Electronic Journal of Polish Agricultural Universities (EJPAU) founded by all Polish Agriculture Universities presents original papers and review articles relevant to all aspects of agricultural sciences. It is target for persons working both in science and industry,regulatory agencies or teaching in agricultural sector. Covered by IFIS Publishing (Food Science and Technology Abstracts), ELSEVIER Science - Food Science and Technology Program, CAS USA (Chemical Abstracts), CABI Publishing UK and ALPSP (Association of Learned and Professional Society Publisher - full membership). Presented in the Master List of Thomson ISI.
Volume 10
Issue 2
Available Online: http://www.ejpau.media.pl/volume10/issue2/art-22.html


Piotr Sobkowicz1, Magdalena Podgórska-Lesiak2
1 Department of Agroecosystems and Green Areas Management, Wroc³aw University of Environmental and Life Sciences, Poland
2 Department of Soil Management and Plant Cultivation, Wroc³aw University of Environmental and Life Sciences, Poland



Mixture is a system of two or more species that interact with each other in different ways. Interspecific competition, resource complementarity and facilitation may shape performance of a mixture and if the phenomena act together, problem occurs with proper interpretation of results of a simple mixture experiment. To study interactions in crop mixtures and their productivity various designs have been used, like proportional substitutive, substitutive, simple additive and unconstrained design. All these designs have some limitations and they do not answer the same questions about interspecific competition and productivity of mixtures. More extended structures such as addition series and bivariate factorial design are probably more appropriate for this purpose but size of field experiments conducted according to these designs limits their use.

Key words: crop mixtures, competition, resource complementarity facilitation, mixture design, RYT, LER.


The idea of sustainable development and sustainable agriculture, that came into existence in early 90s changed attitude in agricultural policies of developed countries enhancing adoption of agricultural practices that are being used in low-input traditional farming [45]. The notion accompanying the implementation of more sustainable ways of agricultural production is to make better use of complex interactions between species in the more diversified agrecosystem without damage to the environment [3,25]. Among potential methods that may be implemented into practice of sustainable agriculture are: growing arable crops in mixtures (intercrops), employing living mulches and reducing herbicide use by proper crop-weed management. In all these methods, two or more plant species grows together in the same field and the potential success of farming relies on appropriate recognizing interactions between the species and on proper management of these interactions.

Interactions between plant species in natural habitats have been investigated in plant ecology and competition in particular gained a vast body of scientific literature because it is thought to be the main process affecting natural communities [5,9]. The process has been researched also in agriculture, particularly in studies on mixtures of arable crops and in crop-weed studies. The most important issue when planning an experiment is a choice of a design that allows to appropriate assessment whether growing two crops in mixture is more advantageous than sole cropping. Yield of grain per unit area is an essential measure of mixture performance in such experiments although it represents only a part of total plant biomass and may not fully reflect the result of competition between species in mixture [59]. The goal may be superior to detailed analysis of plant interactions between crops in mixed stand, because in many cases results of agricultural experiments serve as recommendations for agricultural practice. On the contrary, plant ecology is much more interested in mechanism of competition without concerning competing species as a mixture that produces marketable yield. Effect of competing species on each other and expression the results on per plant basis may be then more informative for ecologist than for agronomist. In spite of different aims, crop mixtures has been used as a base for competition studies in plant ecology and such examples exists in academic books [5,66]. This suggests that crop mixture is a place in which scientific interests of agriculture and plant ecology meet. If interactions between arable crops in mixture are the point of interest, then the ecological theories on the subject should be taken into consideration to draw valid inferences from an experiment.

The objective of the article is to present main interactions between species in mixtures and problems that may arise when designing and interpreting experiments to study these interactions. The article considers only issues that are relevant to agricultural experiments with mixtures of arable crops and crop-weed relationships.


In plant ecology it is believed that species diversity may increase productivity of plant community, though the notion does not have strong experimental evidence and debate on the issue continues [1,22,62]. The gain from the multispecies plant assemblage may come from the way in which different species capture common limiting resources and interact with each other. Due to differences in architecture of root and shoot system, physiology and growth cycle, species in mixture may use greater quantity of a resource or its different fraction or utilize it more efficiently than when grown alone. This is defined as resource complementarity or species complementarity [8,30,67]. Resource complementarity is not an interaction between plants per se. It is rather an attribute of plant community. It may manifest itself in a better performance of the community such as plant biomass or grain yield per unit area, but it cannot be expressed per individual species basis. The other phenomenon that may contribute to a greater productivity of plant community is facilitation. Interspecific facilitation operates when plants of one species positively affect growth of plants of the other species [9,10,32,66]. Complementarity and facilitation to occur require proximity of interacting plants, thus the third phenomenon often interrelated with the two is competition. There are many definitions of competition [26,40,70]. One of the widest is that proposed by Vandermeer [66] who consider competition as “...the process in which two individual plants or two populations of plants interact in such a way that at least one exerts a negative effect on the other”. The definition comprises competition for common limiting resources such as light, water and nutrients (exploitation competition) and interference competition e.g. through allelopathy. Competition between plants of the same species is called intraspecific competition while competition between plants of different species is called interspecific competition.

