Community dynamics in relation to management of heathland vegetation in Scotland

The paper describes studies of post-fire succession in heathland vegetation in N.E. Scotland, dominated by Calluna vulgaris. A preliminary model (Legg, 1978) suggested good agreement between simulation of succession on the basis of a Markov chain and observations of stands at different stages of development after burning, at least in the earlier stages. Vegetation transitions are currently being recorded in permanent plots on burnt areas. First results confirm the view that (a) the post-fire succession has the properties of a Markov process, (b) this type of model remains valid when constructed from records of actual transitions, rather than data obtained by inference from evidence of transition. Comparing successional events in stands where, at the time of burning, the Calluna population was in pioneer-, building-, mature-and degenerate phases, shows that transition matrices generally agree with the Markov hypothesis, but not in the case of stands where Calluna was degenerate when burnt. The composition of establishing vegetation 1 year after fire is not confined to species normally associated with the early stages of succession, but reflects the composition of the stand before burning. Redevelopment after fire is described in terms of an initial floristic composition of species with strategies permitting early re-establishment, selected by the recurrence of the fire factor. Subsequent transitions represent changes in their relative abundance due to differing growth properties and competitive interactions. This interpretation applies only under conditions of recurrent incidence of fire (normally once in 10–15 yr). If fire does not recur, Calluna stands pass into the degenerate phase, where changes in the nature of relay floristics may come into play (e.g. with tree colonization).Nomenelature follows Clapham, Tutin & Warburg (1962) for vascular plants; Smith (1978) for bryophytes.     

A Markov approach for describing post-fire succession of vegetation

Vegetation dynamics in the coastal area of the Seto Inland Sea region in Japan, where wild fires occur frequently, were described using a stationary Markov model. In this region, vegetation types ofMiscanthus-Pleioblastus grassland,Lespedeza-Mallotus scrub,Pinus-Rhododendron forest andCrassocephalum-Erechtites community have been identified, and these show cyclic succession under the influence of fires. The model uses parameters determining fire frequency and rate of successional change to analyze the effect of variation in these parameters on the areal ratio of each vegetation type at equilibrium and on the time taken for one vegetation type to succeed another (elapsed successional time). The effect of fire frequency differs between hypothetical habitats with high and low productivity. A policy for vegetation management in areas of high and low productivity is proposed. The advantages and limitations of applying Markov models to studies of vegetation succession are also discussed.     

Bayesian restoration of a hidden Markov chain with applications to DNA sequencing.

Hidden Markov models (HMMs) are a class of stochastic models that have proven to be powerful tools for the analysis of molecular sequence data. A hidden Markov model can be viewed as a black box that generates sequences of observations. The unobservable internal state of the box is stochastic and is determined by a finite state Markov chain. The observable output is stochastic with distribution determined by the state of the hidden Markov chain. We present a Bayesian solution to the problem of restoring the sequence of states visited by the hidden Markov chain from a given sequence of observed outputs. Our approach is based on a Monte Carlo Markov chain algorithm that allows us to draw samples from the full posterior distribution of the hidden Markov chain paths. The problem of estimating the probability of individual paths and the associated Monte Carlo error of these estimates is addressed. The method is illustrated by considering a problem of DNA sequence multiple alignment. The special structure for the hidden Markov model used in the sequence alignment problem is considered in detail. In conclusion, we discuss certain interesting aspects of biological sequence alignments that become accessible through the Bayesian approach to HMM restoration.     

Directionality and community-level selection

Many ecological community dynamics display some degree of directionality, known as succession patterns. But complex interaction networks frequently tend to non-directional dynamics such as chaos, unless additional structures or mechanisms impose some form of, often fragile or shot-lived, directionality. We exhibit here a novel property of emergent long-lasting directionality in competitive communities, which relies on very minimal assumptions. We model communities where each species has a few strong competitive interactions, and many weak ones. We find that, at high enough diversity, the dynamics become directional, meaning that the community state can be characterized by a function that increases in time, which we call ‘maturity'. In the presence of noise, the community composition changes toward increasingly stable and productive states. This scenario occupies a middle ground between deterministic succession and purely random species associations: there are many overlapping stable states, with stochastic transitions, that are nevertheless biased in a particular direction. When a spatial dimension is added in the form of a meta-community, higher-maturity community states are able to expand in space, replacing others by (exact or approximate) copies of themselves. This leads to community-level selection, with the same maturity function acting as fitness. Classic concepts from evolutionary dynamics provide a powerful analogy to understand this strictly ecological, community-level phenomenon of emergent directionality.     

