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On Horn's Markovian model of forest dynamics with particular reference to tropical forests
Abstract
Horn's markovian model of forest succession is discussed. Interesting dynamic behavior and different domains of attraction arise when the replacement probabilities are density dependent. Explicit light-gap opening and colonization terms are introduced to make the model useful in tropical rain forests dynamics.
... We know of only two instances of the model being extended to incorporate nonlinear transition probabilities. Acevedo (1981) introduced a model of forest dynamics in which the transition probabilities were based on the densities of the different species. However, he concentrated on examining simple two-species communities analytically, rather than a full densitydependent model in detail. ...
... Caswell and Cohen (1995) also used a simple nonlinear model to examine the consequences of facilitation and inhibition in a theoretical community. Here we extend Acevedo's (1981) approach and develop a community model of a shallow reef crest coral assemblage in which the transition probabilities are functions of the cover of the different species groups in the model. In a companion paper (Spencer and Tanner 2008), we also examine a densitydependent continuous time version of the model. ...
... Given the more complex nonlinear nature of the model presented here, however, this may not be the situation. Indeed, the only exception to the general finding of ergodicity was in the more complex model of Acevedo (1981), in which the equilibrium composition was highly dependent on the initial community composition. ...
While density dependence is a popular topic of research in population ecology, it has received much less attention at the community level. Using 27 years of data from Heron Island, on Australia's Great Barrier Reef, we develop a matrix model of coral community dynamics that shows that community-level density dependence does occur and that it is fairly common, being found in 38% of the model parameters for which it was tested. In particular, colonization of free space (through either recruitment or growth of existing colonies) was nearly always density dependent. There were no consistent patterns in the results for mortality, persistence, or species interactions. Most transitions were found to be dependent on the cover of the incoming species group, with only a few dependent on that of the outgoing species group. In addition, few of the transitions representing species interactions were dependent on the amount of free space present, suggesting that the cover of other species does not influence encounters. When these results were combined into a model of community dynamics, it was found that density dependence resulted in a moderate increase in coral cover, which was spread over most species groups. The dynamics of the density-dependent assemblage were also a lot noisier than those of an assemblage without density dependence. Sensitivity analysis indicated that it was density dependence in the colonization probabilities, particularly of encrusting acroporids, bushy Acropora and staghorn Acropora, which had the main influence on the model, although persistence of free space was also important. Transitions representing mortality were only of minor importance, and those representing species interactions were of no importance.
... competing species. When a whole landscape is div-Size-structured ided into many microsites and new plants can only model of trees invade vacant sites, the microsite mosaic model corresponds to the Lotka-Volterra competition equations where every a equals 1. Convergence towards a singular equilibrium point of stable coexistence is almost impossible in the Lotka-Volterra model without difference in a except when an inflow of seeds from an outside seed source, or 'bass term' is allowed (Whittaker & Levin 1977;Acevedo 1981). The special case where each species has exactly the same recruitment/mortality balance mediates randomly changing coexistence, and we must seek other factors which can create and stabilize a singular coexistence point (Levin 1976;Whittaker & Levin 1977). ...
... Model I which allows for the inflow of seeds from outside the simulated area, or the existence of 'bass term', gives less information than Model S concerning the organizing mechanisms of coexistence, but it is a powerful tool for analysing secondary succession in small clearings or tree-fall gaps, where seeds are supplied from surrounding canopies. There are two dynamic numerical models of secondary succession of natural mixed forests, the multi-site-based Markovian model (Horn 1975;Acevedo 1981) and the individual tree-based gap model (Botkin, Janak & Wallis 1972;Shugart & West 1977;Shugart 1984;Huston & Smith 1987). The Markovian model divides the whole landscape into even-sized undifferentiated microsites, corresponding to the crowns of canopy trees. ...
On the basis of permanent plot research in a warm-temperate rain forest in S Japan dominated by evergreen broad-leaved trees, a multi-species model of size-structured tree populations is proposed, which takes into account the effect of one-sided competition. The strength of the density effect due to one-sided competition for light on a forest tree of a given size is expressed in terms of the cumulative basal area of trees irrespective of species larger than this tree in the forest stand. Species differences are expressed in terms of potential size growth rate, susceptibility to suppression of size growth rate, mortality and recruitment rate. A one-dimensional drift equation is applied to express the dynamics of the size distributions of species. The model describes successfully secondary succession in an open system (allowing seed inflow from outside the plot). It also yields a stable coexistence in a closed system (without seed inflow) between three representative tree species, though it takes a long time (20 000 yr) to attain equilibrium. Size structure and one-sided competition for light may stabilize forest tree communities. -from Author
... There exist a few published samples where Markov models are used in the study and modelling of successions in forests at the community level (Knyazkov et al., 1992;Rego et al., 1993;Vinogradov and Shitov, 1994;Logofet and Lesnaya, 2000), see reviews in Balzter (2000) and Logofet (2010). Less rare are those where Markov models are applied at the "tree-by-tree replacement" level (Horn, 1975;Acevedo, 1981;a review in Acevedo et al., 1995), or at the fine scale (Hobbs, 1984;Samoylov and Tarkhova, 1985;Wootton, 2001;Hill et al., 2004;Baasch et al., 2010). ...
... That notion links the predominance of primary species (in terms of species richness) in the BCI old forest with the predominance of small gaps there (Fig. l). Some authors have devised mathematical models of gap creation and regrowth to explain tropical forest composition (Acevedo, 1981) or physical structure (Torquebiau, 1981). Doyle (1981), with the advantage of good autecological data for a species-poor community, closely simulated the dynamics of the montane Luquillo Forest, Puerto Rico. ...
... That notion links the predominance of primary species (in terms of species richness) in the BCI old forest with the predominance of small gaps there (Fig. l). Some authors have devised mathematical models of gap creation and regrowth to explain tropical forest composition (Acevedo, 1981) or physical structure (Torquebiau, 1981). Doyle (1981), with the advantage of good autecological data for a species-poor community, closely simulated the dynamics of the montane Luquillo Forest, Puerto Rico. ...
Examines the nature of treefall gap regimes in terms of parameters of frequency, size, and related phenomena that determine patch character and dimensions in the vegetation mosaic. Plant regeneration behaviour in relation to gaps is discussed, with distinction drawn between primary tree species, pioneer tree species, and understorey plants and lianes. The population or size class structure of many tropical trees is related to the persistent stage in which they await gap opening. Type of persistence involves associated traits that enhance a species' ability to reach maturity in certain situations but restrict its ability to capitalise on all situations - a trade-off which produces a degree of regeneration niche differentiation regarding gap size and sites within gaps that explains to some extent the species richness of tropical forests. Regrowth in gaps is described, with comments on plant sources (previously established plants, seeds arriving before and after gap formation, and ingrowth from the gap margin), and adaptations for gap regeneration such as rapid growth are indicated. Connections are drawn between treefall regimes, species requirements and tropical forest plant community structure. -P.J.Jarvis
... Esses modelos têm sido usados intensivamente em estudos de dinâmica de populações de plantas ou animais em várias regiões do mundo. Alguns exemplos são: a demografia do jack-in-the-pulpit em Nova York (bierzychudek, 1982); dinâmica florestal de uma população de Araucaria numa floresta tropical úmida de Papua Nova Guinea e Nothofagus em floresta montana temperada da Nova Zelândia (Enright & Ogden, 1979); sucessão de térmitas em Gana (Usher, 1979); sucessão florestal na Nova Jersey (horn, 1975); aplicação da Cadeia de Markov em estudos de dinâmica florestal em florestas tropicais (Acevedo, 1981) e a aplicação de Markov para predizer o desenvolvimento de um povoamento florestal (Usher, 1966;bruner & Moser, 1973;Peden et al., 1973;buogiorno & Michie, 1980). Alder (1980) também descreve a matriz de transição como uma possível ferramenta para análise de dados de crescimento e incremento de povoamentos multianos de florestas tropicais mistas. ...
To combine protection and utilization of forest resources in the tropics, the understanding of forest dynamics is essential. It is also important in the definition of strategies for rehabilitation of degraded areas. In Forestry, forest dynamics could be translated as the understanding of recruitment, mortality and biomass increment rates over time. For this study, these rates were estimated based on measurements carried out in 2000 and 2004 over two transects measuring 20 by 2500 m (5 hectares) each, in Manaus region. This paper deals with forest dynamics of a pristine forest based on the probabilistic transition matrix (the first-order Markov Chain) approach. The main objective is to report 4-year (2000 to 2004) changes in the forest structure. Diameter distribution and tree mortality will be projected ahead to 2008 (t+2), based upon a 4-year period of observations completed in 2004 (t+1) and its immediate past in 2000 (t). In terms of fresh aboveground biomass, this site accumulated 8.34 t.ha-1.ano-1. The c 2 test has shown no statistical difference (p = 0.01) between observed diameter frequency and the expected projected by Markov Chain. This result indicates that the Markov Chain approach is a reliable tool to project the forest dynamics on a short-term basis. In 2008, the total number of individuals will have a decrease of 2.7%, and the mortality rate will 15% higher than in 2004.
