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The use of mathematical modelling in allelopathy

Min An

Environmental and Analytical Laboratories and EH Graham Centre for Agricultural Innovation, Charles Sturt University, Wagga Wagga, NSW 2678, Australia. Email


Of the disciplines involved in allelopathy research, mathematical modelling is making increasingly significant contributions. This article discusses the important roles that mathematical modelling has played in allelopathic research, specifically the dose-response phenomenon in allelopathic research and its applications in characterizing allelochemicals from living plants, in residue allelopathy and in data analysis. The contributions that mathematical modelling can make to this discipline are further illustrated.

Media summary

Mathematical modelling can increase our understanding of allelopathy, highlight directions for future research and provide a theoretical framework and insights into the mechanisms of allelopathy phenomena.

Key Words

Mathematical modelling, hormesis, allelopathy, allelochemical(s), plant defence


During the last two decades, the science of allelopathy has attracted a great number of scientists from diverse fields world wide and is now viewed from a multifaceted approach (Putnam and Tang 1986; Chou et al. 1999; Reigosa and Pedrol 2002). This diverse interest has been greatly driven by the prospects that allelopathy holds for meeting increased demands for sustainability in agriculture and quality food production for humans, on reducing environmental damage and health hazards from chemical inputs, minimizing soil erosion, reducing reliance on synthetic herbicides and finding alternatives for their replacement.

Of the disciplines involved in allelopathy research, mathematical modelling is making increasingly significant contributions. Such theoretical contributions range from separating allelopathy from competition (Weidenhamer et al. 1989), characterizing allelopathy and its ecological roles (Goslee et al. 2001), elucidating fundamentals of allelopathy (An et al. 1993), simulating specific cases, ie. plant residue allelopathy (An et al. 1996), to the modelling of effects by external factors, such as density of target plants (Weidenhamer et al. 1989; Sinkkonen 2001).

This article, largely based on our previous modelling work, discusses fundamental issues associated with the dose-response phenomenon in allelopathic research and the important roles that mathematical modelling played, to review the latest developments in this area, and to further illustrate the above- mentioned contributions that mathematical modelling can make to this discipline.

Modelling dose-response relationship (hormesis) in allelopathy

The characteristic responses of an organism to an allelochemical, i.e. stimulation or attraction at low concentrations of allelochemicals and inhibition or repellence as the concentration increases have been well recognised in studies of allelopathy (Figure 1). These phenomena have been widely observed in allelochemicals from living plants, in allelopathic effects from decaying plant residues, and from the gross morphological level to the biochemical level. They have also been widely recognized in other growth-regulating chemicals, including herbicides and even medicines (Calabrese and Baldwin 2003). However, we are still facing the challenge of interpreting such phenomena in allelopathy and their significance is yet to be fully explored.

Being inspired by Chinese Yin/Yang theory, an analogy is employed to interpret such allelopathic manifestations. It is hypothesized that the characteristic biological response to allelochemicals is a result of the character of the allelochemicals themselves. An allelochemical is assumed to have two complementary attributes: stimulation and inhibition. These attributes act in a way that is antagonistic to each other as well as coexisting within the unity of an allelochemical. As a unity of opposites, stimulation and inhibition, although separate attributes, are really part of the same ultimate unity, i.e. stimulation implies inhibition, inhibition embodies stimulation, and each is, therefore, necessary to the other. As concentration changes for a given receptor organism, the relative dominance of stimulation and inhibition by the allelochemical alters. It is such alternation that determines the overall property of the allelochemical. This can only be shown through the biological responses when an allelochemical acts on an organism, and is referred to as the biological property of the allelochemical (as opposed to chemical or physical properties). Therefore the observed allelopathic phenomenon shall depend on the specific allelochemical, the given organism, the environmental conditions in which allelochemicals are produced and released, and in which the organism is grown, etc.

Figure 1. Biological response of an organism to allelochemicals

This hypothesis is mathematically expressed by the following model:

P = 100 + SI

where P represents the biological response to an allelochemical, S and I are biological responses to the stimulatory and inhibitory attributes of the allelochemical respectively, and are expressed in the model by enzyme kinetics (An et al. 1993).

