Two Is Between the Extremes Of many (all) points on a curve and One
What Has Been Done Since Then? In 1978: –Two-point method Purely an empirical finding No theoretical rationale. In 2008: –I now understand why this method works. As a result, it has become possible to describe many other things, including the entire forestry.
A Fundamental Equivalence # of points = # of local parameters = # of growth factors Equations may contain global parameters or constants.
Reinforcement or Opposition? Do the equation terms Reinforce each other? Oppose OR
Current annual increment first increases, then decreases. This fact indicates that factors of growth are opposites. Age Increment Empirical Support
Two Sides Every complex problem has two opposing sides. The key is to recognize both.
… and vice versa Many of our errors can be traced to taking one side for the whole. The rest of my talk is an illustration of these simple points.
A Description of Growth A complete, if coarse, description of growth: where the increase of organism size, dy, during an instant of time, dt, is presented as the product of influences that facilitate the growth and those that check it.
Growth Expansion Basic growth process is Unrestricted cell division The growth rate is proportional to either the number of cells or the size of an organism
Complications The proportion of living cells decreases where k is the coefficient of proportionality and p < 1 is the allometric coefficient specifying the diminishing portion of living tissue.
Growth Decline Unlimited expansion is checked by aging and finite area. The simplest assumption is that growth declines linearly with age: where q is the constant rate of decline. This module predicts the complete termination of growth at age t = 1/q. Afterwards, the growth would be negative, which is not realistic.
Adaptation The linear growth decline is counterbalanced by phenotypic (built-in) adaptation. Deceleration of growth decline is proportional to the current rate of decline:
Adaptation – Growth Integration results in a non-linear growth decline: The requirement that. This module can be viewed as the sum of the infinite number of terms of a Taylor series.
Combined Model of Tree Growth The product of the growth expansion and decline modules unites two opposite trends in a model of tree growth:
Structure of the Growth Model Unlimited cell division Limited volume of live tissue Tree Growth Positive Factors Negative Factors Growth restraints Adaptive alleviation
Empirical Verification The equation favored by foresters for growth modeling: It does not look like the derived model.
Empirical Verification Actually, the Richards equation is identical to the derived model: Integration with specific values of the parameter p produces the Gompertz ( p = 1 ), logistic ( p = 2 ), and Bertalanffy ( p = 2/3 ) equations.
Empirical Verification In the Richards equation a, b, and c are
What Does the Identity Tell Us? The biological processes and their analytical forms have substance. The success of the empirical equations, reflects the fact that they unwittingly express the basic processes of growth. While the processes infuse meaning into the empirical equations, the equations give shape to the processes, making them tangible and operational.
Why Are Two Points Sufficient? Because those opposites are related. Out of three parameters of the Richards equation, only two are independent. Parameter c is global. It is determined by tree structure rather than site quality, growth rates, or tolerance.
Growth curves combined at 50 years Heights of 36 spruce trees of site class 16.4 combined at 50 years Guttenberg, A.R., von. 1915. [Growth and yield of spruce in Hochgebirge.] Franz Deuticke, Wien. 153 p.
Growth curves combined at inflection Rescaled heights and ages of 104 spruce trees combined at the inflection point
Shortcoming of Growth Equations Growth equations cannot reflect variations in stand density.
Density Module: Opposites 1.A total lack of competition among trees and full availability of resources. 2.The extreme competition and density that preclude any growth. The growth model describes the first opposite.
Density Module: Solution Given adaptation, the decrease in volume growth, -dy', is proportional to the product of volume growth and density increase, y'dS, rather than to the density increase dS alone: where S is stand density and m is a parameter.
Tree Growth-Density Model Incorporating the density module into the growth model produces a growth-density model describing tree growth in stands of any age, size, and density:
Stand Growth-Density Model Multiplying the volume growth of average tree, v’, by number of trees produces stand growth: Stand growth in terms of average diameter, age and density:
Forest Management The theory outlined above exposes the inner mechanisms of forest stand dynamics. It is about regularities inferred from past observations. In contrast, forest management is active and forward-oriented; it is prescriptive rather than descriptive.
Two Goals Of Management 1.Preserving the environment, and 2.Meeting the current and future wood products needs of an increasing human population. All the diversity of forest management is made of various combinations of these two opposite goals.
Solution: Spatial Separation Conflicting goals cannot be satisfied at the same time and at the same place But they can be satisfied in different places.
Solution: Spatial Separation Spatial separation reverses the conflict between the goals and makes sustainable intensive management for wood products a prerequisite for the existence of undisturbed forests.
Why not combine some use with some conservation on the same land? Why not thin stands before trees rot? Or leave some patches of native vegetation in the middle of forest plantations and agricultural fields? Maximizing Combined Utility
Because curtailed preservation on the same land would detract from both environmental quality and production. Spatial separation maximizes the combined utility.
How To Manage For Preservation Opposites: Restoration ecology versus the "hands-off" approach. Solution: Leave nature alone.
How To Manage For Preservation Nature is the generator and best manager of biodiversity. It is counterproductive and supercilious to interfere with its eternal work of creation and destruction.
How To Manage Wood Production A promising way to preserve nature is to increase productivity on the portion of land devoted to the second goal of forest management – wood production.