... Each stage in the pathway represents a frame. Frame models can be developed using an empirical approach where the transition from one state to another is deterministic (Chew, 1997) or using a probabilistic approach where transitions are stochastic, such as Markov process (Acevedo, 1981). The timing and direction of the transitions are often quantified from extensive field and simulation data. ...
... aciculata). Microsite-based competition models were introduced by Skellam (1951) and have since become popular as models of competition for space, most notably as models of forest dynamics ( Horn 1975;Acevedo 1981). We developed a new model which allows us to consider the influences of sex expression of individual trees on forest dynamics. ...
The effects of dioecy on community dynamics were examined by using transition matrix models for two dioecious tree species, one a superior competitor with a narrow dispersal range and the other an inferior competitor with a wide dispersal range. The models are based on tree-by-tree replacements in each identical microsite occupied by either male or female canopy trees of the superior competitor and canopy trees of the inferior competitor. Coexistence of the two species is possible not only because of a trade-off between competitive and dispersal abilities but also because of the existence of a competitor gap, which the superior competitor cannot occupy. The competitor gap is created under the male trees of the superior competitor. The inferior competitor occupies the competitor gap because of its wide dispersal range. The relative abundance of the two species depends on the dispersal ability and sex ratios of the superior competitor. The decreasing dispersal ability and the female abundance of the superior competitor increase the competitor gap, which allows the regeneration of the inferior competitor.
... In forests, transitions between saplings and mature trees are usually examined . Based on Horn's approach other forest models were developed (Acevedo, 1981; Miles et al., 1985; Acevedo et al., 1995 ). Post-fire succession in Quercus forests in France was modelled with Markov chains by Rego et al., (1993). ...
A theoretical implementation of Markov chain models of vegetation dynamics is presented. An overview of 22 applications of Markov chain models is presented, using data from four sources examining different grassland communities with varying sampling techniques, data types and vegetation parameters. For microdata, individual transitions have been observed, and several statistical tests of model assumptions are performed. The goodness of fit of the model predictions is assessed both for micro- and macrodata using the mean square error, Spearman’s rank correlation coefficient and Wilcoxon’s signed-rank test. It is concluded that the performance of the model varies between data sets, microdata generate a lower mean square error than aggregated macrodata, and time steps of one year are preferable to three months. The rank order of dominant species is found to be the most reliable prediction achievable with the models proposed.
... Each stage in the pathway represents a frame. Frame models can be developed using an empirical approach where the transition from one state to another is deterministic (Chew, 1997) or using a probabilistic approach where transitions are stochastic, such as Markov process (Acevedo, 1981). The timing and direction of the transitions are often quantified from extensive field and simulation data. ...
A classification of spatial simulation models of fire and vegetation dynamics (landscape fire succession models or LFSMs) is presented. The classification was developed to provide a foundation for comparing models and to help identify the appropriate fire and vegetation processes and their simulation to include in coarse scale dynamic global vegetation models. Other uses include a decision tool for research and management applications and a vehicle to interpret differences between LFSMs. The classification is based on the four primary processes that influence fire and vegetation dynamics: fire ignition, fire spread, fire effects, and vegetation succession. Forty-four LFSMs that explicitly simulated the four processes were rated by the authors and the modelers on a scale from 0 to 10 for their inherent degree of stochasticity, complexity, and mechanism for each of the four processes. These ratings were then used to group LFSMs into similar classes using common ordination and clustering techniques. Another database was created to describe each LFSM using selected keywords for over 20 explanatory categories. This database and the ordination and clustering results were then used to create the final LFSM classification that contains 12 classes and a corresponding key. The database and analysis results were used to construct a second classification key so managers can pick the most appropriate model for their application based on computer resources, available modeling expertise, and management objective.
The bilberry-vs.-cowberry fine-scale dynamics has been modelled by a discrete Markov chain of transitions among the four observable statuses of permanent sample plots: bilberry (Vaccinium myrtillus) alone, cowberry (V. vitis-idaea) alone, both species, and species-free (Logofet and Maslov, 2019). Six successive examinations of spp. presence/absence on 20 × 20 cm quadrats located along permanent transects in a Scots pine boreal forest were initiated in 1980, 26 years after a forest fire in 1954, and repeated in every 5 years, thus scoping the 25 years of post-fire succession (Maslov, 1989). Those data have provided the exact calibration of 5 one-step transition matrices forming a nonautonomous Markov-chain model. Its basic prediction (forward forecast) has been obtained as the limiting distribution of states generated by the geometric average of transition matrices featured increase in the share species coexistence.
Presented in this paper, the alternate modes of prediction are the validation forecast and the backward prediction. The former is implemented as the geometric average of the first 4 one-step matrices and its dominant eigenvector as the limiting distribution of quadrate states. It appears to hit a 5% vicinity of the distribution observed in 2005, thus certifying a success of model validation.
As regards the backward prediction, I propose a technique that realizes the idea of backward prediction by means of reversing the time dimension in the observation data. The ensuing formula for the average backward transition matrix looks somewhat different from that for the forward one but enables calculating the matrix as reliable as the forward formula does. Although the 1955 distribution back-predicted from the observed 1980 one cannot be quantitatively interpreted as a community that might form just 1 year after the fire, it does qualitatively confirm (together with the backward limiting distribution) the directionality revealed earlier in the forward dynamics.
Data from permanent plots in a seasonal forest in the western plains of Venezuela were used to establish the parameters for FACET, a model based on individual trees and sensitive to topographical conditions of the terrain. Due to the large number of species, these were grouped according to shade tolerance (tolerant, intermediate and intolerant) and their maximum height (large, medium and small). Palm species were considered as a separate group. Forest dynamics was simulated for each physiographic position (banco, higher elevation areas formed by old river banks, sub-banco, areas of intermediate elevation and bajío, flooded depressions). Simulation results for 210 years are similar to observed values in banco and sub-banco but not in bajío. After 500 years of simulation, large-tolerant species dominate in bancos, accumulating an index of value (IV) of 27.28 followed by palms with IV= 18.86; similar results are obtained in sub-banco where large-tolerant species dominate with IV= 26.78 and palms accumulate 20.27; which corresponds to the expected composition for a non-perturbed forest. In bajío, large-tolerant species also dominate with IV= 28.74 but are followed by large-intermediate species with 20.27. In conclusion, the FACET model has a great potential to predict the dynamics of tropical forests, except in the bajíos or flooded depressions with drainage problems. Therefore, it is necessary to modify the water balance routine of FACET to achieve a better representation of forest dynamics at these flooded sites.
The allometric relations of diameter and height were analyzed for 34 tree species of the Imataca Forest Reserve, according to the Chapman and Richard equation used in the individual-based ZELIG and FACET models that simulate forest dynamics. The two allometric coefficients employed were calculated by linear regression applied to the logarithmic transform of the equation. The exponential equation was found to explain a large proportion of the variability in height even though the data came from different sites. The relationships will be employed in FACET for future work. The existence of allometric patterns was evaluated after grouping species according to light requirements (shade-intolerant or pioneer and shade-tolerant) and maximum height classes (medium 10-30m, and large >30m), forming four functional groups: medium-pioneer, large-pioneer, medium-tolerant, and large-tolerant. Differences were found in allometric coefficients between the groups, using analysis of variance and discriminant analysis. The differences in the allometric coefficients, taken one at a time, are more significant between pioneer and tolerant than between medium and large. The combination of both coefficients, b2 y b3, emphasizes differences among all groups, specially when using a linear discriminant function, which in addition, will allow predicting to what group a species may belong, given the values of its allometric coefficients.
