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Modelling tropical natural forests dynamics: which level of description?
Abstract
La modélisation de la dynamique des forêts naturelles tropicales peut reposer sur deux niveaux de description : l'arbre ou la distribution en taille des arbres au sein du peuplement. Mes travaux ont porté essentiellement sur le lien entre ces deux niveaux de description, que ce soit d'un point de vue théorique ou appliqué. La théorie de l'agrégation permet ainsi de basculer du niveau détaillé de l'arbre au niveau agrégé de la distribution. Lorsque les interactions entre arbres ne sont pas dépendantes des distances, l'agrégation est asymptotiquement parfaite. Dans le cas contraire, il est essentiel de pouvoir modéliser la relation rétro-active de la dynamique sur la répartition spatiale des arbres. La méthode des moments apporte alors une solution approximative à la question de l'agrégation. La question peut également être abordée dans le sens inverse (désagrégation), le modèle de dynamique étant vu comme un algorithme de simulation d'un processus ponctuel marqué (la marque étant la taille des arbres). Ces questions se transposent pour traiter le cas de la diversité spécifique propre aux forêts tropicales. La théorie de l'agrégation peut à nouveau être mise en oeuvre pour regrouper les multiples espèces en groupes ayant des caractéristiques de dynamique communes.
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Growth and yield simulators may be characterized with regard to the aggregation structure employed. Individual-tree simulators are an example of a passive aggregation approach. Whole-stand models represent an active aggregation structure. Of particular interest is the disaggregative modeling approach, which employs elements of both individual-tree and whole-stand simulators. The apparent resolution of the disaggregative model is at the tree-level; however, the functional resolution is at the stand level. Disaggregation may be achieved with either additive or proportional allocation of growth. Additive allocation may not ensure positive individual-tree projections of growth. Either may be constrained so as to maintain symmetry between the whole-stand projection of growth and the aggregate of predicted tree growth. Traditionally, the disaggregative approach has seen the application of very simple structures for allocation of growth depending on tree dimension alone. However, the concept may be generalized to include traditional individual-tree growth equations constrained so as to act as an allocation of predicted stand growth.
Mémoire de DEA, Université Paris 6, Paris, 81 p.
A procedure for predicting diameter distribution at two points in time in single-species forest stands was developed based on the bivariate SB distribution model. The future diameter distribution is determined from an initial distribution and a specified initial-future diameter relationship. The model developed is theoretically appealing and it performed as well as, and in some cases better than, the established methods of forest diameter distribution prediction such as parameter recovery, percentile prediction, and parameter prediction. Although developed for forest diameter distribution prediction, the methods presented here should be useful in a wide array of problems where initial and future distributions of some size characteristic are needed.
A fundamental characteristic of any biological invasion is the speed at which the geographic range of the population expands. This invasion speed is determined by both population growth and dispersal. We construct a discrete-time model for biological invasions that couples matrix population models (for population growth) with integrodifference equations (for dispersal). This model captures the important facts that individuals differ both in their vital rates and in their dispersal abilities, and that these differences are often determined by age, size, or developmental stage. For an important class of these equations, we demonstrate how to calculate the population's asymptotic invasion speed. We also derive formulas for the sensitivity and elasticity of the invasion speed to changes in demographic and dispersal parameters. These results are directly comparable to the familiar sensitivity and elasticity of population growth rate. We present illustrative examples, using published data on two plants: teasel (Dipsacus sylvestris) and Calathea ovandensis. Sensitivity and elasticity of invasion speed is highly correlated with the sensitivity and elasticity of population growth rate in both populations. We also find that, when dispersal contains both long- and short-distance components, it is the long-distance component that governs the invasion speed - even when long-distance dispersal is rare.
