Forestry An International Journal of Forest Research

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1 Forestry An International Journal of Forest Research Forestry 217; 9, 18 31, oi:1.193/forestry/cpw42 Avance Access publication 24 October 216 A emographic stuy of the exponential istribution applie to uneven-age forests Jeffrey H. Gove* USDA Forest Service, Northern Research Station, 271 Mast Roa, Durham, NH 3824, USA *Corresponing author. Tel.: +(63) ; jgove@fs.fe.us Receive 18 February 216 A emographic approach base on a size-structure version of the McKenrick-Von Foerster equation is use to emonstrate a theoretical link between the population size istribution an the unerlying vital rates (recruitment, mortality an iameter growth) for the population of iniviuals whose iameter istribution is negative exponential. This moel supports the conclusion that the negative exponential istribution as applie to balance natural uneven-age stans is a stable equilibrium moel uner appropriate assumptions. These assumptions inclue constant recruitment of stems into the smallest class that balances total mortality, an a simple relation between per capita mortality an iameter growth. A simple maximum likelihoo-base solution to parameter estimation in the inverse problem is evelope allowing the estimation of recruitment an mortality given reasonable sample of iameters, along with an estimate of population size an iameter growth rate. Two sets of stan ynamics equations are evelope that are base on (i) the form of the unerlying negative exponential istribution, an (ii) more generally, from the erivation of the McKenrick-Von Foerster equation. Applications of the methos an their assumptions are iscusse with regar to both manage an ol growth uneven-age stans. Stans or forests that are close to negative exponential structure an are juge to be reasonably close to steay state will have vital rates that support this moel. In contrast, the negative exponential is likely more important an applicable as a pragmatic target istribution when use in manage forests. Introuction Uneven-age stans an forests are characterize by the presence of trees of at least three ifferent age classes or cohorts (Helms, 1998, p. 193). There is no commonly accepte stanar with regar to what characterizes the cohorts in terms of age (e.g. 5-, 1-, 2- years), as it is epenent on a number of factors such as species composition an site. The important point is that the trees have not all originate from one or two events, creating for example, a simple two-layere (-age) forest; rather, the ifferentiation in ages (an often species) tens to prouce a more complex multi-layere vertical structure. One of the main characteristics of uneven-age forests is the presence of some form of eclining, reverse J-shape iameter istribution. This was eviently first pointe out by e Liocourt (1898). Later, Meyer an Stevenson (1943) emonstrate that a scale form of the negative exponential istribution woul fit the iameter istribution from these stans an forests. Meyer (1952) subsequently expane on this iea, an introuce the iminution quotient, q, base on a geometric progression of the number of trees by iameter class. He attribute this iea of constant q to e Liocourt, though a recent more thorough look at e Liocourt s analysis by Kerr (214) rightly questions this lineage because the progression of number of trees in e Liocourt s ata is not constant throughout all iameter classes. Meyer s (1952) work can be consiere seminal for two reasons. The first is the introuction of a constant q value with the link he mae to the exponential istribution. This has become a stanar in the management of uneven-age stans where some combination of constant q, maximum stan iameter an resiual basal area are chosen (the so-calle BDq approach) as a target conition after harvest. But eviently it also applies in natural forests, such as the virgin northern harwoo hemlock stans escribe in Meyer an Stevenson (1943), where their scale exponential istribution approach fitte the observe iameter istributions well. Because of Meyer s inaccurate attribution of q to e Liocourt, it appears that the iea is actually ue to Meyer himself. The secon classic iea reporte by Meyer (1952) is the concept of balance in uneven-age forests or stans. Meyer ifferentiates between manage an natural stans in his efinition. A manage balance uneven-age forest is one in which the current growth can be remove annually or perioically while maintaining at the same time the structure an initial volume of the forest. He goes on to illustrate this concept an gives appropriate formula base on the exponential-q assumption to ai foresters in achieving such target structures. A secon efinition of a balance forest was also given by Meyer that pertains to ol growth forests: Publishe by Oxfor University Press on behalf of Institute of Chartere Foresters 216. This work is written by (a) US Government employee(s) an is in the public omain in the US. 18

2 A emographic stuy of the exponential istribution applie to uneven-age forests In a balance virgin forest, the current growth is offset by current mortality, an the existing balance between growth an mortality makes it possible for such a forest to perpetuate itself inefinitely. These two concepts, while similar in intent, are actually quite ifferent in reality. In manage forests, balance comes via manipulation, the target iameter istribution is perpetuate by harvests, which artificially return it to the target structure. However, in virgin forests, there is no requirement for human involvement as the forest or stan appears to be in equilibrium. Inee, it can be conjecture that any isturbance in such stans woul at least temporarily upset this equilibrium, after which the original stan structure may or may not be reache again in the exact form as before isturbance, epening on its severity. As note above, Meyer s concept of balance as applie to natural ol growth (virgin) stans is escribing a form of equilibrium or steay-state conition as is commonly encountere in biology, chemistry an other branches of science. Another relate concept from biology an emography is that of a stable istribution. As