Modelling Cooperation between fishermen with a Cellular automaton. A framework for fishing effort spatial dynamics

In many fisheries sharing information between vessels is an important characteristic of fishermen's behaviour rarely modelled or analyzed. A Cellular Automaton is designed in an attempt to understand circumstances that favour group formation. The simulated world is toroidal with a static fishing resource distributed in patches. Movement decisions are random in the case of fishermen in a local scale. After a certain time interval, sharing information is possible between fishermen in a dynamic Moore's neighbourhood of cells at a broader scale and movement to adjacent areas may occur according to a set of rules. The dynamic neighbourhood is a novel concept defined in this work within the framework of Cellular Automata. Decision making by each fisherman is a function of the influence other fishermen (neighbours) exert on them as well as on personal knowledge, to form an opinion of the areas (cells) quality, and take action consequently.     

空间异质性对植物种群动态的影响:三种模拟方法的比较

为了讨论单一物种在异质性景观中的空间传播,将平均场近似模型和偶对近似模型的结果进行对比研究.本研究选择了有代表性的四邻域和八邻域时物种的传播情况,首先运用细胞自动机建立了理想模型,对偶对近似模型和平均场近似模型在全局密度和局域密度固定时随着出生率与死亡率比值变化的结果比较,以细胞自动机模型结果为依据,判断偶对近似与平均场近似哪个结果更加接近细胞自动机模型的结果.通过分析得到四邻域时在近似细胞自动机模型结果时偶对近似的结果优于平均场近似的结果,但是在八邻域时三个模型之间的差异性不再那么明显,偶对近似依然能够很好的预测细胞自动机模型的结果.     

A spatial model of forest dynamics

Effects of spatial processes on temperate deciduous forest structure and dynamics were investigated with a spatial simulator derived from a forest gap model. The multi-species neighborhood model accounted for competitive interactions and endogenous disturbance in the form of small canopy gaps. Simulated and actual spatial pattern of old-growth stands were compared. The 400 yr simulations produced a pattern scale (0.07–0.2 ha patches) similar to that of an actual stand; simulated pattern intensity was greater than actual intensity, however. Distances to nearest neighbor were somewhat similar for trees in the simulated and actual stands; yet the frequency distributions of distance to nearest neighbor values differed substantially. The simulated stand patterns were generally less random than the actual patterns. Spatial pattern changed markedly during the course of simulated succession. Pattern approached a random dispersion in early succession. Intensity peaked at mid-succession (ca. 150 yr) with a hyperdispersed overstory and a strongly clumped understory. Pattern intensity diminished in late succession as a mixed size structure developed. Old-growth patch size was greater than the neighborhood (or gap) size, suggesting the gap-sized areas do not behave independently.     

Analytical methods for predicting the behaviour of population models with general spatial interactions

Many biologists use population models that are spatial, stochastic and individual based. Analytical methods that describe the behaviour of these models approximately are attracting increasing interest as an alternative to expensive computer simulation. The methods can be employed for both prediction and fitting models to data. Recent work has extended existing (mean field) methods with the aim of accounting for the development of spatial correlations. A common feature is the use of closure approximations for truncating the set of evolution equations for summary statistics. We investigate an analytical approach for spatial and stochastic models where individuals interact according to a generic function of their distance; this extends previous methods for lattice models with interactions between close neighbours, such as the pair approximation. Our study also complements work by Bolker and Pacala (BP) [Theor. Pop. Biol. 52 (1997) 179; Am. Naturalist 153 (1999) 575]: it treats individuals as being spatially discrete (defined on a lattice) rather than as a continuous mass distribution; it tests the accuracy of different closure approximations over parameter space, including the additive moment closure (MC) used by BP and the Kirkwood approximation. The study is done in the context of an susceptible-infected-susceptible epidemic model with primary infection and with secondary infection represented by power-law interactions. MC is numerically unstable or inaccurate in parameter regions with low primary infection (or density-independent birth rates). A modified Kirkwood approximation gives stable and generally accurate transient and long-term solutions; we argue it can be applied to lattice and to continuous-space models as a substitute for MC. We derive a generalisation of the basic reproduction ratio, R(0), for spatial models.     

Direction of regeneration waves in grid-based models for forest dynamics

Progressing waves of regeneration are observed in forest ecosystems such as Shimagare fir forests. The patterns generated by lattice models for forest dynamics often show similar waves of disturbance and recovery. This paper introduces a method to detect and quantify the directional movement of these waves. The method is based only on the disturbance times of the sites and allows to distinguish three types of wave patterns: patterns with global direction, patterns with local direction, and patterns without direction. We apply this to several grid-based models for forest dynamics which are evaluated analytically or by simulation. The results reveal a clear distinction of the models which earlier studies were not able to detect.     

