Modeling and analysis of stochastic invasion processes

In this paper we derive spatially explicit equations to describe a stochastic invasion process. Parents are assumed to produce a random number of offspring which then disperse according to a spatial redistribution kernel. Equations for population moments, such as expected density and covariance averaged over an ensemble of identical stochastic processes, take the form of deterministic integro-difference equations. These equations describe the spatial spread of population moments as the invasion progresses. We use the second order moments to analyse two basic properties of the invasion. The first property is permanence of form in the correlation structure of the wave. Analysis of the asymptotic form of the invasion wave shows that either (i) the covariance in the leading edge of the wave of invasion asymptotically achieves a permanence of form with a characteristic structure described by an unchanging spatial correlation function, or (ii) the leading edge of the wave has no asymptotic permanence of form with the length scales of spatial correlations continually increasing over time. Which of these two outcomes pertains is governed by a single statistic, φ which depends upon the shape of the dispersal kernel and the net reproductive number. The second property of the invasion is its patchy structure. Patchiness, defined in terms of spatial correlations on separate short (within patch) and long (between patch) spatial scales, is linked to the dispersal kernel. Analysis shows how a leptokurtic dispersal kernel gives rise to patchiness in spread of a population. Received: 11 August 1997 / Revised version: 22 September 1998 / Published online: 4 October 2000     

Spatial correlations in attribute communities

Community detection is an important tool for exploring and classifying the properties of large complex networks and should be of great help for spatial networks. Indeed, in addition to their location, nodes in spatial networks can have attributes such as the language for individuals, or any other socio-economical feature that we would like to identify in communities. We discuss in this paper a crucial aspect which was not considered in previous studies which is the possible existence of correlations between space and attributes. Introducing a simple toy model in which both space and node attributes are considered, we discuss the effect of space-attribute correlations on the results of various community detection methods proposed for spatial networks in this paper and in previous studies. When space is irrelevant, our model is equivalent to the stochastic block model which has been shown to display a detectability-non detectability transition. In the regime where space dominates the link formation process, most methods can fail to recover the communities, an effect which is particularly marked when space-attributes correlations are strong. In this latter case, community detection methods which remove the spatial component of the network can miss a large part of the community structure and can lead to incorrect results.     

Social personality polymorphism and the spread of invasive species: a model

Ecological invasions are a major worldwide problem exacting tremendous economic and ecological costs. Efforts to explain variability in invasion speed and impact by searching for combinations of ecological conditions and species traits associated with invasions have met with mixed success. We use a simulation model that integrates insights from life-history theory, animal personalities, network theory, and spatial ecology to derive a new mechanism for explaining variation in animal invasion success. We show that spread occurs most rapidly when (1) a species includes a mix of life-history or personality types that differ in density-dependent performance and dispersal tendencies, (2) the differences between types are of intermediate magnitude, and (3) patch connections are intermediate in number and widely spread. Within-species polymorphism in phenotype (e.g., life-history strategies or personality), a feature not included in previous models, is important for overcoming the fact that different traits are associated with success in different stages of the invasion process. Polymorphism in sociability (a personality type) increases the speed of the invasion front, since asocial individuals colonize empty patches and facilitate the local growth of social types that, in turn, induce faster dispersal by asocials at the invasion edge. The results hold implications for the prediction of invasion impacts and the classification of traits associated with invasiveness.     

Spread rate for a nonlinear stochastic invasion

Despite the recognized importance of stochastic factors, models for ecological invasions are almost exclusively formulated using deterministic equations [29]. Stochastic factors relevant to invasions can be either extrinsic (quantities such as temperature or habitat quality which vary randomly in time and space and are external to the population itself) or intrinsic (arising from a finite population of individuals each reproducing, dying, and interacting with other individuals in a probabilistic manner). It has been long conjectured [27] that intrinsic stochastic factors associated with interacting individuals can slow the spread of a population or disease, even in a uniform environment. While this conjecture has been borne out by numerical simulations, we are not aware of a thorough analytical investigation. In this paper we analyze the effect of intrinsic stochastic factors when individuals interact locally over small neighborhoods. We formulate a set of equations describing the dynamics of spatial moments of the population. Although the full equations cannot be expressed in closed form, a mixture of a moment closure and comparison methods can be used to derive upper and lower bounds for the expected density of individuals. Analysis of the upper solution gives a bound on the rate of spread of the stochastic invasion process which lies strictly below the rate of spread for the deterministic model. The slow spread is most evident when invaders occur in widely spaced high density foci. In this case spatial correlations between individuals mean that density dependent effects are significant even when expected population densities are low. Finally, we propose a heuristic formula for estimating the true rate of spread for the full nonlinear stochastic process based on a scaling argument for moments. Received: 19 October 1998 / Revised version: 1 September 1999 / Published online: 4 October 2000     