Graphs, random sums, and sojourn time distributions, with application to ion-channel modeling.

This paper considers the distribution of a sojourn time in a class of states of a stochastic process having finite discrete state space where sojourn times in any individual state are independent and identically distributed, and transitions between states follow a Markov chain. The state space and possible transitions of the process are represented by a graph. Class sojourn time distributions are derived by modifying this graph using 'composition' of states, defining a new Markov chain on the modified graph, and expressing the sojourn time in a composition state as a random sum. Appropriate compositions are chosen according to the possible "cores" of sojourns in the particular class, where a core describes the structure of a sojourn in terms of a single state or a chain in the original graph. Graph methods provide an algorithmic basis for the derivation, which can be simplified by using symmetry results. Models of ion-channel kinetics are used throughout for illustration; class sojourn time distributions are important in such models because individual states are often indistinguishable experimentally. Markov processes are the special case where sojourn times in individual states are exponentially distributed. In this case kinetic parameter estimation based on the observed class sojourn time distribution is briefly discussed; explicit estimating equations applicable to sequential models of nicotinic receptor kinetics are given.     

Moderate disturbances accelerate forest transition dynamics under climate change in the temperate–boreal ecotone of eastern North America

Several temperate tree species are expected to migrate northward and colonize boreal forests in response to climate change. Tree migrations could lead to transitions in forest types, but these could be influenced by several non‐climatic factors, such as disturbances and soil conditions. We analysed over 10,000 forest inventory plots, sampled from 1970 to 2018 in meridional Québec, Canada, to identify what environmental conditions promote or prevent regional‐scale forest transitions. We used a continuous‐time multi‐state Markov model to quantify the probabilities of transitions between forest states (temperate, boreal, mixed, pioneer) as a function of climate (mean temperature and climate moisture index during the growing season), soil conditions (pH and drainage) and disturbances (severity levels of natural disturbances and logging). We further investigate how different disturbance types and severities impact forests' short‐term transient dynamics and long‐term equilibrium using properties of Markov transition matrices. The most common transitions observed during the study period were from mixed to temperate states, as well as from pioneer to boreal forests. In our study, transitions were mainly driven by natural and anthropogenic disturbances and secondarily by climate, whereas soil characteristics exerted relatively minor constraints. While major disturbances only promoted transitions to the pioneer state, moderate disturbances increased the probability of transition from mixed to temperate states. Long‐term projections of our model under the current environmental conditions indicate that moderate disturbances would promote a northward shift of the temperate forest. Moreover, disturbances reduced turnover and convergence time for all transitions, thereby accelerating forest dynamics. Contrary to our expectation, mixed to temperate transitions were not driven by temperate tree recruitment but by mortality and growth. Overall, our results suggest that moderate disturbances could catalyse rapid forest transitions and accelerate broad‐scale biome shifts.     

Models of vegetation dynamics in semi-arid vegetation: Application to lowland Central Otago, New Zealand

Predictions from three conceptual models of the dynamics of semi-arid vegetation (Clementsian succession, alternative stable states and annuation/pulse phenomena) are used to review the available evidence on changes in the vegetation of semi- arid lowland Central Otago, New Zealand. Evidence is presented from Central Otago that corresponds with Clementsian succession and with annuation/pulse phenomena, although there is so far no formal evidence of alternative stable states. A declining- productivity model, which combines aspects of the other models, is also shown to fit the process of vegetation change in vegetation dynamics in Central Otago are insufficient for the employment of management frameworks such as degradation gradient assessment and the state-and-transition model.     