The three phenomena: competition, resource complementarity and facilitation may function in any cereal-legume mixture (Fig. 1):

  1. Competition. In agricultural practice mixture of the species is grown in a dense plant stand thus plants compete intensely for common limiting resources during their growth.

  2. For a low density of plants of a single species increasing the density increases yield per unit area and intraspecific competition becomes more intense, because greater number of individuals compete for the same amount of common limiting resources. In pure stands, increase in the intensity of competition manifests itself by the reduction of the performance of the individual, e.g. biomass of single plant or in agricultural studies reduction of grain weight per plant [42,70]. For sufficiently high range of plant densities, yield per unit area becomes constant because the reduction in plant weight (or grain weight per plant) is inversely proportional to increase in plant density. The interdependence is called the “law of constant final yield” [5]. The law operates in many agricultural fields because recommended for agricultural practice seeding densities of crops termed “optimum seeding densities”, are those that secure maximization of grain yield. Although the law was formed for single species stand, it is probably valid also for mixtures of crops.

  3. Partial complementarity in nitrogen use. There are partially different sources of nitrogen for component species of any cereal-legume mixture. Both components acquire soil and fertilizer nitrogen but when the legume starts to fix atmospheric nitrogen competition for soil and fertilizer sources of N between the species decreases [35].

  4. Facilitation and interference competition. If cereal acquires biologically fixed nitrogen released from roots of legume, it means the legume facilitates growth of the cereal. Though this N transfer has not been definitely proved by researchers [15,28]. Cereal may also obtain biologically fixed N during mixture growth from decaying small roots of the legume [60]. The cereal may facilitate growth of legume providing support for plants of the legume of such type as vetch or pea preventing them from lodging. When grown in pure stands the legumes lodge and are attacked by fungal diseases that leads to partial or almost complete decaying of pods [53]. Thus the more yield of the legume species is destroyed in pure stand the greater is the importance of supporting role of cereal in mixture. Legume plants leaning on cereal plants interfere with their growth and this is an example of direct interference [16]. If the legume plants prevail in mixture it may lead to lodging of the whole stand.

Fig. 1. Interactions between triticale and common vetch in mixture

Although the complex interaction between cereal and legume species is anticipated before seeding of mixture and facilitation of the legume by cereal is visually observed, contribution of each of the phenomenon to yield of mixture is difficult to quantify. When response of a species to growing in mixture is measured, usually a net result, namely competition or facilitation is recorded [10,32]. In research of Sobkowicz [53] net facilitation of vetch by triticale in mixture was found because cereal plants prevented legume plants from lodging and partial grain decaying that occurred in pure stand of vetch as a consequence of frequent rainfall (Fig. 2). It enabled vetch in mixture to outyield vetch in pure stand. Even when plant density of triticale in mixture was doubled, causing greater competition pressure on vetch, facilitation was still the net result for grain yield of the legume. Facilitation is not often sufficiently strong to increase yield of a species in mixture above its yield in pure stand, but if the phenomenon exists in mixture, then their effect may be confounded with the effect of resource complementarity [67]. This was observed in experiment with cereal mixtures, in which oats and triticale were less susceptible to lodging than barley, giving support to the species in mixtures [54]. In consequence, lodging of mixtures was lesser than lodging of pure stand barley and occurred later during growing season. The component species differed also in a length of growing cycle showing partial complementarity in time of resource use in mixtures. These two phenomena overlapped causing an increase in relative yield total (RYT), an index that was originally developed to detect resource complementarity.

Fig. 2. Yields of grain of common vetch in mixture with spring triticale. Pure stand yield of
vetch = 100%. Drawn based on data from Sobkowicz [53]

Supporting function of one species in mixture against lodging of the other is probably most easily visible example of facilitation and may partially explain advantage of a mixture over sole cropping. The next example is protection from insect pests or fungal diseases provided by one component species for other component in mixture. This kind of interaction is an example of indirect facilitation [9] and has been found in crop mixtures [23,68,71].

Interaction between components in mixture of three species is far more complex and difficult to examine, because species affect each other directly and indirectly. In such a mixture indirect facilitation may occur when species A outcompetes species B and the latter is unable to compete intensely with species C. Thus species A facilitates growth of species C indirectly [6,67]. This is what is expected from living mulch, the small plant species (e.g. clover) that produces plant cover in the bottom of cereal canopy. It is assumed the plants of living mulch producing shade over the soil protects the cereal from competition from weeds that emerge later in the growing season [61]. On the other hand, plants of living mulch should not compete intensely with the cereal but experiments show the goal is difficult to attain and competition from living mulch may reduce cereal yield significantly [2,61,74].