We describe and apply a correspondence between two major modeling approaches to forest dynamics: transition markovian models and Sap models or JABOWA-FORET type simulators. A transition model can be derived from a gap model by defining states on the basis of species, functional roles, vertical structure, or other convenient cover types. A gap-size plot can be assigned to one state according to dominance of one of these cover types. A semi-Markov framework is used for the transition model by considering not only the transition probabilities among the states, but also the holding times in each transition. The holding times are considered to be a combination of distributed and fixed time delays. Spatial extensions are possible by considering collections of gap-size plots and the. proportions of these plots occupied by each state. The advantages of this approach include: reducing simulation time, analytical guidance to the simulations, direct analytical exploration of hypothesis and the possibility of fast computation from closed-form solutions and formulae. These advantages can be useful in the simulation of landscape dynamics and of species-rich forests, as well as in designing management strategies. A preliminary application to the H. J. Andrews forest in the Oregon Cascades is presented for demonstration.
The Usher matrix model for stage-grouped populations is studied by using a renewal-equation approach. Recurrent formulas are provided for computing the survival function, which depends on both stage and age. It is used to derive the renewal equation. The stable asymptotic behavior of the population is characterized by its stage-by-age structure, its asymptotic multiplication rate, the generation length, and the mean age at childbirth. The last two quantities play a prominent part in the sensitivities of the multiplication rate to the parameters, which are derived explicitly and used to estimate the sampling variance of the asymptotic multiplication rate. These results generalize the standard stable population theory. A forestry example is provided.
Stand growth modelling based on single tree responses to their surroundings requires a description of the spatial structure of a stand. While such detailed information is rarely available from field measurements, a method to create it from more general stand variables is needed. A marked Gibbs point potential theory combined with Markov chain Monte Carlo (MCMC) random process was used to create a spatial configuration for any given number of trees. The trees are considered as charges rejecting each other and building 'potential energy'. As an analogue of the potential energy in physical systems, the potential of a stand is defined in terms of size-dependent tree-to-tree interactions that can be thought of as related to resource depletion and competition. The idea that bigger trees induce larger potentials brings 3-dimensional effects into the system. Any feasible spatial structure is a state of the system, and the related potential can be calculated. The probability that a certain state occurs is assumed to be a decreasing function of its potential. Because more regular structures have lower potentials, by adjusting the steepness of the probability distribution the spatial structure can be allowed to have a lot of randomness (naturally regenerated stands) or forced to be very regular (planted stands). The MCMC algorithm is a numerical method of finding stand configurations that correspond to the expected level of the potential, given the size distribution of trees and the shape of the probability density function. The method also allows us to take into account spatial variation in the terrain. Some spots can be defined to have lower basic potential than others (ditch, planting furrow, etc.) in order to create areas of higher than average stocking density. A preliminary test of the method was conducted on two measured stands. The results suggest that the method could provide an efficient and flexible means of mimicking variable stand structures.
We tested the null hypothesis that diameter class transitions over a 40-year period (1940–1980) in an even-aged mixed Pinusresinosa Ait. – Pinusstrobus L. stand represent a stationary Markov process. Transition probabilities were constructed from growth data on 202 trees ≥ 5.1 cm DBH in a 0.4-ha permanent plot. For each species, a diameter distribution in 1980 was predicted with the stationary Markov model using the 1940–1950 transition probability matrix. The predicted and observed 1980 distributions were significantly different for P. resinosa (χ ² = 31.67, p < 0.01), but not P. strobus (χ ² = 7.86, not significant). In P. resinosa, growth rates of trees in the same diameter class at different points in time declined with increasing stand age. This violates the assumption of stationarity. Growth did not decline in most diameter classes in P. strobus. The model also assumes that differences in the competitive histories of trees of the same size do not affect transition probabilities. The average growth rates of trees in different crown classes within two diameter classes were significantly different for P. resinosa, but not for P. strobus. Both assumptions of the model were violated in the case of P. resinosa. The.model was not rejected for P. strobus because it is a midtolerant species capable of relatively constant growth in the understory. However, this study included only the middle period of stand development when growth rates are relatively constant. These results indicate that more biologically realistic modelling approaches which incorporate growth declines with stand age and past competitive effects should be pursued.
