Introduction
In a previous paper (1), published here on this same website in 2018, we used the work of Thornley (1990) as a milestone in the discussion about initial exponential plant growth, because:
- It was widely quoted and seemed highly appreciated.
- It had a thorough discussion of most functions that describe plant growth
- It had a statement that Hunt (1982) has proven that a plant grows exponentially in the very beginning (plants are known to lose later this ability, changing to linear growth until the growth decreases almost to zero. A sigmoid curve emerges for the entire growth)
We already questioned an exponential growth in the beginning in the mentioned paper. We showed with a lot of other data (2, 3, 4, 5), the first dating back to 1981, that this is not the case. Actually, the reason for this website is an unsuccessful journey through the institutes with this insight since 1981.
The discussion is still actual after 1990. In a new edition of Thornley (2000) the statement that Hunt (1982) showed plants to grow exponentially in early growth, is still there. In 1999 Birch put forward a new sigmoid equation, “unique because it always tends towards exponential growth at small sizes or low densities, unlike the Richards equation, which only has this characteristic in part of its range”. So, he was still expecting initial exponential growth. In 2008 Damgaard uses this function with this argumentation.
A second milestone in the (initial exponential) plant growth
The work of Paine et al. (2011), only recently discovered by the author, can be regarded as a second milestone because:
- It discusses again most functions describing growth
- It makes a statement against exponential growth and dismisses the models assuming this.
That would be a support for the mission of this website, but there is an hesitation in the article: It states “In the unlikely event that neither environmental nor physiological factors slow the of biomass acquisition, the exponential model may be appropriate (for example, in the initial portion of a plant’s lifespan)”. The earlier mentioned data require a more firm statement: plants grow NOT exponentially right from day one. Plants show second order polynomial growth.
It is important because it will lead to more fundamental insight in plant growth and especially how a plant is regulating its growth on a higher level, e.g. how a plant is balancing its growth machinery and non-growth machinery under constant conditions. We showed that a simple model, based on well-known equations (6), can produce second order polynomial growth. The non-growth machinery was given the function of competition (avoidance) ability and a simple assertion about the balance between this competition (avoidance) ability machinery and the growth machinery did the rest. Next, we showed (7, 8, 9) in simulation experiments, as simple as possible, that growing according this assertion can have an evolutionary advantage.
The work of Gachocki et al. (2022) was, on the first sight, a candidate for a third milestone: They compare again a lot of functions with focus on non-linear functions, even non-parametric curve-fitting. But it says nothing about initial exponential growth. So, the question vanishes in 2022 without a firm answer. This answer is, once again, important as a starting point in further reasoning, see above.
Notice, that the search for control parameters on a higher level is normal, see the ideas of ontogeny of Hunt (1982), or architecture (e.g. Fourcaud et al., 2008 with emphasis on source/sink relations). It avoids the obligation to describe plant growth in chemical reactions, physical parameters, etc., which require a lot of detail knowledge, and give insights on higher abstraction (generalization) levels.
Towards the next milestone: linearity versus non-linearity
The question is how to proceed. Non-linearity will put forward as the next thing to do in analysis. It is made big by the success in analysing thermodynamical far from equilibrium systems and made possible by huge advances in mathematics and computer technology. See for use in plant growth analysis again the work of Paine et al. (2011). But non-linearity has a well-known disadvantage, as non-linear relations are more complex and difficult to explain than linear.
So, do not focus too soon on non-linearity. It is not necessary, as the experiments and reasoning mentioned above showed a lot of linearities, see the following paper:
The use of linearity and non-linearity in plant growth analysis.
References
Birch, C.P.D., 1999. A new generalized logistic sigmoid growth equation compared with the Richards growth equation. Annals of Botany 83, 713.
Blackman, V.H. 1919. The compound interest law and plant growth. Annals of Botany, os-33, 353–360.
Damgaard, C. and J. Weiner, 2008. Modelling the growth of individuals in crowded plant populations. Journal of Plant Ecology, Volume 1, Issue 2, Pages 111–116
Fourcaud, T.; Zhang, X.: Stokes, A.; Lambers, H..; Körner, C., 2008. Plant Growth Modelling and Applications: The Increasing Importance of Plant Architecture in Growth Models. Annals of Botany, Volume 101, Issue 8, Pages 1053–1063,
Gachoki, P.; Muraya, M.; Njoroge, G., 2022. Modelling Plant Growth Based on Gompertz, Logistic Curve, Extreme Gradient Boosting and Light Gradient Boosting Models Using High Dimensional Image Derived Maize (Zea mays L.) Phenomic Data. American Journal of Applied Mathematics and Statistics. 2022, 10(2), 52-64.
Hunt, R., 1982. Plant Growth Curves: The Functional Approach to Plant Growth Analysis. Edward Arnold, London.
Paine, C.E.T.; Marthews, T.R.; Vogt, D.R.; Purves, D.; Rees, M.; Hector, A.; Turnbull, L.A., 2011. How to fit nonlinear plant growth models and calculate growth rates: an update for ecologists. Methods in Ecology and Evolution. Volume 3, Issue 2 p. 245-256
Thornley, J.H.M. and I.B. Johnson, 1990. Plant and Crop Modelling: A mathematical Approach to Plant and Crop Physiology. Clarendon, Oxford.
Thornley, J.H.M. and I.B. Johnson, 2000. Plant and Crop Modelling: A mathematical Approach to Plant and Crop Physiology. The Blackburn Press, Caldwell, New Jersey.