The assumption of many experiments with mixtures of species is that they should resemble situation met in agricultural field. To investigate productivity of mixtures of two arable crops grown for grain the experiments are conducted at plant densities that ensure maximum grain yield, in other words they are conducted at optimum plant (or seed) densities of crops. Simple two-species mixture may comprise half of the pure stand optimum plant density of each species. It means the ratio of components used to form mixture is 0.5/0.5. Because recommended optimum plant densities of crops often vary, this produces density of mixture different from pure stand densities of the components. The design has been used in experiments with mixtures of small grain species [68] and in cereal-legume mixtures [35,46,57]. It may be called after Jolliffe [38] a proportional replacement series design or proportional substitutive design with 0.5/0.5 ratio of components (Table 1). If optimum pure stand densities of the two species are equal then mixture density will also be the same, resulted in 0.5/0.5 substitutive design (0.5/0.5 replacement series design). Substitutive design has been used in experiments with mixtures of small grain species and in experiments with mixtures of different cultivars of one species [14,36,72]. Other than 0.5/0.5 proportions of the species in mixture are also used in proportional substitutive design and in substitutive design but sum of the proportions must always equal unity. Different proportions are employed particularly if the objective of a study is to find the most productive proportion of seeded species in mixture [46,48,50]. In proportional substitutive design it results in different plant densities of composed mixtures, while in substitutive design, density of all mixtures is constant. Both designs are suitable to investigate yield advantage of mixtures with two or more components [4,7,14,36]. According to review made by Gibson et al. [24] substitutive design has been frequently used in plant ecology for competition experiments. It has been also used in crop-weed research [19] and to build models of competition between species [31,51].

Table 1. Basic designs used in 2-species mixture experiments


Number of plants of species A and B in pure stands

Composition of mixture of species A and B

Proportional substitutive

a b

m = pa + (1 – p)b


a = b

m = pa + (1 – p)b

Simple additive

a = b or a b

m = a + b


a = b or a b

m = pa + qb; p + q  1

a – number of plants of species A
b – number of plants of species B
m – number of plants in mixture of species A and B
p – proportion of plants from pure stand of species A taken to mixture with B
q – proportion of plants from pure stand of species B taken to mixture with A

Following the idea of similarity of the experiment to the situation in agroecosystem, dominant weed species that emerges in established crop stand in the farmer’s field adds their plants to the plants of the standing crop. This may imply the most appropriate for study of competition will be simple additive design. In the design crop and weed species are grown in pure stands, and mixture is formed by addition pure stand plant densities of both species [44]. In simple additive design only one mixture may be formed with 1:1 ratio of components [52]. The design can be modified if the point of interest is only influence of the weed on crop without inverse effect. Then the weed species may be employed in several plant densities in mixture, but without corresponding pure stands, giving partial additive design [20,24]. This modification of simple additive design is also used when in cereal stand other crop species is interseeded to produce living mulch [2].

Composition and density of plants of two crops in mixture may not comply with additive or substitutive rule producing certain type of design that may be termed unconstrained design. In the design, two crops are grown at the optimum pure stand plant densities and mixture composition is created arbitrary. In other words, sum of proportions of pure stand plant densities of species used to form such mixture is different from unity. The design is used in agricultural practice when one crop in mixture is of the particular interest of a grower. For instance, if cereal species gives high and stable yield in mixture and the main aim is to secure high yield of cereal-legume mixture the design may be used. The cereal is then grown in mixture at full or near full pure stand plant density with a fraction of the legume [11]. The percentage of components in mixtures of cereals and cultivars of pea prepared for seeding may be established based on several plant traits of the components and anticipated interactions between them [47]. It results in different than unity proportion of species taken from pure stands to compose mixtures. Unconstrained design has also been used in crop-weed research [19].

There are two other designs for studies of interactions between plants that gained attention of researchers. They have been termed the response surface designs [18,34]. These are: addition series design, and bivariate factorial design (Fig. 3). Addition series design is substitutive design repeated at different total plant densities. It has been used to study competition in mixtures composed of species of similar optimum plant densities such as small grains [37,39,54,58]. The design is suitable to find optimum seeding rate for mixture or most productive proportion of component species if different ratios of component species are used in mixture. Another kind of addition series design is based on proportional substitutive design for species that differ in optimum plant densities such as barley and pea [27].

Fig. 3. Response surface designs used in experiments with crop mixtures. Open circle – plant density in pure stand. Closed circle – plant density in mixture

Bivariate factorial design is composed of two species that are grown at two or more pure stand plant densities, and of all possible additive mixtures that are formed based on the densities [8]. In its simplest form, it comprises two pure stand densities of plants of each species and four additive mixtures [29,55]. Like addition series design it may be suitable to study competition and to find density of mixture and ratio of components that gives the highest yield advantage of the mixture over sole crops. Because pure stands of species vary independently in the bivariate factorial design, it is more suitable than addition series design for mixtures of crops that distinctly vary in commercial densities such as cereals and legumes [8].