Ce document passe en revue un certain nombre de méthodes et de modèles utilisés pour décrire et simuler la dynamique des peuplements hétérogènes dont les forêts denses humides constituent un exemple emblématique. On y insiste sur les problèmes liés (i) à la définition, la délimitation et la caractérisation de l’objet étudié, (ii) à l’interaction entre processus démographiques et (iii) aux contraintes spatiales qui régulent les mécanismes biologiques (compétition, densité du peuplement).
On rappelle l’existence et les caractéristiques des modèles de production globaux, des modèles démographiques en temps discret, régulés ou non par la densité du peuplement, des modèles de trouées basées sur une stratification verticale du peuplement, des modèles de mosaïque, sans ou avec interaction spatiale, et des modèles de croissance individuels.
The management of a renewable resource, classified by size classes, is considered. A mathematical model which predicts the stable structure of such a resource in developed. Data on the percentage recruitment of organisms from one class to the class above and data on the regeneration of young organisms are used as elements of a matrix. The largest real latent root of this matrix gives the maximum exploitation, and the latent vector associated with this root gives the stable structure. The model is illustrated by reference to a Scots pine forest in Inverness-shire. An iterative process for finding the latent root and vector is described. This matrix method is compared with a geometric method.
The development of an analytical theory of plant demography has been frustrated by problems of complexity related to spatial heterogeneity in population structure and limiting resource (Schaffer and Leigh 1976, Pacala and Silander 1985, Pacala 1988), asymmetric competition (large individuals control a disproportionate share of the resource base) (Harper 1977, Weiner 1985, 1986), and time-dependent parameters such as birth and death rates (Clark 1991a). The problem of applying to plants the population models developed for mobile organisms, such as animals or plankton in chemostats, has long been recognized (Schaffer and Leigh 1976, Pacala 1989). Plant recruitment, growth, and mortality respond to extremely localized resource levels rather than to landscape averages.
For analysing field data as well as for modelling purposes it is useful to classify tree species into a few functional types. In this paper a new aggregation of tree species of the dipterocarp rain forests in Sabah (Borneo), Malaysia, is developed. The aggregation is based on the two criteria successional status and potential maximum height. Three classes of successional status (early, mid and late successional species) and five classes of potential maximum heights ([greater-than-or-equal] 5 m, 5–15, 15–25, 25–36, > 36 m) lead to a combination of 15 functional types. The criteria of the developed classification are chosen to suit for applications with process-based models, such as FORMIX3 and FORMIND, which are based on photosynthesis production as the main process determining tree growth. The concept is universal and can easily be applied to other areas. With this new method of grouping a more realistic parametrization of process-based rain forest growth models seems to be possible.
Short-run and long-run effects of current management regimes in Southeast Asia were evaluated for their economic returns and diversity of tree size and species. A two-species (dipterocarp and non-dipterocarp), seven size-class growth model was constructed to agree in the steadystate with data from old-growth lowland dipterocarp forests in Peninsular Malaysia. Three options for Southeast Asian forest management systems were evaluated and compared to constrained optimum regimes. The evaluation criteria were the Shannon-Wiener index to measure diversity, minimum number of trees in any species-size class, soil rent, forest value, internal rate of return, and annual yield. Regimes were compared in the steady-state (long-run) and during convergence (short-run) of stands of different initial conditions to the steady-state. Regimes that gave the highest economic performance or maximum diversity in the steady-state also did so during conversion, for stands with either high or low initial basal area. To a great extent, the ability to harvest heavily and early in the conversion period determined the economic performance of a management regime. Among the regimes studied, a good compromise between economics and diversity was to cut mostly 30- to 40-cm trees of dipterocarps and non-dipterocarps every 10 years. This would maintain some trees in all size and species classes. Financial returns would be comparable to those of other investments in Malaysia and similar to the highest yield under current management regimes, but tree diversity would be much higher.