Principles of modelling vegetation dynamics in the successional time scale are discussed. Global models of biosphere predict shifts of bioclimatic zones in response to climate changes by means of corresponding shifts in a global space of climatic indices designed in a special way. On the contrary, the localization principle lays a local geobotanic knowledge into the models, namely, the knowledge of conceptual scheme for, and regularities in, those plant successions which proceed in the area under study. The Markov-property principle (postulate) arises once a particular scheme of succession is determined, thereafter random Markov chains serve a convenient tool to describe in formal terms how phytocoenoses change in time. The ergodicity principle (hypothesis) is used whenever the behaviour of a temporal (successional) series is judged from the analogy with that of the spatial (ecological) one or vice versa, and this kind of reasoning is generalized in the fundamental mathematical notion of ergodicity. The homogeneity principle, i.e. independence of time in the transition probabilities of the chain, gives rise to a mathematical property in common of such models, namely, convergence to a stable limit distribution, thus meeting the major paradigm of succession theory: regular movement from pioneer stages to the stable (poly)climax one. However, the traditional time-homogeneous models (with constant transition probabilities) accept actually the hypothesis of environmental invariance, which can hardly be accepted when the matter is about long-term prediction in the successional time scale. Models of new generation, the inhomogeneous chains, in which the transition probabilities are designed as functions of the key factors of the environment influencing the course of succession, now loose the algebraic elegance of their homogeneous prototypes but gain sensivity to real (climatic, in particular) parameters of the environment. The problem to model successions under global changes is thus reduced to the task to scale the global climate scenario down to the level of local key factors. In the gnosiological perspective, these models signify a move from phenomenology to the causality principle, which the homogeneous description has been a priori ruling out. The theoretical principles are exemplified with a practice in modelling successions in the forest-steppe zone.
All landscapes are to some extent patchy. The biological heterogeneity of communities on patchy landscapes reflects the time scales of local biotic interactions and abiotic disturbance, the time and space scales of dispersal, and (especially) the interaction of these scales. To investigate these factors, we examine here a simple model that provides a framework for building models of patchy communities directly from hypotheses about time scales. The model has numerous applications (Caswell and Cohen, 1991, in preparation); here we focus on the interplay of competition and disturbance as well as the kinds of biological heterogeneity that can be maintained by that interplay.
The study of plants could be undertaken at essentially every level of organization within biology, from that within a cell to the entire biosphere. To cover even the main theoretical questions on these diverse levels and the mathematical approaches used to analyse them would require several volumes. My objective here is to consider a small subset of the work that has been done, dealing only with the levels normally taken as being part of the purview of ecology and touching somewhat on a few more applied problems in agriculture. I will not discuss statistical analyses of plant community assemblages nor most aspects of plant-animal interactions, such as pollination biology (Real, 1983). Biophysical approaches were reviewed earlier (Gross, 1986b). Background references should be consulted for further details (e. g. France and Thornley, 1984; Givnish, 1986a; Gross and Miura, 1986; Jean, 1984; Rose and Charles-Edwards, 1981).
RÉSUMÉ
Dans le cadre de recherches menées sur la spatialisation des modèles écologiques, un modèle de simulation de la dynamique forestière a été conçu. L'objectif est d'observer l'émergence d'un comportement global du peuplement quand seules des règles locales de compétition pour les ressources du milieu sont utilisées; ces règles déterminent les processus de croissance, de mortalité et de régénération des individus.
Pour simuler les interactions spatiales entre les individus, un prototype utilisant une architecture en réseau d'automates cellulaires a été développé : l'espace forestier est divisé de façon régulière en cellules et chaque cellule peut contenir un arbre de chaque espèce; l'état de chaque individu (espèce, âge, diamètre à hauteur de poitrine, hauteur, …) est remis à jour à chaque pas de temps (une année) en fonction de son état et de celui de ses voisins au pas de temps précédent. Le modèle simule l'évolution d'un peuplement composé d'espèces héliophiles (bouleau blanc: Betula papyrifera) intermédiaires (bouleau jaune: Betula alleghaniensis) et sciaphiles (hêtre à grandes feuilles: Fagus grandifolia).
Dans l'article suivant sont présentés différents types de modèles de dynamique forestière et particulièrement le modèle à base de trouée JABOWA (Botkin et al., 1972) et le modèle spatialisé de Wissel (Wisset, 1991). Notre modèle, qui s'efforce de relier ces deux approches, est ensuite décrit, puis des résultats de simulation sont analysés et des perspectives de développement futur du travail sont données.
Gap-phase regeneration of trees was described for the first 5-6 yr of regrowth in 30 treefall gaps (20-705 m^2) in tropical moist forest on Barro Colorado Island, Panama. Trees were classified as pioneers (saplings found only in gaps) or primary species (saplings found in gaps and in the understory of mature forest). In most of the gaps studied, stem densities rose rapidly after gap formation, then levelled off or declined by years 3-6. This pattern was particularly marked in some large gaps (>150 m^2), where pioneers attained high densities, then experienced heavy mortality. Stem density of primary species did not vary with gap size. In large gaps the mean rate of growth in height was greater for pioneers than for primary species, size-class distribution broadened more for pioneers than for primary species, and early recruits of both regeneration types grew faster than later ones. Gap formation fosters regeneration of pioneer and primary species and, in this forest, produces patches that differ markedly in tree population dynamics, species composition, and growth rate.
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.
Canonical analysis of time-series data identified the stationary Markov chain as a plausible mathematics for the recovery process in burned heathland and exploited mixed forests. The process appeared robust and highly deterministic. The results indicated that a steady state may be attained in less than a decade in heathland with continued grazing, but may require more than a century in the mixed forest after exploitation ceased. These contradict, in part, findings by others who could not justify, based on field estimates of transition probabilities and axiomatic tests, the stationary Markov chain as a valid model.
Recent history plays an important role in the physiology, behavior, and ecology of individuals, and in the dynamics of populations and assemblages of species. In this paper, we examine the impact of history on the species composition of intertidal reef corals, by comparing simulation models that incorporate four different levels of knowledge about the recent past (over a time scale of 1-27 yr). The models are Markov or semi-Markov transition probability matrix models, based on rates of colonization, persistence, and species replacement measured from a long-term study spanning three decades at Heron Island, Great Barrier Reef. Rates of colonization (transitions from free space) varied 20-fold for different species groups, while mortality (transitions to free space) ranged fivefold, reflecting a wide range of life histories among the coral assemblage. Virtually all species groups could undergo reciprocal transitions (e.g., from A to B, and B to A) in a single time interval, indicating the lack of a single competitive dominant that was capable of outcompeting all or most other species. Transition probabilities changed markedly as a function of history. For most species groups, the probability of persisting (i.e., ''transitions'' from A to A) increased with time. Thus, a colony that had occupied space for some time was generally more likely to continue to do so than a new arrival. This result is consistent with an escape in size for older colonies from mortality agents such as competition and predation. However, three species groups showed the opposite pattern. Algae, Pocilloporid corals, and fragile tabular Acroporn showed marked increases in transitions to free space after 3-5 yr, reflecting a more ephemeral suite of life history traits. Similarly, free space that had recently been generated had a higher rate of colonization than substratum that had been unoccupied for some time. These results falsify a major assumption of standard first-order models, i.e., that transition probabilities are constant, and that history is irrelevant. Although the changes in transition probabilities as a function of history were often striking, the four different models we employed show only minor variation in community composition in both transitory and climax (equilibrium) phases. Thus, while recent history was important in determining transition probabilities, it had little effect on community dynamics and structure in this system. This discrepancy is due to the rapid turnover of corals and algae on shallow reef crests, where only a small proportion of colonies survive long enough to display effects of history. All models agreed that the length of time required for this system to reach an equilibrium community structure is far longer than the observed interval between recurrent disturbances from tropical cyclones.
One of the major questions in ecology is, what controls the structure of communities? We used projection matrix models to examine community dynamics and patterns of succession. The inputs of the model are transition probabilities of species replacements that were measured repeatedly during a long-term (1962-1989) study of diverse coral assemblages on Heron Island, Great Barrier Reef. Transitions varied strikingly among species and sites, reflecting differences in recruitment, growth, longevity (persistence), and the rate of replacement of one species by another. Species that had a poor ability to persist (e.g., algae and Pocilloporid corals) were generally good colonists. The observed number of transitions expressed as a proportion of the maximum number possible provides an index of the complexity of interactions in an assemblage, analogous to the concept of connectance in food-web analysis. Transitions occurred to and from nearly every species group, indicating that there was no competitive dominant in this system. We use the models in simulations to track transitory changes in species abundance and community composition following a major disturbance (e.g., due to a cyclone or outbreak of crown-of-thorns starfish). Some species showed a rapid initial increase followed by a decline to lower equilibrium levels, while others increased smoothly to a generally higher equilibrial abundance. The length of time required to reach a climax assemblage using the same matrix recurrently (approximate to 20 yr) is far greater than the observed interval between major disturbances, supporting nonequilibrium theories of coral reef communities. Climax assemblages were highly diverse and varied in composition from site to site. The ''intermediate disturbance hypothesis'' does not fully predict successional changes in these shallow-water coral assemblages since diversity remained very high at equilibrium (i.e., long after a major disturbance), Competitively inferior species were not eliminated because routine mortality ensured that some space always remained available for colonization. We also present a novel method for quantifying the relative importance of each species interaction to community composition and the rate of succession, based on a sensitivity analysis of the transition matrix. The analysis shows that the importance of a species to the dynamics of a community may be unrelated to its abundance at equilibrium, with some rare species groups having a greater impact than more common ones. Sensitivity analysis of this type will provide a powerful means of identifying ''keystone'' species in complex assemblages where experimental manipulation of each species is impossible.