The illustration of the model is shown in Figures 2-5. It can be seen from the illustrations that the model simulates well the biological response to allelochemicals and is in a good agreement with a wide range of experimental data taken from the literature (An et al. 1993). The model provides a useful platform for analysing experimental data, predicting allelopathic effects in practice, and for further theoretically exploring the fundamentals of allelopathy matters, which are discussed in the next sections.

Figure 2. The response of linseed in radicle length to benzylamine (An et al. 1993).

Figure 3. Effect of castanospermine on root growth of lettuce (An et al. 1993).

Figure 4. The response of wild oats (A. ludoviciana) in total number emerged to wheat straw leachate (An et al. 1993).

Figure 5. The response of α-amylase activity to scopolamine (An et al. 1993).

Modelling dynamics of allelochemicals from living plants in the environment

Defence agents, allelochemicals or allelopathins, play an important role in allelopathic interactions or plant defence and act as important ecological mechanisms (Rice 1984). The allelopathic characteristic of an allelochemical is defined as a biological property of the allelochemical, as opposed to its physical or chemical properties (An et al. 1993). The content of allelochemicals in living plants, and the fate and dynamics of such compounds in the environment are important aspects in the study of allelopathy.

An et al. (2003) developed a mechanistic model, by applying the concept of a diffusion process, which integrated earlier scattered research information with present knowledge. This model assembled a generalized picture of allelochemical production in living plants with the fate of allelochemicals and their dynamics in the environment, and also explored the possible ecological significance of plant allelopathy. Through their modelling work it is proposed that there are two kinds of allelochemical production in a plant. These are dictated by age and plant stress and are reflected by the corresponding dynamics in the environment. Generally, allelochemical content in living plants declines with age after an early initial maximum, and there is a corresponding later fate in the environment, while periodic production may occasionally be a special case (An et al. 2003). By combining this model with the above-mentioned dose-response model, these authors demonstrated the possible existence of corresponding periodic dynamics in the environment; successfully simulated the response fluctuation of receiver plants to allelochemicals; theoretically interpreted allelopathic results as reported in the literature; and attributed such results to the periodic production of allelochemicals. (Figures 6-7). The combined model has helped us to understand why the results from allelopathic research are disparate when conducted at different stages of plant growth and development, and why results fluctuate as experiments proceed. The model has also helped to reduce confusion on allelopathy and suggest directions for future research.

Figure 6. Simulated fluctuation of responses of receiver plants to periodic dynamics of allelochemicals in the environment, which is described by the combination of allelochemical – biological response model (An et al. 1993) and the model (An et al. 2003). P is response of test plants to allelochemicals, % control; t is time course of donor plant growth, arbitrary unit (An et al. 2003).

Figure 7. Simulated periodic production of an allelochemical under constant stress. Ap is allelochemical concentration in plant; t is time course of plant growth, both in arbitrary units. X1, X2, X3 are concentrations at an equilibrium (An et al. 2003).

Modelling Allelopathy caused by plant residues during their decomposition

Plant residues undergoing decomposition may have allelopathic effects on the growth and yield of subsequent plants. Such effects have been attributed to the phytotoxic chemicals leached from the residues together with toxins produced by micro-organisms during residue decomposition. The potential phytotoxicity is dependent on numerous factors that together govern the rate of residue decomposition, the net rate of active allelochemical production and the subsequent degrees of phytotoxicity (An et al. 2002).

A mechanistic model was constructed to simulate such allelopathic phenomena (An et al. 1996), and later further developed by including intrinsic and extrinsic factors to examine these phenomena under much wider conditions (An et al. 2002). This model, combined with the dose-response model, formed a new model, which provides an integrated view of the allelopathic pattern of plant residues during decomposition, in terms of both the response of a receiver plant and allelochemical dynamics in the environment (Figure 8). The new model indicates two aspects to phytotoxicity: stimulation and inhibition. The extent of each over the whole course of residue decomposition is not balanced. The most severe inhibition occurs at the early stages of residue decomposition. Phytotoxicity proceeds from stimulation to inhibition at this stage and reaches its maximum inhibition soon after decomposition starts. Compared with the whole decomposition course, this stage is relatively short. At later stages of residue decomposition, the inhibition due to phytotoxicity is declining while stimulation gradually emerges. In general, over the whole course of residue decomposition in terms of agronomic practice, inhibition dominates residue phytotoxicity, which declines with increasing decomposition time.

