# Spruce forest stands in a stationary state

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## Abstract

We present stationarity criteria for forest stands, and establish embodiments using a Norwegian empirical stand development model. The natural stationary states only slightly differ from the outcome of long-term simulations previously implemented using the same empirical model. Human interference in terms of diameter-limit cutting is introduced. Consequently, stationary states differing from the natural one appear. Standing volume, growth and monetary value appear low but the financial return rate may be significant. Volume yield and financial return clearly contradict each other, the former arising from harvesting large trees, the latter from frequent removal of small trees. An exponential tree size distribution does not appear to comply with the stationarity criterion.

## Keywords

Growth Plenterwald (selection cutting) Recruitment Stand development Yield## Introduction

Forest trees often are produced in growth cycles, including terminal harvesting and artificial or natural regeneration (Kuusela 1961; Pearse 1967; Goodburn and Lorimer 1999). However, cyclical production is not the only option. It is also possible to maintain a continuous stand cover (Buongiorno et al. 1995; Pukkala et al. 2009, 2010; Pukkala 2016; Valkonen et al. 2017). Several studies have suggested that continuous-cover forestry has particular benefits (Hyytiäinen et al. 2004; Chang and Gadow 2010; Tahvonen 2011; Buongiorno et al. 2012). Financial sustainability has been investigated in terms of maximized net present value of future proceeds (Tahvonen 2016; Rämö and Tahvonen 2015, 2016; Tahvonen and Ramo 2016; Sinha et al. 2017). As a special case of continuous-cover process, a stationary system may appear which displays some kind of a demographic equilibrium (Schütz 1975, 1997, 2006; Brzeziecki et al. 2016). In principle, a stationary stand may develop naturally, provided the system has enough time for transient effects to level off. However in several cases, transient times may be long and it is not easy to find naturally developed stationary forests (Bollandsås et al. 2008; Aakala et al. 2009; Brzeziecki et al. 2016).

It has been postulated that in a natural state, the appearance frequency of trees would decrease exponentially as a function of tree size (de Liocourt 1898; Kerr 2014; Picard and Gasparotto 2016). However, we are not aware of any criterion of stationarity that would specifically produce exponential distributions. Exponential tree size distribution within a forest stand may be approached through specially designed harvesting schedules (Helliwell 1997; Pukkaa 2016; Schütz et al. 2016). However appears that such tailored systems are not stationary but in a kind of transient state (O’hara et al 2007).

In this paper, stationarity conditions for the size distribution of forest trees are established. Systems fulfilling a stationarity criterion are discussed because of their conceptual simplicity and practical implementability.

The established steady-state equations are parametrized using a Norwegian empirical model for the growth and mortality of spruce trees, as well as recruitment of new trees (Bollandsås et al. 2008). Tree size distribution, total basal area, total volume as well as the financial value of trees in a natural stationary state are examined for three stand fertilities. Naturally dying trees and their financial value are discussed, provided they can be harvested.

We resist the temptation to implement any multi-objective optimization or any other treatment with limited tractability and comprehensibility. Instead, a few simple stationary harvesting patterns are introduced. Human interference affects size distribution, total basal area, total volume, as well as the financial value of the trees. Harvesting also produces a financial return. The interplay of harvest yield and financial return rate is discussed. Finally, the possibility of applying longer harvesting cycles, resulting as non-stationary states, is discussed.

## Materials and methods

*i*, \(Id5(D_{i} )\) is the probability that a tree survives and grows into the next diameter class, and \(m(D_{i} )\) is mortality. Such a stationarity criterion is rather generic and appears in a variety of contexts. For forest stands, we established the criterion independently before finding out that it has been several times mentioned by Schütz, and applied by several researchers (Schütz 1975; Coomes et al. 2003; Kohyama et al. 2003; Schütz 2006; Muller-Landau et al. 2006).

*i*as positive natural numbers in Eq. (1),

*i*–1 becomes ill defined with the smallest value

*i*=1 because there is no description of existing trees smaller than those in the smallest diameter class. In other words, for the smallest diameter class, we need a boundary condition:

*R*corresponds to the number of trees recruited into the smallest diameter class

*i*–1 to class

*i*.

## Results

### Properties of the stationary state

^{2}ha

^{−1}. An interesting feature is that the stationary size distributions are not monotonically decreasing in Fig. 1. In terms of Eqs. (1)–(5), with large trees, the growth rate related to Eq. (3) decreases as a function of increasing tree size. This induces crowding of appearance frequencies in large diameters. Crowding is partially compensated by increased mortality according to Eq. (4). Compensation is almost complete in the case of the highest fertility, but clearly incomplete with lower fertilities. However, in the largest diameter classes, mortality becomes high enough to reduce the appearance frequency of trees regardless of site fertility.

