Pattern formation triggered by rare events: lessons from the spread of rabies.

Understanding of large-scale spatial pattern formation is a key to successful management in ecology and epidemiology. Neighbourhood interactions between local units are known to contribute to large-scale patterns, but how much do they contribute and what is the role of regional interactions caused by long-distance processes? How much long-distance dispersal do we need to explain the patterns that we observe in nature? There seems to be no way to answer these questions empirically. Therefore, we present a modelling approach that is a combination of a grid-based model describing local interactions and an individual-based model describing dispersal. Applying our approach to the spread of rabies, we show that in addition to local rabies dynamics, one long-distance infection per 14000 km2 per year is sufficient to reproduce the wave-like spread of this disease. We conclude that even rare ecological events that couple local dynamics on a regional scale may have profound impacts on large-scale patterns and, in turn, dynamics. Furthermore, the following results emerge: (i) Both neighbourhood infection and long-distance infection are needed to generate the wave-like dispersal pattern of rabies; (ii) randomly walking rabid foxes are not sufficient to generate the wave pattern; and (iii) on a scale of less than 100 km x 100 km, temporal oscillations emerge that are independent from long-distance dispersal.     

The intermediate distance hypothesis of biological invasions

Biological invasions are a worldwide phenomenon, but the global flows between native and alien regions have rarely been investigated in a cross‐taxonomic study. We therefore lack a thorough understanding of the global patterns of alien species spread. Using native and alien ranges of 1380 alien species, we show that the number of alien species follows a hump‐shaped function of geographic distance. We observe distinct variations in the relationship between alien species exchanges and distance among taxonomic groups, which relate to the taxa‐specific dispersal modes and their pathways of introduction. We formulate a simple statistical model, combining trade volume and biogeographic dissimilarity, which reproduces the observed pattern in good agreement with reported data and even captures variations among taxonomic groups. This study demonstrates the universality of the intermediate distance hypothesis of alien species spread across taxonomic groups, which will help to improve the predictability of new alien species arrivals.     

Patterns of patchy spread in multi-species reaction–diffusion models

Spread of populations in space often takes place via formation, interaction and propagation of separated patches of high species density, without formation of continuous fronts. This type of spread is called a ‘patchy spread’. In earlier models, this phenomenon was considered to be a result of a pronounced environmental or/and demographic stochasticity. Recently, it was found that a patchy spread can arise in a fully deterministic predator–prey system and in models of infectious diseases; in each case the process takes place in a homogeneous environment. It is well recognized that the observed patterns of patchy spread in nature are a result of interplay between stochastic and deterministic factors. However, the models considering deterministic mechanism of patchy spread are developed and studied much less compared to those based on stochastic mechanisms. A further progress in the understanding of the role of deterministic factors in the patchy spread would be extremely helpful. Here we apply multi-species reaction–diffusion models of two spatial dimensions in a homogeneous environment. We demonstrate that patterns of patchy spread are rather common for the considered approach, in particular, they arise both in mutualism and competition models influenced by predation. We show that this phenomenon can occur in a system without a strong Allee effect, contrary to what was assumed to be crucial in earlier models. We show, as well, a pattern of patchy spread having significantly different speeds in different spatial directions. We analyze basic features of spatiotemporal dynamics of patchy spread common for the reaction–diffusion approach. We discuss in which ecosystems we would observe patterns of deterministic patchy spread due to the considered mechanism.     