The change in all-sized dbh distribution with stand age was simulated using data from secondary stands of different ages after clear-felling in warm-temperate rain forests in Yakushima Island, southern Japan. The cumulative basal area [cm2 m-2] of all trees larger than a given dbh x [cm]was used as an index of one-sided density effect, which primarily regulates the growth rate and mortality of the tree at x. The relative growth rate in dbh was expressed by a negative linear function of both the cumulative basal area and logarithmic dbh, irrespective of stand age. Mortality was found to be positively related to the cumulative basal area. From these empirical relations, the change in dbh distribution during the course not only of stand development after clear-felling, but also of gap regeneration within primary forests, was successfully simulated using the continuity equation, eliminating 'stand age factors' from the model. Results of the simulation satisfied the - 3 2 power law of self-thinning.
Modellers usually define groups of species to deal with tree species diversity in tropical rainforests. They may be ecological groups with a biological interest or commercial categories with a practical interest. At M'Baïki station in the Central African Republic, seven groups of species are defined at the intersect of five ecological groups and five commercial categories. The ecological groups were obtained by a cluster analysis based on species diameter growth rates, their mortality rates and recruitment rates. A forest dynamics matrix model was then built to assess the relevance of the groups. The stationary state of the stand is well predicted by the model except for one group that appears to pool species with contrasted ecological behaviours.
1 - Variables aléatoires. Théorème de la limite centrale
2 - Processus aléatoires. Théorème de Wiener-Khintchine
3 - Thermodynamique linéaire des processus irréversibles (1) : affinités et flux
4 - Thermodynamique linéaire des processus irréversibles (2) : réponse linéaire
5 - Effets thermoélectriques
6 - Description statistique des systèmes physiques classiques et quantiques
7 - Fonctions de distribution réduites
8 - Equation de Boltzmann
9 - Etat d'équilibre d'un gaz dilué
10 - Coefficients de transport
11 - De l'équation de Boltzmann aux équations hydrodynamiques
12 - Phénomènes de transport dans les solides (1)
13 - Phénomènes de transport dans les solides (2)
14 - Processus de diffusion des électrons
15 - Introduction aux fonctions de réponse
16 - Introduction aux fonctions de corrélation
17 - Théorie de la réponse linéaire (1) : réponse
18 - Théorie de la réponse linéaire (2) : susceptibilité et relaxation
19 - Théorie de la réponse linéaire (3) : théorème de fluctuation-dissipation
20 - Relaxation diélectrique
21 - Théorie quantique du transport électronique
22 - Perturbations non mécaniques
23 - Diffusion des neutrons
24 - Diffusion de la lumière par un fluide normal
25 - Mouvement brownien (1) : modèle de Langevin
26 - Mouvement brownien (2) : équation de Fokker-Planck
For the past 20 years, forest modellers have largely used the Weibull distribution to summarize diameter distributions and to predict timber volume in a forest at clearfell time. The wider family of Burr distributions is investigated here as an alternative to the Weibull distribution. Using data from 20 permanent sample plots of Pinus radiata, we compare Burr and Weibull fits, showing that the Burr distribution enhances precision by roughly 13%. The effectiveness of the Burr distribution for prediction is found to be similar to that of the Weibull distribution.
Spatial patterns of trees >10 cm dbh were studied in a 4—ha forest plot on St. John, United States Virgin Islands, by measuring nearest—neighbor distance for 28 tree species and mapping 5 uncommon species. Tree species diversity was high, with an even distribution of dominance (28 species out of 132 individuals, and H' = 4.37). A clumped distribution pattern with short intertree distances was prevalent. All but five species had average nearest neighbor distances of <25 m and most were <10 m. Indices of dispersion based on variance to mean rations for the 16 most abundant species suggested clumped distributions for 12 of the species, and only one regularly distributed species. Uncommon mapped species were located in distinct clumps of approximately 700—1300 m ² , also with small average nearest—neighbor distances. The combination of high species diversity with many clumped distribution patterns and low interindividual distances for both common and uncommon species, suggests that mechanisms other than species—specific predators are important in understanding tree diversity differences between temperate and tropical forests. The importance of microhabitats is suggested. Available published evidences indicates a prevalence of clumped and a paucity of regular distribution patterns for tree species in tropical forests.