Figure 8. Simulating dynamics of residue phytotoxicity. P is phytotoxicity and t is decomposition time, both in arbitrary units (An et al. 1996).

Such modelling analysis has great application potential in the management of plant residues for weed control and in overcoming the negative effects of plant residues to meet the increased demand for conservation and no-tillage farming systems by means of managing the inhibition and stimulation periods. For example, with knowledge of residue decay dynamics, and by analysing risk situations with respect to residue retention, a crop manager may avoid the inhibitory period of decaying residues, thus minimising their negative effects and await the stimulatory effect to crop plants, thereby enhancing the benefits of residue retention on the soil. By extending the inhibitory period of decaying residues and enhancing its effects, weeds may be controlled (An et al. 1998). A successful example in applying this principle in practice is presented in this Proceedings by An et al. (2005a).

Modelling density-dependent chemical interference

Sinkkonen (2001) further developed the above dose-response model by combining it with the density-dependent graphic model of Weidenhamer et al. (1989) to form a new model to describe plant response to change in phytochemical concentration in those cases of density-dependent phytotoxicity. According to this extension model, direct chemical interference is density-dependent. With increasing target plant density, the effects of phytochemicals are diluted. As a result, inhibition is the most probable outcome in density-dependent phytochemical interactions at low target plant densities, but phytotoxic effects often become stimulatory as target plant density increases (Sinkkonen 2001). The author also claimed that his extension of the dose-response model by An et al. (1993) is useful when estimating whether the response of plants to direct chemical interference can be distinguished from pure competition (Sinkkonen 2001), an important but often-argued issue in allelopathy research. The same author concluded that, based on the examples presented in his paper, the dose- response model by An et al. (1993) is suited for modelling plant responses to density-dependent chemical interference. He also made suggestions to further modify the model equations in some cases. These include the cases where an environmental factor, such as adsorption of phytochemicals by soil particles or degradation of phytochemicals by soil micro-organisms, alters the model in predictable way; and to add a density-dependent stress factor to the model, which would change the power of stimulation and inhibitory attributes at different plant densities. However, Sinkkonen also cautioned that every modification must be based on empirical data showing a distinctive pattern that can be modelled (Sinkkonen 2001).

Later, by combining the model by An et al. (1996), Sinkkonen (2003) extended the density-dependent model (Sinkkonen 2001) to describe residue allelopathy at different densities of growing plants. While the original residue allelopathy model predicts inhibitory effects in most cases, the new density-dependent extension of the residue allelopathy model predicts that the density of target plants determines whether or not inhibition occurs. According to the new model, the intensity of inhibition decreases and the final stimulatory period begins earlier if target plant density increases (Sinkkonen 2003). The author claimed that combining the effects of density-dependency with the residue allelopathy model enhances our understanding of chemical interference. In addition, the new model may partially explain why several field studies have not observed chemically driven inhibitory effects similar to those observed in laboratory experiments (Sinkkonen 2003).

Whole-range data assessment and inhibition index

Based on the typical biological responses of an organism to allelochemicals, An et al. (2005b) developed concepts of whole-range assessment and inhibition index for better analysing allelopathic data. The method is concise and comprehensive, and makes data grouping and multiple comparisons simple, logical, and possible. It improves data interpretation, enhances research outcomes, and is a statistically efficient summary of the plant response profiles. See An et al. (2005b) in this Proceedings for the details.