Spatio-temporal variation in Markov chain models of subtidal community succession

In this paper we ask whether succession in a rocky subtidal community varies in space and time, and if so how much affect that variation has on predictions of community dynamics and structure. We describe succession by Markov chain models based on observed frequencies of species replacements. We use loglinear analysis to detect and quantify spatio‐temporal variation in the transition matrices describing succession. The analysis shows that space and time, but not their interaction, have highly significant effects on transition probabilities. To explore the ecological importance of the spatio‐temporal variability detected in this analysis, we compare the equilibria and the transient dynamics among three Markov chain models: a time‐averaged model that includes the effects of space on succession, a spatially averaged model that include the effects of time, and a constant matrix that averages over the effects of space and time. All three models predicted similar equilibrium composition and similar rates of convergence to equilibrium, as measured by the damping ratio or the subdominant Lyapunov exponent. The predicted equilibria from all three models were very similar to the observed community structure. Thus, although spatial and temporal variation is statistically significant, at least in this system this variation does not prevent homogeneous models from predicting community structure.     

Covariate Adjustment of Event Histories Estimated from Markov Chains: The Additive Approach

Markov chain models are frequently used for studying event histories that include transitions between several states. An empirical transition matrix for nonhomogeneous Markov chains has previously been developed, including a detailed statistical theory based on counting processes and martingales. In this article, we show how to estimate transition probabilities dependent on covariates. This technique may, e.g., be used for making estimates of individual prognosis in epidemiological or clinical studies. The covariates are included through nonparametric additive models on the transition intensities of the Markov chain. The additive model allows for estimation of covariate-dependent transition intensities, and again a detailed theory exists based on counting processes. The martingale setting now allows for a very natural combination of the empirical transition matrix and the additive model, resulting in estimates that can be expressed as stochastic integrals, and hence their properties are easily evaluated. Two medical examples will be given. In the first example, we study how the lung cancer mortality of uranium miners depends on smoking and radon exposure. In the second example, we study how the probability of being in response depends on patient group and prophylactic treatment for leukemia patients who have had a bone marrow transplantation. A program in R and S-PLUS that can carry out the analyses described here has been developed and is freely available on the Internet.     

Markov and Semi‐Markov Switching Linear Mixed Models Used to Identify Forest Tree Growth Components

Summary Tree growth is assumed to be mainly the result of three components: (i) an endogenous component assumed to be structured as a succession of roughly stationary phases separated by marked change points that are asynchronous among individuals, (ii) a time‐varying environmental component assumed to take the form of synchronous fluctuations among individuals, and (iii) an individual component corresponding mainly to the local environment of each tree. To identify and characterize these three components, we propose to use semi‐Markov switching linear mixed models, i.e., models that combine linear mixed models in a semi‐Markovian manner. The underlying semi‐Markov chain represents the succession of growth phases and their lengths (endogenous component) whereas the linear mixed models attached to each state of the underlying semi‐Markov chain represent—in the corresponding growth phase—both the influence of time‐varying climatic covariates (environmental component) as fixed effects, and interindividual heterogeneity (individual component) as random effects. In this article, we address the estimation of Markov and semi‐Markov switching linear mixed models in a general framework. We propose a Monte Carlo expectation–maximization like algorithm whose iterations decompose into three steps: (i) sampling of state sequences given random effects, (ii) prediction of random effects given state sequences, and (iii) maximization. The proposed statistical modeling approach is illustrated by the analysis of successive annual shoots along Corsican pine trunks influenced by climatic covariates.     

长白山次生林演替过程中土壤动物群落的变化

通过对长白山白河局不同林龄的次生林土壤动物的组成进行比较。探讨了土壤动物群落随森林采伐更新演替的变化以及演替阶段(增长期、过渡期和稳定期)．研究表明,不同类群土壤动物的演变趋势是腐食类群数量相对较多,其变化趋势接近植被的变化规律,均是经过上升、下降,最后达到稳定;植食类群演替前期在整个群落中所占比例较高,随着演替年代的增加,比例呈下降的趋势;捕食类群经过100年左右的演替,数量明显增加;从昆虫类群组成来看,林龄较短者,未稳定的类群数较多,林龄较长者,未稳定的类群较少．     

Analysis of Longitudinal Data of Epileptic Seizure Counts – A Two‐State Hidden Markov Regression Approach