a classic emographic example, Keyfitz (1977, p. 47) in a iscussion of age-class structure notes that the stable population has meaning as a set of proportions, not of absolute numbers. This efinition is echoe in the evelopment of size-structure moels (VanSickle, 1977a) to be iscusse shortly. Thus, we can think of a constant form of probability ensity function (PDF) representing the iameter istribution as stable, if its shape oes not change over time. However, this is only half of the efinition, because we woul like a criterion that allows us to etermine whether the population size is growing, eclining or staying constant (in equilibrium). A general efinition of a population in equilibrium (Keyfitz, 1977, p. 17; Nisbet an Gurney, 1982, p. 22) is that input must equal output: births (regeneration) must equal eaths (whether from natural mortality or harvesting). It follows from this input output balance that the population size, N, must be constant such that uner exponential growth, N t = rn implies r =, the population size neither increases nor ecreases over time (e Roos, 1997, p. 155; VanSickle, 1977a), where r is the intrinsic growth rate of the population. Thus, a stable equilibrium (steay state, stationary) istribution for a population is one where the shape of the istribution oes not change over time, nor oes the size of the population itself. Meyer s concept of balance is similar in spirit then to the more well-efine iea of a stable equilibrium structure. Meyer faile to explicitly inclue recruitment (births) into the efinition, though one coul argue that he implicitly was incluing these in his current growth. It will remain unknown whether he also ha the iea of a constant population size, N, inmin; butwewillconjec- ture that balance sensu Meyer, can be interprete as equivalent to the stable equilibrium. Because these ieas constitute a balance law, they can be formalize mathematically into a ynamical system, not just one that relies on the resultant iameter istribution as will be shown shortly. The tools that allow one to o this for the negative exponential or any other istribution come from both emography an population biology, the two being closely linke in this area. They istinguish the vital rates of apopulation recruitment, growth an mortality as the critical parameters; an we assume that the vital rates of a population are known or can be estimate, similarly with the population size, N. In general, the vital rates can be moele as functions of age or size (or both). In particular, for iameter istributions in uneven-age stans, size (physiological age, p-age, VanSickle, 1977a) is the appropriate attribute. Dynamical moels that couple the vital rates of a population with the size structure are known as size-structure population moels, an the istribution itself is referre to as the size-structure istribution (SSD) (Botsfor et al., 1994), or more generally, physiologically structure population moels (e Roos, 1997; VanSickle, 1977a). The theory of size-structure population moels provies both a ynamical an equilibrium solution for the population istribution in terms of the vital rates. Thus, for example, proviing a way to etermine whether the negative exponential istribution is inee a stable equilibrium istribution, an the conitions uner which this is true. This is an important point to resolve as the stability of the exponential istribution has recently been questione (Lopez Torres et al., 212). This paper concentrates on a partial ifferential equation (PDE) approach, because this metho leas to a clear close-form negative exponential solution, an provies other insights into the characteristics of the vital rates that are require for such a solution. It also provies relate probablistic results, such as the mortality probability ensity function, which is also linke to the vital rates. Matrix moels are a complementary an often simpler approach, but lack the nice closeform results that the PDE solution can provie; however, they are often simpler to work with in general. The main results an examples are given in the text, with erivations in Supplementary ata. In what follows it is often convenient to rop the extra stable qualifier when the context is obvious an simply speak in terms of an equilibrium, steay state or stationary conition as synonymous. In aition, the terms exponential istribution, negative exponential istribution an q istribution shoul be consiere synonymous. Demographic methos Preliminaries It was note that the general moel for the size-structure of a population can be efine in terms of a balance law, where inputs equal outputs, an growth moves the iniviuals through the iameter istribution. When the population is in a steay state, growth maintains the shape of the istribution, but prior to the population reaching a steay state, growth or changes in any of the vital rates may change the shape of the population istribution. The population istribution is given in terms of a continuous numbers ensity, n (, t), the number of iniviuals at time t of iameter ( number cm 1 ). Note that this is a loose interpretation, since the numbers ensity is akin to a scale probability ensity (see, e.g. e Roos, 1997, p. 138) with, e.g., t u N = n(, t) ( 1) l being the number of trees in the iameter interval [, ] at time t; thus, Nt = n(, t), or at equilibrium, simply N. In what follows = an = unless state otherwise or more generally an unattainable asymptotic maximum iameter (e.g. e Roos, 1997, p. 142) to etermine the theory, but in practice it woul be some maximum iameter limit. l u 19