Neutral biogeography of phylogenetically structured interaction networks

Little is known about how biogeographic processes affect the dynamics of species interactions in space and time, although it is widely accepted that they drive community assemblage. In functional interactions, such as pollination and seed dispersal, species that share common ancestry tend to retain a common number of interactions and interact with similar sets of species, a pattern more commonly observed for animals than plants. On the one hand, the most coherent explanation for the phylogenetic structure of pollination and seed dispersal networks is that species retain ecological traits over evolution, which would cause the conservation of interaction partners. On the other hand, fundamental processes of biodiversity, such as dispersal and evolutionary rates seem to have important roles shaping the observed phylogenetic structure of mutualistic networks, but no model has been created to study the effect of these processes in the phylogenetic structure of mutualistic interactions. Here, we developed a stochastic simulation model to study the evolution of two interacting groups of species, which evolve independently over the same geographical domain. In our model, individuals of the same interaction group share ecological traits, whereas individuals of different trophic groups are ecologically distinct. We show that even in the absence of ecological differences between individuals, and disregarding any conservation of phenotypical and phenological traits between species, the interplay of dispersal and speciation is still a major driver of complex phylogenetic structure of functional interactions, such as pollination and seed dispersal.     

Interactions Between Pattern Formation and Domain Growth

In this paper we develop a theoretical framework for investigating pattern formation in biological systems for which the tissue on which the spatial pattern resides is growing at a rate which is itself regulated by the diffusible chemicals that establish the spatial pattern. We present numerical simulations for two cases of interest, namely exponential domain growth and chemically controlled growth. Our analysis reveals that for domains undergoing rapid exponential growth dilution effects associated with domain growth influence both the spatial patterns that emerge and the concentration of chemicals present in the domain. In the latter case, there is complex interplay between the effects of the chemicals on the domain size and the influence of the domain size on the formation of patterns. The nature of these interactions is revealed by a weakly nonlinear analysis of the full system. This yields a pair of nonlinear equations for the amplitude of the spatial pattern and the domain size. The domain is found to grow (or shrink) at a rate that depends quadratically on the pattern amplitude, the particular functional forms used to model the local tissue growth rate and the kinetics of the two diffusible species dictating the resulting behaviour.     

Vegetation pattern formation in a fog-dependent ecosystem

Vegetation pattern formation is a striking characteristic of several water-limited ecosystems around the world. Typically, they have been described on runoff-based ecosystems emphasizing local interactions between water, biomass interception, growth and dispersal. Here, we show that this situation is by no means general, as banded patterns in vegetation can emerge in areas without rainfall and in plants without functional root (the Bromeliad Tillandsia landbeckii) and where fog is the principal source of moisture. We show that a simple model based on the advection of fog-water by wind and its interception by the vegetation can reproduce banded patterns which agree with empirical patterns observed in the Coastal Atacama Desert. Our model predicts how the parameters may affect the conditions to form the banded pattern, showing a transition from a uniform vegetated state, at high water input or terrain slope to a desert state throughout intermediate banded states. Moreover, the model predicts that the pattern wavelength is a decreasing non-linear function of fog-water input and slope, and an increasing function of plant loss and fog-water flow speed. Finally, we show that the vegetation density is increased by the formation of the regular pattern compared to the density expected by the spatially homogeneous model emphasizing the importance of self-organization in arid ecosystems.     

Competitive interactions change the pattern of species co‐occurrences under neutral dispersal

Non‐random patterns of species segregation and aggregation within ecological communities are often interpreted as evidence for interspecific interactions. However, it is unclear whether theoretical models can predict such patterns and how environmental factors may modify the effects of species interactions on species co‐occurrence. Here we extend a spatially explicit neutral model by including competitive effects on birth and death probabilities to assess whether competition alone is able to produce non‐random patterns of species co‐occurrence. We show that transitive and intransitive competitive hierarchies alone (in the absence of environmental heterogeneity) are indeed able to generate non‐random patterns with commonly used metrics and null models. Moreover, even weak levels of intransitive competition can increase local species richness. However, there is no simple rule or consistent directional change towards aggregation or segregation caused by competitive interactions. Instead, the spatial pattern depends on both the type of species interaction and the strength of dispersal. We conclude that co‐occurrence analysis alone may not able to identify the underlying processes that generate the patterns.     