Local facilitation, bistability and transitions in arid ecosystems

Arid ecosystems are liable to undergo sudden discontinuous transitions from a vegetated to a desert state as a result of human pressure and climate change. A predictive framework about the conditions under which such transitions occur is lacking. Here, we derive and analyze a general model describing the spatial dynamics of vegetation in arid ecosystems considering local facilitation as an essential process. We investigate the conditions under which continuous or discontinuous transitions from a vegetated to a desert state are likely to occur. We focus on arid ecosystems but our approach is sufficiently general to be applied to other ecosystems with severe environmental conditions. The model exhibits bistability and vegetation patchiness. High local facilitation decreases the risk of discontinuous transitions. Moreover, for arid ecosystems where local facilitation is a driving process, vegetation patchiness indicates proximity to a transition point, but does not allow distinguishing between continuous and discontinuous transitions.     

Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure

We describe a mathematically exact method for the analysis of spatially structured Markov processes. The method is based on a systematic perturbation expansion around the deterministic, non-spatial mean-field theory, using the theory of distributions to account for space and the underlying stochastic differential equations to account for stochasticity. As an example, we consider a spatial version of the Levins metapopulation model, in which the habitat patches are distributed in the d-dimensional landscape Rd in a random (but possibly correlated) manner. Assuming that the dispersal kernel is characterized by a length scale L, we examine how the behavior of the metapopulation deviates from the mean-field model for a finite but large L. For example, we show that the equilibrium fraction of occupied patches is given by p(0)+c/L(d)+O(L(-3d/2)), where p(0) is the equilibrium state of the Levins model and the constant c depends on p(0), the dispersal kernel, and the structure of the landscape. We show that patch occupancy can be increased or decreased by spatial structure, but is always decreased by stochasticity. Comparison with simulations show that the analytical results are not only asymptotically exact (as L-->infinity), but a good approximation also when L is relatively small.     

Fractal properties of forest spatial structure

The definition of fractal dimension of natural objects, which enables to deal with scale dependence of fractal dimension is discussed. Abrupt changes of fractal dimension of spatial structure of terrestrial ecosystems are considered in the context of hierarchical paradigm. On this ground the procedure is proposed for segmentation of a territory, which takes into account the scale dependence of spatial variability of ecological parameters. Using remotely sensed data — normalized difference vegetation index (NDVI) and thermal radiation in the infrared band — fractal dimensions and critical scales are evaluated for different forest types with the help of software, developed for this purpose. The results obtained corroborate the potentialities of fractal approach in ecology. These methods and results can be used for discrimination of remotely sensed data; but further investigations, including detailed comparison of fractal characteristics of remotely sensed forest images with results of on-site field studies are necessary to validate them.     

Probing the effects of the well-mixed assumption on viral infection dynamics

Viral kinetics have been extensively studied in the past through the use of spatially well-mixed ordinary differential equations describing the time evolution of the diseased state. However, emerging spatial structures such as localized populations of dead cells might adversely affect the spread of infection, similar to the manner in which a counter-fire can stop a forest fire from spreading. In a previous publication [Beauchemin, C., Samuel, J., Tuszynski, J., 2005. A simple cellular automaton model for influenza A viral infections. J. Theor. Biol. 232(2), 223-234], a simple two-dimensional cellular automaton model was introduced and shown to be accurate enough to model an uncomplicated infection with influenza A. Here, this model is used to investigate the effects of relaxing the well-mixed assumption. Particularly, the effects of the initial distribution of infected cells, the regeneration rule for dead epithelial cells, and the proliferation rule for immune cells are explored and shown to have an important impact on the development and outcome of the viral infection in our model.     

Spatial pattern analysis in forest dynamics: deviation from power law and direction of regeneration waves

Grid-based models have been used to understand spatial heterogeneity of the vegetation height in forests and to analyze spatio-temporal dynamics of the forest regeneration process. In this report, we present two methods of identifying lattice models when spatio-temporal data are given. The first method detects directionality of regeneration waves based on the timing of local disturbance events. The second evaluates the forest pattern by recording the fraction of high and low vegetation areas at multiple spatial scales. We illustrate these methods by applying them to patterns generated using three simple stochastic lattice models: (1) two-state model, distinguishing sites with high and low vegetation, (2) three-state model, in which sites can be in an additional disturbed state, and (3) Shimagare model, which considers a continuous range of states. The combination of the two methods provides efficient means of distinguishing the models. The first method has a more direct ecological meaning, while the second is useful when forest data are limited in time.     

