Self-organized dynamics in spatially structured populations

Self-organization and pattern formation represent the emergence of order in temporal and spatial processes. Self-organization in population ecology is gaining attention due to the recent advances concerning temporal fluctuations in the population size of dispersal-linked subunits. We shall report that spatially structured models of population renewal promote the emergence of a complex power law order in spatial population dynamics. We analyse a variety of population models showing that self-organization can be identified as a temporal match in population dynamics among local units, and how the synchrony changes in time. Our theoretical results are concordant with analyses of population data on the Canada lynx.     

Reconstructing the dynamics of unobserved variables in spatially extended systems

Attractor reconstruction using embedding techniques is a widely used tool when analysing data from real systems. It allows reconstruction of the system dynamics from only one observable and is thus extremely powerful. We show here that this reconstruction is also possible from spatially coupled systems. We use a common host–parasitoid model as an example as ecological systems are virtually always spatially extended. Additionally, data from ecological systems has often only one observable, e.g. population density, from a potentially much higher-dimensional system. Singular value decomposition is used to show the existence of a functional relationship mapping the time delayed coordinates of one variable to the full spatially coupled system. We investigate the effects of noise and indicate two important spatial scales. Finally, we illustrate that a reconstruction can be obtained from a system that is only partially sampled.     

Evolutionary dynamics of rhizopine within spatially structured rhizobium populations

Symbiosis between legumes and nitrogen-fixing bacteria is thought to bring mutual benefit to each participant. However, it is not known how rhizobia benefit from nodulating legume hosts because they fix nitrogen only after becoming bacteroids, which are terminally differentiated cells that cannot reproduce. Because undifferentiated rhizobia in and around the nodule can reproduce, evolution of symbiotic nitrogen fixation may depend upon kin selection. In some hosts, these kin may persist in the nodule as viable, undifferentiated bacteria. In other hosts, no viable rhizobia survive to reproduce after nodule senescence. Bacteroids in these hosts may benefit their free-living kin by enhancing production of plant root exudates. However, unrelated non-mutualists may also benefit from increased plant exudates. Rhizopines, compounds produced by bacteroids in nodules and catabolized only by related free-living rhizobia, may provide a mechanism by which bacteroids can preferentially benefit kin. Despite this apparent advantage, rhizopine genotypes are relatively rare. We constructed a mathematical model to examine how mixing within rhizobium populations influences the evolution of rhizopine genotypes. Our model predicts that the success of rhizopine genotypes is strongly dependent upon the spatial genetic structure of the bacterial population; rhizopine is more likely to dominate well-mixed populations. Further, for a given level of mixing, we find that rhizopine evolves under a positive frequency-dependent process in which stochastic accumulation of rhizopine alleles is necessary for rhizopine establishment. This process leads to increased spatial structure in rhizobium populations, and suggests that rhizopine may expand the conditions under which nitrogen fixation can evolve via kin selection.     

Multivariate recurrence plots for visualizing and quantifying the dynamics of spatially extended ecosystems

Few methods for quantifying the dynamics of temporal processes are readily applicable to spatially extended systems when equations governing the motion are unknown. The objective of this paper is to illustrate how the MRP-RQA (multivariate recurrence plot-recurrence quantification analysis) approach may serve to characterize ecosystems driven by both deterministic and stochastic forces. The strength of the MRP-RQA approach resides in its independence from constraining assumptions regarding outliers, noise, stationarity and transients. Its utility is demonstrated by means of two spatiotemporal series (summer and spring datasets) of light intensity variations in an old growth forest ecosystem. Results revealed qualitative differences in homogeneity, transiency, and drift typologies between the MRPs derived from each dataset. RQA estimates of determinism and Kolmogorov entropy supported the idea that mixed chaotic–stochastic dynamics may be common in mesoscale forest habitats. Advantages and inconveniences of the MRP-RQA approach are also discussed in the more general context of monitoring ecosystems.     