Application of nonlinear regression models in allelopathy

Backed by an ancient statement that ‘the poison is in the dose’ by physician Paracelsus some 500 years ago, Belz et al. (2005) experimentally reinforced the recognition of dose-response relationships in allelopathy. Their efforts were not just limited to the confirmation of such phenomena, but also to the exploring of allelopathy’s fundamentals and practical utilization, aided by the mathematical modelling of hormesis.

They transferred the methodology of dose-response experiments in weed science and data analysis using log-logistic model and other non-linear regression methods on curve parallelism, ED50, and curve slopes into the allelopathy field. They screened over one hundred wheat cultivars against a test species for their allelopathic potential, compared the responses by synthetic allelochemicals, and verified the dose-responses by density-dependent phytotoxicity of allelochemicals produced and released by living plants. They concluded that dose-response studies as used in bioassays in other biological sciences, are an appropriate method for analysing allelopathic interactions between living plants. The four-parameter log-logistic model (or its peaked expansion) adequately described most of the observed dose-response patterns and provided a valuable tool for various approaches and comparative studies in allelopathy. The potential application of their research is that it can be used to identify the primary cause of observed allelopathic interactions, to point to the mode of action of allelochemicals, and to preselect cultivars with allelopathic traits based upon allelochemicals with a different mode of action.

Interpretation of the function of allelopathy from a mathematical modeling perspective

One important concept, derived from the operation of the above dose-response model, is that a plant always contains a certain amount of allelochemical whether under stress or not.

Under normal conditions, allelochemicals in a plant may be inactive and concentrations may be relatively stable. However, as environment conditions become stressful for plant growth, the allelochemical content dramatically increases. Stressful conditions include abnormal radiation, mineral deficiencies, water deficits, temperature extremes, and attack by pathogens and predators.

According to Cruickshank and Perrin (1964), a similar conclusion was proposed by Muller and Borger in 1939, i.e. 'phytoalexin theory' of disease resistance, which proposed that phytoalexins are metabolites, which only form or become active when a parasite comes in contact with the host cells.

Plants have evolved means of adjusting the environment in their favour. It is well known that plants have developed physical means, such as the cuticle and trichomes, for their defence. Recently, it has been recognized that allelochemicals may also be employed in the defence systems of plants (Lovett and Ryuntyu 1992; Bais et al. 2003). Plants may defend themselves by means of such chemicals in several ways. Phenolics, particularly flavonoids, are considered to protect plants from UV radiation (McClure 1975). Under stressful conditions, such as drought or insufficient nutrients, allelochemicals may inhibit the growth of other plants and favour the producer (Kuo et al. 1989). Allelochemicals, such as phenolics or their oxidation products, may affect the digestive enzymes of insects or fungal enzymes (Woodhead 1981). Also, allelochemicals may cause the rapid death of a few cells in plants under disease attack, confine the pathogen to a restricted area and thus minimize the damage by the pathogen. Recently by integrating ecological, physiological, biochemical, cellular, and genomic approaches Bais et al. (2003) demonstrated that Centaurea maculosa (spotted knapweed), an invasive species in the western United States, displaces native plant species by exuding the phototoxin (-)-catechin from its roots, which in susceptible species triggers a wave of reactive oxygen species (ROS) initiated at the root meristem that leads to a Ca2+ signalling cascade triggering genome-wide changes in gene expression and, ultimately, death of the root system.

Allelopathy may act as a defensive system in plants (Lovett and Ryuntyu 1992). Visible allelopathic effects or increase of allelochemical contents in plants may be the results of operation of this system under stress. Its main purpose is to protect plants from stress and to keep an ideal or normal growth environment for plants. While under ideal conditions there are no allelopathic effects occurring, the allelochemicals are inactive and the plant content is stable. It is known that plants produce numerous allelochemicals, each of which (or a combination) may have different functions against different stress factors. Stress as referred to here, has a broad definition, which includes those external constraints, such as water deficits, mineral deficiencies, temperature extremes, abnormal radiation, herbivores feeding and disease any of which disturb normal plant growth and reduce the rate of dry matter production.