3 Forestry The vital rates are require for the etermination of the moel. Diameter growth, t= g(, t) ( cm yr 1 ) is assume to be annual in time step, as is per-capita mortality m (, t) ( yr 1 ). The recruitment rate, R t ( number yr 1 ), epens on the per capita birth rate, b(, t) ( yr 1 ), at which iniviuals of p-age give birth to those of p-age (VanSickle, 1977a) an is given by the renewal equation R = b(, t) n(, t) ( 2) t bu bl n(, t) m()n(, t)δ where R t is total recruitment an b(, t) = for [ bl, bu ], the iameter limits between which reprouction occurs. VanSickle shows that (2) can be recast to the bounary conition R = g(, t) n(, t) ( 3a) t or, at equilibrium, the recruitment is constant over time, i.e. R = g( ) n( ) ( 3b) Regarless of the form (2) (3b), recruitment is the require input into the system that counterbalances mortality an supplies the fresh pool of iniviuals to support the movement of trees through the istribution. A balance equation states that changes to the target state variable in the system (e.g. numbers, mass, volume) must come from known inputs (sources) an exit via known outputs (sinks) (Gurney an Nisbet, 1998, p. 1; Soetaert an Herman, 29, p. 18). Using this concept, one can state a simple balance equation for the population growth rate as N = bu ( ) ( ) b, t n, t m t (, t ) n (, t ) ( 4 ) bl At equilibrium (r = implies N t = uner exponential population growth) this can be recast as a constraint on the system that ensures an exact balance between inputs an outputs; viz., bu bl b(, t) n(, t) = m(, t) n(, t) ( 5) or, uner constant recruitment an mortality (M) R = MN ( 6) Continuing with this concept of balance, one can erive a iscrete balance equation for population size through iameter given the vital rates for the population. Let Δ be a very small increment in iameter, an similarly let Δt be a very small time increment. Then the balance equation for the net change the ifference between the number of trees at times t an t +Δt in a class of with Δ is (e.g. e Roos, 1997, p. 141) n(, t + Δ t)δ n(, t)δ = m( ) n(, t)δδt +[ g ( l) n ( l, t) g ( +Δ n ) ( +Δ t, )]Δ t ( 7) Here, to make things simpler, the vital rates are assume to be time-invariant, epening only on iameter. The relationship l l g( l )n( l,t) l Δ in (7) is illustrate in Figure 1; the change in the numbers ensity for a small increment of iameter is compose of movement into the class (ingrowth) an movement out (upgrowth) plus mortality. Finally, at equilibrium, (5) or(6) ensures that the system is in perfect balance, sensu Meyer. The iscrete balance equation (7) has very practical applications, not the least of which is the coupling of population vital rates with the iameter istribution. Also slight algebraic rearrangement leas to an accounting equation use to grow stems in uneven-age moels as will be shown in the following sections. The continuous size-structure moel The iscrete balance equation (7) can be easily recast into the PDE (e.g. e Roos, 1997, p. 141) yieling the size-structure moel of the flow of iniviuals first introuce by VanSickle (1977a) n (, t) [ gtnt (, ) (, )] + = m (, tn ) (, t) ( 8) t A simplification of this equation was first suggeste by McKenrick (1925) in the stuy of vital statistics for human populations. It was later inepenently introuce as a balance equation in the stuy of cell populations by Von Foerster (1959). In both cases, age was use so that g (, t) = 1. Various forms of this equation are use in other fiels where, for example, it is also known as the transport equation with ecay (Olver, 214, Ch. 2; Picar et al., 21). Other variations to this moel exist; for example, it has also been extene to two inepenent variables, age an size, by Sinko an Striefer (1967). Because mortality is a sink term there must be an accompanying bounary conition, e.g. (3a), as a source term for input into the system. Finally, an initial conition n (,) = n ( ) can be specifie when not working with the steay-state form, making this an initialbounary value problem. While the iscrete form is convenient for unerstaning the genesis of (8) an for etermining iscrete projection equations, the PDE form allows the evelopment of a more formal u g( u )n( u,t) Figure 1 The flow through the system at a small time interval Δt; asan example, the numbers ensity, n (, t), is given by the exponential istribution. 2

4 A emographic stuy of the exponential istribution applie to uneven-age forests mathematical analysis, particularly for the steay-state istribution. At equilibrium the time ifferential term n (, t) t is zero, an applying the prouct rule to the size ifferential with some algebra yiels the equilibrium numbers ensity in terms of the population vital rates R m ( ) n ( ) = exp > g ( ) g ( ) R = ( ) ( ) g S 9 ( ) where recruitment is constant as in (6). Furthermore, the exponential term is a survival function, S ( ), giving the probability that a tree survives from size to size (VanSickle, 1977a; e Roos, 1997, p. 133). In some cases, the survival integral can be etermine analytically yieling a closeform solution to the numbers ensity. In others, where either mortality or growth are more complicate functions, the integral is unyieling to analytic solution, but can be integrate numerically. Finally, note from (1) that the population size is given by N = n( ) ( 1) uner the equilibrium istribution (9). The negative exponential special case Because our concern is with balance in uneven-age stans uner the exponential istribution moel, it is natural to inquire what form the growth an mortality functions can take in orer to yiel an exponential istribution when applie to (9). In other wors, we woul like a resultant negative exponential numbers ensity with corresponing PDF n ( ) = Nψ exp( ψ) ( 11) p ( ) = ψexp( ψ) ( 12) For example, Muller-Lanau et al. (26, their supplementary ata) showe that when all vital rates are constant, i.e. g ( ) = G, m ( ) = M an Rt = R, then the solution to (9) istheexponential istribution. However, there is a more general solution to the question, with constant rates being a special case. It is straightforwar to emonstrate (Supplementary ata section S.1) that given a iameter growth function g,ifthemortality ( ) function is constraine to the form m ( ) = ψg ( ) g ( ) ( 13) where g ( ) = g ( ) (e.g. VanSickle, 1977a, 1977b, 1977c), then the resulting numbers ensity will be negative exponential in form with rate parameter ψ (Dr Anré M. e Roos, pers. comm.). Rearranging (13), we have in general that the rate parameter for the size-structure exponential istribution is m ( ) + g ( ) ψ = g ( ) ( 14) which implies that the right-han sie is a constant for all. It is trivial to show that in the case of constant rates, (14) reuces to M ψ = ( 15) G since g ( ) = in this special case. The pool of possible iameter growth