In any experiment in which interspecific competition is being studying information is needed about a response of one species to competition from another species in mixture. To measure this, several indices were developed [69]. A widely accepted and simple index in ecological and agricultural studies is the relative yield (RY). It is the species yield per unit area (biomass or grain) in mixture divided by its yield in pure stand (see explanations under Table 3). It shows yield of a species in mixture in proportion to the pure stand yield of the species. It is important however that research based on different designs, consider different aspects of competition. Keddy [40] and Sackville Hamilton [49] show the problem concerns experiments that are conducted according to substitutive or additive design. In substitutive design for 0.5/0.5 mixture, RY lower than 0.5 for a species means that competition from plants of the component species in mixture is greater than the competition from plants of the same species in pure stand, while RY greater than 0.5 means the opposite. It is essential that substitutive design compares interspecific competition with intraspecific competition. For instance, if 500 plants per m2 of oats and wheat are grown for grain in pure stands then 250 plants of oats that grow in mixture with 250 plants of wheat may perceive plants of wheat as stronger, weaker or equal competitors than its own 250 plants in pure stand. If RYs of oats and wheat equal 0.5 then the species have the same competitive abilities. That means interspecific competition equals intraspecific competition for each species. The result (RY = 0.5 for both species) may be achieved also when only 2 plants·1m-2 of oats and 2 plants·1m-2 of wheat are grown in pure stands and substitutive mixture is created with one plant of each species. It is questionable whether competition or any other interaction between the cereal plants grown at such low density in pure and mixed stand exists. If the species are not competing then the yield is directly proportional to plant number per unit area giving relative yields of 0.5 for each species. Hence, conclusion that species have equal competitive abilities is then false. On the other hand, use of such low plant densities in studies with mixtures of small grains species grown for grain is probably rare. The conclusion about equal competitive abilities between species may also be false when species were seeded at optimum densities but competition is investigated in early stages of plant growth. In the situation plants may be too small to interact with each other.

Relative yields of species were used also to determine response of a species to competition from the other species in proportional substitutive design [35,46]. The design is used only in agricultural research to determine advantage of a crop mixture over sole crops in terms of grain yield thus the species in the experiments are grown at optimum plant densities competing intensely in pure and mixed stands. Because plant density per unit area of each sole crop and density of a mixture vary in the design, it is not fully approved as valid for studying competition [38]. On the other hand, it would be artificial to use substitutive design for mixture experiment with agricultural plant species varying distinctly in optimum plant densities. For instance, one total density, a requirement for substitutive design, used for wheat, field beans and its mixture would differ from optimum plant density of at least one of the species. Comparison between species and mixture to investigate productivity would be then invalid. The rationale behind proportional substitutive design is that it uses maximum grain yield equivalent achieved at optimum plant density of each species but not plant density equivalent (as in substitutive design). Thus mixture in proportional substitutive design is grown at a plant density termed by Connolly et al. [18] as “functional density” in contrast to “demographic density” that is used in substitutive design.

Simple additive design that is now probably most accepted design for competition experiments with wild plants [21,24,33]. It measures interspecific competition irrespective of intraspecific competition and unlike both substitutive designs it detects competition between species for any density of plants and at any growth stage. Thus it is impossible to confound equal competitive abilities of two species with an absence of competition using the design. If there is no competition between species in mixture, yield of each species in additive mixture is the same as in pure stand and RY of each of the species equals unity. If species are in competition and have the same competitive abilities, RYs of the two species equal each other and their values are less than one. Thus, the design is suitable for competition studies even when each species is represented by single plant in pure stand, and mixture contains only two plants.

Comparison between substitutive and simple additive design is presented in Table 2 on the assumption the competing species are identical. The assumption underlines basic differences in what is measured in both designs. The example comprises two scenarios and in each of them the species do not interact, compete with moderate intensity or compete intensely. In the first scenario, interspecific competition is measured at different total plant densities. It is expected that with increasing density intensity of competition between species increases. The second scenario assumes high but constant density of plants of a mixture with competition measured at different stages of plant growth. It is supposed the increase in plant size during growing season should increase competition intensity due to intensified mutual shading and due to overlapping of root systems zones [64]. For both scenarios substitutive design shows the same result at each level of competition intensity. It demonstrates the species have the same competitive abilities what is consistent with the assumption above that species are identical (they RYs are the same) but says nothing about competition intensity as a result of changes in plant density or plant size. In addition it confounds equal competitive abilities of the species with lack of competition when the plant density is too low or growth stage is too early for competition to occur. Simple additive design is more precise. As substitutive design it shows the competitive abilities of the species are equal but for each plant density or growth stage also shows if competition between species exists or not. When a competition study comprises different plant densities or different growth stages simple additive design is suitable to demonstrate if the increase in plant density per unit area or enlargement of plant size intensifies competition. In the example more intense competition is expressed by decrease in value of RY of both species.