Modelling the impact of colonisation on genetic diversity and differentiation of forest trees: interaction of life cycle,pollen flow and seed long-distance dispersal

It was shown previously that the long lifespan and juvenile phase of trees strongly attenuate founder effects during colonisation in a diffusive dispersal model. However, this model yielded too slow a colonisation rate in comparison with palynological data for temperate forest trees. Since rare long-distance dispersal events have been shown to increase considerably colonisation rates in population dynamics models, we investigate here the impact of long-distance dispersal on within-population diversity (H(S)) and among-population differentiation (F(ST)) during the colonisation process. We use a stochastic approach and compare several dispersal strategies, ranging from very rare dispersal events of large amplitude to more frequent events of smaller amplitude. Using a simulation approach, which takes into account tree life-history traits, we show that long-distance dispersal events increase colonisation speed, and yield much larger founder effects in comparison with the diffusive model. The two models that include intermediate- and long-distance dispersal events show stronger deviations from experimental F(ST) values during and at the end of the colonisation process than the model with more frequent events of smaller dispersal variance. Furthermore, the introduction of a high level of pollen flow has a much more limited impact on models that include long-distance dispersal than on a diffusive dispersal model. The relatively high H(S) values that were obtained in all models are discussed according to the assumed mutation rate and effective population size. This study is an example of how observed genetic data can provide additional evidence on the best demographic model for a given species or group of species.     

The effects of species pool,dispersal and competition on the diversity–productivity relationship

Aim The diversity–productivity relationship is a controversial issue in ecology. Diversity is sometimes seen to increase with productivity but a unimodal relationship has often been reported. Competitive exclusion was cited initially to account for the decrease of diversity at high productivity. Subsequently, the roles of evolutionary history (species pool size) and dispersal rate have been acknowledged. We explore how the effects of species pool, dispersal and competition combine to produce different diversity–productivity relationships. Methods We use a series of simulations with a spatially explicit, individual‐based model. Following empirical expectations, we used four scenarios to characterize species pool size along the productivity gradient (uniformly low and high, linear increase and unimodal). Similarly, the dispersal rate varied along the productivity gradient (uniformly low and high, and unimodal). We considered both neutral communities and communities with competitive exclusion. Results and main conclusions Our model predicts that competitive interactions will result in unimodal diversity–productivity relationships. The model often predicts unimodal patterns in neutral communities as well, although the decline in richness at high productivity is less than in competing communities. A positive diversity–productivity relationship is simulated for neutral communities when the species pool size increases with productivity and the dispersal rate is high. This scenario is probably more widespread in nature than the others since positive diversity–productivity relationships have been observed more frequently than previously expected, especially in the tropics and for woody species. Our simulated effects of species pool, dispersal and competition on diversity patterns can be linked to empirical observations to uncover mechanisms behind the diversity–productivity relationship.     

What causes conspecific plant aggregation? Disentangling the role of dispersal,habitat heterogeneity and plant–plant interactions

Spatial patterns of plant species are determined by an array of ecologica factors including biotic and abiotic environmental constraints and intrinsic species traits. Thus, an observed aggregated pattern may be the result of short‐distance dispersal, the presence of habitat heterogeneity, plant–plant interactions or a combination of the above. Here, we studied the spatial pattern of Mediterranean alpine plant Silene ciliata (Caryophyllaceae) in five populations and assessed the contribution of dispersal, habitat heterogeneity and conspecific plant interactions to observed patterns. For this purpose, we used spatial point pattern analysis combined with specific a priori hypotheses linked to spatial pattern creation. The spatial pattern of S. ciliata recruits was not homogeneous and showed small‐scale aggregation. This is consistent with the species’ short‐distance seed dispersal and the heterogeneous distribution of suitable sites for germination and establishment. Furthermore, the spatial pattern of recruits was independent of the spatial pattern of adults. This suggests a low relevance of adult‐recruits interactions in the spatial pattern creation. The difference in aggregation between recruits and adults suggests that once established, recruits are subjected to self‐thinning. However, seedling mortality did not erase the spatial pattern generated by seed dispersal, as S. ciliata adults were still aggregated. Thus, the spatial aggregation of adults is probably due to seed dispersal limitation and the heterogeneous distribution of suitable sites at seedling establishment rather than the presence of positive plant–plant interactions at the adult stage. In fact, a negative density‐dependent effect of the conspecific neighbourhood was found on adult reproductive performance. Overall, results provide empirical evidence of the lack of a simple and direct relationship between the spatial structure of plant populations and the sign of plant–plant interactions and outline the importance of considering dispersal and habitat heterogeneity when performing spatial analysis assessments.     