functions is large, though it is not immeiately clear that any posite growth function woul yiel a mortality function constraine by (13) that is equally appropriate to the ata. For this reason, an because constant rates are simple an can be valiate with existing ata, much of the remainer of this paper focuses on this special case, though an example with a more general growth moel uner (13) will be revisite later. Uner the constant rates scenario, the initial form of the solution to (9) is n R M exp G G 16 ( ) = ( ) Integrating (16) over [, ) verifies (6), an substituting this result into (16) for R yiels n M G ( ) = exp ( 17) N M G or iviing through by N, the PDF is p M G M G ( ) = exp ( 18) an finally, using (15) yiels (11) an (12), respectively. These results lea irectly to the following relationship for q uner the size-structure moel with constant rates q M G w = exp ( 19) where w (cm) is the iameter class with. Much like the survival term in (9), the interpretation of (17) with rate parameter (15) can intuitively be escribe in terms of a survival function. If mortality is small compare with growth, then ψ is small an the numbers ensity (17) will have a slow ecline with a heavy tail since many trees survive to the larger iameter classes. Alternatively, if mortality is large compare with growth then so is ψ, consequently trees ie quickly in the istribution, which is characterize by many small trees an few trees surviving to the larger size classes. The inverse problem Up to this point, it has been assume that the vital rate parameters for a population are known, perhaps from permanent growth ata. However, this will not always be the case, an estimates of both recruitment an mortality are more ifficult to come by than those of annual growth, which can be euce 21

5 Forestry from, e.g. increment cores. Therefore, it is of interest to look at the exponential version of the SSD (18) to see how one might estimate a subset of the vital rate parameters. This is often calle the inverse problem in the literature, because we are backing out the unerlying ynamics from an observe istribution. The alternate forwar problem fins the istribution from the known ynamics (Banks et al., 1991; Metz an Diekmann, 1986; van Straalen, 1986; Woo, 1997). In our case, the forwar problem is known since we know the form of the equilibrium istribution, the inverse problem requires a metho to estimate any unknown vital rate parameters. Uner the special case of constant rates, the relationship for the exponential rate parameter in (15) shows that one of the two rates, growth or mortality, must be known as there are infinitely many combinations of this ratio that will yiel the same value of ψ. As note above, it is assume that growth has been estimate from fiel ata because it is more reaily obtainable in practice. Because (18) is a PDF, any statistical metho that is useful for estimating parameters for ensities can be use. In this case, maximum likelihoo (ML) willbeusetofina mortality estimator using the observe iameter istribution an the sample-base estimate of growth, Ĝ. In general, the ML estimator (MLE) for the negative exponential istribution is ψˆ = D 1,where = n D n i = 1 i for a sample of n trees. Substituting this into (15) yiels(supplementary ata section S.2) ˆ Gˆ M = D ( 2) as an estimator for mortality given estimates of growth an average iameter. Similar reasoning can be employe for the more general case (14). That is, if ψ is estimate from the arithmetic mean of the sample iameters, then (13) provies the estimator for mortality base on ψ an some fitte growth function, where g ( ) an m ( ) share the same parameter values. In either case, mortality can be estimate from the growth function an sample average iameter for the population uner stuy. Now the steay-state balance equation (6) provies an estimator for recruitment as long as an estimate of the population size (e.g. number of trees ha 1 ), ˆN, is available; i.e. Rˆ = MN. ˆˆ The growth estimate will normally contain some uncertainty but for simplicity, we will assume that the effects ontheestimateofmortalityanrecruitmentarenegligibleas these errors woul presumably be minor in relation to the error in D from a sample of small n; an a rigorous assessment of the effects of these sources of error is beyon the scope of the current stuy. The extension of this estimator for recruitment uner the more general assumption (13) isgiven in Supplementary ata section S.5, an emonstrate in the examples. Given iameters partitione into iscrete classes, then from (1) let N = n(, t) ( 21) it, u i li be the number of trees in the ith iameter class [ li, ui] of with w at time t. Then the general iscrete movement equations for the time interval [ t, t +Δ] t (i.e. Δt 1 for annual growth) follow N = N + ingrowth upgrowth mortality ( 22) it, + 1 it, for all iameter classes i = 1,, n. Here ingrowth is growth in number of trees into the ith class from class i 1 an is equivalent to recruitment, R, for the lowest class. Likewise, upgrowth is the growth in number of trees from class i to i + 1. Note that we assume without loss of generality that a tree will not grow more than one iameter class uring the growth perio: i.e. GΔt w. This assumption is in accorance with Usher (1966) for the matrix approach. There are at least two ifferent specific solutions to the movement equations (22) base on irect partitioning of the ensity, or on the iscrete balance equation (7). In the following, the istribution is assume to be in steay state, so the timeinvariant numbers ensity, n ( ) replaces n (, t). As a consequence, the stan at time t + 1 will exactly equal the stan at time t if the movement equations are correct. Movement via ensity partitioning Note that when the growth rate is constant, every tree will grow GΔt cm per year. Thus, in each iameter class i, the only trees that can move up one class are those in the interval [ Gi, ui] where Gi = ui GΔt (Figure 2a). Since the steay-state PDF is known up to an estimate of the vital rates, the number of trees in this interval is given by pf it, N = N p( ) i = 1,, n t u i u i G i = n ( ) i= 1,, n ( 23) G i where N t is the total number of trees in the stan at time t. Therefore, the accounting equations (22) become pf it, + 1 it, it, it, N = N + R N MN i = 1 pf pf it, i 1, t i, t it, = N + N N MN i = 2,, n ( 24) The type of movement given in (23) has eviently been recognize previously (e.g. Kohyama an Takaa, 1998; Picar et al., 28). In what follows, we will refer to (23) an (24) as the approximate accounting/movement equations. Discrete movement equations The general form of iscrete movement or accounting equations for uneven-age stan projection in terms of numbers have been known for ecaes (e.g. Aams an Ek, 1974). Movement via the PDE A secon version of (22) can be erive irectly from the balance equation (7), which will yiel the exact ynamics when the iameter istribution is in equilibrium. First, let 22