Gradual regime shifts in spatially extended ecosystems

Ecosystem regime shifts are regarded as abrupt global transitions from one stable state to an alternative stable state, induced by slow environmental changes or by global disturbances. Spatially extended ecosystems, however, can also respond to local disturbances by the formation of small domains of the alternative state. Such a response can lead to gradual regime shifts involving front propagation and the coalescence of alternative-state domains. When one of the states is spatially patterned, a multitude of intermediate stable states appears, giving rise to step-like gradual shifts with extended pauses at these states. Using a minimal model, we study gradual state transitions and show that they precede abrupt transitions. We propose indicators to probe gradual regime shifts, and suggest that a combination of abrupt-shift indicators and gradual-shift indicators might be needed to unambiguously identify regime shifts. Our results are particularly relevant to desertification in drylands where transitions to bare soil take place from spotted vegetation, and the degradation process appears to involve step-like events of local vegetation mortality caused by repeated droughts.     

Neutral evolution in spatially continuous populations

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.     

Informed dispersal, heterogeneity in animal dispersal syndromes and the dynamics of spatially structured populations

There is accumulating evidence that individuals leave their natal area and select a breeding habitat non-randomly by relying upon information about their natal and future breeding environments. This variation in dispersal is not only based on external information (condition dependence) but also depends upon the internal state of individuals (phenotype dependence). As a consequence, not all dispersers are of the same quality or search for the same habitats. In addition, the individual's state is characterized by morphological, physiological or behavioural attributes that might themselves serve as a cue altering the habitat choice of conspecifics. These combined effects of internal and external information have the potential to generate complex movement patterns and could influence population dynamics and colonization processes. Here, we highlight three particular processes that link condition-dependent dispersal, phenotype-dependent dispersal and habitat choice strategies: (1) the relationship between the cause of departure and the dispersers' phenotype; (2) the relationship between the cause of departure and the settlement behaviour and (3) the concept of informed dispersal, where individuals gather and transfer information before and during their movements through the landscape. We review the empirical evidence for these processes with a special emphasis on vertebrate and arthropod model systems, and present case studies that have quantified the impacts of these processes on spatially structured population dynamics. We also discuss recent literature providing strong evidence that individual variation in dispersal has an important impact on both reinforcement and colonization success and therefore must be taken into account when predicting ecological responses to global warming and habitat fragmentation.     

Pollen dispersal in spatially aggregated populations

We perform a theoretical study of effective pollen dispersal within plant populations exhibiting intraspecific spatial aggregation. We simulate nonuniform distributions of individuals by means of a Poisson cluster process and use an individual-based spatially explicit model of pollen dispersal to assess the effects of different aggregation patterns on the effective pollen pool size (N(ep)) and the axial variance of pollen dispersal (sigma (p)). Results show clear interactions between clumping and both N(ep) and sigma (p), whose precise form and intensity depend on the relative spatial scale of aggregation to pollen dispersal range. If clump size is small relative to dispersal range, clumping results in lower N(ep) and sigma (p) than in randomly distributed populations. Interestingly, by contrast, aggregation may actually enlarge N(ep) and has minimum impact on sigma (p) if clump size is near or above the scale of dispersal. High intraclump to global density ratios enhance the sensitivity of both N(ep) and sigma (p) to clumping, while leptokurtic pollen dispersal generates sharper reductions of both N(ep) and sigma (p) for small clump sizes and stronger increments of N(ep) for larger clump sizes. Overall, our results indicate that isolation-by-distance models in plants should not ignore the effects of intraspecific spatial aggregation on effective dispersal.     

Evolution of reproductive effort in viscous populations: the importance of population dynamics

I present two ecological models for the evolution of reproductive effort in viscous populations with empty sites. In contrast with previous studies, I show that limited dispersal needs not have a positive effect on the evolutionarily stable allocation of resources to fecundity versus survival. Rather, depending on the feedback between the trait and the population dynamics, population viscosity may have no effect or even lead to a decrease in the evolutionarily stable reproductive effort when individuals can degrade their environment during their lifetime. I show that the different evolutionary outcomes can be explained by the asymmetry in the level of kin competition resulting from investing into juveniles or into adults.     