Stem diameter can be converted to trunk volume in a variety of ways. The relationship provided by Rämö and Tahvonen (2015) is applied for eutrophic spruce stands. The commercially utilizable trunk volume in trees of different diameter classes is given in Fig. 2. The total commercial stand volume per hectare is 237, 282 and 324 m^{3} ha^{−1} for the three fertility classes.

^{−1}for the three fertility classes (Fig. 3).

In a stationary state, the commercial trunk volume remains constant. Correspondingly, in the absence of harvesting, the reduction of the volume of living trees through death equals volumetric growth in Eq. (1). The stem volume in dying trees within any diameter class is shown in Fig. 5. The total amount of growth per hectare during a five-year period is 6, 12, and 20 m^{3} for the three fertility classes. Correspondingly, the annual growth varies from 1.2 to 4.0 m^{3} ha^{−1}.

^{3}ha

^{−1}would be considered small. However, in small-scale forestry, such harvesting operations may well be feasible. In Fig. 6, the commercial stumpage value of trees dying within any five-year period is 336, 645, and 1092 Euros for the three fertility classes.

It is of interest to compare the stumpage values of dying trees to the total monetary value of standing trees. Provided the five-year growth can be technically harvested and then yields the expected stumpage value, the annualized return of the capital becomes 0.53%, 0.85% and 1.26% for the three different site fertility classes. Obviously, gaining the full stumpage value requires that the harvester is a professional capable of identifying dying trees before they suffer any deterioration in commercial value.

### Human interference

In Figs. 7, 8 and 9, the total number of trees exceeding the diameter of 50 mm, as well as basal area and total volume, increase as a function of cutting limit diameter. This occurs regardless of the fact that tree recruitment as well as mortality depend on total basal area according to Eqs. (4) and (5). The number of small trees is greater in Fig. 7 with a small cutting limit diameter than the number of small trees in the natural state illustrated in Fig. 1. The number of trees in Fig. 7 approaches the total number of trees appearing in Fig. 1 in the natural stationary state as the cutting diameter limit increases. The total volume of trees in Fig. 9 approaches the total volume appearing in Fig. 2 in the natural stationary state, as the cutting diameter limit increases.

There is a significant number of trees in Fig. 1 as well as in Fig. 7 in the case of a large cutting diameter limit. Similarly, the total volume is significant in Figs. 2 and 9 in the case of a large cutting diameter limit. However, at small cutting diameter limits, the number of trees, as well as the standing volume are rather low in Figs. 7 and 9. The same applies to basal area in Fig. 8.

^{3}·ha

^{−1}, and yielding harvest volumes of only 1.3–3.0 m

^{3}ha

^{−1}for any five-year period.

There is another local maximum in the capital return rate curve at a harvesting diameter limit of 200 mm. This also is restricted by a small number of stems (151–210 ha^{−1}), a small standing volume (13–20 m^{3} ha^{−1}) and a small volumetric yield in harvesting within any 5-year period (4.4–10.2 m^{3} ha^{−1}). With a harvesting diameter limit of 450 mm, the five-year yield would be greater (10.5–26.7 m^{3} ha^{−1}) but the capital return rate much lower (1.7–3.1% per year).

### Applications with longer harvesting intervals

Figure 10 shows that volumetric yields in harvesting repeated every five years are low, even in the case of the highest fertility and high cutting limit diameters. Therefore, it might be feasible to apply longer harvesting intervals. A question arises whether the forestry still can be considered stationary rather than periodic, and the obvious answer is “no”. However, periodic harvesting may be a natural extension of stationary forestry.

Starting with a stationary state with the boundary condition that all trees greater than a particular cutting diameter limit are nonexistent, equations (1)–(5) are valid to determine the number of trees in any diameter class, with the exception that the number of trees in diameter classes above the diameter cutting limit is zero. If Eqs. (1) and (2) are then neglected, equations (3)–(5) may further modify the number of trees in any diameter class.

^{3}ha

^{−1}at the initial state, and its further development strongly depends on site fertility. A volumetric harvesting yield of 20 m

^{3}ha

^{−1}would require a harvesting interval from 8 to 18 years, depending on site fertility.

The distinction between total volume and volume in trees over 200 mm diameter is reduced with time, i.e., the volume of small trees is reduced (Fig. 13). This is due to reduced recruitment along with increasing basal area according to Eq. (5). Consequently, the system does not immediately return to the initial state at the instant of diameter-limit cutting.