6 A emographic stuy of the exponential istribution applie to uneven-age forests (a) n(,t) (b) n(,t) G G li 1 li ui li 1 li ui Figure 2 An illustration of growth projection using (a) approximate an (b) exact rules for the numbers ensity, n (, t) n ( ), for the system in equilibrium as given by the exponential istribution. The shae areas of with = GΔt show the ensity that shifts from one class to the next ue to growth uner the two projection methos (refer to the text for more explanation). then pe it, ui N = GΔ t n( ) i =,, n ( 25) pe pe it, + 1 it, it, i 1, t i, t N = N MN + N N i = 1,, n ( 26) where recruitment is given by (3b) when i =, since u. The quantity efine in (25) is a rectangle of with GΔt an height given by the point estimate of the numbers ensity at the upper limit of the class. These ynamics are illustrate in Figure 2b. Note that only a portion of the ensity in a particular block of with GΔt is shown to move as compare with (23) an Figure 2a. The approximate ynamics intuitively may seem correct at first glance. This is because the same geometric progression of trees given by q that preserves the ensity relationship from one class to the next also preserves any portion of ensity; for pf l + Δ + example, N q i G t 1 it, G n( ), where q l G is the iminution i + 1 coefficient for a class of with w = GΔt. From this the recruitment for the approximate metho is given by + GΔt R = qg n ( ). However, R > R, where R is given by (3b), which is the correct amount of recruitment to balance mortality M = RNfrom (6). As a consequence of the balance in (5) an (6), if recruitment is given by R, then a new estimate of mortality is require to balance the source sink relationship of the system; this is given by M = RN yieling M > M. However, this now leas to a new exponential istribution with rate parameter ψ = M G as in (15). Thus, while the ynamics given by (24) are now balance for the original exponential istribution, the require vital rates R an M now apply to an exponential istribution with rate ψ an not the original istribution with rate ψ in (15). The conclusion is that too much recruitment is require for the equilibrium istribution to be in balance using the approximate ynamics as illustrate in Figure 2a. However, because the amount of recruitment given in (3b) is proportionately the same (with appropriate q ajustment) as the ingrowth into each class, an because recruitment balances with mortality (6), the system is exactly balance an remains so when projecte using (26). All of this can be euce geometrically by careful examination of Figure 2. It shoul be note that the ifferences ( R R) an ( M M) may be small in practice, epening on the population, yieling small ifferences in growth. In aition, for a istribution that is not in equilibrium an oes not exactly follow the negative exponential, either of the two systems may project better than the other. A conservation law: The fact that equations (26) efine the exact movement of trees is no mathematical coincience. The systems (5) an (8) efine a conservation law with source sink balance in general, not only in the case of constant rates. The PDE (8) gives the ifferential form of the conservation law, which is at equilibrium when the time ifferential is zero as previously mentione; this latter case applies to the exponential istribution. The integral form of the conservation equation with source sink terms assuming time-invariant vital rates is (LeVeque, 22, p. 22). t u l u n (, t) = ( ) ( ) gnt, l u mn ( ) (, t) ( 27) l This equation states that the change in the numbers ensity between any.b.h. limits [ l, u] is equal to minus the flux through the enpoints minus the mortality (sink). In Supplementary ata section S.3 it is emonstrate that this equation leas irectly to the master movement equation (22) an provies the link between this an (26) with general growth equation component to the flux, leaing to (25) in the special case of constant growth. It is important to unerstan that the equilibrium system efinition given here for the exponential istribution means that iniviuals are conserve through time because of the source sink balance (5). This again correspons to the intrinsic rate of increase being zero as previously note. 23

7 Forestry The mortality ensity It may seem reasonable to posit the assumption that if the mortality an growth functions are exactly proportional over all iameters, then the result is a negative exponential istribution that can be use to moel the tree iameter istribution. In other wors, let not negative exponential, but whose mortality ensities are (Supplementary ata section S.4). The mortality ensity (3) shoul be compare with the equilibrium numbers ensity in (9), where the recruitment scale factor (constant) in the latter has been replace by the mortality (shaping) function. Integrating (3) gives the probability of a tree ying in, e.g., a given size range as ψ = m ( ) g ( ) ( 28) M(, ) = m( ) ( 31) l u u l be the negative exponential rate parameter uner this assumption. Then, this result leas immeiately to the probability ensity m ( ) m p ( ) = exp ( ) ( ) g ( ) g ( ) 29 which is equivalent to (12) uner (28), with a presume numbers ensity following from multiplication by N as in (11). However, it will be emonstrate in what follows that the interpretation of (29) leaing to a numbers ensity is incorrect in general uner the assume ynamics in (8) for two reasons. First, (28) only