Local adaptation,dispersal evolution,and the spatial eco‐evolutionary dynamics of invasion

Local adaptation and dispersal evolution are key evolutionary processes shaping the invasion dynamics of populations colonizing new environments. Yet their interaction is largely unresolved. Using a single‐species population model along a one‐dimensional environmental gradient, we show how local competition and dispersal jointly shape the eco‐evolutionary dynamics and speed of invasion. From a focal introduction site, the generic pattern predicted by our model features a temporal transition from wave‐like to pulsed invasion. Each regime is driven primarily by local adaptation, while the transition is caused by eco‐evolutionary feedbacks mediated by dispersal. The interaction range and cost of dispersal arise as key factors of the duration and speed of each phase. Our results demonstrate that spatial eco‐evolutionary feedbacks along environmental gradients can drive strong temporal variation in the rate and structure of population spread, and must be considered to better understand and forecast invasion rates and range dynamics.     

The common patterns of nature

We typically observe large‐scale outcomes that arise from the interactions of many hidden, small‐scale processes. Examples include age of disease onset, rates of amino acid substitutions and composition of ecological communities. The macroscopic patterns in each problem often vary around a characteristic shape that can be generated by neutral processes. A neutral generative model assumes that each microscopic process follows unbiased or random stochastic fluctuations: random connections of network nodes; amino acid substitutions with no effect on fitness; species that arise or disappear from communities randomly. These neutral generative models often match common patterns of nature. In this paper, I present the theoretical background by which we can understand why these neutral generative models are so successful. I show where the classic patterns come from, such as the Poisson pattern, the normal or Gaussian pattern and many others. Each classic pattern was often discovered by a simple neutral generative model. The neutral patterns share a special characteristic: they describe the patterns of nature that follow from simple constraints on information. For example, any aggregation of processes that preserves information only about the mean and variance attracts to the Gaussian pattern; any aggregation that preserves information only about the mean attracts to the exponential pattern; any aggregation that preserves information only about the geometric mean attracts to the power law pattern. I present a simple and consistent informational framework of the common patterns of nature based on the method of maximum entropy. This framework shows that each neutral generative model is a special case that helps to discover a particular set of informational constraints; those informational constraints define a much wider domain of non‐neutral generative processes that attract to the same neutral pattern.     

Global patterns in the shape of species geographical ranges reveal range determinants

Aim Do species range shapes follow general patterns? If so, what mechanisms underlie those patterns? We show for 11,582 species from a variety of taxa across the world that most species have similar latitudinal and longitudinal ranges. We then seek to disentangle the roles of climate, extrinsic dispersal limitation (e.g. barriers) and intrinsic dispersal limitation (reflecting a species’ ability to disperse) as constraints of species range shape. We also assess the relationship between range size and shape. Location Global. Methods Range shape patterns were measured as the slope of the regression of latitudinal species ranges against longitudinal ranges for each taxon and continent, and as the coefficient of determination measuring the degree of scattering of species ranges from the 1:1 line (i.e. latitudinal range = longitudinal range). Two major competing hypotheses explaining species distributions (i.e. dispersal or climatic determinism) were explored. To this end, we compared the observed slopes and coefficients of determination with those predicted by a climatic null model that estimates the potential range shapes in the absence of dispersal limitation. The predictions compared were that species distribution shapes are determined purely by (1) intrinsic dispersal limitation, (2) extrinsic dispersal limitations such as topographic barriers, and (3) climate. Results  Using this methodology, we show for a wide variety of taxa across the globe that species generally have very similar latitudinal and longitudinal ranges. However, neither neutral models assuming random but spatially constrained dispersal, nor models assuming climatic control of species distributions describe range shapes adequately. The empirical relationship between the latitudinal and longitudinal ranges of species falls between the predictions of these competing models. Main conclusions We propose that this pattern arises from the combined effect of macroclimate and intrinsic dispersal limitation, the latter being the major determinant among restricted‐range species. Hence, accurately projecting the impact of climate change onto species ranges will require a solid understanding of how climate and dispersal jointly control species ranges.     