A complex analytical approach is presented with examples for the analysis of structured phytosociologal data tables. The full text is downloadable from my webpage: https://sites.google.com/site/statisticalecology/
When one is attempting to simulate the dynamic response of the
vegetation over short timescales (days to as long as years), it is
efficacious to ignore the change in the vertical and the horizontal
structure of natural vegetation. At longer timescales, for example, the
timescales associated with monitoring and predicting interannual
variation in the vegetation, structural dynamics can strongly alter the
process-based predictions of such features as ecosystem productivity.
There are ecosystem models that have the capability of simulating
structural change, and these models have been tested to some degree
using large-scale remotely sensed information. This paper reviews types
of ecological modeling approaches used for evaluations of terrestrial
vegetation change at large-area and longer-than-annual timescale. It
then provides examples of results from models that simulate structural
change. Finally, examples of ecosystem dynamics that change with spatial
scale are presented.
One of the most interesting and vexing problems in ecology is how distinctly different communities of plants and animals can occur in the same ecosystem. The theory of these systems, known as multiple stable states, is well understood, but whether multiple stable states actually exist in nature has remained a hotly debated subject. This book provides a broad and synthetic critique of recent advances in theory and new experimental evidence. Modern models of systems with multiple stable states are placed in historical context. Current theories are covered in a rigorous fashion with the specific goal of identifying testable predictions about multiple stable states. The book provides a more synthetic, more critical, and broader analysis of multiple stable states in natural ecosystems than any previous review. By making the theory more transparent and the analysis of the evidence more comparative, the book broadens the discussion about multiple stable states, leading to a more general consideration of the interplay between theory and experiment in community ecology and environmental management.
Our goal is to model and simulate forest dynamics in order to observe large-scale characteristics of the forest as they emerge from local competitive interplay among individuals. This objective can be reached by developing a spatial gap model. Spatialisation can be achieved from a space- or individual-oriented perspective. Because the choice of one approach over the other leads to consequences affecting the simulated dynamics that are difficult to assess a priori, we have decided to consider both and have thus developed two models: a cellular automata network and an individual-based model. Both models are simulated and respective temporal and spatial dynamics are analysed. Although simulation results are similar, some differences highlight underlying assumptions of each conceptual approach. The manner in which the two models complement each other is all the more interesting in that each one has its particular assets and can be used for different perspectives of development.
Models based on transition matrices have recently been developed that described the dynamics of populations in patchy habitats. The models are historically related to transition matrix models of plant succession and like these were developed empirically from data describing abundance changes within patches. The theoretical development of transition matrix models, however, has received little attention.A single-species model is developed that considers a population inhabiting a system of homogeneous patches where any number of individuals may occupy a patch. Certain existing transition matrix population dynamics models can be viewed as an approximation to this model and, for the single-species case, are shown to have similar dynamical behaviour. This provides some theoretical justification for the form of the empirical models. The dynamical behaviour of possible two-species models is discussed; analogous n-species models are complex and their behaviour must be studied through simulations.
In an earlier paper, Woolhouse and Harmsen developed a non-homogeneous Markov chain model that described the dynamics of a one-prey-two-predator acarid complex inhabiting apple foliage. This paper describes the development of an analogous model that incorporates a second prey taxon that has a significant effect on the dynamics of the acarid complex. The model considers transitions among 16 states (defined as the possible combinations of high/low abundances of the four taxa in a leaf sample) for individual trees over two summer seasons. The transition probabilities are related to temperature and to the abundances of the four taxa. The model provides a reasonable description of changes in mite abundances during the two seasons and proves capable of predicting abundance changes through two further seasons with considerable accuracy, particularly for the two prey taxa. More general predictions regarding the effects of differing abundances of the other taxa on one of the prey taxa (tetranychids — a major apple pest) are also discussed.
Trema micrantha individuals that survived until years 8-9 in the Panamanian forest were recruited (reached 1m tall) only in the 1st year after gap formation. This species grows very fast (up to 7m yr-1). After 8-9yr it occurred in only the largest gaps (>376m2). Cecropia insignis survivors were recruited mostly in the 1st year after gap formation, but some survived from years 2 and 3. This species did not grow as fast (up to 4.9m yr-1) as Trema. It occurred in the larger gaps (>215m2). Miconia argentea survivors were recruited mostly in years 2 and 3, but some survivors were recruited as late as 7 yr after gap formation. This species had the slowest growth of the 3 species studied (up to 2.5m yr-1) and colonized gaps from the largest to one of only 102m2. The gap-phase regeneration behaviour of these species forms a gradient in terms of recruitment, growth and gap-size requirement. Such dissimilarity could maintain some diversity of gap colonizers, but apparent overlap in regeneration behaviour among these and other pioneer species suggests that chance also promotes their coexistence.-from Author
In the context of a simple mathematical model, we derive several mechanisms whereby plant species can coexist in a community without differing in their trophic niches (their relations with habitats, resources and exploiters). The model is based on the dynamics of species turnover in microsites, and incorporates localized competition, non-uniform seed dispersal and aspects of spatiotemporal environmental heterogeneity. These factors, which are not included in most standard competition models, allow stable coexistence of trophically equivalent species due to:
(a)
Differences in life-history ‘strategy’.
(b)
Input of seeds from nearby habitats (spatial Mass Effect).
(c)
Differences in demographic responses to environmental fluctuations (temporal Mass Effect).
(d)
Turnover in species composition between different habitat patches.
Quantitative descriptive studies are presented, demonstrating the occurrence of vegetation patterns predicted on the basis of the hypothesized mechanisms. We also review previously proposed mechanisms that would allow trophically equivalent species to coexist, and explore the theoretical and methodological implications of recognizing coexistence mechanisms independent of trophic niche differentiation. In particular, we propose that these mechanisms contribute to the dissimilarity of within-community replicate samples and the maintenance of many rare species in plant communities.
Land base models in regional forest resource supply analyses project area changes in forest vegetation and land use. Models of forest vegetation dynamics are classified according to the basic modeling unit (canopy gap and forest); land use dynamics models are classified according to technique (inventory–descriptive, normative, and positive). Relationships among models used in analyzing timber supply are reviewed, including necessary links between models of forest vegetation dynamics and land use dynamics.
Introduces a demographic model for a local population of sessile marine invertebrates that have a pelagic larval phase, eg Balanus glandula. The processes in the model are the settling of larvae onto empty space, and the growth and mortality of the settled organisms. The rate of settlement per unit of unoccupied space is assumed to be determined by factors outside of the local system. The model predicts the number of animals of each age in the local system through time. The principal result is that the growth of the settled organisms is destabilizing. There is always a steady state where recruitment balances mortality, but growth can interfere with recruitment and can destabilize this steady state, provided also that the settlement rate is sufficiently high. The model suggests that 2 distinct pictures of population structure result, depending on settlement rate. In the high settlement limit, the intertidal landscape is a mosaic of cohorts, punctuated with occasional gaps of vacant substrate. In the low settlement limit, the intertidal landscape has vacant space and organisms of all ages mixed together, and spatial variation in abundance is caused by microgeographic variation in settlement and mortality rates. -from Authors
A classification of spatial simulation models of fire and vegetation dynamics (landscape fire succession models or LFSMs) is presented. The classification was developed to provide a foundation for comparing models and to help identify the appropriate fire and vegetation processes and their simulation to include in coarse scale dynamic global vegetation models. Other uses include a decision tool for research and management applications and a vehicle to interpret differences between LFSMs. The classification is based on the four primary processes that influence fire and vegetation dynamics: fire ignition, fire spread, fire effects, and vegetation succession. Forty-four LFSMs that explicitly simulated the four processes were rated by the authors and the modelers on a scale from 0 to 10 for their inherent degree of stochasticity, complexity, and mechanism for each of the four processes. These ratings were then used to group LFSMs into similar classes using common ordination and clustering techniques. Another database was created to describe each LFSM using selected keywords for over 20 explanatory categories. This database and the ordination and clustering results were then used to create the final LFSM classification that contains 12 classes and a corresponding key. The database and analysis results were used to construct a second classification key so managers can pick the most appropriate model for their application based on computer resources, available modeling expertise, and management objective. Published by Elsevier B.V.
