Coexistence under positive frequency dependence

Emergence of power-law distributions in self-segregation reaction-diffusion processes

Many natural or human-made systems encompassing local reactions and diffusion processes exhibit spatially distributed patterns of some relevant dynamical variable. These interactions, through self-organization and critical phenomena, give rise to power-law distributions, where emergent patterns and structures become visible across vastly different scales. Recent observations reveal power-law distributions in the spatial organization of, e.g., tree clusters and forest patch sizes. Crucially, these patterns do not follow a spatially periodic order but rather a statistical one. Unlike the spatially periodic patterns elucidated by the Turing mechanism, the statistical order of these particular vegetation patterns suggests an incomplete understanding of the underlying mechanisms. Here, we present a self-segregation mechanism, driving the emergence of power-law scalings in pattern-forming systems. The model incorporates an Allee-logistic reaction term, responsible for the local growth, and a nonlinear diffusion process accounting for positive interactions and limited resources. According to a self-organized criticality (SOC) principle, after an initial decrease, the system mass reaches an analytically predictable threshold, beyond which it self-segregates into distinct clusters, due to local positive interactions that promote cooperation. Numerical investigations show that the distribution of cluster sizes obeys a power law with an exponential cutoff.

Pervasive within-species spatial repulsion among adult tropical trees

For species to coexist, performance must decline as the density of conspecific individuals increases. Although evidence for such conspecific negative density dependence (CNDD) exists in forests, the within-species spatial repulsion it should produce has rarely been demonstrated in adults. In this study, we show that in comparison to a null model of stochastic birth, death, and limited dispersal, the adults of dozens of tropical forest tree species show strong spatial repulsion, some to surprising distances of approximately 100 meters. We used simulations to show that such strong repulsion can only occur if CNDD considerably exceeds heterospecific negative density dependence-an even stronger condition required for coexistence-and that large-scale repulsion can indeed result from small-scale CNDD. These results demonstrate substantial niche differences between species that may stabilize species diversity.

Rapid evolutionary repair by secondary perturbation of a primary disrupted transcriptional network

The discrete steps of transcriptional rewiring have been proposed to occur neutrally to ensure steady gene expression under stabilizing selection. A conflict-free switch of a regulon between regulators may require an immediate compensatory evolution to minimize deleterious effects. Here, we perform an evolutionary repair experiment on the Lachancea kluyveri yeast sef1Δ mutant using a suppressor development strategy. Complete loss of SEF1 forces cells to initiate a compensatory process for the pleiotropic defects arising from misexpression of TCA cycle genes. Using different selective conditions, we identify two adaptive loss-of-function mutations of IRA1 and AZF1. Subsequent analyses show that Azf1 is a weak transcriptional activator regulated by the Ras1-PKA pathway. Azf1 loss-of-function triggers extensive gene expression changes responsible for compensatory, beneficial, and trade-off phenotypes. The trade-offs can be alleviated by higher cell density. Our results not only indicate that secondary transcriptional perturbation provides rapid and adaptive mechanisms potentially stabilizing the initial stage of transcriptional rewiring but also suggest how genetic polymorphisms of pleiotropic mutations could be maintained in the population.

Females face more positive plant-soil feedback and intersexual competition under adequate nitrogen conditions compared to males in Populus cathayana

Plant-soil feedback (PSF) and competition influence plant performance, community structure and functions. However, how nutrient availability affects the interaction of PSF, sexual competition and coexistence in dioecious plants is poorly understood. In this study, the strengths of PSF and sexual competition, and their responses to nutrient availability were assessed in dioecious Populus cathayana using a garden experiment. We found that PSF reduced but did not eliminate the inequal sexual competition at low nitrogen (N) availability. Intersexual competition and nutrient limitation induced more negative PSF, which promoted sexual coexistence. PSF and competition were rather related to sexual dimorphism. Female plants experience more positive PSF and intersexual competition under adequate N conditions compared to males; the contrary was true with low N supply. Furthermore, the stability of root exudate networks and soil nutrient availability reflects the possibility of sexual coexistence regulated by PSF. Intersexual interaction promote more stable root exudate profiles and more saccharide secretion at low N supply. Meanwhile, the increased soil N and P mineralization in females with cultivated males explained the possible coexistence between females and males at low nutrient availability. Thus, these results indicate that soil biota can mitigate differences in sexual competitiveness and improve the stability of root exudate networks, consequently promoting sexual coexistence at low nutrient availability.