A flow network model for animal movement on a landscape with application to invasion

Animal movement, whether for foraging, mate-seeking, predator avoidance, dispersal, or migration, is a fundamental aspect of ecology that shapes spatial abundance distributions, genetic compositions, and dynamics of populations. A variety of movement models have been used for predicting the effects of natural or human-caused landscape changes, invading species, or other disturbances on local ecology. Here we introduce the flow network—a general modeling framework for population dynamics and movement in a metapopulation representing a network of habitat sites (nodes). Based on the principles of physical transport phenomena such as fluid flow through pipes (Pouiselle’s Law) and analogously, the flow of electric current across a circuit (Ohm’s Law), the flow network provides a novel way of modeling movement, where flow rates are functions of relative node pressures and the resistance to movement between them. Flow networks offer the flexibility of incorporating abiotic and biotic conditions that affect either pressures, resistance, or both. To illustrate an application of the flow network, we present a theoretical invasion scenario. We consider the effects of spatial structure on the speed of invasion by varying the spatial regularity of node arrangement. In the context of invasion, we model management actions targeting nodes or edges, and consider the effects on speed of invasion, node occupation, and total abundance. The flow network approach offers the flexibility to incorporate spatial heterogeneity in both rates of flow and site pressures and offers an intuitive approach to connecting population dynamics and landscape features to model movement.     

Effects of landscape structure on the invasive spread of black cherry Prunus serotina in an agricultural landscape in Flanders, Belgium

Analysing invasive spread from a landscape ecological perspective forms an important challenge in plant invasion ecology. The present study examines the effects of landscape structure on the spatial and temporal dynamics of an expanding black cherry Prunus serotina population within a rural landscape in Flanders, Belgium, carrying a dense network of interconnected hedgerows. The study area, 251 ha in size, harboured a total of 2962 P. serotina individuals. The population was characterised by a negative exponential age distribution, a high growth rate and an early and continuous reproduction throughout the species' life cycle. The historical rate of spread of the species through the hedgerow network progressively increased with time, especially during the last decade. Spatial point pattern analysis revealed that the individuals had a significantly clustered distribution pattern and were spatially aggregated around seed sources, hedgerow intersections and roosting trees. Logistic regression analysis confirmed the effect of landscape structure on P. serotina occurrence, suggesting directional long distance dispersal by avian dispersal vectors, resulting in a differential seed pressure throughout the hedgerow network due to the preference of dispersing birds for roosting in structurally rich hedgerow with large trees near hedgerow intersections. Hence, the distribution of P. serotina in agricultural landscapes was strongly mediated by dispersal processes. Furthermore, decreasing spatial aggregation along the species life cycle, with especially seedlings and saplings being significantly aggregated while adult individuals were mostly distributed at random, and a relative outward shift of seedling recruitment curves with time indicate density dependent mortality, probably caused by intraspecific competition.     

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.      

Spatial vs. Temporal Features in ICA of Resting-State fMRI – A Quantitative and Qualitative Investigation in the Context of Response Inhibition