A study of forest succession was conducted in an 11 ha old-growth beech-maple stand located at the northern limit of the temperate deciduous forest of North America. Eventual changes in dominance were simulated from three transition matrix models based on understory composition or observed replacement in forest gaps. All the models suggested that the forest composition is not currently at equilibrium. If the transition tendency persists, American beech (Fagus grandifolia Ehrh.) will sharply increase in abundance at the expense of sugar maple (Acer saccharum Marsh.). Possible mechanisms responsible for this replacement are discussed.
Patterns of tree species replacement in a Picea-Abies forest, determined by several different methods, are compared and the methods are assessed. Methods are grouped as either understory-based or gap-based estimates of replacement. The understory-based methods characterize canopy-understory interactions with spatial statistics, sapling density measurement, sapling frequency measurement, and successor sapling identification beneath live canopy trees. The gap-based methods include sapling density measurement, sapling frequency measurement, and successor sapling identification in tree-fall gaps. Methods except those based on frequency indicate a strong trend of replacement of all canopy species by Abies. Understory-based methods may underestimate canopy recruitment of intolerant trees, while gap-based methods relying on sapling density or frequency may overestimate recruitment of intolerant trees. Estimates based on the selection of successor saplings in the understory or in gaps are reliable. Gap successor estimates consider the process of gap capture and are useful in analyses of forest dynamics.
Both spatial and temporal variability in recruitment probabilities can lead to coexistence in gap-phase regenerating forests which would otherwise tend to be dominated by fewer species. Using modified Markov models, the potential roles were examined of temporal variability and differential mortality rates among species in the dynamics of a forest for which spatial variability has been rejected as a strong factor leading to coexistence. Differential longevity modifies results obtained from a simple Markov model: it exerts a strong influence on the equilibrium species composition, on the rate of community change and on the time a community requires to reach equilibrium. Simulations with varying transition probabilities mimicked a changing climate, producing four main results: 1. Unless the duration of climate states is very long or very short, forest composition is in a continual state of disequilibrium. 2. Species vary in their response times to changing climate. 3. The mean abundance of each species under a varying climate scenario is different from that expected from the mean climate state. 4. The rare, long-lived species was favored by climatic fluctuations at the expense of more common shorter lived species. Differential mortality rates provide an equilibrium-based mechanism for coexistence, and temporally fluctuating recruitment probabilities a non-equilibrium mechanism. Composition could be maintained by differential longevity among species and climatic fluctuations allowing periodic recruitment of the less common species.
Treefall gaps sampled for size, age, and woody vegetation in 1977 were resampled in 1981. Large gaps had more and larger stems than did small gaps; however, basal area and number of stems per unit area remained fairly constant over the range of gap sizes encountered. Overall, most species showed little change in their relative importance values in gaps of different sizes or ages. However, the four main canopy species differed in their responses. Acer saccharum Marsh, reached maximum importance in small gaps, Liriodendron tulipifera L. and Fraxinus americana L. in large ones. As gaps aged, Fraxinus decreased in importance while Fagus grandifolia Ehrh. gained in importance for all gap sizes. For about half the gaps studied, some new adjacent disturbance had occurred between 1977 and 1981.
The present review gives an account of the applicability of mathematical modelling in ecological succession studies. The ability
of particular model types to solve problems of both theory and management is discussed. The Markovian models are found to
be useful for short term predictions, but of very limited value for theoretical considerations. Finally, the predictability
of successional pathways is discussed. It is argued that the less we understand about processes in vegetation dynamics, the
more we will see the course of succession as random and unpredictable.
We consider a general class of Markov population models formulated as stochastic difference equations. The population density is shown to converge either to 0, to +, or to a unique stationary distribution concentrated on (0, +), depending on the signs of the mean log growth rates near 0 and +. These results are applied to the Watkinson-MacDonald bottleneck model of annual plants with a seedbank, extended to allow for random environmental fluctuations and competition among co-occurring species. We obtain criteria for long-term persistence of single-species populations, and for coexistence of two competing species, and the biological significance of the criteria is discussed. The lamentably few applications to the problem at hand of classical limit-theory for Markov chains are surveyed.
Data on the presence of a number of vegetation states (defined in terms of species dominance in areas of 1010 cm) and transition probabilities were derived from permanent quadrats in a number of recently burned heath stands. Data were taken from a species-rich community, a species-poor type and a high-level Calluna-Eriophorum bog. Simple Markovian models were constructed using these data, and the model predictions were compared with known or expected trends. Models for species-rich heath yielded poor simulations of expected trends since matrices derived from data for the first years after fire did not contain sufficient information on transitions to states important later in the developmental sequence. Model results for the simpler species-poor and bog communities were more satisfactory and simulated expected trends. In these types all species recovered quickly after fire and less rearrangement of species abundances took place. Maximum likelihood statistics carried out on the transition matrices produced inconclusive results for the species-rich and species-poor types, but indicated that the data from the Calluna-Eriophorum bog approximated a first-order time-homogeneous Markov chain. It was concluded that Markov models lack predictive ability except in relatively simple systems, but that they may be useful in illustrating variations in short-term community dynamics.
We give the first mathematically rigorous proof that disturbances allow competing species to coexist. This work provides a mathematical framework to explain the existence of fugitive species and the role played by disturbances in increasing or decreasing the biodiversity of ecosystems. We study modifications of the metapopulation model for patchy environments proposed by Caswell and Cohen (1990, 1991). For the one- and two-species models we give necessary and sufficient conditions on the parameters for the existence of a non-trivial equilibrium solution, which is shown to be always globally stable.
Transition matrices have been commonly used as models of ecological successions but not as models of non-successional population dynamics. Existing transition matrix models are usually homogeneous Markov chains, and make no attempt to incorporate non-stationarity, non-linearity or second-order behaviour despite the fact that ecological transition data generally show these features. This paper develops a transition matrix model to describe the changes in abundance of three mite populations coexisting on apple foliage. It is a predator-prey system, comprised of zetzellid and phytoseiid predators and tetranychid prey. Transitions among eight states (defined as the possible presence/absence combinations of the three populations in a leaf sample) over 14-day intervals were recorded for individual trees in an orchard in 1982 and 1983. The transition probabilities were found to be non-stationary and to be related to temperature and to the three overall population densities. A non-homogeneous Markov chain model was developed in which the transitions probabilities were functions of these three variables. This is the first application of such a model to population dynamics. The model accurately describes changes in the three population abundances through the summer season, although a test of its predictive potential using 1984 data gave equivocal results. Simulations of the effects of varying temperature and predator abundance on tetranychid abundance, however, agreed well with field observations. The advantages of transition matrix models in general are discussed.
A linkage between the two major modeling approaches to forest dynamics, transition Markovian models and jabowa-foret type simulators, is generated by developing a compact model of forest dynamics. This patch transition model utilizes functional roles instead of species. The roles or types are based on the regeneration and mortality characteristics of tree species; specifically, the requirements of canopy gaps for regeneration and the capacity to create canopy gaps upon death. A gap-size plot can be assigned to each of a set of states defined according to dominance of one of the roles. Transition probabilities among these states and mean holding times in each transition lead to semi-Markovian analytical calculations of the stationary state probabilities. Forest dynamics, as the proportions of total canopy space occupied by each role in a collection of gap-size plots, can be analyzed and simulated using a chain of first-order differential equations to emulate the distributed time-delays. Additional fixed time-delays in the transition of every pair of states is also included to account for long latencies. In addition to simplifying the simulations, the resulting model can also utilize available results of the theory of semi-Markov processes; and therefore, can provide analytical guidance to the simulations, the feasibility of direct exploration of hypothesis and the possibility of fast computation from closed-form solutions and formulae. These advantages can especially be useful in the simulation of landscape dynamics and species-rich tropical forests.
Many simple plant communities in extremely arid regions of the Sonoran Desert consist essentially of 2 shrub species, Ambrosia dumosa and Larrea tridentata. The majority of space is devoid of perennial plant cover, and Ambrosia readily colonize this open space. Recruitment of Larrea occurs less frequently and is largely limited to areas beneath the canopies of large, older Ambrosia individuals. The dynamics of these systems can be modeled as Markov chains. The dynamics of species-rich plant communities in less xeric parts of the Sonoran Desert appear to be qualitatively similar. One shrub species in particular, A. deltoidea, is capable of colonizing open space, whereas for most other shrub, tree and succulent species, recruitment is largely limited to areas beneath canopies of other plants, especially canopies of A. deltoidea. Although seemingly more complex, a digraph model showing the transitions among states possesses the same fundamental characteristics as does a digraph of the simple two-species system. The mechanisms underlying these dynamics include factors that influence seed distributions, germination success and patterns of post-germination mortality. Even in the simple two-species systems, it is likely that no individual mechanism such as competition can be singled out as the predominant determinant of dynamics and, hence, of the patterns of community structure. -from Author
In the context of a simple mathematical model, we derive several mechanisms whereby plant species can coexist in a community without differing in their trophic niches (their relations with habitats, resources and exploiters). The model is based on the dynamics of species turnover in microsites, and incorporates localized competition, non-uniform seed dispersal and aspects of spatiotemporal environmental heterogeneity. These factors, which are not included in most standard competition models, allow stable coexistence of trophically equivalent species due to:
(a)
Differences in life-history ‘strategy’.