A model for the management of renewable resources is developed, a variant of a model previously given by Usher [1966] which posed one question-is the manager faced with one or several solutions? The mathematical development here shows that there is only one solution of the model that is biologically meaningful. This is associated with the only latent root of the matrix which is greater than unity. This dominant latent root has a latent vector such that all the elements can be chosen as positive. The result is based on a study of the derivative of the model, and from this a rapid method of solving the model is described. Data for a Scots pine forest are given.
The management of a renewable resource, classified by size classes, is considered. A mathematical model which predicts the stable structure of such a resource is developed. Data on the percentage recruitment of organisms from one class to the class above and data on the regeneration of young organisms are used as elements of a matrix. The largest real latent root of this matrix gives the maximum exploitation, and the latent vector associated with this root gives the stable structure. The model is illustrated by reference to a Scots pine forest in Inverness-shire. An iterative process for finding the latent root and vector is described. This matrix method is compared with a geometric method.
Within the geoscience community the estimation of natural resources is a challenging topic. The difficulties are threefold: Intitially, the design of appropriate models to take account of the complexity of the variables of interest and their interactions. This book discusses a wide range of spatial models, including random sets and functions, point processes and object populations. Secondly,the construction of algorithms which reproduce the variability inherent in the models. Finally, the conditioning of the simulations for the data, which can considerably reduce their variability. Besides the classical algorithm for gaussian random functions, specific algorithms based on markovian iterations are presented for conditioning a wide range of spatial models (boolean model, Voronoi tesselation, substitution random function etc.) This volume is the result of a series of courses given in the USA and Latin America to civil, mining and petroleum engineers, as well as to gradute students is statistics. It is the first book to discuss geostatistical simulation techniques in such a systematic way.
[Related software at https://github.com/ogarciav/rsde]
A model and an estimating procedure for predicting the height growth of even-aged forest stands was developed as part of a methodology for modelling stand growth in forest plantations [García, 1979, in Mensuration and Management Planning of Exotic Forest Plantations, FRI Symposium No. 20, D. A. Elliot (ed.), 315-353. New Zealand Forest Research Institute]. The data consists of heights measured at several ages in a number of sample plots. The ages and the number of measurements may differ among plots, and the measurements may not be evenly spaced in time. The height-growth model is assumed to have some parameters that are common to all plots and others that are specific to individual plots. In addition to random environmental variation affecting the growth, there are random measurement errors.
The height growth is modelled by a stochastic differential equation in which the deterministic part is equivalent to the Bertalanffy-Richards model (von Bertalanffy, 1949, Nature 163, 156-158; 1957, Quarterly Review of Biology 32, 217-231; Richards, 1959, Journal of Experimental Botany 10, 290-300). The model also includes a component representing the measurement errors. Explicit expressions for the likelihood function are obtained. All the parameters are estimated simultaneously by a maximum likelihood procedure. A modified Newton method that exploits the special structure of the problem is used.
Key words: Richards model; Von Bertalanffy model; Maximum likelihood estimation; Time series; Statistical computing; Optimization.