This paper discusses a two‐state hidden Markov Poisson regression (MPR) model for analyzing longitudinal data of epileptic seizure counts, which allows for the rate of the Poisson process to depend on covariates through an exponential link function and to change according to the states of a two‐state Markov chain with its transition probabilities associated with covariates through a logit link function. This paper also considers a two‐state hidden Markov negative binomial regression (MNBR) model, as an alternative, by using the negative binomial instead of Poisson distribution in the proposed MPR model when there exists extra‐Poisson variation conditional on the states of the Markov chain. The two proposed models in this paper relax the stationary requirement of the Markov chain, allow for overdispersion relative to the usual Poisson regression model and for correlation between repeated observations. The proposed methodology provides a plausible analysis for the longitudinal data of epileptic seizure counts, and the MNBR model fits the data much better than the MPR model. Maximum likelihood estimation using the EM and quasi‐Newton algorithms is discussed. A Monte Carlo study for the proposed MPR model investigates the reliability of the estimation method, the choice of probabilities for the initial states of the Markov chain, and some finite sample behaviors of the maximum likelihood estimates, suggesting that (1) the estimation method is accurate and reliable as long as the total number of observations is reasonably large, and (2) the choice of probabilities for the initial states of the Markov process has little impact on the parameter estimates.     

Simulated dynamics of succession in a North American subtropical Prosopis savanna

Abstract. A transition matrix model was used to explore the dynamics, rate and potential extent of changes in landscape vegetation patterns on a southern Texas Prosopis savanna. Transitions between seven vegetation classes were determined for the periods 1941–1960 and 1960–1983 on aerial photographs of three sites. During these periods, the sites were heavily grazed by cattle and were fire-free. Vegetation states assessed in grids of 20 m x 20 m cells superimposed on photographs ranged from grass-dominated to woody plant-dominated. The 1941–1960 period (denoted DRY) was characterized by prolonged drought, whereas annual rainfall during the 1960–1983 period (denoted WET) was typically normal to above-normal. The 1941 landscape consisted of herbaceous zones (6% of cells), woodland (50% of cells) and savanna parkland (44% of cells with grass/woody plant mixtures). The woodland state was the most stable, with probabilities of no change being 0.970 and 0.873 in WET and DRY periods, respectively. The herbaceous state was least stable, with corresponding values of 0.074 and 0.353. Past and future landscape structure was modelled by randomly selecting DRY or WET transitions at 20 year time steps. The model was run under a series of rainfall scenarios where the probability of selecting the WET transition matrix (P[WET]) ranged from 0 (DRY always chosen) to 1 (WET always chosen). Historical records indicate P[WET] has approximated 0.3 to 0.4 in the region. The rate of succession to states of greater woody cover increased as P[WET] increased. Forward simulations based on P[WET] > 0.2 suggest the present landscape is unstable and will develop into a closed-canopy woodland within the next 180 years, assuming the processes operating between 1941 and 1983 continue (e.g. grazing by cattle and lack of fire). Reverse simulations concur with historical observations and projections derived from woody plant growth rates in other studies and suggest that 200 to 300 yr BP these landscapes contained a substantially greater proportion of cells dominated by grassland or grassland with scattered woody plants (43 to 74%) than was present in 1983 (19%). Based upon elapsed time between predicted past and future steady states, succession from open savanna to closed-canopy woodland may occur in ca. 400 to 500 yr for P(WET) ≥ 0.33. Arresting or reversing the projected trend may require changes in climate and/or changes in livestock grazing and land management practices. The approaches employed in this study illustrate how time series maps, aerial photographs and satellite imagery can be analyzed and used to interpret, project and reconstruct local and regional changes in ecosystem structure. Difficulties and limitations associated with the use of Markov chains to model succession are identified and discussed.     