(b)
Input of seeds from neaby habitats (spatial Mass Effect).
(c)
Differences in demographic responses to environmental fluctuations (temporal Mass Effect).
(d)
Turnover in species composition between different habitat patches.
Quantitative descriptive studies are presented, demonstrating the occurrence of vegetation patterns predicted on the basis of the hypothesized mechanisms. We also review previously proposed mechanisms that would allow trophically equivalent species to coexist, and explore the theoretical and methodological implications of recognizing coexistence mechanisms independent of trophic niche differentiation. In particular, we propose that these mechanisms contribute to the dissimilarity of within-community replicate samples and the maintenance of many rare species in plant communities.
Recent interest in the ecology and evolution of metapopulations and conservation of fragmented populations has stimulated the development of models that combine patch and population dynamics in tropical forests. One approach uses matrix models that are actual metapopulation or multi-regional demographic models. Another approach uses computer simulations to model forest succession based on the behavior of individual trees. We review applications of both types of models and suggest new combined modelling approaches.
Los datos de parcelas permanentes, establecidas en un bosque estacional de los llanos occidentales venezolanos, se usaron para establecer los parámetros de FACET, un modelo basado en árboles individuales y sensible a las características topográficas del terreno. Debido a su alto número, las especies arbóreas se agruparon de acuerdo con su tolerancia a la sombra (tolerantes, intermedias e intolerantes) y su altura máxima (grandes, medianas y pequeñas). Las palmas fueron consideradas como un grupo separado. La dinámica forestal fue simulada para cada posición fisiográfica (banco, sub-banco y bajío). Los resultados de la simulación para un periodo de 210 años se aproximan a la dinámica esperada tanto para banco como sub-banco, no así para bajío. Tras una simulación de 500 años, en banco las tolerantes grandes dominan el área, acumulando un índice de valor (IV) de 27,28 seguidas por las palmas con un IV= 18,86; igual ocurre en sub-banco donde las tolerantes grandes predominan con un IV= 26,78 y las palmas acumulan 20,27; lo que se corresponde a la composición esperada para un bosque sin perturbaciones. En bajío, las tolerantes grandes también mantienen el primer lugar con un IV= 28,74 pero las siguen las intermedias grandes con 20,27. En conclusión, el modelo FACET tiene gran potencialidad para predecir la dinámica de bosques tropicales, excepto en las áreas de bajío con problemas de drenaje. En consecuencia, es necesario introducir modificaciones en la rutina de balance de agua en el suelo para lograr una mejor representación de la dinámica forestal en esos sitios.
Las relaciones alómetricas diámetro-altura fueron analizadas en 34 especies arbóreas de la Reserva Forestal Imataca, según la ecuación exponencial de Chapman y Richard, utilizada por los modelos de base individual ZELIG y FACET, que simulan la dinámica de crecimiento del bosque. Los coeficientes de la relación alométrica para cada especie fueron calculados por regresión lineal aplicada a la transformación logarítmica de la ecuación exponencial. Se determinó que la relación exponencial explica una proporción elevada de la variabilidad de altura para las especies estudiadas aunque se utilizaron datos de parcelas en localidades diferentes. Las relaciones resultantes serán incorporadas al modelo FACET en trabajos posteriores. Se evaluó la existencia de patrones alométricos, agrupando las especies según los requerimientos de luz (intolerantes o pioneras y tolerantes a la sombra) y por clases de altura (medianas con 10-30m, grandes de >30m), conformando cuatro grupos funcionales: pioneras medianas, pioneras grandes, tolerantes medianas y tolerantes grandes. Se encontraron diferencias entre las alometrías de los cuatro grupos utilizando análisis de varianza y análisis discriminante. Las diferencias en los coeficientes alométricos, tomados uno a la vez, son más significativas entre pioneras y tolerantes que entre las grandes y medianas. La combinación de ambos coeficientes, b2 y b3, permite diferenciar todos los grupos, especialmente usando una función discriminante, que además permitirá predecir la pertenencia de otras especies a cada grupo dados sus valores de coeficientes alométricos.
This chapter is offered in response to the oft-asked question: Where are all the large trees, if this is a virgin tropical forest? Here I will document the size-class distribution of trees in a tropical wet forest and examine several possible reasons for low densities of large trees. On the basis of factual and theoretic considerations of tropical forest dynamics, several hypotheses relevant to the successful colonization of disturbed areas are offered.
The purpose of this short paper is to explain how a projection matrix model has been used to describe the population dynamics of a tropical tree species (Hartshorn, 1972). A detailed description of model development and some potential uses of the model will be presented. The tree Pentaclethra macroloba (Willd.) Ktze. (Mimosaceae) is a large canopy species dominating areas of tropical wet forest in the Atlantic lowlands of Costa Rica.
Neotropical forest dynamics are reviewed by focusing on four questions: (1) What is a mature neotropical forest? (2) How long does it take to attain maturity? (3) How important are gaps to species regeneration? and (4) What are the important equilibrium processes in neotropical forest dynamics? The absence of regeneration of dominants has often been used as a distinguishing feature of late secondary forest; however, the abundance of shade-intolerant species in mature forest suggests that local absence of regeneration is an inadequate criterion for distinguishing between late secondary and mature forest. Recent studies estimate forest turnover rates of 75-150 years, indicating tropical forests are much more dynamic than thought previously. The dependence on gaps by almost half of the 320 tree species in a Costa Rican wet forest for successful regeneration illustrates the importance of gaps in tropical forest dynamics. Factors important in determining which species successfully colonize a gap are: time of gap occurrence; proximity and dispersal of seeds; size of gap; substrate conditions; and plant-herbivore interactions.
Previously it was shown that reproductive-cycle parameters such as time to maturity, ovulation interval, gestation period, duration of regression, duration of nonreproductive lactation period, and the like, can be incorporated into population models rather easily through the use of a simple network approach. In this paper, the network approach is extended to include the same types of reproductive parameters when their values are not necessarily fixed, but may vary randomly from one member of a population to the next and/or for a given member from one time to the next. It is shown that linear transforms of the parameter distribution functions can be incorporated directly into the network models and that analysis of the resulting dynamics follows in a straightforward manner, the characteristic dynamical equation being obtainable by inspection with Mason's algorithm and the roots of the equation being obtainable by direct analysis in simple cases or by well-established numerical methods in complicated cases. The roots themselves can be interpreted directly in terms of dominant patterns of population growth and deduced propensity of the population to sustain oscillations triggered by external stimuli. In the case of a simple natality cycle with gamma, negative binomial, and binomial distributions of maturation times, it is shown that the dominant growth pattern approximates rather closely that expected for a nonrandom maturation time equal to the mean of the distribution, and that the propensity to sustain population oscillations decreases markedly both with increasing standard deviation and with increasing (positive) skewness in the distribution.
Theory and observation indicate that natural multi-species assemblies of
plants and animals are likely to possess several different equilibrium
points. This review discusses how alternate stable states can arise in
simple 1- and 2-species systems, and applies these ideas to grazing
systems, to insect pests, and to some human host-parasite systems.
A high number of tree species, low density of adults of each species, and long distances between conspecific adults are characteristic of many low-land tropical forest habitats. I propose that these three traits, in large part, are the result of the action of predators on seeds and seedlings. A model is presented that allows detailed examination of the effect of different predators, dispersal agents, seed-crop sizes, etc. on these three traits. In short, any event that increases the efficiency of the predators at eating seeds and seedlings of a given tree species may lead to a reduction in population density of the adults of that species and/or to increased distance between new adults and their parents. Either event will lead to more space in the habitat for other species of trees, and therefore higher total number of tree species, provided seed sources are available over evolutionary time. As one moves from the wet lowland tropics to the dry tropics or temperate zones, the seed and seedling predators in a habitat are hypothesized to be progressively less efficient at keeping one or a few tree species from monopolizing the habitat through competitive superiority. This lowered efficiency of the predators is brought about by the increased severity and unpredictability of the physical environment, which in turn leads to regular or erratic escape of large seed or seedling cohorts from the predators.