Using network models to approximate spatial point-process models

Spatial effects are fundamental to ecological and epidemiological systems, yet the incorporation of space into models is potentially complex. Fixed-edge network models (i.e. networks where each edge has the same fixed strength of interaction) are widely used to study spatial processes but they make simplistic assumptions about spatial scale and structure. Furthermore, it can be difficult to parameterize such models with empirical data. By comparison, spatial point-process models are often more realistic than fixed-edge network models, but are also more difficult to analyze. Here we develop a moment closure technique that allows us to define a fixed-edge network model which predicts the prevalence and rate of epidemic spread of a continuous spatial point-process epidemic model. This approach provides a systematic method for accurate parameterization of network models using data from continuously distributed populations (such as data on dispersal kernels). Insofar as point-process models are accurate representations of real spatial biological systems, our example also supports the view that network models are realistic representations of space.     

Periodic solutions for a population dynamics problem with age-dependence and spatial structure

Using a linear model with age-dependence and spatial structure we show how a periodical supply of individuals will transform an exponentially decaying distribution of population into a non-trivial asymptotically stable periodic distribution. Next we give an application to an epidemic model.     

Spatio-temporal patterns of tree community dynamics in a tropical forest fragment in South-east Brazil

The tree community (dbh > 5 cm) of a fragment of tropical montane semi-deciduous forest in South-east Brazil was repeatedly surveyed over a 19-year period in order to assess spatial and temporal patterns of dynamics. The surveys took place in 1987, 1992, 1996, 2001, and 2006 in a grid of 126 20 × 20 m permanent plots covering almost the entire fragment (5.8 ha). Overall patterns indicated that a self-thinning process has taken place in the fragment since 1992. Community dynamics varied in space and time, with most dynamics highly spatially clustered. With exception of mortality rates, there were no changes in the spatial patterns of community dynamics through time. No relation between edges and dynamics variables was found. Most species with increasing density and basal area were shade-bearers, while most decreasing species were canopy light demanders and pioneers.     

Analytic approximation of spatial epidemic models of foot and mouth disease

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been made in developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods for approximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R₀, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrary number of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies in the ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability by deriving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networks and the effect of spatial correlations.     

A simple cellular automaton model for influenza A viral infections

Viral kinetics have been extensively studied in the past through the use of spatially homogeneous ordinary differential equations describing the time evolution of the diseased state. However, spatial characteristics such as localized populations of dead cells might adversely affect the spread of infection, similar to the manner in which a counter-fire can stop a forest fire from spreading. In order to investigate the influence of spatial heterogeneities on viral spread, a simple 2-D cellular automaton (CA) model of a viral infection has been developed. In this initial phase of the investigation, the CA model is validated against clinical immunological data for uncomplicated influenza A infections. Our results will be shown and discussed.     

不同松林内马尾松毛虫种群动态的特征

通过对1992-2001年5种不同混交林型的马尾松林内各代幼虫发生量,受害面积和有虫株率的系统调查,研究了各种不同林型的松林对马尾松毛虫种群动态的影响。结果表明,不同树种混交的马尾松林对马尾松毛虫的种群动态有较大的影响。马尾松毛虫在马尾松纯林中发生量多,为害最严重。暴发周期最短;在与杉木等混交的马尾松林中发生次之;而在与阔叶树混交的马尾松林中平均每株虫口数少,发生与危害率及有虫株率均较其它林型小,不易引起松毛虫暴发为害。     

Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion

In this work we analyze the large time behavior in a nonlinear model of population dynamics with age-dependence and spatial diffusion. We show that when t+ either the solution of our problem goes to 0 or it stabilizes to a nontrivial stationary solution. We give two typical examples where the stationary solutions can be evaluated upon solving very simple partial differential equations. As a by-product of the extinction case we find a necessary condition for a nontrivial periodic solution to exist. Numerical computations not described below show a rapid stabilization.This work was partially supported by the Centre National de la Recherche Scientifique through ATP 95939900     

The effects of spatial heterogeneity in population dynamics

The dynamics of a population inhabiting a heterogeneous environment are modelled by a diffusive logistic equation with spatially varying growth rate. The overall suitability of an environment is characterized by the principal eigenvalue of the corresponding linearized equation. The dependence of the eigenvalue on the spatial arrangement of regions of favorable and unfavorable habitat and on boundary conditions is analyzed in a number of cases.Research supported by National Science Foundation grant #DMS 88-02346