But if there is a constant rate of mortality, space/resources will always become available into which trees could continue to grow? Hence at some point, growth rate will equal the rate at which resources become available (mortality rate). A bit simplistic I know, but maybe something to think about?

Seems for such equilibrium it will took long period. The tree can live for a couple of hundreds to 1000s years; seedling survival rate (or mortality rate) could be considered but does it really make a big difference in limiting the biomass addition in the forest?

Another source of freeing up resources might be disturbance. While I do not have any specific papers at hand, there is plenty of literature on the role of disturbance and how it can influence resource availability. Eg. Intermediate disturbance hypothesis etc.

Danswell, I agree intermediate level of disturbances help in increasing species diversity as a disturbance open opportunity for species to colonize the newly available area. Primarily herbs occupy such available areas and for trees it took long time to establish. Natural disturbances are rare while human induced disturbances are frequent and we consider later one as disturbance while performing ecological work. Human disturbances and increasing in the species diversity as a result of disturbance are important. But my concern is determining the growth threshold or the saturation condition after reaching which tree biomass in a particular piece of land (let's say in one hectare of land) do not increase significantly, with a condition of absence of external interference or other forces.

we can estimate the threshold of forest growth by mathematical equation which have asymptote line as the estimated threshold. Historical forest growth data could be fitted by those equation so the threshold could be estimated. Some suggested equations are sigmoid or modified sigmoid, gompertz curve, etc.

Effendi, this sounds interesting. Could you refer some publication? Also in the absence of historical forest growth is it possible to estimate? I assume with dendrochronological work annual growth pattern could be estimated. Is it scientific to use such data to define forest growth and a model to explain likely threshold of forest growth?

I have some example for the case of fiber length growth in every growth ring. The equations were suitable for fitting the growth of fiber length. As we know, in juvenile period, cell dimension was changing significantly and become relatively constant in mature period. These cell characteristic were similar with curve characteristic. I can send you example of my data processing in my work via e-mail. Please give me your e-mail address.,

[email protected]

To answer to your questions, let take them in order:

1. Yes, there are limits of tree growth. Such limits may be established using growth models, generally based on regressions. For example: The growth model of Prodan (Forstliche Biometrie, Munich-Bonn-Wien, 1961) is quit simple and is expressed by eq: log (growth) = bo+b1*log(Age)+b2*(Age). This equation has all the properties of a growth curve, i.e: if Age = 0, Growth = 0; Age tents to infinity - growth tends to zero; has a maximum which can be considered as threshold and which can be expressed by the report between b1 and b2.

There are also very interesting models and I think you have to address to forest biometry (see Prodan, Mitscherlich, Backman, Assman, Anuchin, and so on)

2. The concept is true. A very simple conceptual model is expressed by Zeide who starts from Reineke concept of Stand density index. Search for Boris Zeide, 2005, "How to measure stand density", Trees 19, p 1-14. The author give also a growth model and explains the concept of self -thinning which is similar with that you said.

3&4 There are a lot of publications. You have to refers at stand composition (species) which are in your target.

A dendrochronological basis may be useful to find a relationship which express the growth in diameter. Because the growth of a tree is explained by the concomitant growth of diameter and height and because there is a closed relationship between those variables you must apply a known height - diameter relationship for the target species.

One of the limits to trees ever growing in height is the maximum height to which water in a capillary can moved against gravity. For a given capillary pore size, water can only be drawn up to a fixed maximum height. In fact, the click sounds associated with when water in capillaries are broken have been recorded in trees. It follows that trees cannot grow taller than the height to which water within their xylem can be moved upwards.

Clem

Costinel, equation log(growth) = b0+b1*log(Age)+b2*(Age) is not suitable for estimating the growth of tree. The growth is not zero when the age is zero but it is exp(b0+b1*ln(0)). Since ln (0) has no rational solution so zero is asymptote line. The other asymptote line of above equation is infinity when the growth become zero. So equation above would result an infinity value for age limit of tree. Does it mean tree never die?

I suggest this equation:

N(t+1)=Nt+Nt*(a+b*Nt+c*Nt^2) or N(t+1)=Nt+Nt*(a+b*ln(Nt)+c*(ln(Nt))^2.

Nt is tree dimension or volume of the standing tree or biomass at (t) years old, while N(t+1) is tree dimension or volume of standing tree or biomass at (t+1) years old.

The solution of both equations above could be easily obtained by data analysis using Microsoft Excel.

Both equations could result two, one, or none asymptote line depend upon the data. If the data come from juvenile (young) period, none asymptote is found because the solution will be imaginary number. If the data come from juvenile (young) and mature period two or one asymptote line could be found which could be considered as the threshold of tree dimension or biomass or volume of standing tree in the forest.

Unfortunately, i have not found any publication which confirm my idea above. I have used above equation to fit cell dimension of wood at every growth (annual) ring, weight of cow each year in the farm, and length of bamboo internode. All of them give very good result. The coefficient of determination were higher than 90%. I will publish the result in my next publication.