The evolution of disease resistance and tolerance in spatially structured populations

THE UBIQUITOUS CHALLENGE FROM INFECTIOUS DISEASE HAS PROMPTED THE EVOLUTION OF DIVERSE HOST DEFENSES, WHICH CAN BE DIVIDED INTO TWO BROAD CLASSES: resistance (which limits pathogen growth and infection) and tolerance (which does not limit infection, but instead reduces or offsets its negative fitness consequences). Resistance and tolerance may provide equivalent short-term benefits, but have fundamentally different epidemiological consequences and thus exhibit different evolutionary behaviors. We consider the evolution of resistance and tolerance in a spatially structured population using a stochastic simulation model. We show that tolerance can invade a population of susceptible individuals (i.e., neither resistant nor tolerant) with higher cost than resistance, even though they each provide equivalent direct benefits to the host, because tolerant hosts impose higher disease burden upon vulnerable competitors. However, in spatially structured settings, tolerance can invade a population of resistant hosts only with lower cost than resistance due to spatial genetic structure and the higher local incidence of disease around invading tolerant individuals. The evolution of tolerance is therefore constrained by spatial genetic structure in a manner not previously revealed by nonspatially explicit models, suggesting mechanisms that could maintain variation or limit the occurrence of tolerance relative to resistance.     

Imitative behavior in nonuniformly spatially distributed populations

The theory of imitative behavior, developed previously, is applied to the case of two social groups which are separated spatially. If the information of each group as to the behavior of the other is complete, the case reduces to that of a single group. When any information is lacking at all, the two groups are independent. If we have two mutually exclusive behaviorsA andB, all four combinationsAA, AB, BA, andBB are possible. If the mutual information gradually increases from zero, then for a certain value of it, the group which is more informed about the behavior of the other will change to that behavior if it did not already exhibit it. If for constant information the size of the group increases, then above a certain threshold value, the larger group imposes its behavior on the smaller.     

Intra-deme molecular diversity in spatially expanding populations

We report here a simulation study examining the effect of a recent spatial expansion on the pattern of molecular diversity within a deme. We first simulate a range expansion in a virtual world consisting in a two-dimensional array of demes exchanging a given proportion of migrants (m) with their neighbors. The recorded demographic and migration histories are then used under a coalescent approach to generate the genetic diversity in a sample of genes. We find that the shape of the gene genealogies and the overall pattern of diversity within demes depend not only on the age of the expansion but also on the level of gene flow between neighboring demes, as measured by the product Nm, where N is the size of a deme. For small Nm values (< approximately 20 migrants sent outwards per generation), a substantial proportion of coalescent events occur early in the genealogy, whereas with larger levels of gene flow, most coalescent events occur around the time of the onset of the spatial expansion. Gene genealogies are star shaped, and mismatch distributions are unimodal after a range expansion for large Nm values. In contrast, gene genealogies present a mixture of both very short and very long branch lengths, and mismatch distributions are multimodal for small Nm values. It follows that statistics used in tests of selective neutrality like Tajima's D statistic or Fu's F(S) statistic will show very significant negative values after a spatial expansion only in demes with high Nm values. In the context of human evolution, this difference could explain very simply the fact that analyses of samples of mitochondrial DNA sequences reveal multimodal mismatch distributions in hunter-gatherers and unimodal distributions in post-Neolithic populations. Indeed, the current simulations show that a recent increase in deme size (resulting in a larger Nm value) is sufficient to prevent recent coalescent events and thus lead to unimodal mismatch distributions, even if deme sizes (and therefore Nm values) were previously much smaller. The fact that molecular diversity within deme is so dependent on recent levels of gene flow suggests that it should be possible to estimate Nm values from samples drawn from a single deme.     

Evolutionary games for spatially dispersed populations

An evolutionary game model is developed that incorporates both spatial dispersion and density effects in the evolutionary dynamic. It is shown that a stable equilibrium (e.g. an evolutionarily stable strategy) of the non-dispersed frequency dynamic becomes a stable equilibrium of the larger system if population density stabilizes at these fixed frequencies. It is also shown, by example, that other equilibria, whose frequencies change from one location to another, may appear when dispersal rates are relatively small.Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A6187Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A7822     

Bifurcations and phase transitions in spatially extended two-member hypercycles

Mounting theoretical and experimental evidence indicates that the success of molecular replicators is strongly tied to the local nature of their interactions. Local dispersal in a given spatial domain, particularly on surfaces, might strongly enhance the growth and selection of fit molecules and their resistance to parasites. In this work the spatial dynamics of a simple hypercycle model consisting of two molecular species is analysed. In order to characterize it, both mean field models and stochastic, spatially explicit approaches are considered. The mean field approach predicts the presence of a saddle-node bifurcation separating a phase involving stable hypercycles from extinction, consistently with spatially explicit models, where an absorbing first-order phase transition is shown to exist and diffusion is explicitly introduced. The saddle-node bifurcation is shown to leave a ghost in the phase plane. A metapopulation-based model is also developed in order to account for the observed phases when both diffusion and reaction are considered. The role of information and diffusion as well as the relevance of these phases and the underlying spatial structures are discussed, and their potential implications for the evolution of early replicators are outlined.     

