# 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 $Latex formula$ represents the state of a plant characteristic at time $Latex formula$, for example the biomass of a plant, instantaneous relative growth rate can be expressed as

$Latex formula$.

Since relative growth rate is equivalent to the derivative of $Latex formula$ with respect to time $Latex formula$, studying the relative growth of $Latex formula$ is equivalent to studying the absolute growth of $Latex formula$ .

In empirical studies, we commonly deal with discrete time, e.g. $Latex formula$, which are our scheduled survey days or years. The period between two discrete instances of time can be denoted by $Latex formula$ with $Latex formula$. For simplification we can now set $Latex formula$ and $Latex formula$.

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

$Latex formula$.

In forestry, $Latex formula$ 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, $Latex formula$:

$Latex formula$

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.

# Creativity – where do we get it from?

Haven’t we all experienced this? – We can sit for hours in our offices trying to come up with a brilliant new idea, to solve a problem, to shape an important text or to find a nagging error in our computer code. And then out of nowhere, once we have set off to go home or for a coffee – there is the solution, straight and neat. Apparently when we let things go and don’t pursue them, the solution comes to us. Intriguingly there is an old story in the Welsh legends of the Mabinogi describing how a king tried to pursue a beautiful woman on horseback, but the faster he rode the larger the distance between them became. Finally the king figured out that he had to ride more slowly, not faster. This did the trick and he eventually caught up with the lady of his heart.

Often even a small change of perspective helps when you stand up,  go for a short walk or talk to a colleague in the corridor. Recently I read that someone had reviewed where researchers said they regularly have their best ideas. Not surprisingly the various locations mentioned rarely include the office.

Obviously we are sharing this experience with many others who have creative professions outside research and higher education.

How can we make better use of such flashes of creativity? Is there something wrong about our offices? Is it the noise, the disturbance or is it simply the change of locations and situations that fuel our creativity? Probably not easy to say and quite dependent on everyone’s personality. In any event, the culture of occasionally working from home, at other universities and abroad is certainly something that stimulates research output.  Quite frequently I even experience fits of creativity in airport cafes and while travelling in planes and trains packed with people. Also blocking other activities such as teaching and administration opens up windows of quality research time that can be used to think things thoroughly through – a rare commodity in this day and age.

Increasingly I am enjoying the chats I am sharing with my staff and other colleagues at lunch, coffee break or on the corridor. After each of them many research ideas appear in a new light, thoughts have become deeper and above all – a renewed flow of inspiration and love for my research field has filled my heart and on this wave of energy I get carried away to new shores.

An intriguing question in this context is, if we can actually “teach” creativity to our students or can we just inspire and promote it?

It is probably an important part of our research quest to find for ourselves what works best for us. Still, there may be some common “laws” and “principles” that work for many and that we can adopt to improve our research culture. I am curious to discover more of them as I am experimenting with myself.

# Where to start?

As this is my first posting on this webblog, I am starting it with a “smooth” thought. Forest Sciences as a scientific field have seen many changes since I have been a student in the 1990s.  It used to be a world of its own, pretty much uncontested and a microcosm of general science including such fields as for example history, law and politics on top of natural sciences. Forest Biometrics was one of the scientific fields, much respected both by students and academics, and usually honoured by a university chair.  Professorships in Forest Biometrics were usually responsible for the mathematical and statistical education of students, for consultation and for quantitative research.

Since then Forest Sciences have been absorbed by natural sciences, environmental science or natural resource management. This worldwide development was often coupled with a re-naming of forestry faculties.  This process of change has most likely not come to an end and is tied into university politics favouring basic rather than applied sciences.

In the current constellation, (Forest) Biometrics (also referred to as mathematical or computational forestry, mathematical natural-resource science, see Cieszewski and Strub, 2009 in MCFNS) is seen as a field of basic science by some and as an applied study area by others. In quite a few university chairs, Forest Biometrics is viewed as a synonym of statistics and all teaching and research is orientated towards it. Others have a broader approach and include mathematical topics, plant growth analysis and modelling along with other subjects such as sampling and forest inventory.  Causton and Venus (1981) for example wrote in their book “The Biometry of Plant Growth”: “We, however, take the view that biometry is a subject in its own right. The aspects of biology requiring quantitative study should form an integral part of biometry, and not merely dismissed once the problem has been put into quantitative form and attention turned to mathematical and statistical theory and methods.”  In his book “Mathematics of Life” Stewart (2012) is of a similar opinion when he writes “Mathematics is being used not just to help biologists manage their data, but on a deeper level to provide significant insights into the science itself, to help explain how life works. Biomathematics is not merely a new application for existing mathematical methods. You can’t just pull an established mathematical technique off the shelf and put it to use: it has to be tailored to fit the question. Biology requires – indeed demands – entirely new mathematical concepts and techniques, and it raises new and fascinating problems for mathematical research.”  I find this view quite agreeable, since Forest Biometrics in my opinion should be about interdisciplinary work bringing biology/ecology and mathematics/statistics together. Forest biometricians are meant to act as mediators between mathematical  statistics and forest science able to speak and understand both “languages”. On the websites of my Chair (http://www.slu.se/mat-stat-forest) you can see a few examples. Naturally, it is thrilling and re-assuring to see that there is a great diversity of research visions for Forest Biometrics. I am convinced that we need this pluralism of ideas to make real progress in quantitative research.  This topic is also considered in Joel E. Cohen’s essay “Mathematics is biology’s next microscope, only better; biology is mathematic’s next physics, only better from 2004 (PLoS Biol. 2, e439)”, which apparently has become a sort of proverb in biomathematics.

Cieszewski and Strub (2009) among others also pointed out that the advances in computer technology form another important column of research in Forest Biometrics in the same way as this technological development as resulted in other specialised fields such as Computational Physics and Computational Genetics.

As someone who has recently taken up a University Chair in Forest Biometrics, I am wondering how others – whether they are in a similar situation or not – feel about this?