How to Manage Wood Production Aside from expensive and not-always- rewarding site alteration, this can be done by –minimizing interspecific competition and –optimizing intraspecific competition.
Interspecific Competition Among the various benefits ascribed to biodiverse forests are: –higher productivity, –beauty, and –stable dynamics. None of these claims is not supported by evidence.
Interspecific Competition In fact, interspecific competition is one of the most harmful factors of tree life. –kills many trees –prevents others from reaching their growth potential. Interspecific competition should be minimized.
Intraspecific Competition Current consensus: thinning can redistribute growth from smaller to larger stems but not increase its amount “As long as the site is fully occupied (trees making their full use of available resources), the species will produce the same amount of wood per year at various densities. Whether there are many small trees or fewer large trees, a similar wood volume is produced” (Spurr and Barnes 1980: p.376).
Empirical Basis of Consensus Relationship between standing volume and volume increment by Langsaeter (1941) Langsaeter's curve
A Problem with the Consensus Stand growth is a function of average diameter, age and density:
Optimal Density = m Differentiating this equation with respect to density and setting the derivative equal to zero allows us to determine the density at which the current stand volume growth reaches maximum.
Optimal And Normal Densities m = 678±39 For undisturbed permanent plots (control and initial measurements) of the Monticello study, normal density is 638±16 When the difference in diameter is taken into account, the highest growth is observed in stands of high, and not medium, density
Volume Growth and Density Relationship between volume growth (m 3 /ha) and current density for loblolly pine stands –equal age (20 years), –diameter (25 cm), and –site index (20 m) (base=25 years).
Sum of Growth Maxima Is Not Maximum If volume growth is maximum at the highest current density, then this density should maximize the total yield over rotation. But this inference contradicts forestry experience, which tells us that moderately dense stands are more productive.
Inverse Relationship An inverse relationship exists between tree size and average density. Although, at a given moment, normal density does produce maximum growth, when it is maintained over an extended period, the same density suppresses diameter and, as a result, reduces volume growth and volume, itself.
Inverse Relationship For this reason, the sum of maxima at each moment does not produce the maximum of final harvest.
Growth-Density Model and Langsaeter's Curve Langsaeter's curve bundles the effect of tree size together with that of current density. As a result, the position of the optimum is misplaced toward middle densities. The model is not restricted to stands of the same age and site as is Langsaeter's curve. It includes the terms reflecting these variables and is applicable to even-aged stands of any age and site.
Growth-Density Model and Langsaeter's Curve = normal density
Optimization of Stand Density Trajectory For centuries, foresters have been searching for a single level of optimal density to be maintained throughout stand life. But nobody has proven that keeping density at 15 years the same as at 35 years would maximize harvest. Now the challenge is to find an optimal trajectory of current density.
Conflicting Requirements To maximize average tree size, we need the lowest density. To maximize stand volume growth, we need the highest density. How do we minimize the negative side of density (small size) and maximize its positive side (maximum volume of trees with a given size)?
Resolving the Conflict The number should be the minimum that assures the density sufficient to maximize financial returns by harvest time. Such a prescription can be called the minimum number- maximum yield (minimax) strategy. Albeit unknown in forestry, it is not new: for millennia, farmers have grown only the plants they intend to harvest. Keep the number of trees per unit area constant.
Advantages of Minimax Besides maximum returns from final harvest, minimax has several other benefits: –saving on planting and pre-commercial thinning; –minimization of root rot, insect infestation, and other risks associated with high density; –sturdy well-spaced trees with laterally symmetrical crowns which reduces damage from ice, wind, and other hazards. - before the trees close their canopies, up to 90% of the land can be used for other purposes.
Disadvantages of Minimax Minimax is one extreme. It may maximize final yield and profit in theory, but it cannot be applied without some compromises because of the following problems: –establishment mortality; –the lack of selection; –interspecific competition; –wood quality; –forfeiting intermediate harvest; –rectangularity.
The Science of Forestry Basic Opposites Theory of Forest Stand Dynamics Forest Management Proximate aim: Passive description of what is Active prescription of what should be Ultimate aim: TruthGood Output: General lawsSpecific forward- oriented actions
Opposites of Stand Dynamics Stand Dynamics Open growth of single trees Negative factors Growth restraints Adaptive alleviation Positive factors Unlimited cell division Limited volume of live tissue Confined growth of competing trees SizeNumber Outcome: Stand Density Adaptive relief Resource limitations
Management Opposites Forest Management Preservation Action: None Maximum profit Environmental controlGenetic modification Site ReturnsCosts Action: Improve when it pays Competition Intraspecific Interspecific Action: Minimize Life-long Current Optimum = Maximum High at the end Low at the beginning Action: Keep the number of trees constant
Knowledge: Means or End? Along with practical utility, the science of forestry refines research methods. Do we use our mind to understand things and improve our standard of living or we study things to clarify our thinking? As with much else, science is a synthesis of these means and ends.
THE 1-2-1 METHOD It considers two opposite explanations simultaneously. The method organizes research into cycles containing three basic steps. Defining a problem Exposing two oppose explanations Locating a solution The name–1-2-1 method–refers to the sequence of one problem, two explanations, and one solution.