[See https://www.researchgate.net/publication/265969578_A_Site_Index_Model_for_Lodgepole_Pine_in_British_Columbia for recent work]
[On line at http://www.jstor.org/stable/2531339]
EXAMPLES OF SPATIAL POINT PATTERNS INTRODUCTION TO POINT PROCESSES Point Processes on R^d Marked Point Processes and Multivariate Point Processes Unified Framework Space-Time Processes POISSON POINT PROCESSES Basic Properties Further Results Marked Poisson Processes SUMMARY STATISTICS First and Second Order Properties Summary Statistics Nonparametric Estimation Summary Statistics for Multivariate Point Processes Summary Statistics for Marked Point Processes COX PROCESSES Definition and Simple Examples Basic Properties Neyman-Scott Processes as Cox Processes Shot Noise Cox Processes Approximate Simulation of SNCPs Log Gaussian Cox Processes Simulation of Gaussian Fields and LGCPs Multivariate Cox Processes MARKOV POINT PROCESSES Finite Point Processes with a Density Pairwise Interaction Point Processes Markov Point Processes Extensions of Markov Point Processes to R^d Inhomogeneous Markov Point Processes Marked and Multivariate Markov Point Processes METROPOLIS-HASTINGS ALGORITHMS Description of Algorithms Background Material for Markov Chains Convergence Properties of Algorithms SIMULATION-BASED INFERENCE Monte Carlo Methods and Output Analysis Estimation of Ratios of Normalising Constants Approximate Likelihood Inference Using MCMC Monte Carlo Error Distribution of Estimates and Hypothesis Tests Approximate MissingData Likelihoods INFERENCE FOR MARKOV POINT PROCESSES Maximum Likelihood Inference Pseudo Likelihood Bayesian Inference INFERENCE FOR COX PROCESSES Minimum Contrast Estimation Conditional Simulation and Prediction Maximum Likelihood Inference Bayesian Inference BIRTH-DEATH PROCESSES AND PERFECT SIMULATION Spatial Birth-Death Processes Perfect Simulation APPENDICES History, Bibliography, and Software Measure Theoretical Details Moment Measures and Palm Distributions Perfect Simulation of SNCPs Simulation of Gaussian Fields Nearest-Neighbour Markov Point Processes Results for Spatial Birth-Death Processes References Subject Index Notation Index
Matrix models of forest dynamics rely on four hypotheses: independence hypothesis, Markov’s hypothesis, Usher’s hypothesis, and temporal homogeneity hypothesis. We investigate the consequences of relaxing Markov’s hypothesis, allowing the state of the tree at time t to depend on its states at time t−1 and t−2. The methodology for building and testing the relevance of second-order matrix model is thus proposed. The derivation of second-order transition probabilities turns to be sensitive to the width of the diameter classes. A strategy for choosing diameter classes is proposed. A second-order matrix model is then built for a tropical rain-forest in French Guiana. A different behaviour is detected between small (dbh ≤30 cm) and large trees, the smaller trees being more sensitive to their past history: small trees that have well grown have a tendency to grow well again, and small trees that have not grown tend to have a higher probability to die. The widths of the diameter classes that are selected are much less than the widths usually retained, that favour first-order selection.
An algorithm is presented to provide a linkage between stand-level models and individual-tree models. The algorithm is based on the assumptions that stand-level estimates of basal area, trees per unit area, and average diameter are correct and that the projected individual trees must be adjusted to equate to the stand-level estimates. The form of the individual-tree model is not important so long as initial and projected tree lists are provided. It is further assumed that the relative growth between individuals is to be maintained and the adjusted survivals are restricted to be less than or equal to 100%. Simplification of the general algorithm are also developed. These simplifications take the form of dropping the survival equation from the individual-tree model and dropping the average diameter estimate from the stand-level model. Examples are provided of a bimodal diameter distribution.
The Handbook of Stochastic Methods covers systematically and in simple language the foundations of Markov systems, stochastic differential equations, Fokker-Planck equations, approximation methods, chemical master equations, and quatum-mechanical Markov processes. Strong emphasis is placed on systematic approximation methods for solving problems. Stochastic adiabatic elimination is newly formulated. The book contains the "folklore" of stochastic methods in systematic form and is suitable for use as a reference work.
A common goal in functional type research is to find a useful classification that defines the dynamic behaviour of groups of species in relation to environmental variation. Long‐term data sets on the dynamics of forests are difficult to obtain; thus, it would be useful if more readily available data, such as that on morphological and life history characters, could be used to derive groups that reflect the dynamics of the species. We used a 30‐yr data set on the dynamics of subtropical rainforests in Australia to derive classification based on the dynamics of the species and compared this classification with groups of species derived by other approaches. Functional types were derived for ca. 80 tree species using subjective, deductive and data‐driven approaches. The subjective classification used was a pioneer to late successional grouping. The deductive classification was an extension of the vital attribute approach.