satisfies (14) in the case where mortality an growth are constant (i.e. g ( ) = ). Secon, uner the sizestructure vital rate ynamics, the ensity in (29) is a special case of the probability ensity for mortality uner assumption (28). Thus, it woul be erroneous to infer a numbers ensity from this PDF in the sense of (9), again with the exception of the special case of constant vital rates. Most sources on survival methos erive the mortality ensity from the survival function in terms of age-structure population moels (Keyfitz, 1977, p. 5; Elant-Johnson an Johnson, 198, p.5).however,e Roos (1997, p. 134) has expane this erivation to size-structure moels, with survival S ( ), yieling the general form of the probability ensity for mortality m ( ) m ( ) m( ) = exp ( 3) g ( ) g ( ) It is important to note that m( ) is the general form for the mortality ensity for any growth an mortality function regarless of whether assumption (28) hols or not; i.e. it is not necessary for mortality an growth to be proportional, proportionality is a special case yieling a negative exponential mortality ensity of the form (29). In other wors, (3) applies equally to vital rates that o not give rise to negative exponential istributions; i.e. when (14) oes not hol. The mortality ensity arises from the ynamics for the survival function, S ( ) t (e.g. e Roos, 1997, p. 134), while the equilibrium numbers ensity arises from (8) with time ifferential zero; the ifferent respective ynamics unerlying the two ensities reinforces that they can not be equivalent in general. Thus,thevitalrateynamicswillnot,ingeneral,support(29) as an equilibrium ensity useful in moeling the iameter istribution. Because of this lack of corresponence at the most basic level, it is straightforwar to evise examples with vital rates that follow (28) an prouce numbers ensities that are The relation between total mortality,, as etermine from the numbers ensity, the ensity for mortality, (3), an recruitment is given in general as = mn ( ) ( ) m ( ) = R ( ) g ( ) S = R M(, ) = R ( 32) where the support for iameter is [, ), an can be finite as will be emonstrate shortly. Once again, there is conservation of iniviuals through the balance between total mortality an recruitment in this general (not necessarily negative exponential) case. Application examples The examples that follow illustrate many of the concepts presente in the previous sections. The equations that have been presente for the iscrete version of the PDE an the corresponing matrix formulation iscusse later (Supplementary ata section S.7) can be shown to be equivalent, an therefore yiel the same results when applie to ata, either simulate or actual. Therefore, only one set of results encompassing both methos is presente. Exact exponential In this section, the results are applie to known exponential istributions in orer to explore the estimation of vital rates an application of growth ynamics. Effects of sample size on estimation of vital rates First, a set of simulations were esigne to emonstrate the convergence of the MLE for mortality uner constant rates as sample size, n, increases given the assumption state earlier that growth rate an population size are assume known. The results for recruitment will parallel these, since recruitment is estimate as R ˆ = MN. ˆ Figure 3a presents three ifferent exponential istributions all with the same vital rates with the exception of mortality. As note earlier, higher mortality naturally leas to a steeper ecline in the ensity, all else being equal. Figure 3b illustrates the convergence of (2) with increasing sample size. Likewise, the relative root mean square error 24

8 A emographic stuy of the exponential istribution applie to uneven-age forests (a) f() (b) Estimate Mortality (M) ^ bh Sample size (n) (c) RMSE Mortality (%) () Capture Rate (%) Sample size (n) Sample size (n) Figure 3 The effect of sample size, n, on the MLE of mortality, ˆM with true M =.3, G =.5 an R = 62 for all panels. Panel (a) illustrates three exponential istributions with M =.3 (soli),.1 (ashe),.2 (ot-ashe). Panels (b) an (c) show estimate mortality an relative RMSE, respectively. Lastly, the confience interval capture rates are shown in (). Panels (b) () are base on 1 Monte Carlo raws for each n. 1 n ( RMSE % = ( M ˆ M) 2 n), is given in Figure 3c, with a M i= 1 minimum of 4.5% at n = 5. The final panel, Figure 3, shows the percentage capture rate for confience intervals aroun the exponential mean (15). Because the only estimate parameter in the mean is ˆM, these results can be applie to mortality. The convergence results for ˆM an the confience interval convergence seem to suggest that a sample size of at least n = 1 is necessary to estimate mortality with 95% coverage, while n = 2 woul be better, an this estimate woul have an associate error estimate of 7 1% of the true unknown mortality for the tract (Figure 3c). These results are the same for all three exponential istributions epicte, an thus will be consistent for any other (constant) vital rates chosen. However, it shoul be note that somewhat ifferent results will occur in repeate simulation runs, an epen on the number of Monte Carlo samples rawn, so these figures shoul be use simply as rough but reasonable guielines. Stan projection comparisons The exact an approximate ynamics as compute from (26) an (24), respectively, are compare in Table 1. The RMSE for the exact ynamics is zero, while that for the approximate ynamics is.3 trees ha 1 (for the classes shown). Note the slight unerestimation of number of trees in the smallest iameter class uner the approximate ynamics. This is attributable to the amount of recruitment (6), which is exact for exact ynamics, but slightly too small for approximate ynamics, as note earlier. This slight unerestimation affects only the first iameter class, the remaining classes are overestimate ue to the extra trees coming into the class (Figure 2) until the ifference between the methos becomes very small in the largest classes. This example illustrates the small magnitue of the ifferences overall between the two approaches. Therefore, if the iameter istribution follows an exact exponential with constant vital rates, either metho coul be use to project the stan in practical applications. Virgin harwoos Meyer an Stevenson (1943) stuie virgin beech birch maple hemlock (northern harwoo) stans in northern Pennsylvania. They presente growth increment ata by iameter class for six ifferent tracts ranging in size from 6 to 6 ha. These authors also presente iameter istributions for eight ifferent structural types, which were not necessarily equivalent with the original six tracts (Meyer an Stevenson, 1943, p. 48). However, as note by the authors the growth ata were only slightly ifferent for the six tracts; therefore, for illustration of the methos, in the following analysis the growth for stans A F was applie to the average iameter istributions for structural types A F as if they were inee from the same areas. Hereafter 25