Structure in plant–animal interaction assemblages

We present a comprehensive approach to detect pattern in assemblages of plant and animal species linked by interactions such as pollination, frugivory or herbivory. Simple structural models produce gradient, compartmented or nested patterns of interaction; intermediate patterns between a gradient and compartments are possible, and nesting within compartments produces a combined model. Interaction patterns can be visualized and analyzed either as matrices, as bipartite networks or as multivariate sets through correspondence analysis. We argue that differences among patterns represent outcomes of distinct evolutionary and ecological processes in these highly diversified assemblages. Instead of choosing one model a priori, assemblages should be probed for a suite of patterns. A plant–pollinator assemblage exemplifies a simple nested pattern, whereas a plant–herbivore assemblage illustrates a compound pattern with nested structures within compartments. Compartmentation should reflect coevolutionary histories and constraints, whereas differences in species abundance or dispersal may generate nestedness.     

Dispersal, Environmental Correlation, and Spatial Synchrony in Population Dynamics

Many species exhibit widespread spatial synchrony in population fluctuations. This pattern is of great ecological interest and can be a source of concern when the species is rare or endangered. Both dispersal and spatial correlations in the environment have been implicated as possible causes of this pattern, but these two factors have rarely been studied in combination. We develop a spatially structured population model, simple enough to obtain analytic solutions for the population correlation, that incorporates both dispersal and environmental correlation. We ask whether these two synchronizing factors contribute additively to the total spatial population covariance. We find that there is always an interaction between these two factors and that this interaction is small only when one or both of the environmental correlation and the dispersal rate are small. The interaction is opposite in sign to the environmental correlation; so, in the normal case of positive environmental correlation across sites, the population synchrony will be lower than predicted by simply adding the effects of dispersal and environmental correlation. We also find that population synchrony declines as the strength of population regulation increases. These results indicate that dispersal and environmental correlation need to be considered in combination as explanations for observed patterns of population synchrony.     

Spatiotemporal patterns provoked by environmental variability in a predator–prey model

The emergence of spatiotemporal patterns in the distribution of species is one of the most striking phenomena in ecology and nonlinear science. Since it is known that spatial inhomogeneities can significantly affect the dynamics of ecological populations, in the present paper we investigate the impact of environmental variability on the formation of patterns in a spatially extended predator–prey model. In particular, we utilize a predator–prey system with a Holling III functional response and introduce random spatial variations of the kinetic parameter signifying the intrinsic growth rate of the prey, reflecting the impact of a heterogeneous environment. Our results reveal that in the proximity of the Hopf bifurcation environmental variability is able to provoke pattern formation, whereby the coherence of the patterns exhibits a resonance-like dependence on the variability strength. Furthermore, we show that the phenomenon can only be observed if the spatial heterogeneities exhibit large enough regions with high growth rates of the prey. Our findings thus indicate that variability could be an essential pattern formation mechanism of the populations.     