A modern two-layer hypothesis helps resolve the 'savanna problem'

For over a century, deep roots have been assumed to allow trees to avoid competition with grasses (i.e., the two-layer hypothesis). Yet, in part because it remains difficult to measure water uptake in the field, there has been a shift in savanna ecology away from the two-layer hypothesis and towards alternative explanations of tree-grass coexistence. Here, we combine hydrologic tracer experiments and soil water flow models to demonstrate how the distribution of active roots affects water uptake across a range of savanna conditions. Grass roots were shallower and provided pre-emptive access to enough soil water to allow nearly continuous grass cover, but slightly deeper roots provided trees with more total water under most conditions. This 'some water now or more water later' tradeoff varied with precipitation amount, soil texture, and tree and grass relative root abundance in ways that helped explain tree and grass landscape abundance.

Role of bacterial persistence in spatial population expansion

Bacterial persistence, tolerance to antibiotics via stochastic phenotype switching, provides a survival strategy and a fitness advantage in temporally fluctuating environments. Here we study its possible benefit in spatially varying environments using a Fisher wave approach. We study the spatial expansion of a population with stochastic switching between two phenotypes in spatially homogeneous conditions and in the presence of an antibiotic barrier. Our analytical results show that the expansion speed in growth-supporting conditions depends on the fraction of persister cells at the leading edge of the population wave. The leading edge contains a small fraction of persister cells, keeping the effect on the expansion speed minimal. The fraction of persisters increases gradually in the interior of the wave. This persister pool benefits the population when it is stalled by an antibiotic environment. In that case, the presence of persister enables the population to spread deeper into the antibiotic region and to cross an antibiotic region more rapidly. Further we observe that optimal switching rates maximize the expansion speed of the population in spatially varying environments with alternating regions of growth permitting conditions and antibiotics. Overall, our results show that stochastic switching can promote population expansion in the presence of antibiotic barriers or other stressful environments.

A nucleation framework for transition between alternate states: short-circuiting barriers to ecosystem recovery

The theory of alternate stable states provides an explanation for rapid ecosystem degradation, yielding important implications for ecosystem conservation and restoration. However, utilizing this theory to initiate transitions from degraded to desired ecosystem states remains a significant challenge. Applications of the alternative stable states framework may currently be impeded by a mismatch between local‐scale driving processes and landscape‐scale emergent system transitions. We show how nucleation theory provides an elegant bridge between local‐scale positive feedback mechanisms and landscape‐scale transitions between alternate stable ecosystem states. Geometrical principles can be used to derive a critical patch radius: a spatially explicit, local description of an unstable equilibrium point. This insight can be used to derive an optimal patch size that minimizes the cost of restoration, and to provide a framework to measure the resilience of desired ecosystem states to the synergistic effects of disturbance and environmental change.

The geometry of habitat fragmentation: Effects of species distribution patterns on extinction risk due to habitat conversion