Independent component analysis (ICA) can identify covarying functional networks in the resting brain. Despite its relatively widespread use, the potential of the temporal information (unlike spatial information) obtained by ICA from resting state fMRI (RS-fMRI) data is not always fully utilized. In this study, we systematically investigated which features in ICA of resting-state fMRI relate to behaviour, with stop signal reaction time (SSRT) in a stop-signal task taken as a test case. We did this by correlating SSRT with the following three kinds of measure obtained from RS-fMRI data: (1) the amplitude of each resting state network (RSN) (evaluated by the standard deviation of the RSN timeseries), (2) the temporal correlation between every pair of RSN timeseries, and (3) the spatial map of each RSN. For multiple networks, we found significant correlations not only between SSRT and spatial maps, but also between SSRT and network activity amplitude. Most of these correlations are of functional interpretability. The temporal correlations between RSN pairs were of functional significance, but these correlations did not appear to be very sensitive to finding SSRT correlations. In addition, we also investigated the effects of the decomposition dimension, spatial smoothing and Z-transformation of the spatial maps, as well as the techniques for evaluating the temporal correlation between RSN timeseries. Overall, the temporal information acquired by ICA enabled us to investigate brain function from a complementary perspective to the information provided by spatial maps.     

Space to choose: network analysis of social preferences in a captive chimpanzee community, and implications for management

Social network analysis (SNA) is rapidly gaining popularity in primatology, but its application to the management of zoo-housed primates has been largely overlooked. Here I use SNA techniques to explore the social structure of chimpanzees (Pan troglodytes) housed in the new "Budongo Trail" exhibit at Edinburgh Zoo, UK. Given that individuals have extensive space (2332 m(2)), and access to several interconnected exhibit sections, I test the hypothesis that individuals are able to choose to interact with specific social partners. Spatial association and social interaction data were recorded from 400 focal watches on 11 individuals, and association, affiliative, and agonistic networks were constructed. Matrix correlations showed that individuals who spent time in close proximity were likely to affiliate with one another, but spatial association did not predict the frequency of agonistic encounters. Cluster analysis revealed significantly distinct sub-groups in the affiliative network (but not association or agonistic networks) in line with maternal kinship. Overall my findings support the hypothesis that the Budongo Trail exhibit facilitates the expression of social preferences, and suggests that SNA can be a useful tool to study zoo primates when proximity between individuals is not forced (i.e. in large, modern exhibits). Now that I have validated a set of SNA methods for this community, they can be used to trace changes in social dynamics over a longer time period, and ultimately assist zoo staff in their management decisions.     

基于种间竞争模型的毒杂草入侵空间分布模拟研究

以天然草场中毒杂草与可食牧草为研究对象,在种间竞争的生态数学模型的基础上引入入侵扩散因素建立毒杂草入侵扩散模型。采用元胞自动机理论将竞争模型扩展到空间网络进行模拟研究,分析毒杂草属的空间分布类型,为毒杂草的控制提供数据支持。研究表明:(1)在入侵扩散作用下,毒杂草与可食牧草的共存平衡点由一个增加为两个,增加了共存的可能性;(2)入侵扩散作用影响了毒杂草种群的空间分布特征,减少了种群空间分布的聚集程度。     

PARTITIONING THE EFFECTS OF SPATIAL ISOLATION,NEST HABITAT,AND INDIVIDUAL DIET IN CAUSING ASSORTATIVE MATING WITHIN A POPULATION OF THREESPINE STICKLEBACK

Assortative mating is measured as a phenotypic or genotypic correlation between mates. Although biologists typically view assortative mating in terms of mate preference for similar partners, correlations between mates can also arise from phenotypic spatial structure arising from spatial isolation or habitat preferences. Here, we test whether diet‐assortative mating within an ecologically variable population of threespine stickleback results from small‐scale geographic isolation or microhabitat preference. We find evidence for assortative mating in the form of a positive correlation between mated pairs’ diets (measured using stable isotopes). Stable isotopes reveal diet differences between different nesting areas and among individuals using different nest habitat within a nesting area. This spatial segregation of diet types should generate some assortative mating, but is insufficient to explain the observed assortment strength. Significant male–female isotope correlations remain after controlling for spatial variables. We therefore conclude that sticklebacks’ diet‐assortative mating arises from additional behavioral preference. More generally, our results illustrate the point that spatial segregation can only drive appreciable levels of phenotypic assortative mating when environment‐phenotype correlations are parallel and strong in both sexes. Consequently, intraspecific assortative mating may typically entail mating preferences rather than just spatial cosegregation of phenotypes.     