Theory predicts a rapid transition from niche-structured to neutral biodiversity patterns across a speciation-rate gradient

A central challenge in community ecology is to predict patterns of biodiversity with mechanistic models. The neutral model of biodiversity is a simple model that appears to provide parsimonious and accurate predictions of biodiversity patterns in some ecosystems, even though it ignores processes such as species interactions and niche structure. In a recent paper, we used analytical techniques to reveal why the mean predictions of the neutral model are robust to niche structure in high diversity but not low-diversity ecosystems. In the present paper, we explore this phenomenon further by generating stochastic simulated data from a spatially implicit hybrid niche-neutral model across different speciation rates. We compare the resulting patterns of species richness and abundance with the patterns expected from a pure neutral and a pure niche model. As the speciation rate in the hybrid model increases, we observe a surprisingly rapid transition from an ecosystem in which diversity is almost entirely governed by niche structure to one in which diversity is statistically indistinguishable from that of the neutral model. Because the transition is rapid, one prediction of our abstract model is that high-diversity ecosystems such as tropical forests can be approximated by one simple model—the neutral model—whereas low-diversity ecosystems such as temperate forests can be approximated by another simple model—the niche model. Ecosystems that require the hybrid model are predicted to be rare, occurring only over a narrow range of speciation rates.     

Detecting local diversity‐dependence in diversification

Whether there are ecological limits to species diversification is a hotly debated topic. Molecular phylogenies show slowdowns in lineage accumulation, suggesting that speciation rates decline with increasing diversity. A maximum‐likelihood (ML) method to detect diversity‐dependent (DD) diversification from phylogenetic branching times exists, but it assumes that diversity‐dependence is a global phenomenon and therefore ignores that the underlying species interactions are mostly local, and not all species in the phylogeny co‐occur locally. Here, we explore whether this ML method based on the nonspatial diversity‐dependence model can detect local diversity‐dependence, by applying it to phylogenies, simulated with a spatial stochastic model of local DD speciation, extinction, and dispersal between two local communities. We find that type I errors (falsely detecting diversity‐dependence) are low, and the power to detect diversity‐dependence is high when dispersal rates are not too low. Interestingly, when dispersal is high the power to detect diversity‐dependence is even higher than in the nonspatial model. Moreover, estimates of intrinsic speciation rate, extinction rate, and ecological limit strongly depend on dispersal rate. We conclude that the nonspatial DD approach can be used to detect diversity‐dependence in clades of species that live in not too disconnected areas, but parameter estimates must be interpreted cautiously.     

Active colonization dynamics and diversity patterns are influenced by dendritic network connectivity and species interactions

Habitat network connectivity influences colonization dynamics, species invasions, and biodiversity patterns. Recent theoretical work suggests dendritic networks, such as those found in rivers, alter expectations regarding colonization and dispersal dynamics compared with other network types. As many native and non‐native species are spreading along river networks, this may have important ecological implications. However, experimental studies testing the effects of network structure on colonization and diversity patterns are scarce. Up to now, experimental studies have only considered networks where sites are connected with small corridors, or dispersal was experimentally controlled, which eliminates possible effects of species interactions on colonization dynamics. Here, we tested the effect of network connectivity and species interactions on colonization dynamics using continuous linear and dendritic (i.e., river‐like) networks, which allow for active dispersal. We used a set of six protist species and one rotifer species in linear and dendritic microcosm networks. At the start of the experiment, we introduced species, either singularly or as a community within the networks. Species subsequently actively colonized the networks. We periodically measured densities of species throughout the networks over 2 weeks to track community dynamics, colonization, and diversity patterns. We found that colonization of dendritic networks was faster compared with colonization of linear networks, which resulted in higher local mean species richness in dendritic networks. Initially, community similarity was also greater in dendritic networks compared with linear networks, but this effect vanished over time. The presence of species interactions increased community evenness over time, compared with extrapolations from single‐species setups. Our experimental findings confirm previous theoretical work and show that network connectivity, species‐specific dispersal ability, and species interactions greatly influence the dispersal and colonization of dendritic networks. We argue that these factors need to be considered in empirical studies, where effects of network connectivity on colonization patterns have been largely underestimated.