^{3}ha

^{−1}initially and its further development strongly depends on site fertility. A volumetric harvesting yield of 20 m

^{3}ha

^{−1}would require harvesting interval from five to 13 years, depending on site fertility.

Comparing Figs. 13 and 15 shows that the net growth rate is 40–50% higher if the cutting limit is 300 mm instead of 200 mm, with a boundary condition of 20 m^{3} ha^{−1} harvesting yield. Considering also Figs. 14 and 16, the capital return rate is 100% higher with the 200 mm cutting limit, with the boundary condition of 20 m^{3} ha^{−1} harvesting yield.

## Discussion

The number of trees in a natural stationary forest appears small in comparison to non-stationary forest ecosystems (Figs. 1, 7; Pukkala 2006; Tahvonen 2011; Rämö and Tahvonen 2015; Lundqvist 2017; Sinha et al. 2017; Valkonen et al. 2017). The growth rate is less than that reported for comparative sites in non-stationary forests (Figs. 5, 10; Pukkala 2006; Tahvonen 2011; Lundqvist et al. 2013; Drössler et al. 2014; Rämö and Tahvonen 2015; Valkonen et al. 2017; Sinha et al. 2017; Lundqvist 2017). This is directly due to Eqs. (3)–(5). The recruitment rates given by Eq. (5) appear rather slow, inducing stationary systems with a small number of trees. On the other hand, the basal area and standing volume become significant, provided the stationary state is not disrupted (Figs. 2, 8, 9). Basal areas however, are less than those in long-term simulations (Bollandsås et al. 2008). This may be related to two issues of equations (1)–(5). Firstly, age does not contribute to mortality in Eq. (4). Secondly, Eqs. (1) and (2) have not been applied in long-term simulations, possibly resulting as a transient state at simulated high stand age.

Non-natural stationary states satisfying Eqs. (1) and (2) may develop under anthropogenic influences. One example is the boundary condition of repeated diameter-limit harvesting. Volumetric yield and financial return rate clearly appear to contradict each other. The highest financial return rates are achieved with frequent harvesting of small trees, resulting in a small amount of standing volume, along with rather small volumetric growth (Figs. 7, 8, 9, 10, 11, 12, 13, 14, 15, 16). On the other hand, a greater amount of capital in standing trees increases growth and yield but inevitably reduces capital return rate (Figs. 7, 8, 9, 10, 11, 12, 13, 14, 15, 16).

It is customary thinking within the field of forestry that trees should be grown as long as the capital return rate exceeds an external alternative rate of return or “opportunity cost”. In the mind of the Author, this is incorrect. From a financial perspective, the capital return rate should be maximized within any production process. A justification is that an alternative practice within forestry itself, providing a higher capital return rate, may form an opportunity cost. The capital returns in this paper have been discussed in a purely operative basis. This is justified since bare land, not being a consumable, is not subject to amortizations. Proper allocation of capital between industries would require some kind of a scenario of real estate appreciation which is outside the scope of this study.

The empirical models (3)–(5) in this study describe growth, mortality and recruitment in a statistical sense. Significant scattering beyond modelled trends appears in any dataset (Bollandsås et al. 2008). Consequently, some amount of uncertainty in the present results is obviously related to the reliability of the models. However, qualitatively, the appearance of slow recruitment, in accordance with Eq. (5) appears to agree with several reports (Lundqvist 1993; Newbery et al. 2004; Pukkala et al. 2010; Vlam et al. 2016). There are also observations indicating a higher rate of recruitment (Lundqvist 1991; Lundqvist and Nilson 2007).

Few stationary structures have been found in old-growth forests (Newbery et al. 2004; Brzeziecki et al. 2016; Vlam et al. 2016). Recruitment often not being sufficient, the number of trees tends to decrease and the age of dominant trees to increase (Pukkala et al. 2010; Brzeziecki et al. 2016; Lundqvist 1993). Such a transitory situation apparently may endure several centuries, possibly close to a millennium (Aakala et al. 2009; Pukkala et al. 2010; Brzeziecki et al. 2016). During such a period, a variety of disturbances may appear, interfering with the development of a stationary state.

Obviously the recruitment of seedlings can be increased by artificial or seminatural regeneration (Busing 1994; Goodburn and Lorimer 1999; Pukkala 2006; Pyy et al. 2017). Such actions easily lead to periodic forestry, instead of stationary forestry. The prospects for high volumetric and monetary yield rates favor periodic forestry. The high financial return rate of capital bound within the process appears to be the significant benefit of stationary forestry.

## Notes

### Acknowledgements

Open access funding provided by University of Eastern Finland (UEF) including Kuopio University Hospital.

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