Invasion Waves in Populations with Excitable Dynamics

Whilst the most obvious mechanism for a biological invasion is the occupation of a new territory as a result of direct ingress by individuals of the invading population, a more subtle “invasion” may occur without significant motion of invading individuals if the population dynamics in a predator prey scenario has an “excitable” character. Here, “excitable” means that a local equilibrium state, either of coexistence of predator and prey, or of prey only, may, when disturbed by a small perturbation, switch to a new, essentially invaded state. In an invasion of this type little spatial movement of individuals occurs, but a wave of rapid change of population level nevertheless travels through the invaded territory. In this article we summarise and review recent modelling research which shows that the macroscopic features of these invasion waves depend strongly on the detailed spatial dynamics of the predator–prey relationship; the models assume simple (linear) diffusion and pursuit-evasion, represented by (non-linear) cross-diffusion, as examples. In the context of plankton population dynamics, such waves may be produced by sudden injections of nutrient and consequent rapid increase in plankton populations, brought about, for example, by the upwelling caused by a passing atmospheric low pressure system.     

Global hot spots of biological invasions: evaluating options for ballast-water management

Biological invasions from ballast water are a severe environmental threat and exceedingly costly to society. We identify global hot spots of invasion based on worldwide patterns of ship traffic. We then estimate the rate of port-to-port invasion using gravity models for spatial interactions, and we identify bottlenecks to the regional exchange of species using the Ford-Fulkerson algorithm for network flows. Finally, using stochastic simulations of different strategies for controlling ballast-water introductions, we find that reducing the per-ship-visit chance of causing invasion is more effective in reducing the rate of biotic homogenization than eliminating key ports that are the epicentres for global spread.     

Spatial Autocorrelation of Genotypes under Directional Selection

The spatial distributions of genetic variation under selection-mutation equilibrium within populations that have limited dispersal are investigated. The results show that directional selection with moderate strength rapidly reduces the amount of genetic structure and spatial autocorrelations far below that predicted for selectively neutral loci. For the latter, homozygotes are spatially clustered into separate areas or patches, each consisting of several hundred homozygotes. When selection is added the patches of the deleterious homozygotes are much smaller, in the range of 25 to 50 individuals. Selection also reduces temporal correlations. Also investigated are the effects of random replacement processes, such as mutation, immigration, and long-distance migration, on spatial and temporal correlations. The detection of natural selection through spatial pattern analysis is discussed, and applied to data from populations of the morning glory, Ipomoea purpurea.     

Joint Effects of Habitat Heterogeneity and Species’ Life-History Traits on Population Dynamics in Spatially Structured Landscapes

Both habitat heterogeneity and species’ life-history traits play important roles in driving population dynamics, yet there is little scientific consensus around the combined effect of these two factors on populations in complex landscapes. Using a spatially explicit agent-based model, we explored how interactions between habitat spatial structure (defined here as the scale of spatial autocorrelation in habitat quality) and species life-history strategies (defined here by species environmental tolerance and movement capacity) affect population dynamics in spatially heterogeneous landscapes. We compared the responses of four hypothetical species with different life-history traits to four landscape scenarios differing in the scale of spatial autocorrelation in habitat quality. The results showed that the population size of all hypothetical species exhibited a substantial increase as the scale of spatial autocorrelation in habitat quality increased, yet the pattern of population increase was shaped by species’ movement capacity. The increasing scale of spatial autocorrelation in habitat quality promoted the resource share of individuals, but had little effect on the mean mortality rate of individuals. Species’ movement capacity also determined the proportion of individuals in high-quality cells as well as the proportion of individuals experiencing competition in response to increased spatial autocorrelation in habitat quality. Positive correlations between the resource share of individuals and the proportion of individuals experiencing competition indicate that large-scale spatial autocorrelation in habitat quality may mask the density-dependent effect on populations through increasing the resource share of individuals, especially for species with low mobility. These findings suggest that low-mobility species may be more sensitive to habitat spatial heterogeneity in spatially structured landscapes. In addition, localized movement in combination with spatial autocorrelation may increase the population size, despite increased density effects.     