Many biological populations breed seasonally and have nonoverlapping generations, so that their dynamics are described by first-order difference equations, Nt+1 = F (Nt). In many cases, F(N) as a function of N will have a hump. We show, very generally, that as such a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes." We give a detailed account of the underlying mathematics of this process and review other situations (in two- and higher dimensional systems, or in differential equation systems) where apparently random dynamics can arise from bifurcation processes. This complicated behavior, in simple deterministic models, can have disturbing implications for the analysis and interpretation of biological data.
There have been few studies of succession in species-rich tropical forest, and most of these have focused on the early pioneer stages. This report is an analysis of late secondary succession in the forest on Barro Colorado Island (BCI), a 15Km2 island in Gatun Lake, Panama Canal Zone.
As defined, the modeling procedure is quite broad. For example, the chosen compartments may contain a single organism, a population of organisms, or an ensemble of populations. A population compartment, in turn, could be homogeneous or possess structure in size or age. Likewise,
the mathematical statements may be deterministic or probabilistic in nature, linear or nonlinear, autonomous or able to possess memory. Examples of all types appear in the literature. In practice, however, ecosystem modelers have focused upon particular types of model constructions. Most analyses seem to treat compartments which are nonsegregated (populations or trophic levels) and homogeneous. The accompanying mathematics is, for the most part, deterministic and autonomous. Despite the enormous effort which has gone into such ecosystem modeling, there remains a paucity of models which meets the rigorous &! validation criteria which might be applied to a model of a mechanical system. Most ecosystem models are short on prediction ability. Even some classical examples, such as the Lotka-Volterra predator-prey scheme, have not spawned validated examples.
The Cambridge Botanical Expedition to Nigeria, 1947-48, studied the pattern of distribution of species, the regeneration, and the stability of Rain Forest, particularly in relation to the theory of `mosaic' structure. The Okomu Forest Reserve, a compartment of which was studied by transects, lies on Benin sands and has a rainfall of 80-100 in.; it has been lightly exploited for at least forty years. On well-drained `plateau' sites `Gully-margin Forest', presumably an edaphic variant, is distinguishable both floristically and physiognomically from the more extensive forest which was mainly studied. The plateau-forest is a patchwork of the following serally-related phases, which are more distinct physiognomically than floristically: (a) `High Forest' with continuous middle and lower storeys and numerous emergents. (b) `Broken High Forest', differing from (a) in being interrupted by slight gaps. (c) and (d) `Tall' and `Low Closed Scrub' with a continuous climber-blanketed canopy at 6-12 m. and 3-6 m. respectively and fewer emergents than in (a). (e) `Open Scrub', as (c) and (d), but interrupted by many open glades containing light-demanding herbs, and much frequented by elephants. Scrub has been greatly extended by tree-felling, but it is also formed by wind and is a normal component of unexploited forest. Of the three storeys, that of the emergents is best characterized floristically; 21 per cent of its stems are species which are very strong light-demanders that are scarcely represented in the two lower storeys, 45 per cent are species which are abundant in the lower storey but scarcely represented in the middle storey, and 32 per cent are species which increase progressively in abundance from the emergent to the lower storey. There are striking variations in the abundance of particular species even over distances of a few kilometres, and within the 200 ha. which were sampled by transects there are some species which are randomly distributed, others which are patchily distributed on a large scale though randomly distributed on a small scale, others which are widely distributed but tend to occur in small patches, etc. In general, emergents are distributed more nearly at random than are members of the two lower storeys and the tendency to aggregation increases with decreasing size; nevertheless, some species of medium or small size appear to be randomly distributed, and others of comparable size and abundance are patchily distributed. Very few pairs of species (only Lovoa klaineana with Khaya ivorensis, Macrolobium macrophyllum with Barteria fistulosa, B. fistulosa with Grewia coriacea, and G. coriacea with Xylopia quintasii) could be shown to tend to associate, as they should if floristically well-defined communities were present. Thus despite the existence of a mosaic of physiognomically distinct phases and despite the patchy distribution of certain species, it was not possible to detect any well-marked mosaic of species. Of the emergent species nearly half are wind-dispersed and an equal number are animal-dispersed; in the lower storeys the proportion of wind-dispersed species is small and that of animal-dispersed species high. Seeds with a short life and rapid germination are frequent, but seeds with prolonged dormancy are not rare, especially amongst the animal-dispersed species, and perhaps amongst the members of the understorey. Seedlings of most of the woody species are probably present in the forest, but many are extremely local in their distribution, tending to occur in small patches, e.g. near mature trees. Broadly speaking, all sizes of the shade-tolerant members of the lower and middle storeys are adequately represented, suggesting that they may be regenerating continuously and maintaining the existing population; nevertheless, dead trees of some of these species (e.g. Anonidium, Trichilia prieuriana) are very rare, and dead trees of other species (e.g. Scottellia, Strombosia spp.) are abundant, suggesting that their regeneration may be discontinuous. In contrast, in most of the emergent species stems of medium size tend to be less abundant than large-sized stems, even when small sizes and seedlings are abundant. Some figures for rate of growth of trees of various sizes suggest that this paucity of middle sizes cannot always be explained by fast growth of the deficient size-classes; the alternative explanation must be that of discontinuous recruitment. Seedlings of even the more shade-tolerant emergents survive only in or near small gaps in the canopy. The number of small individuals of emergent species per unit area increases with the degree of degradation of the forest; in High Forest they are certainly not numerous enough to maintain the existing stocking of mature emergents. In the scrub phases they are about twice as numerous and more species are represented, but even here it is doubtful whether they are numerous enough to reconstitute a forest as rich in emergents as that now existing, assuming that the scrub will revert to High Forest. Some emergent species seem to be represented by a disproportionately large number of standing dead trees; dead individuals of other abundant emergent species were not recorded. The death-rate increases rapidly with increasing size above about 7 ft. girth. There is thus much to suggest that the composition of the forest is tending to change, particularly in respect of the emergents. Fragments of pottery and charcoal in the soil show that the land has been farmed. It is probable that the farming was relatively recent rather than prehistoric, but it cannot have been less than about 200 years ago, and it is in accordance with all the evidence to regard the forest as secondary forest of this age which is now breaking up. There are reasons for suspecting that no matter whether scrub is a normal phase in the progression or not, the forest of the true climatic climax might have few large trees, and perhaps fewer species and a simpler structure.
1. This paper describes an extensive survey of the composition of the canopy in two sample areas taken from 1 mile2 (2.5 km2) of relatively uniform, undisturbed Lowland Dipterocarp forest in the Malay Peninsula. This survey was designed to find out what factors control the distribution of species within such a forest. 2. A brief introduction is given to Malayan forest types based mainly on the work of Wyatt-Smith; this shows that a number of types of forest are already recognized and that the distribution of these, even within the complex of Lowland Dipterocarp forest, is related broadly to habitat. 3. Jengka Forest Reserve is situated to the east of the main watershed of the Malay Peninsula between 150 and 250 ft (45 and 75 m) above sea level. The survey area lies on Triassic shales and sandstones which form a yellow latosol. The climate is typically equatorial with a well-distributed rainfall, though there is evidence that evaporation exceeds precipitation for some months in dry years. The forest is typical in structure of Malayan Lowland Dipterocarp forest and is, as far as can be ascertained, undisturbed by human interference past or present. 4. Two surveys were made. `A' was a grid sample amounting to 20 ha in which all trees reaching the canopy (mainly those over 3 ft (91 cm) girth) were enumerated and mapped. `B' was a block of 24 ha in which all individuals reaching the canopy of the nineteen commonest species were enumerated and mapped. 5. A total species list and an analysis of the basal area and number of trees per family was prepared for `A'. This shows that the Dipterocarpaceae is the most important family by any criterion. The sample contained 2773 trees of 375 species, 139 genera and fifty-two families. The average basal area was 24.2 m2/ha. 6. The network of narrow swamps which traverse the area (amounting to 4.6% of the total) have a significantly different flora from the rest. 7. The eighteen commonest trees in `B' were mapped and thirteen of these were analysed by the partition of variance to detect pattern. Some species were found to be randomly distributed while others showed significant aggregations of various dimensions. The most marked pattern for several species measured approximately 200 m across. The degree of aggregation seemed to be related to means of dispersal and size of fruit. 8. The distribution maps of the same thirteen species showed that smaller trees were generally grouped round larger trees but occasionally occurred in isolated positions, presumably forming foci for new aggregation. 9. Covariance analysis of the thirteen commonest species for pattern showed no groups or pairs of species which were consistently positively or negatively associated with one another at all sizes of aggregation. The degree and the sign of the association between any pair of species frequently changed as the size of the aggregations changed. 10. Normal and inverse association analysis were applied to the distribution of 184 species in 192 plots each of 20 × 60 m. Normal association analysis revealed those species which were known to have indicator value for swamp conditions. In the inverse analysis there appeared a number of groups of species which could be interpreted from present knowledge of the ecology of Malayan Lowland forest. These groups contained mainly the rarer species of the survey area. The groups containing the commoner species did not correspond with the results of the covariance analysis. 11. Further analysis of the data show that a plot of at least 2 ha is necessary to obtain a coefficient of similarity of 50% for canopy tree species between two samples of this forest and that Williams' Index of Diversity (α) increases with increasing size of plot. Calculations made from the proportion of gap in the forest and from the distribution of girth classes both lead to the view that any piece of forest may persist, between gap phases, for an average of 200-400 years. 12. An hypothesis is advanced to account for the kinds of distribution of species that were found. The `rarer' species are thought to occur in associated groups, the distribution of which depends on changes of soil and micro-environment. These occur within a matrix of commoner species whose distribution is determined more by the relations between flowering, fruiting, agency of dispersal and the formation of gaps (i.e. by chance) than by their relation to small changes of environment. 13. The evidence of the behaviour of Dipterocarpus, Koompassia and Shorea leprosula suggest that the forest is not in equilibrium and that the balance of some of the more important species in it has changed within the last 100 years.