Severe hindrance of viral infection propagation in spatially extended hosts

The production of large progeny numbers affected by high mutation rates is a ubiquitous strategy of viruses, as it promotes quick adaptation and survival to changing environments. However, this situation often ushers in an arms race between the virus and the host cells. In this paper we investigate in depth a model for the dynamics of a phenotypically heterogeneous population of viruses whose propagation is limited to two-dimensional geometries, and where host cells are able to develop defenses against infection. Our analytical and numerical analyses are developed in close connection to directed percolation models. In fact, we show that making the space explicit in the model, which in turn amounts to reducing viral mobility and hindering the infective ability of the virus, connects our work with similar dynamical models that lie in the universality class of directed percolation. In addition, we use the fact that our model is a multicomponent generalization of the Domany-Kinzel probabilistic cellular automaton to employ several techniques developed in the past in that context, such as the two-site approximation to the extinction transition line. Our aim is to better understand propagation of viral infections with mobility restrictions, e.g., in crops or in plant leaves, in order to inspire new strategies for effective viral control.     

Dynamics and genealogy of strains in spatially extended host-pathogen models

We examine the dynamics of evolution in a generic spatial model of a pathogen infecting a population of hosts, or an analogous predator-prey system. Previous studies of this model have found a range of interesting phenomena that differ from the well-mixed version. We extend these studies by examining the spatial and temporal dynamics of strains using genealogical tracing. When transmissibility can evolve by mutation, strains of intermediate transmissibility dominate even though high-transmissibility mutants have a short-term reproductive advantage. Mutant strains continually arise and grow rapidly for many generations but eventually go extinct before dominating the system. We find that, after a number of generations, the mutant pathogen characteristics strongly impact the spatial distribution of their local host environment, even when there are diverse types coexisting. Extinction is due to the depletion of susceptibles in the local environment of these mutant strains. Studies of spatial and genealogical relatedness reveal the self-organized spatial clustering of strains that enables their impact on the local environment. Thus, we find that selection acts against the high-transmissibility strains on long time-scales as a result of the feedback due to environmental change. Our study shows that averages over space or time should not be assumed to adequately describe the evolutionary dynamics of spatially distributed host-pathogen systems.     

Host resistance and coevolution in spatially structured populations

Natural, agricultural and human populations are structured, with a proportion of interactions occurring locally or within social groups rather than at random. This within-population spatial and social structure is important to the evolution of parasites but little attention has been paid to how spatial structure affects the evolution of host resistance, and as a consequence the coevolutionary outcome. We examine the evolution of resistance across a range of mixing patterns using an approximate mathematical model and stochastic simulations. As reproduction becomes increasingly local, hosts are always selected to increase resistance. More localized transmission also selects for higher resistance, but only if reproduction is also predominantly local. If the hosts disperse, lower resistance evolves as transmission becomes more local. These effects can be understood as a combination of genetic (kin) and ecological structuring on individual fitness. When hosts and parasites coevolve, local interactions select for hosts with high defence and parasites with low transmissibility and virulence. Crucially, this means that more population mixing may lead to the evolution of both fast-transmitting highly virulent parasites and reduced resistance in the host.     

The dynamics of hierarchical age-structured populations

An age-structured population is considered in which the birth and death rates of an individual of age a is a function of the density of individuals older and/or younger than a. An existence/uniqueness theorem is proved for the McKendrick equation that governs the dynamics of the age distribution function. This proof shows how a decoupled ordinary differential equation for the total population size can be derived. This result makes a study of the population's asymptotic dynamics (indeed, often its global asymptotic dynamics) mathematically tractable. Several applications to models for intra-specific competition and predation are given.