The Correlation Structure of Local Neuronal Networks Intrinsically Results from Recurrent Dynamics

Correlated neuronal activity is a natural consequence of network connectivity and shared inputs to pairs of neurons, but the task-dependent modulation of correlations in relation to behavior also hints at a functional role. Correlations influence the gain of postsynaptic neurons, the amount of information encoded in the population activity and decoded by readout neurons, and synaptic plasticity. Further, it affects the power and spatial reach of extracellular signals like the local-field potential. A theory of correlated neuronal activity accounting for recurrent connectivity as well as fluctuating external sources is currently lacking. In particular, it is unclear how the recently found mechanism of active decorrelation by negative feedback on the population level affects the network response to externally applied correlated stimuli. Here, we present such an extension of the theory of correlations in stochastic binary networks. We show that (1) for homogeneous external input, the structure of correlations is mainly determined by the local recurrent connectivity, (2) homogeneous external inputs provide an additive, unspecific contribution to the correlations, (3) inhibitory feedback effectively decorrelates neuronal activity, even if neurons receive identical external inputs, and (4) identical synaptic input statistics to excitatory and to inhibitory cells increases intrinsically generated fluctuations and pairwise correlations. We further demonstrate how the accuracy of mean-field predictions can be improved by self-consistently including correlations. As a byproduct, we show that the cancellation of correlations between the summed inputs to pairs of neurons does not originate from the fast tracking of external input, but from the suppression of fluctuations on the population level by the local network. This suppression is a necessary constraint, but not sufficient to determine the structure of correlations; specifically, the structure observed at finite network size differs from the prediction based on perfect tracking, even though perfect tracking implies suppression of population fluctuations.     

Hydrological Networks and Associated Topographic Variation as Templates for the Spatial Organization of Tropical Forest Vegetation

An understanding of the spatial variability in tropical forest structure and biomass, and the mechanisms that underpin this variability, is critical for designing, interpreting, and upscaling field studies for regional carbon inventories. We investigated the spatial structure of tropical forest vegetation and its relationship to the hydrological network and associated topographic structure across spatial scales of 10–1000 m using high-resolution maps of LiDAR-derived mean canopy profile height (MCH) and elevation for 4930 ha of tropical forest in central Panama. MCH was strongly associated with the hydrological network: canopy height was highest in areas of positive convexity (valleys, depressions) close to channels draining 1 ha or more. Average MCH declined strongly with decreasing convexity (transition to ridges, hilltops) and increasing distance from the nearest channel. Spectral analysis, performed with wavelet decomposition, showed that the variance in MCH had fractal similarity at scales of ∼30–600 m, and was strongly associated with variation in elevation, with peak correlations at scales of ∼250 m. Whereas previous studies of topographic correlates of tropical forest structure conducted analyses at just one or a few spatial grains, our study found that correlations were strongly scale-dependent. Multi-scale analyses of correlations of MCH with slope, aspect, curvature, and Laplacian convexity found that MCH was most strongly related to convexity measured at scales of 20–300 m, a topographic variable that is a good proxy for position with respect to the hydrological network. Overall, our results support the idea that, even in these mesic forests, hydrological networks and associated topographical variation serve as templates upon which vegetation is organized over specific ranges of scales. These findings constitute an important step towards a mechanistic understanding of these patterns, and can guide upscaling and downscaling.     

Pair-edge approximation for heterogeneous lattice population models

To increase the analytical tractability of lattice stochastic spatial population models, several approximations have been developed. The pair-edge approximation is a moment-closure method that is effective in predicting persistence criteria and invasion speeds on a homogeneous lattice. Here we evaluate the effectiveness of the pair-edge approximation on a spatially heterogeneous lattice in which some sites are unoccupiable, or "dead". This model has several possible interpretations, including a spatial SIS epidemic model, in which some sites are occupied by immobile host-species individuals while others are empty. We find that, as in the homogeneous model, the pair-edge approximation is significantly more accurate than the ordinary pair approximation in determining conditions for persistence. However, habitat heterogeneity decreases invasion speed more than is predicted by the pair-edge approximation, and the discrepancy increases with greater clustering of "dead" sites. The accuracy of the approximation validates the underlying heuristic picture of population spread and therefore provides qualitative insight into the dynamics of lattice models. Conversely, the situations where the approximation is less accurate reveals limitations of pair approximation in the presence of spatial heterogeneity.