Dear Effendi, OK, I could agree with your observation. In fact the form of the eq. I presented is a linearly transformation of a more general eq which is Growth = a*[T^b]*exp[-F(T)] where T = age, F(T) a polynomial function depending also of age. The form presented is used to fit experimental data. As you can see, analysing this form that I said about zero point and asymptotic trend is true. You may be right asking about "infinite life" of tree (or stand), but we have - till now - no precisely limits concerning aged mortality. In the complex system that is the forest, a tree dyes more being killed by disturbances - as Danswell suggests - than becoming old. A limit of age mortality become very difficult to establish, so that "infinite life" - in the present day stage of knowledge - might be accepted. That asymptotic trend to zero in the decline curve of growth is explained - as in the case of human beings - by the "wear" of the whole organism of tree.

Concerning your suggested eq., they are used by auxology and dendrometry of trees and stands. Try to find in libraries books and papers written by Prodan, Mitscherlich, Backman, Assman, Anuchin which put the basis of stand growth science. I might also recommend a lot of Romanian books, but they are in Romanian.

"I'll be back" - to use a very known expression.

As I promised, I'm back.

In fact the Effendi's equations are nothing else than a general eq. of the so called current or annual percentage increment which expression is I(ncrement) = [N(t+1)-N(t)]/N(t)=F(t). The first form of F(t) = a+b*Nt+c*Nt^2 is the m of the first term of Prodan's eq. and is used in Romania by Giurgiu, V - 1967 - Studiul cresterilor la arborete (study of growth in stands), Ed Agrosilvica, Bucuresti (as an example). The second form F(t) = a+b*ln(Nt)+c*(ln(Nt))^2 is the Backman function. These function are largely used to build yield tables for trees and stands.

To Clem: the hypothesis was verified indeed, but: what are we doing with the difference in pressure between crown level - generated by transpiration and other photosynthetic processes - and roots level ? Because not only capillary forces are involved in drawing up sap.

Hello Costinel, that is a good question. But is transpiration a consequence of capillary tension alone? Does transpiration from leaves actually exert a force differential between the leaves and roots? I do not know the answer to that.

Plant physiology not being one of my strong skill, I dare to issue an explanation: transpiration is the consequence of two main processes. One is the tension created by heat (see evaporation) the other is the necessity of the plant itself to keep a balance between its own water content (needed to live = 1.2% arround) and the necessity to delivery (98-99%) the main transport agent which take the nutrients from the environment and bring them in the leaf laboratory, in the water cycle (knows plant water cycle? perhaps !).

Concerning your second question, if you accept Transpiration Pull and Cohesion-tension Theory, yes, the transpiration exert a differential force between the leaves and roots.

Seem my original question was left far behind! I do not think we need to look at plant physiology in detail for my question. My question is related to biomass growth and carrying capacity of the occurrence site of the species. Resources are limited so there must be some threshold of increment of biomass beyond, which theoretically biomass would not get increased unless external force change the forest structure or mortality make space available. Is it possible to get such threshold with tree ring study? In general ring width get narrower in later years than in early year of plant life. Climate and water availability have a strong relation to tree ring width in spite of that is it possible to find growth threshold or near to such condition?.

Dear Sailesh, You will find answers analysing the file attached. It is derived from The Romanian Auxologic Model which is the basis of Yield tables used in sylviculture. The graphs will give the answers you looking for. Try to analyze for many species and you will see the differences.

Dear Costinel, Thank you for the dataset. I am interested to check and analyze your dataset to find out answer to my question.

Dear Sailesh,

This seems like a simple question, but the literature on this topic is abundant and not easy to summarize. I think Jeremy Lichstein et al. summarized many of the relevant issues well in a book chapter (citation below), where the author's use forest inventory and analysis plot data from the USA to try to answer this question. This book has several other relevant chapters, particularly those written by Christian Wirth. I have found it very helpful.

- Lichstein JW, Wirth C, Horn HS, Pacala SW (2009) Biomass chronosequences of United States forests: implications for carbon storage and forest management. In: Wirth C, Gleixner G, Heimann M (eds.) Old-growth forests: function, fate and value. Ecological Studies vol. 207, Springer-Verlag, Berlin, Heidelberg pp 301-341

Another way to look at this question is to look at models researchers have constructed to try to simulate the processes you're discussing (there are many). One that I am familiar with is the LANDIS-II simulation model, that has a model for succession and changes in forest biomass described by Scheller et al (citation below).

- Scheller, R.M. and Mladenoff, D.J. 2004. A forest growth and biomass module for a landscape simulation model, LANDIS: design, validation, and application. Ecological Modelling 180:211-229.

One other approach is to analyze historical forest data, when this is possible. This can often reveal unexpected carrying capacities for forest ecosystems that we cannot observe in current forests, because they have been subject to so much human disturbance and are typically of a younger age distribution than has been historically possible. This paper (citation below) is a good example.

- Rhemtulla JM, Mladenoff DJ, Clayton MK (2009) Historical forest baselines reveal potential for continued carbon sequestration. Proc Natl Acad Sci 106:6082-6087

I hope some of these references help.

P.S. If you are working with tree ring-width data to address this question, just remember that this data can be tricky. Ring-widths get narrower as a tree grows in diameter because the tree has to lay down wood over the surface area of a cylinder of increasing radius and height. As a result, narrow rings on a large-diameter tree represent much more wood biomass than the same ring-width on a small diameter tree, and trends in wood biomass increment vs. tree diameter are often very different from trends in ring-width (often opposite in sign!).

Thank you Jane! Your answer explains a lot.