9 Forestry the tract-structural type combinations are referre to simply as stans. The relevant stan attributes an parameter estimates are presente in Table 2. Attributes base on iameters such as average iameter, D, an all estimate parameters such as ˆM, were calculate by shifting the iameter istribution such that the lower limit of the smallest class aligne to zero. The growth ata were igitize from Meyer an Stevenson s figure 4 an an average iameter growth, Ĝ, was calculate for both harwoo an hemlock components for each stan in accorance with Table 1 Number of trees per hectare by 4-cm iameter class for a stan with q 1.5, N = 124, G =.5, M =.5 an R = 65.b.h. (cm) N t + exact Table 2 Stan parameters an estimate vital statistics for six virgin northern harwoo stans from Meyer an Stevenson (1943) (see Figure 4) a Stan N t 1 N t+ 1 approximate Note: The larger iameter classes totaling three trees are not shown for clarity. Parameters A B C D E F Area (ha) N ( ha 1 ) D (cm) G ˆ ( cm yr 1 ) ˆM ( yr 1 ) ˆR ( ha 1yr ) ψˆ ( cm 1 ) RMSE N pf ( ha 1 ) RMSE N pe ( ha 1 ) Note: The table entries are roune, therefore calculating R ˆ = NM ˆ from the entries will only approximate the given value (true, also roune) in each stan. a Parameters are base on shifte iameters where applicable (see text for etails). the theory presente previously. It shoul be note that the growth ata for both species components of each stan were essentially constant with minor ranom fluctuations by iameter class. A weighte mean of the species components of growth was use for the analysis in each stan base on the percentage of hemlock in the structural type (Meyer an Stevenson, 1943, their table 7). The RMSE values presente in Table 2 clearly emonstrate that both the approximate an exact ynamics are able to reprouce the original stan istribution to a egree of accuracy in the hunreths of a tree per hectare. This is corroborate graphically in Figure 4, where the ifference between the original stan ata an the projecte stans is inistinguishable. Note that the negative exponential istribution estimate from the parameters in Table 2 is also presente for reference. Stans A C are quite close to the theoretical q istribution, while stans D F are slightly less so. However, since the RMSE s are of the same magnitue for all stans, it is encouraging that both sets of ynamics are able to project the stan to a reasonable egree of accuracy even when they are not necessarily exponential, but only close. This analysis further emonstrates that the estimation of mortality by the MLE (2), an subsequently recruitment, from empirical average growth rates an iameter istributions is reasonable for stans of the nature escribe here, an is therefore vali not only for ata that conform to an exact exponential istribution as presente in the previous section, but for empirical ata that are close to exponential. There are two extra sources of error in this analysis beyon those normally associate with such ata (e.g. measurement, sampling): the first is the arbitrary matching of tract growth with structural type istributions, the Trees per hectare A B C D E F DBH (cm) Figure 4 Observe iameter istributions (ots) for six virgin northern harwoo stans from Meyer an Stevenson (1943) with approximate (soli) an exact (ashe) ynamics applie base on the parameters in Table 2. The ML fit to each stan are shown for comparison (otashe)

10 A emographic stuy of the exponential istribution applie to uneven-age forests Numbers Density: n() DBH CR Growth & Mortality Probability of Mortality & Survival DBH DBH Figure 5 The equilibrium numbers ensity (left) with N = 1, R 46.3, ψ =.1, m =, γ =.2, η =.4 an = 2 uner assumption (13) a truncate negative exponential numbers ensity; an (top right) the riving CR growth (soli), mortality (ashe) an growth erivative (ot-ashe) functions, with (bottom right) probability of survival (otte) enote at the upper limit of each iameter class (soli ots) an probability ensity for mortality (bar/ashe) (note that the first an last.b.h. classes carry more ensity in both histograms because they are wier than the other classes). secon is igitization error associate with the growth trens. The accuracy of the moel results, even with these two extra sources of error, hints at a reassuring robustness with respect to the equilibrium moeling approach. Generalize growth an mortality example This section presents an example using the general growth an mortality relation (13) where growth follows a Chapman- Richars (CR) moel. The CR function for iameter growth is given by (Pienaar an Turnbull, 1973) m 1 η m γ g ( ) = γ < = ( 33) η with erivative g ( ) = ηmm 1 γ ( 34) where m, η an γ are moel parameters to be estimate from the ata. The CR moel has an asymptotic maximum iameter limit given above, which will therefore yiel a numbers ensity in the form of a truncate exponential istribution. It is important to note that not all parameter combinations for (33) 1 are vali uner the constraint (13) because mortality must be positive. Therefore, any parameter combinations that result in g ( ) > ψg ( ) are inamissible. The class of amissible forms uner CR appears quite restricte. Moreover, it is ifficult to etermine exactly what parameter values will fulfill this constraint for the CR moel ue to the interaction of the parameters. However, trial an error shows that a ecreasing slightly curvilinear or linear function are possible special cases (along with constant rates), where the linear case is emonstrate below. Figure 5 shows the results of a simple example where mortality is etermine by (13) with ψ =.1. The CR parameters for the example are m =, γ =.2, η =.4, yieling = 2 cm. These parameter values result in the following two linear equations for growth an mortality g ( ) = η γ=.4.2 m ( ) = ψη+ γ ψγ=.6.2 The first an last iameter classes shown in both histograms are wier than the nominal w = 1 cm with for the other classes, an hence there is more ensity shown in each of these classes. Recruitment can not be calculate from relation (6), since mortality is not constant in this example. However, it can be 27