Land-use changes, which cause loss, degradation, and fragmentation of natural habitats, are important anthropogenic drivers of biodiversity change. However, there is an ongoing debate about how fragmentation per se affects biodiversity in a given amount of habitat. Here, we illustrate why it is important to distinguish two different aspects of fragmentation to resolve this debate: (a) geometric fragmentation effects, which exclusively arise from the spatial distributions of species and habitat fragments, and (b) demographic fragmentation effects due to reduced fragment sizes, and/or changes in fragment isolation, edge effects, or species interactions. While most empirical studies are primarily interested in quantifying demographic fragmentation effects, geometric effects are typically invoked as post hoc explanations of biodiversity responses to fragmentation per se. Here, we present an approach to quantify geometric fragmentation effects on species survival and extinction probabilities. We illustrate this approach using spatial simulations where we systematically varied the initial abundances and distribution patterns (i.e., random, aggregated, or regular) of species as well as habitat amount and fragmentation per se. As expected, we found no geometric fragmentation effects when species were randomly distributed. However, when species were aggregated, we found positive effects of fragmentation per se on survival probability for a large range of scenarios. For regular species distributions, we found weakly negative geometric effects. These findings are independent of the ecological mechanisms which generate nonrandom species distributions. Our study helps to reconcile seemingly contradictory results of previous fragmentation studies. Since intraspecific aggregation is a ubiquitous pattern in nature, our findings imply widespread positive geometric fragmentation effects. This expectation is supported by many studies that find positive effects of fragmentation per se on species occurrences and diversity after controlling for habitat amount. We outline how to disentangle geometric and demographic fragmentation effects, which is critical for predicting the response of biodiversity to landscape change.

Eco-Evolutionary Buffering: Rapid Evolution Facilitates Regional Species Coexistence despite Local Priority Effects

Inhibitory priority effects, in which early-arriving species exclude competing species from local communities, are thought to enhance regional species diversity via community divergence. Theory suggests, however, that these same priority effects make it difficult for species to coexist in the region unless individuals are continuously supplied from an external species pool, often an unrealistic assumption. Here we develop an eco-evolutionary hypothesis to solve this conundrum. We build a metacommunity model in which local priority effects occur between two species via interspecific interference. Within each species there are two genotypes: one is more resistant to interspecific interference than the other but pays a fitness cost for its resistance. Because of this trade-off, species evolve to become less resistant as they become regionally more common. Rare species can then invade some local patches and consequently recover in regional frequency. This "eco-evolutionary buffering" enables the regional coexistence of species despite local priority effects, even in the absence of immigration from an external species pool. Our model predicts that eco-evolutionary buffering is particularly effective when local communities are small and connected by infrequent dispersal.

Population structure reduces benefits from partner choice in mutualistic symbiosis

Mutualistic symbioses are key drivers of evolutionary and ecological processes. Understanding how different species can evolve to interact in mutually beneficial ways is an important goal of evolutionary theory, especially when the benefits require costly investments by the partners. For such costly investments to evolve, some sort of fitness feedback mechanism must exist that more than recoups the direct costs. Several such feedback mechanisms have been explored both theoretically and empirically, yet we know relatively little of how they might act together, as they probably do in nature. In this paper, I model the joint action of three of the main mechanisms that can maintain interspecific cooperation in symbioses: partner choice by hosts, population structure amongst symbionts and undirected rewards from hosts to symbionts. The model shows that population structure reduces the benefit from partner choice to hosts. It may help or hinder beneficial symbionts and create positive or negative frequency dependence depending on the nature of host rewards to the symbiont. Strong population structure also makes it less likely that host choosiness and symbiont cooperation will be jointly maintained in a population. The intuition behind these results is that all else being equal, population structure reduces local variation available to the host to choose from. Thus, population structure is not always beneficial for the evolution of cooperation between species. These results also underscore the need to do full analyses of multiple mechanisms of social evolution to uncover the interactions between them.

Predator Perspective Drives Geographic Variation in Frequency-Dependent Polymorphism

Color polymorphism in natural populations can manifest as a striking patchwork of phenotypes in space, with neighboring populations characterized by dramatic differences in morph composition. These geographic mosaics can be challenging to explain in the absence of localized selection because they are unlikely to result from simple isolation-by-distance or clinal variation in selective regimes. To identify processes that can lead to the formation of geographic mosaics, we developed a simulation-based model to explore the influence of predator perspective, selection, migration, and genetic linkage of color loci on allele frequencies in polymorphic populations over space and time. Using simulated populations inspired by the biology of Heliconius longwing butterflies, Cepaea land snails, Oophaga poison frogs, and Sonora ground snakes, we found that the relative sizes of predator and prey home ranges can produce large differences in morph composition between neighboring populations under both positive and negative frequency-dependent selection. We also demonstrated the importance of the interaction of predator perspective with the type of frequency dependence and localized directional selection across migration and selection intensities. Our results show that regional-scale predation can promote the formation of phenotypic mosaics in prey species, without the need to invoke spatial variation in selective regimes. We suggest that predator behavior can play an important and underappreciated role in the formation and maintenance of geographic mosaics in polymorphic species.