Arthropod orientation in choice chambers as a Markov chain

ABSTRACT. A Markov model of arthropod locomotor behaviour in choice chambers is presented. In the model, the compartments of the choice chamber and the movements of the animals from one compartment to another are treated as the states and the state transitions of a Markov chain, respectively. The model allows one to calculate the transition probabilities and the equilibrium distribution of animals in a choice chamber from direct measurements of displacement speeds and boundary turning reactions in each compartment. The compartment preferences (i.e. the proportion of time spent in each compartment) in two-compartment choice chambers were found to be strongly correlated with the predictions of the model. Klinokinesis is suggested to have negligible significance in some of the experiments reviewed. An equation is given to evaluate quantitatively the relative strength of kinetic and tactic components in the overall preference in the choice chamber; this assessment has not previously been possible.     

Changing importance of environmental factors driving secondary succession on molehills

Question: Species composition during secondary succession is influenced by a number of factors, such as soil moisture, disturbance timing and surrounding vegetation. How does the importance of these factors change over the course of succession? Methods: We set up a full‐factorial block design using molehills differing in (a) disturbance timing, (b) soil moisture and (c) composition of surrounding vegetation, and recorded the cover of all species present on the molehills over 3 years. M1ultiple regression analyses on the dissimilarity matrices of community composition and of environmental factors were applied for each of five age classes of molehills to estimate the effect of the single factors at different stages of succession. Results: The timing of disturbance did not significantly affect community composition at any stage of succession. In contrast, the effects of soil moisture and surrounding vegetation changed significantly over time, with moisture being more important at earlier stages of succession and surrounding vegetation at later stages. Conclusion: The importance of environmental factors for species composition change significantly over the course of secondary succession. Instead of aggregating the effects of environmental factors over time, future studies should consider underlying dynamics of recolonization more comprehensively.     

Detecting Memory and Structure in Human Navigation Patterns Using Markov Chain Models of Varying Order

One of the most frequently used models for understanding human navigation on the Web is the Markov chain model, where Web pages are represented as states and hyperlinks as probabilities of navigating from one page to another. Predominantly, human navigation on the Web has been thought to satisfy the memoryless Markov property stating that the next page a user visits only depends on her current page and not on previously visited ones. This idea has found its way in numerous applications such as Google''s PageRank algorithm and others. Recently, new studies suggested that human navigation may better be modeled using higher order Markov chain models, i.e., the next page depends on a longer history of past clicks. Yet, this finding is preliminary and does not account for the higher complexity of higher order Markov chain models which is why the memoryless model is still widely used. In this work we thoroughly present a diverse array of advanced inference methods for determining the appropriate Markov chain order. We highlight strengths and weaknesses of each method and apply them for investigating memory and structure of human navigation on the Web. Our experiments reveal that the complexity of higher order models grows faster than their utility, and thus we confirm that the memoryless model represents a quite practical model for human navigation on a page level. However, when we expand our analysis to a topical level, where we abstract away from specific page transitions to transitions between topics, we find that the memoryless assumption is violated and specific regularities can be observed. We report results from experiments with two types of navigational datasets (goal-oriented vs. free form) and observe interesting structural differences that make a strong argument for more contextual studies of human navigation in future work.     

Estimating local interaction from spatiotemporal forest data, and Monte Carlo bias correction

We point out a general problem in fitting continuous time spatially explicit models to a temporal sequence of spatial data observed at discrete times. To illustrate the problem, we examined the continuous time Markov model for forest gap dynamics. A forest is assumed to be apportioned into discrete cells (or sites) arranged in a regular square lattice. Each site is characterized as either a gap or a non-gap site according to the vegetation height of trees. The model incorporates the influence of neighboring sites on transition rate: transition rate from a non-gap to a gap site increases linearly with the number of neighbors that are currently in the gap state, and vice versa. We fitted the model to the spatiotemporal data of canopy height observed at the permanent plot in Barro Colorado Island (BCI). When we used the approximate maximum likelihood method to estimate the parameters of the model, the estimated transition rates included a large bias-in particular, the strength of interaction between nearby sites was underestimated. This bias originated from the assumption that each transition between two observation times is independent. The interaction between sites at local scale creates a long chain of transitions within a single census interval, which violates the independence of each transition. We show that a computer-intensive method, called Monte Carlo bias correction (MCBC), is very effective in removing the bias included in the estimate. The global and local gap densities measuring spatial aggregation of gap sites were computed from simulated and real gap dynamics to assess the model. When the approximate likelihood estimates were applied to the model, the predicted local gap density was clearly lower than the observed one. The use of MCBC estimates, suggesting a strong interaction between sites, improved this discrepancy.     