Two data sets were used for the data‐based classifications, one based on morphological, life history and phenological characters (morphological data) readily available from taxonomic descriptions and another based on long‐term observations on the establishment, growth and death of all individuals on permanent plots (dynamic data).
SAHN (Sequential, Agglomerative, Hierarchical and Nested) clustering techniques were used for the numerical classifications. There was some similarity between the classification based on dynamic characters and the subjective and deductive classifications. The classification based on the readily available morphological characters showed less similarity with other classifications. However, the morphological data could be used to predict group membership in the dynamic classification using discriminant analysis with 87% accuracy. Thus, it appears that surrogate classifications might be found to describe the dynamics of the subtropical rainforest site. Further exploration and testing at other sites is required, especially to link the functional classification to specific perturbations.
A density-dependent matrix growth model was constructed for the dry woodlands of Uganda basing on material collected from 42 sample plots with 7904 trees. The model was based on functions for individual tree upgrowth and mortality, and area-based ingrowth, with explicatory variables representing tree size, stand density and stand structure. The trees were pooled into three species groups basing on ecological and morphological criteria. For all groups, parameter estimates for tree size and stand density were found highly significant (p
Transformation into seminatural managed forests and sustained use of wood is a major goal to maintain nature-oriented forest ecosystems in the tropics and subtropics. In the present study a transition matrix was used to model the development of an uneven-aged subtropical humid forest of Rio Grande do Sul (Brazil) which was remeasured three times. The approach consists of an estimation of the probability matrix of diameter classes and the incorporation of recruitment and mortality from this forest. Different harvest strategies are elaborated within a sustainable management approach with the objective of sawtimber production (goal-diameters of 60 and 80cm of the valuable species, respectively). The application of a 15-year cutting cycle with a basal area removal of 7m2 (or ∼14m2 in a 30-year cutting cycle) could be proven to be a sustainable intervention strategy.
The method of pair correlation function has been modified to allow treatment of spatial distributions when the data are scattered over a number of separate sample plots. The method provides an opportunity to detect significant deviations from a random pattern, which is not possible for separate individual plots. Two sets of data have been studied: (i) a Norway spruce stand and, (ii) a mixed oak–beech forest. Larger spruces have a greater radius of spatial correlation than do smaller spruces. The arrangement of these trees has a negative correlation, with a radius of ca. 2.2–2.4 m. In spite of a decrease of density over 30 yr, the radius remained practically the same. Mutual arrangements of oaks and beeches depend on the dimensions of the beeches. Larger beeches have a larger radius of spatial correlation with the oak stems.
A deterministic nonlinear matrix growth model of naturally regenerated loblolly pine (Pinus taeda L.) stands (Lin, C.-R., Buongiorno, J., Prestemon, J., Skog, K., 1998. Growth model for uneven-aged Loblolly pine stands: simulations and management implications. FPL-RP-569. USDA ForServ., For. Products Lab., Madison, pp. 13.) predicts that in about 3 centuries, pines are replaced by more shade-tolerant hardwood species. Pines are also absent in the deterministic steady state. We modified this model to recognize the effects of small-scale, high-frequency disturbances on stand growth. Stochastic simulations with the same initial condition as the deterministic simulation gave very different results. With stochastic disturbances pines and hardwoods always co-existed and in the steady state there was about the same basal area in pines as in hardwoods. As a result, the stochastic nonlinear model predicted much higher species diversity than the deterministic one. In contrast, a linear stochastic matrix growth model would predict the same expected steady state as the linear deterministic model. Thus, nonlinearities in stochastic forest growth models may change considerably the predictions of the expected future stand state, relative to those of the nonlinear deterministic counterpart. In this study, significant differences remained between the expected stochastic and the deterministic predictions even after scaling down the stochastic term by a factor of 2–10.