Table 2. Relative yield (RY) and relative yield total (RYT) calculated for substitutive and simple additive design on the assumption that the species produce the same yield, have the same competitive abilities and show no resource complementarity in mixture

Scenario 1:
different densities of plants per unit area, full size of plants

Scenario 2:
different growth stage of plants
at high plant density per unit area

Competition intensity

Substitutive design 0.5/0.5

Simple additive design

Very low plant density

very early growth stage

no competition

RYa = RYb = 0.5
RYT = 1

RYa = RYb = 1
RYT = 2

Density of plants not sufficient to maximize yield

plants start to compete due to increase in size

moderate intensity

RYa = RYb
0.5 < (RYa, RYb) < 1
1 < RYT < 2

High density, the law of “constant final yield” operates in pure stands and in mixture

full size of plants, the law of
“constant final yield” operates in pure stands and in mixture


RYa = RYb = 0.5
RYT = 1

RYa – relative yield of species A
RYb – relative yield of species B
RYT – relative yield total

Table 3. Yields and indices in different mixture designs


Yield of species
in pure stand

of design*

Proportion of species from pure stands taken to form mixture

Yield of species
in mixture

Yield of mixture
































































































*sub – substitutive, add – simple additive, unc – unconstrained
RY – relative yield RYa = (Yab/ Yaa)
RYT – relative yield total RYT = RYa + RYb = (Yab/ Yaa) + (Yba/ Ybb)
LER – land equivalent ratio LER = RYa + RYb = (Yab/ Yaa) + (Yba/ Ybb)
A/E – actual/expected yield ratio A/E = M/[(pYaa + qYbb)/(p + q)]
M – yield of mixture
p – proportion of seeding rate from pure stand of species A taken to compose mixture with species B
q – proportion of seeding rate from pure stand of species B taken to compose mixture with species A
Yaa – yield of species A in pure stand
Ybb – yield of species B in pure stand
Yab – yield of species A in mixture with species B
Yba – yield of species B in mixture with species A
RYa – relative yield of species A
RYb – relative yield of species B

The rule of adding plant densities from pure stands to form mixture is probably most promising in experiments on competition between crop and weed. The mixture may exactly reflect situation from agricultural practice, where emerging weed adds its plants to plants of a crop and start to compete for resources. In this type of experiments crop is grown at optimum plant density, while plant density of a weed is usually similar to the weed density observed in agricultural field. One may wish however to increase plant density of crop in order to suppress weed species and to examine if higher plant density prevents the crop from significant decrease in grain yield. Thus any manipulation of density of plants in pure stands usually changes the effect of the species on each other in additive mixture. Problem arises, when simple additive design is to be used for measuring competition between two crops. For a crop mixture of such species as cereals and legumes it is difficult to apply the same logic as for crop-weed mixture, if favoring any of two species is not intended. Choosing high plant density of one component in pure stand and low density of the second component, enhances competitive ability of the former [29,55]. For instance, if one crop is grown at optimum plant density and the second is represented by only one plant per m2 in pure stand, relative decrease in yield in additive mixture will be greater for the second species than for the first. For mixture of two small grain cereals the most neutral will be then using the same plant density in pure stands. Ratio of pure stand densities of other crops destined to form additive mixture should correspond with the ratio of their optimum plant densities recommended for agricultural practice. The species should be grown in pure stands below its optimum plant density, otherwise mixture will be overcrowded and prone to lodging.

The second difficulty with additive design is that each species responds not only to the identity of other species, represented by a set of characteristics of its plants, but also to increased total density of plants in mixture. It is widely recognized the yield components, such as productive tillering or number and weight of grain per inflorescence reduces with increasing plant density. When analyzing the traits in additive mixture a decrease in their values are expected because of the denisity-dependence phenomenon. This poses a problem when comparing the results with those from substitutive design, in which species in pure stands and in mixture are grown at the same total plant density. In substitutive design, if intraspecific competition equals interspecific competition for a species, its plant traits are not expected to change in mixture.

More problematic is the investigation of interspecific competition in unconstrained design. Apart from differences in plant densities between pure stands and mixture (as in proportional substitutive design) the sum of the proportions of pure stand plant densities used to form mixture do not equal unity. The response of a species to competition from another species in the mixture depends then on too many variables to make valid inferences about interspecific competition when using the design in the simplest form.

The basic designs discussed above have one disadvantage. They employ one set of plant densities in pure stands to form one mixture with usually one proportion of components, that limits considerably interpretation of results [49]. They are however rarely used in this simplest form. Usually an additional treatment is applied in such experiments that allows examining then changes in competitiveness of the mixture components with a treatment change. In the experiment of Sobkowicz [53] triticale and vetch were grown in pure stands and mixture was formed according to unconstrained design. Different planting arrangements of species in mixture, were employed that permitted to observe changes in the balance between competitive and facilitative effects of the species on each other. Paolini et al. [44] using simple additive design examined changes in competition between sugar beet and two weed species affected by different time of N fertilizer application.