Data on tree species composition and population structure are used as a source of ecological information on the species-rich forest of Barro Colorado Island, Panama Canal Zone (BCI). Thirteen stands, representing both the young and old forest on the Island, were sampled using 10 m @? 20 m quadrats for all individuals @?2.5 cm dbh. Over 300 species were encountered and most were identified. The data are evaluated for interpreting late secondary succession (>60 yr), detecting soil-vegetation patterns, and for yielding autecological information. Successional status was estimated by placing species with similar population structure patterns into groups, and then calculating the abundance of species groups that seemed indicative of successional status. Five population structure patterns were recognized. The results suggest that the older BCI forest is not climax after at least 130 yr of succession. Both principal components analysis and a Bray-Curtis type ordination were used to determine whether ecological patterns on BCI are reflected by species composition. Forest age was best reflected by an ordination of the larger tree species. Species found only in the young forest or only in the older forest are identified. Stands with a distinctive gley soil were not segregated on the ordination. Species diversity increases most rapidly during the first 15 yr of succession, but continues to increase slowly after 65 yr. The Shannon-Wiener diversity measure averaged 4.8 (log base 2). Some data suggest that wind-caused canopy gaps are important for the persistence of several species in the older forest, e.g., Cecropia sp.
The mussel Mytilus californianus is a competitive dominant on wave-swept rocky intertidal shores. Mussel beds may exist as extensive monocultures; more often they are an everchanging mosaic of many species which inhabit wave-generated patches or gaps. This paper describes observations and experiments designed to measure the critical parameters of a model of patch birth and death, and to use the model to predict the spatial structure of mussel beds. Most measurements were made at Tatoosh Island, Washington, USA, from 1970-1979. Patch size ranged at birth from a single mussel to 38 m^2; the distribution of patch sizes approximates the lognormal. Birth rates varied seasonally and regionally. At Tatoosh the rate of patch formation varied during six winters from 0.4-5.4% of the mussels removed per month. The disturbance regime during the summer and at two mainland sites was 5-10 times less. Annual disturbance patterns tended to be synchronous within 11 sites on one face of Tatoosh over a 10-yr interval, and over larger distances (16 km) along the coastline. The pattern was asynchronous, however, among four Tatoosh localities. Patch birth rate, and mean and maximum size at birth can be used as adequate indices of disturbance. Patch disappearance (death) occurs by three mechanisms. Very small patches disappear almost immediately due to a leaning response of the border mussels (0.2 cm/d). Intermediate-sized patches (
A stochastic model for stand projection is developed from which difference equations are derived for expectations, variances, and covariances of tree counts. Results are applied to the problems of assessing effects of errors in input and transition parameters on stand projections and of measuring uncertainty in short-term predictions. A worked example illustrates the calculations of expected stand numbers and their variances over a 12-year span and compares them with inventory data. Forest Sci. 19:303-314.
The modeling methodology includes recognition of cover-states which are similar to successional stages. The general form of the model is ordinary linear differential equations using considerations of stand dynamics and sylvics to determine rates of change. Output from an example model for 250 years of succession in the western Great Lakes Region in the absence of fire, epidemics, and management is presented and discussed. Changes in the model necessary to include management, fire, and epidemics are indicated. The conditions under which the model should simulate succession are stated explicitly. Forest Sci. 19:203-212.
Pre-1976 literature on spatial population models is reviewed. Populations that are doomed to local successional extinction may survive globally due to dispersal in a variable environment.
The paper reports results of a long-term (1964–1974) investigation on permanent study sites in natural forest ecosystems of the Tilio-Carpinetum and the Pino-Quercetum in the Bialowieza Forest. The influence of decaying logs and root craters was investigated. It was found that the main causes of uprooting were the spring and autumn winds. Wind direction and the position of logs lying on the ground are correlated. Picea is most susceptible to uprooting by winds. Almost one half of the trees of this species are alive at the moment of uprooting.
By mapping changes in the distribution of uprooted trees on a permanent area in time, a balance of the change over in a 10-year period was determined. It appeared that the decomposition is slower than accumulation. From this, it was concluded that the stand is in a phase of natural thinning. In the study site, compartments were disinguished with various degrees of change in the number of uprooted trees, and the consequences of differentiation and constant transformation of the biotope and biocenosis by the occurrence of uprooted trees and by their decay are described.
The way phenomena or processes evolve or change in time is often described by differential equations or difference equations. One of the simplest mathematical situations occurs when the phenomenon can be described by a single number as, for example, when the number of children susceptible to some disease at the beginning of a school year can be estimated purely as a function of the number for the previous year. That is, when the number x
n+1, at the beginning of the n + 1st year (or time period) can be written $${x_{n + 1}} = F({x_n}),$$ (1.1) where F maps an interval J into itself. Of course such a model for the year by year progress of the disease would be very simplistic and would contain only a shadow of the more complicated phenomena. For other phenomena this model might be more accurate. This equation has been used successfully to model the distribution of points of impact on a spinning bit for oil well drilling, as mentioned if [8, 11] knowing this distribution is helpful in predicting uneven wear of the bit. For another example, if a population of insects has discrete generations, the size of the n + 1st generation will be a function of the nth. A reasonable model would then be a generalized logistic equation $${x_{n + 1}} = r{x_n}[1 - {x_n}/K].$$ (1.2)
The problem of oscillations in populations with separate generations is considered mathematically. The inadequacy of inferring the existence of sustained oscillations from an equilibrium value −1 of the slope of the reproductive curve, is pointed out. It is contended that these oscillations, when they exist, are to be explained as limit cycles and criteria for the existence and stability of the cycles are presented.
Competition, secondary succession, and changes in vegetation accompanying changes in elevation have been successfully reproduced by a computer simulation of a mixed-species, uneven-aged forest ecosystem of north-eastern United States. As much as the data and understanding allow, this simulation has a conceptual basis. Changes in the state of the simulated forest are a function of the present state plus random components. The program predicts a peak standing crop for the forest approximately 200 years following clearcutting, with a relatively stable, but fluctuating, species composition at subsequent periods. Designed to be used in the Hubbard Brook Ecosystem Study, the program provides output in a form compatible with the original vegetation survey of that study and allows a flexible interaction between the user and the simulator.
The design of the life cycle of pin cherry (Prunus pensylvanica L.), a successional species common on disturbed sites throughout much of the northern hardwood and boreal forest ecosystems, assures that its occurrence is integrated into the pattern of disturbance in the climax ecosystem. The combination of buried seed strategy and the mobility offered through avian consumption of fruits ensures reasonably large populations of buried, viable seeds in the soils of forests well after the disappearance of pin cherry from a particular site. Soil sampling data indicate that sufficient numbers of viable pin cherry seeds reside in the soils of second-growth forests in central New Hampshire to account for the dense stands frequently observed after cutting or burning. Further, germination of these buried seeds is apparently triggered by some factor(s) associated with formation of a large gap. By age 25 or 30, when pin cherry individuals are dying rapidly, sufficient numbers of seeds have been produced and disseminated in a dormant condition for the cycle to renew itself with the subsequent occurrence of major disturbance. In high density stands pin cherry grows rapidly, with early attainment of canopy closure (high leaf area index), and rapid attainment of high values of net annual production and nutrient accumulation.
