# Relative growth rates

It happens from time to time, doesn’t it,  that you are digging into what you believe is a specific set of limited methods and suddenly you uncover a whole universe that stretches over a vast number of research fields.  Recently this happened to me again when preparing a review on the concept of relative growth that is related to the important concept of allometry.

The first time I came across this concept was when attending lectures in forest growth and yield given by Prof. Günter Wenk at Dresden University as a visiting student back in 1992. Later in 1995 I re-programmed his forest stand model as a young research assistant and from 2000 to 2008 we developed a close friendship and Prof. Wenk mentored some of my research activities at Bangor University in North Wales.

Relative growth rate (RGR) is simply absolute growth rate divided by the corresponding size variable. Assuming that function $y(t)$ represents the state of a plant characteristic at time $t$, for example the biomass of a plant, instantaneous relative growth rate can be expressed as

$p(t) = \frac{y'(t)}{y(t)} = \frac{\text{d}y}{\text{d}t} \times \frac{1}{y(t)} = \frac{\text{d}}{\text{d}t} \text{log}\; y(t)$.

Since relative growth rate is equivalent to the derivative of $\text{log}\; y(t)$ with respect to time $t$, studying the relative growth of $y(t)$ is equivalent to studying the absolute growth of $\text{log}\; y(t)$ .

In empirical studies, we commonly deal with discrete time, e.g. $t_1, t_2, .., t_n$, which are our scheduled survey days or years. The period between two discrete instances of time can be denoted by $\Delta t = t_k – t_{k-1}$ with $k = 2, .., n$. For simplification we can now set $y(t_k) = y_k$ and $p(t_k) = p_k$.

For empirical data observed at discrete times we can now calculate the mean relative growth rate as

$\overline{p}_k = \frac{\text{log}\; y_k – \text{log}\; y_{k – 1}}{t_k – t_{k-1}} = \frac{\text{log}\; (y_k / y_{k – 1})}{t_k – t_{k-1}}$.

In forestry, $\overline{p}_k$ is also known as mean periodic relative increment, though the concept has not often been used in this field. Relative growth rates are always useful, when the initial size of organisms varies. Then relative growth rates allow a better comparison. This reminds us of the analysis of covariance with initial size as covariate and indeed the two ideas are related. Still, relative growth rates are also size dependent and this can sometimes cause problems in plant growth research.

It is quite amazing to see how many different fields have independently used the concept of relative growth rate, developed their own separate terminology and modelling approaches. For example, a characteristic derived from RGR is the efficiency index, also referred to as growth coefficient and growth multiplier, $M_k$:

$M_k =\frac{y_k}{y_{k – 1}} = e^{p_k \cdot \Delta t} \\$

The growth coefficient or growth multiplier plays a crucial role in projecting future growth based on relative growth rates and has been “re-invented” several times in various separate fields of application.

The vast amount of publications from different subject areas on this topic calls for a standardisation of notation and terminology. They also in way suggest that there are many more similar research topics that would benefit from a more systematic approach. The use of relative growth rates is widespread in general plant growth science but less common in forest science. Interestingly Brand et al. (1987) mention in their paper in Annals of Botany that growth analysis (involving relative growth rates) fills a gap in crop yield research between strictly mechanistic studies of plant physiology and strictly empirical studies of growth and yield. ## By Arne Pommerening

My background is in forest science with a PhD in forest biometrics (from Göttingen University (Germany) and a Habilitation in forest biometrics (from BOKU University Vienna (Austria). For eleven years I have been working in the fields of quantitative forest management and quantitative ecology at Bangor University (North Wales, UK) before working for a short while in Switzerland. Since 2014 I work as a Professor in Mathematical Statistics Applied to Forest Science at the Swedish University of Agricultural Sciences (SLU in Umeå and my research areas include woodland structure analysis and modelling, spatio-temporal dynamics of plant point patterns, individual-based modelling with a focus on plant interactions, plant growth analysis, methods of quantifying and monitoring biodiversity and the analysis of human behaviour of selecting trees. Much of my research is computer-based using simulation experiments and my research is strongly interdisciplinary and international.