Markov modelling of immunological and virological states in HIV-1 infected patients

The purpose of this study was to evaluate the evolution of HIV infected patients and to bring out some significant factors associated with this pathology. The main criteria revealing the State of illness is viral load measurement (VL). However the CD4 lymphocytes also represent an important marker as these reflect the State of the immune reservoir. Many studies have been carried out in this field and different models have been proposed with a view to a better understanding of this disease. Multi State Markov models defined in terms of CD4 counts, or in terms of viral load, have proved to be very useful tools for modelling HIV disease progression. The model we have developed in this study is based on both the CD4 lymphocytes counts and VL. Markov models are characterized by transition intensities. In this paper we explored several structures in succession. First, we used a homogeneous continuous time Markov process with four states defined by crossed values of CD4 and VL in a given patient at a given time. Then, the effect of certain covariates on the infection process was introduced into the model via the transition intensity functions, as with a Cox regression model. Since the hypothesis of homogeneity may be unrealistic in certain cases, we also considered piecewise homogeneous Markov models. Finally, the effects of covariates and time were combined in a piecewise homogeneous model with a covariate. We applied these methods to data from 1313 HIV-infected patients included in the NADIS cohort.     

DNA damage caused by ionizing radiation.

A survey is given of continuous-time Markov chain models for ionizing radiation damage to the genome of mammalian cells. In such models, immediate damage induced by the radiation is regarded as a batch-Poisson arrival process of DNA double-strand breaks (DSBs). Enzymatic modification of the immediate damage is modeled as a Markov process similar to those described by the master equation of stochastic chemical kinetics. An illustrative example is the restitution/complete-exchange model. The model postulates that, after being induced by radiation, DSBs subsequently either undergo enzymatically mediated restitution (repair) or participate pairwise in chromosome exchanges. Some of the exchanges make irremediable lesions such as dicentric chromosome aberrations. One may have rapid irradiation followed by enzymatic DSB processing or have prolonged irradiation with both DSB arrival and enzymatic DSB processing continuing throughout the irradiation period. Methods for analyzing the Markov chains include using an approximate model for expected values, the discrete-time Markov chain embedded at transitions, partial differential equations for generating functions, normal perturbation theory, singular perturbation theory with scaling, numerical computations, and certain matrix methods that combine Perron-Frobenius theory with variational estimates. Applications to experimental results on expected values, variances, and statistical distributions of DNA lesions are briefly outlined. Continuous-time Markov chains are the most systematic of those radiation damage models that treat DSB-DSB interactions within the cell nucleus as homogeneous (e.g., ignore diffusion limitations). They contain virtually all other relevant homogeneous models and semiempirical summaries as special cases, limiting cases, or approximations. However, the Markov models do not seem to be well suited for studying spatial dependence of DSB interactions, which is known to be important in some situations.     

Estimating Behavioral Transition Rates: Problems and Solutions

Continuous time Markov chain (CTMC) models offer ethologists a powerful tool. The methods are based on well-established procedures for estimating the rates at which one state (e.g. resting) changes to some other set of states (e.g. feeding, fighting, etc.). Unfortunately, ethological data typically differ in a very critical manner from the type of data to which these methods are usually applied: ethological data are usually heavily censored in the sense that each behavioral state shows frequent transitions to several other possible states. This occurs when several competing processes can each end a bout.
We used computer simulation of various behavioral models with known transition rates to investigate the unknown performance of four of the most popular statistical tests for screening data prior to application of CTMC models; this included a modification of one of these tests derived under the assumption of random censoring. Two of the four tests failed completely and would result in rejection of nearly all data even if the model did fit the assumptions of the CTMC methods. Only Barlow's total-time-on test performed with an acceptable α error rate under all conditions. None of the tests were particularly effective at detecting certain types of departures from the CTMC assumptions.
Guidelines are given as to how much confidence should be attached to apparent changes in transition rates.