effect of the same topological arrangement more profit was also generated when the level of N, P, K and planting density was constant in the comparison; and the maximum relative efficiencies of land and income were 1.29 and 2.01, respectively.

The greatest gain ($30 472.00 ha^-1) in the topological arrangement an alternating one groove corn bean was obtained with the combination 60-45-15 kg N-P-K and 40 500 of maize plants and 12.5-37.5-7.5 kg NPK and 105 000 bean plants per hectare MIAF. In this topological arrangement the highest relative efficiency soil was 1.27 and its respective relative gain efficiency was 1.58. Both were associated with high corn grain yield (7.09 t 0.5 ha^-1) and a decrease in bean yields.