11 Forestry shown (Supplementary ata section S.5) that for any growth an mortality moel an analogous result is that R = Nm ( 35) where m is efine as an expecte mortality rate in the Supplementary ata. In this example, one can verify that R 46.3 stems ha 1yr 1 when the population size is N = 1 stems ha 1. Discussion The subject of this paper was introuce with a iscussion of the two ifferent, generally accepte concepts of balance in uneven-age forests or stans, an how the stricter efinition of this concept can be interprete in the more general sense of a stable steay-state structure. The subsequent emographic analysis showe that the negative exponential istribution oes inee fulfill the requisite assumptions for this type of structure. However, the analysis also emonstrates that the exponential istribution is also a strict moel in terms of the unerlying ynamics: (i) one recruite tree must replace each tree lost to mortality ue to the equilibrium assumption, (ii) mortality rate must follow (13), which implies a possible restrictiononmoreflexible growth moel forms such as the CR. Now, of course, forest stans are not exactly eterministic systems, so the inputs an outputs may not balance exactly each year, rather on average this nees to be true. Likewise, as we have emonstrate with the virgin northern harwoo ata, the iameter growth oes not nee to be exactly equal among size classes in the constant rate case in orer for the theory to work, some reasonable amount of variability is allowe (Meyer an Stevenson s figure 4), but any eparture from the mean that oes not fulfill the mortality constraint (13), woul lea to a non-exponential (though perhaps still reverse J-shape) structural moel base on (9). Inee, ue to the rigiity of the exponential assumptions with regar to vitalrates,itissomewhatreassuring that it oes inee seem to apply to actual virgin stans such as those in Meyer an Stevenson (1943). Thus, we have emonstrate that, even with the restrictive assumptions in vital rates, the negative exponential istribution can be a vali moel for natural unisturbe stans. In aition, knowing iameter growth with a reasonably large sample of iameters alone on such forests is eviently sufficient information to estimate the remaining vital rates of mortality an recruitment, the latter provie a goo estimate of population size, N, is also available. Manage stans The negative exponential or q istribution is simply a mathematical moel. This moel has been use extensively in uneven-age management, presumably in large part ue to its simplicity, an the iea that manage uneven-age stans shoul structurally mimic natural unisturbe stans that are foun to be in equilibrium. However, Meyer (1952) clearly istinguishe two ifferent conceptualizations of balance uneven-age stans as iscusse earlier. The eluciation of the moel in terms of the couple ynamics an structural component iscusse to this point have been in terms of a stable steay-state istribution where there has been no option for harvest, since the balance is maintaine through three components to the flux: inflow from recruitment, outflow from mortality an flow through the size classes ue to growth (Figure 1). Intervention into the stan in terms of harvesting is not accommoate in the PDE moel presente thus far. If we aopt Meyer s efinition of balance for manage stans, then there is nothing more to be one, because one woul simply pick a target structure an harvest the growth as previously iscusse. This is the pragmatic view of uneven-age management, but the stan is not balance in the sense of conservation of iniviuals an therefore eparts from the mathematical ieals of a theoretical conservation equation, unlike the equilibrium setting. Inee, in the stable-steay state, there are no extra trees to harvest. It is, however, possible to exten the results presente thus far to incorporate harvest into the moel, an this may be of some interest, even if only for peantic reasons. This can be one in a way that oes conform to a stable steay-state balance moel. However, as will be emonstrate, to retain the balance the moel must be precisely followe (with some leeway for stochasticity as mentione above) to keep the stan in this equilibrium state. The numbers ensity moel formulation extening the stable equilibrium theory to harvesting is shown below, etails for several other components to the overall moel are foun in Supplementary ata section S.6. Again, the following concentrates on the special case of constant vital rates as a tractable example. In the special case of constant vital rates harvest, h(, t), is also a constant per capita rate, just like mortality, i.e. h(, t) = H. Aing harvest, the equilibrium numbers ensity, (9) an (17), becomes + + n ( ) = N M H M H exp ( 36) G G + = N M H G M + H ( ) G exp 37 respectively. The integral form in (36) is presente to reinforce that harvest, like mortality, is a component in the survival function. Thus, if it is not constant, then it is iameter-epenent, h( ), an therefore must effectively steal some trees from mortality while both retaining the same form in (13) in orer to preserve the negative exponential moel. To further illustrate, take the simplest case where trees less than a prescribe iameter limit are not harveste, while larger trees are. That is, efine the inicator function if < ( ) = 1 if H H ( 38) where H is the lower iameter limit for harvesting. Then the numbers ensity (36) can be written as 28