Allelopathy may have two functions in a plant: phytotoxicity and autotoxicity. Within the range of its capability to overcome stress, allelopathy is inhibitory to others except a producer. Under severe stress, which is beyond the adjusting capability of the plant, such as massive attack by pathogens or severe mineral deficiencies, allelopathy may act autotoxically to decrease the population of the producer, which may be the best survival strategy for producer plants under unfavourable conditions (Chou 1989). It has been observed that residues from plants grown under stressful conditions are more highly phytotoxic than under normal conditions (Mason-Sedun and Jessop 1989). Chou (1983) also noted that the roots of rice seedlings under water-logged and oxygen-deficient conditions developed abnormally, and the plants tended to produce growth-inhibiting substances, resulting in decrease of yields. When the unfavourable conditions were removed by providing a good drainage system so that phytotoxins could be leached out of the soil, a significant yield increase of up to 40% occurred.

It is well documented that the concentration of secondary plant compounds in plant tissue is determined by the plant's genetic make-up in combination with its interaction with environmental conditions during growth (Bell and Charlwood, 1980; Wu et al., 2003). Therefore, it is not surprising that allelopathic potentials, like other genetic characteristics, vary with and within species, and that this may reflect the extent of the plant's defence capability. For example, levels of phenolic acids in healthy plants of Sorghum bicolor differ considerably among cultivars. Cultivars with the highest normal phenolic levels are the most resistant to insect attack (Woodhead 1981).

Lovett (1982) and Putnam and Tang (1986) indicated that allelopathic characteristics are more likely to occur in crop predecessors or 'wild types' that have evolved in the presence of allelopathic and competitive influences from other species. If allelopathy acts as a defence reaction to stress, then human interference, such as irrigation or the application of fertilizers and pesticides, may help to overcome stress for plants, and hence currently used cultivars have diminished or reduced allelopathic capacity.

Allelochemical contents in plants vary with experimental conditions. Woodhead (1981) reported that laboratory- and field-grown sorghum phenolics follow similar patterns, but that values for all field-grown plants are much higher than for the corresponding laboratory plants. This may be taken to imply that the ideal environment for plant growth is relative, and plants are always under some degree of stress. For example, Dicosmo and Towers (1984) pointed out that in plant cell cultures altered secondary metabolism implies some kind of stress even when conditions seem to be optimum. Even though under no apparent stress, plants may contain a certain amount of allelochemical. The equilibrium point, at which no allelopathic effects occur (i.e. when stimulation and inhibition are equal), is likely to vary with growth conditions. Therefore, it is not surprising that allelochemical concentrations at the equilibrium point of one condition may show allelopathic effect on the same test species under different conditions. This may help to explain the argument that allelopathic effects are observed under conditions of no stress.

Concluding remarks

Practically, scientists are often asked questions such as " what period of time must elapse between the first and second crops in a given region so that inhibition of the second crop does not occur?" and "what rate of the residue from a certain crop should be left on the soil in order to avoid residue phytotoxicity and bring benefits to the soil?"

To answer such questions, scientists have to conduct individual experiments under a set of controlled conditions. However, if the conditions on which experiments are based are changed, the whole procedure has to start again. It is obvious that such an approach is time-consuming, high cost and low accuracy.

The question naturally emerges, "is there any other approach which can overcome these limitations?" Mathematical modelling offers a positive answer. By means of such an approach, scientists can synthesize present information as it is obtained, and provide quantitative predictions for different conditions.

However, the role of mathematical modelling is not limited to that of a prediction tool. From the discussion of the previous sections, it is clear that mathematical modelling works, and, combined with other disciplines, has contributed to increasing our understanding of allelopathy, has helped establish the fundamentals of allelochemical function, has highlighted directions for future research by integrating scattered information, generalising the phenomenon observed in fields and laboratories, and has provided a theoretical framework and insights into the mechanisms of allelopathy phenomena. It can be stated that current mathematical modelling in allelopathy are somewhat rudimentary and more in-depth conceptual treatments are likely to yield greater understanding of allelopathy.


Special thanks to the Organising Committee of 4th World Congress on Allelopathy for the kind invitation of being an invited speaker for this Congress.


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