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. 2003 Jan;1(1):37-50.
doi: 10.1080/15401420390844456.

Mathematical Modeling of Allelopathy. III. A Model for Curve-Fitting Allelochemical Dose Responses

De Li Liu  1 , Min An, Ian R Johnson, John V Lovett
Affiliations

Mathematical Modeling of Allelopathy. III. A Model for Curve-Fitting Allelochemical Dose Responses

De Li Liu et al. Nonlinearity Biol Toxicol Med. 2003 Jan.

Abstract

Bioassay techniques are often used to study the effects of allelochemicals on plant processes, and it is generally observed that the processes are stimulated at low allelochemical concentrations and inhibited as the concentrations increase. A simple empirical model is presented to analyze this type of response. The stimulation-inhibition properties of allelochemical-dose responses can be described by the parameters in the model. The indices, p% reductions, are calculated to assess the allelochemical effects. The model is compared with experimental data for the response of lettuce seedling growth to Centaurepensin, the olfactory response of weevil larvae to alpha-terpineol, and the responses of annual ryegrass (Lolium multiflorum Lam.), creeping red fescue (Festuca rubra L., cv. Ensylva), Kentucky bluegrass (Poa pratensis L., cv. Kenblue), perennial ryegrass (L. perenne L., cv. Manhattan), and Rebel tall fescue (F. arundinacea Schreb) seedling growth to leachates of Rebel and Kentucky 31 tall fescue. The results show that the model gives a good description to observations and can be used to fit a wide range of dose responses. Assessments of the effects of leachates of Rebel and Kentucky 31 tall fescue clearly differentiate the properties of the allelopathic sources and the relative sensitivities of indicators such as the length of root and leaf.

Keywords: allelochemicals; allelopathy; dose-response relationship; inverted U-shape response; mathematical modelling; stimulation.

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Figures

Figure 1.
Figure 1.
A hypothetical allelochemical dose-response curve. Rm is the maximum stimulating peak, Dm is the dose that gives the stimulating peak, D0 is the dose that gives no effect and D50 is the dose that gives 50% reduction of untreated control yield.
Figure 2.
Figure 2.
Details of k = 0, 1, 2, and 3 in the procedures of curve-fittings (the solid lines) of Equation (6) to observations (solid dots) in the responses of the lettuce root length to Centaurepensin (Stevens and Merril 1985).
Figure 3.
Figure 3.
Equation (6) fitted to the data of Selander et al. (1976) with the responses of weevils to α-terpineol; (A): plotting D against R, (B): plotting g(D) against R.
Figure 4.
Figure 4.
Statistics in number of ln(D+1) transformations of the data of Buta and Spaulding (1989).
See this image and copyright information in PMC

References

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    1. An M, Johnson IR, Lovett JV. Mathematical modelling of allelopathy I. Phytotoxicity caused by plant residues during decomposition. Allelopathy J. 1996;3:33–42.
    1. An M, Liu DL, Johnson IR, Lovett JV. Mathematical modelling of allelopathy II. The dynamics of allelochemicals from living plants in the environment. Ecol Model. 2003 (in press)
    1. Behrens R. Quantitative determination of triazine herbicides in soils by bioassay. Residue Rev. 1970;32:355–369. - PubMed
    1. Brain P, Cousens R. An equation to describe dose responses where there is stimulation of growth at low doses. Weed Res. 1989;29:93–96.

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