Response surface designs such as addition series and bivariate factorial design examine interspecific competition at a range of plant densities that makes the analysis more comprehensive. Inferences derived from such studies are not confined then to one total plant density as in substitutive design or one proportion of species in mixture as in simple additive design. In addition series design it is impossible to confuse equal competitive abilities of component species with an absence of competition if the range of used densities is sufficiently wide. For a constant ratio of seeded components in mixture, addition series design shows changes in competition abilities of species with changes in total plant density of a mixture [17]. In the design relative yields are calculated for each density series as in substitutive design [54]. In bivariate factorial design the indices are calculated for each combination of species in mixture as in simple additive design [8]. Although the designs seem to be most accurate to study both: interspecific competition and yield advantage of mixtures over pure stands, problems may arise when additional experimental factor e.g. fertilization is to be implemented. The levels of the factor multiply by pure stand and mixture treatments enlarging size of such experiment. This is probably the cause the designs are not frequently used in agricultural studies.

Yield advantage of mixture
In general, there are no strict rules according to which mixture should be composed if mixture productivity is concerned because various objectives are set for mixture growing. For example, one may be interested in composition of a mixture that produces almost full pure stand yield of only one crop, with additional yield of the second species. According to Willey [73] the situation is typical for agriculture in India where the main crop in mixture is cereal grown for food. Composing mixture in that manner is also reasonable when living mulch is interseeded into a main crop. In both cases, the most important comparison is between yield of the main crop in mixture and its yield in pure stand. In many experiments however no of component species in mixture is favored. It is rather expected that combination of component species in mixture will be more productive than the species grown as sole crops. Here comparison is more difficult because yield of two-species mixture have to be compared with yields of both component species grown alone. For this purpose, mixture is usually composed according to proportional substitutive, substitutive or unconstrained design.

The advantage of mixture may come from resource complementarity, facilitation or from both phenomena. To measure resource complementarity between species relative yield total (RYT) is frequently used in substitutive design, which is the sum of relative yields of the species (see explanations under Table 3). This is probably the most frequently used index in mixture research and biodiversity studies. The index shows to what extent competing species in mixture differs in resource use for their growth. When they compete for all resources there is no resource complementarity and the value of RYT equals unity meaning no mixture advantage over sole cropping. This is presented in Table 2 for moderate and intense competition situations. When species do not interact because of low plant density or early growing stage, RYT also equals unity but no inference about resource complementarity is possible in these situations. Hence, substitutive design is flawed in this respect because it confounds lack of resource complementarity with lack of interaction between species. On the other hand most if not all research looks for resource complementarity in agricultural mixtures grown at optimum plant densities and based on final yields of grain or biomass and then the weakness of the design has no meaning. When partial complementarity in resource use by the species occurs then RYT is greater than unity in substitutive design. Although the index was intended to measure resource complementarity, its value may be biased due to other interactions between species, particularly facilitation [38,41,54,67]. It is important to note that recently RYT has been used in this collective meaning as index to measure complementarity and facilitatation because both phenomena may increase its value [22].

For substitutive design, it is thought the advantage of a mixture over sole cropping is unambiguously proved if mixture yield is significantly greater than yield of each of its component species grown as sole crops. This is termed “transgresssive overyielding” by Trenbath [65]. According to Fridley [22] the probability that mixture “transgressively overyields” is higher if the species produce similar yields in pure stands, than if their yields in pure stands strongly differ. He shows a hypothetical example of the situation in which for the same value of RYT, mixture may or may not “transgressively overyield” only due to the yield differences between sole crops. Similar example is presented in Table 3 for case 1 and 3. In both situations there is the same resource complementarity (RYT = 1.10), while only in case 1 mixture “transgressively overyields”.

In most cases yield of mixture of two crops grown at optimum plant densities lies between yields of its pure stand components and a proper assessment of mixture productivity is not simple. If yields of species in mixture are unknown, one method to evaluate productivity of such mixture is to divide its yield by weighted mean of pure stand yields of mixture components. Weighted mean is calculated according to proportion of pure stand seeding rates of component species used for seeded mixture. The ratio was called the “actual” to “expected” yield of mixture (A/E) by Jokinen [36] (see explanations under Table 3). If the ratio is greater than unity the result is sometimes interpreted as a mixture advantage over sole cropping. Indeed, for a farmer mixture is then a safer choice than probability of choice the lower yielding sole crop for growing. Nevertheless, lack of information on yields of species in mixture limits deeper interpretation of such result. If the information is available then A/E greater than unity may indicate competitive dominance of more productive pure stand component species in mixture but without yield advantage. This is demonstrated in Table 3 for case 2. The dominant species performs relatively better in mixture than in pure stand while subordinate species performs proportionally worse. In substitutive designs RYT equals then unity, thus showing no complementarity in resource use by component species and no mixture advantage over sole cropping. Hector [30] calls the situation as a “positive relationship between diversity and productivity”, because higher yielding species in pure stand dominates the mixture.

Yield advantage of a mixture in terms of RYT does not necessary mean the mixture yield exceeds pure stand average yield. This is presented in case 3, when RYT equals 1.10 but the A/E equals 1. Mixture yield is then lower than mixture yield in case 2 in which there was no complementarity in resource use between species (RYT = 1 and A/E = 1.04). If a farmer is interested only in yield of mixture regardless of species proportion in the yield, he will choose probably higher yielding mixture (case 2).

Presented interpretation is also true for proportional substitutive design. Relative yields and relative yield total were used also for the design [35,46].

Simple additive design is in most cases invalid for investigation of yield advantage of mixtures, although relative yield total, a measure of resource complementarity may also be used in the design [52]. As it is shown in Table 2 values of RYT in additive design vary along with different competitive situations that contradict assumption on species similarity (thus lack of complementarity in use of resources) in the example. This is because using the design it is difficult to distinguish resource complementarity from “incomplete use of resources” by the species [63]. If species in pure stands are grown at too low plant densities to achieve maximum yields they are not able to utilize available resources completely. Thus yield of additive mixture has to be higher at least because of adding plant densities of pure stands. To measure “real” resource complementarity in simple additive design, sole crops should be grown at the densities assuring constant final yield, when yields are independent of plant density. The two situations may be illustrated by hypothetical examples, assuming the crops taken to additive mixture have identical competitive abilities, show no complementarity in resource use and both are grown at the same plant densities in pure stands, as it is shown in case 4 and 5 in Table 3. It was also assumed in the example, that maximum attainable yield (constant final yield) is 6 t·ha-1. In case 4 the species in pure stands are grown at plant densities that are too low for yield maximization, while in case 5 the plant densities in pure stands are sufficient to achieve maximum yield. In the example only relative yield total in case 5 shows true, namely no resource complementarity that is consistent with the assumption about similarity of the species.

When planning an experiment according to simple additive design in order to detect resource complementarity, it is rather impossible to assume, that chosen plant densities for pure stands will assure maximum yield. Thus in any experiment, if the densities are lower, the meaning of RYT in simple additive design is ambiguous [49]. If species differ in resource use in a mixture experiment then RYT may show then collective result, namely, lack of full resource utilization in pure stands and “real” resource complementarity between species in mixture.

The structure of simple additive design in which pure stand seeding rates or pure plant densities are added to form mixture limits its use as a method for searching yield advantage of mixtures. Using the design to measure mixture advantage over sole crops in agricultural research is then restricted probably to experiments in which such species as clover is used as a living mulch in cereal stand. The core of the method of crop growing is to maintain full yield of a main crop (cereal) and to keep a ground cover at the bottom of canopy of the cereal. Composing the mixture according to additive rule seems to be logic because cereal plants are expected to exert a strong competititve pressure on plants of living mulch. In such research, density of plants of the cereal is usually the same as recommended for sole cropping [2]. The problem with assessing yield advantage of a mixture in simple additive design may be of secondary importance in crop-weed studies, in which only competition process is investigated.

Searching for methods to assess yield advantage of a mixture over pure stands resulted in developing another popular index – the land equivalent ratio (LER) [43,73]. The equation to calculate the index is the same as for RYT (see explanations under Table 3), but relative yields of species are often termed partial LERs or PLERs [4,8]. While RYT is being used in ecological and agricultural research, land equivalent ratio is applied only in mixture experiments with arable crops. For LER no assumption concerning rule according to which the mixture should be composed is necessary, hence LER may be used in each experimental design. It is frequently used in proportional substitutive design instead of RYT [4,56]. Only LER but not RYT can be used in unconstrained design [11,12,53]. The index was also used in additive design [13]. Basic difference between RYT and LER lies in interpretation, and the latter index should be more conservatively interpreted than RYT [38]. Land equivalent ratio does not try to indicate underlying causes for yield advantage of a mixture over pure stands such as resource complementarity or facilitation. It is defined as the combined area of land under two sole crops that gives the same production of each component as one unit of land under mixture of the components. LER greater than unity means simply that more land is needed for sole crops than for mixture. For example, in Table 3, RYT may be replaced by LER for cases 1-5 that changes interpretation of the results. Then for case 3, to produce 1.8 t of grain of species A and 3.2 t of grain of species B by 1 ha of mixture, the areas needed in pure stands are: 0.3 for species A and 0.8 ha for species B. This gives total 1.1 ha of land under pure stands and indicates the yield advantage of the mixture. Higher than unity LER value is due to high yield of species B in mixture. It should be noted however that in case 3, A/E ratio is 1 showing the mixture yield equals sole crop average.

If facilitative effect of one species on performance of the second species in mixture is strong, the value of LER may be extremely high. This was reported by Sobkowicz [53] for the experiment with triticale-vetch mixture and the result was due to high relative grain yield of vetch. The species produced very low yield of grain in pure stand because it lodged and its pods partially decayed due to frequent rainfalls before harvest. In mixture, vetch yielded more grain than in pure stand being protected from lodging by triticale. Mean LER for all mixtures in the experiment was 2.56, although no mixture transgressively overyielded and two mixtures with LER > 2, yielded significantly less grain than the pure stand triticale. Two examples from that experiment are presented in Table 3, as cases 6 and 7, which differ in initial ratio of plants of triticale in seeded mixture. The cereal was far more productive species than vetch and yield of mixture depended mainly on yield of the cereal. Reduced ratio of triticale plants from 0.60 (case 6) to 0.30 (case 7) in seeded mixture lowered mixture yield but value of LER was increased due to increased yield of vetch in the mixture. The experiment supports observations of Mead and Willey [43] that the low pure stand yields may contribute more to increase LER value than high yield of mixture.

The example shows that function and interpretation of LER in an experiment is similar to that of RYT. Simply, LER accumulates the same phenomena like resource complementarity and facilitation despite of design of a mixture experiment. Indeed, from the time LER was originated interpretation of the index has been evolving into that applied for RYT.

More complex structures represented by addition series design or bivariate factorial design make comparison between yields of mixtures and pure stands more reliable. Both designs may use such a range of plant densities in their structure that allow to find optimum plant densities for each pure stand species and each mixture in a given experiment. For any simple design containing only two sole crops and one mixture (e.g. 0.5/0.5 substitutive design) optimum plant density of mixture may be different than density of mixture imposed by the rules of the design. Spitters [59] suggests that if mixture is better able to use common limiting resources, its optimum plant density may be higher.

In each experiment conducted according to addition series or bivariate factorial design both indices, RYT and LER may be used and the function of each index is then different. Relative yield total with relative yields are calculated for each mixture with its corresponding pure stands to evaluate competition and complementarity, while LER assesses mixture advantage over sole cropping. In denominators of relative yields of species in LER equation, maximum pure stand yields of the species found in the experiment should be used. Hence in an experiment conducted according to the designs, for each mixture, denominators in LER equation are constant [8,58]. If the experiment shows there is no change in pure stand yield with increasing density of plants (or seeding rates) of a species then yield from the lowest density should be applied for LER equation.


  1. Mixture of arable crops is a system in which species interact with each other in various ways but separation and quantification of these interactions is difficult. To date, most experiments have shown a collective effect of competitive and positive interactions between crops on mixture performance.

  2. Response of a species to presence of other species in mixture can be investigated according to different experimental designs but choice of a design should be dependent on research objectives. Substitutive and proportional substitutive design are adequate to study competition and yield advantage of a mixture over sole crops when experiments are conducted at standard plant densities of crops and when experimental data come from grain or final biomass yields. Using these designs for studying competition at low plant densities or at early phases of plant growth, when there is uncertainty if competition operates in mixture is questionable. For this purpose, a simple additive design is suitable, however it does not answers the same questions on competition as both substitutive designs. Simple additive design poses problem with choice of plant density of crops in pure stands as reference points. If resource complementarity and yield advantage of a mixture over sole crops are to be studied use of simple additive design is severely limited. Most of the drawbacks are meaningless if the design is employed to study competition between crop and weed, being probably the most suitable design for this kind of research. Interpretation of results should be most cautious in experiments conducted according to an unconstrained design, when crop densities vary independently in pure stands and in mixture.

  3. Assessing the advantage of a mixture over sole cropping is difficult if one wants more than just to compare yields of mixture with yields of pure stands. Even if only this is required, it is not simple when yield of mixture lies between yields of sole crops. Such popular index as A/E ratio may be useful but it does not show if the mixture is beneficial in terms of resource use by component species or due to facilitation. On the contrary, even if both phenomena occur in mixture causing an increase in value of such indices as RYT or LER, it does not mean the mixture yield is higher than sole crop average. In other words, there is no single measure that definitely shows superiority of mixture. To answer question, if mixture is better choice than sole cropping other criteria may help, for example, a comparison of feeding or monetary value of mixture yield with value of yields of its components grown in pure stands.


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The research was supported by project: “The Second Scholarship Programme for PhD Students of Wroc³aw University of Environmental and Life Sciences” financed by European Union from European Social Fund and by state funds in the The Integrated Regional Operational Programme.


Accepted for print: 4.06.2007

Piotr Sobkowicz
Department of Agroecosystems and Green Areas Management, Wroc³aw University of Environmental and Life Sciences, Poland
pl. Grunwaldzki 24a
50-363 Wroc³aw
email: piotr.sobkowicz@up.wroc.pl

Magdalena Podgórska-Lesiak
Department of Soil Management and Plant Cultivation,
Wroc³aw University of Environmental and Life Sciences, Poland
pl. Grunwaldzki 24a, 50-363 Wroc³aw, Poland

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