Incorporating the soil environment and microbial community into plant competition theory

Plants affect microbial communities and abiotic properties of nearby soils, which in turn influence plant growth and interspecific interaction, forming a plant-soil feedback (PSF). PSF is a key determinant influencing plant population dynamics, community structure, and ecosystem functions. Despite accumulating evidence for the importance of PSF and development of specific PSF models, different models are not yet fully integrated. Here, we review the theoretical progress in understanding PSF. When first proposed, PSF was integrated with various mathematical frameworks to discuss its influence on plant competition. Recent theoretical models have advanced PSF research at different levels of ecological organizations by considering multiple species, applying spatially explicit simulations to examine how local-scale predictions apply to larger scales, and assessing the effect of PSF on plant temporal dynamics over the course of succession. We then review two foundational models for microbial- and litter-mediated PSF. We present a theoretical framework to illustrate that although the two models are typically presented separately, their behavior can be understood together by invasibility analysis. We conclude with suggestions for future directions in PSF theoretical studies, which include specifically addressing microbial diversity to integrate litter- and microbial-mediated PSF, and apply PSF to general coexistence theory through a trait-based approach.

Ecology: Tribal Warfare Maintains Microbial Diversity

When two tribes of Myxococcus bacteria attack each other, the most numerous usually wins. Established colonies can therefore resist invaders by outnumbering them. This shows how positive frequency dependence can maintain diversity across spatially structured environments.

Positively Frequency-Dependent Interference Competition Maintains Diversity and Pervades a Natural Population of Cooperative Microbes

Positively frequency-dependent selection is predicted from theory to promote diversity in patchily structured populations and communities, but empirical support for this prediction has been lacking. Here, we investigate frequency-dependent selection among isolates from a local natural population of the highly social bacterium Myxococcus xanthus. Upon starvation, closely related cells of M. xanthus cooperate to construct multicellular fruiting bodies, yet recently diverged genotypes co-residing in a local soil population often antagonize one another during fruiting-body development in mixed groups. In the experiments reported here, both fitness per se and strong forms of interference competition exhibit pervasive and strong positive frequency dependence (PFD) among many isolates from a centimeter-scale soil population of M. xanthus. All strains that compete poorly at intermediate frequency are shown to be competitively dominant at high frequency in most genotype pairings during both growth and development, and strongly so. Interference competition is often lethal and appears to be contact dependent rather than mediated by diffusible compounds. Finally, we experimentally demonstrate that positively frequency-dependent selection maintains diversity when genotype frequencies vary patchily in structured populations. These results suggest that PFD contributes to the high levels of local diversity found among M. xanthus social groups in natural soil populations by reinforcing social barriers to cross-territory invasion and thereby also promotes high within-group relatedness. More broadly, our results suggest that potential roles of PFD in maintaining patchily distributed diversity should be investigated more extensively in other species.

Spatial heterogeneity in soil microbes alters outcomes of plant competition

Plant species vary greatly in their responsiveness to nutritional soil mutualists, such as mycorrhizal fungi and rhizobia, and this responsiveness is associated with a trade-off in allocation to root structures for resource uptake. As a result, the outcome of plant competition can change with the density of mutualists, with microbe-responsive plant species having high competitive ability when mutualists are abundant and non-responsive plants having high competitive ability with low densities of mutualists. When responsive plant species also allow mutualists to grow to greater densities, changes in mutualist density can generate a positive feedback, reinforcing an initial advantage to either plant type. We study a model of mutualist-mediated competition to understand outcomes of plant-plant interactions within a patchy environment. We find that a microbe-responsive plant can exclude a non-responsive plant from some initial conditions, but it must do so across the landscape including in the microbe-free areas where it is a poorer competitor. Otherwise, the non-responsive plant will persist in both mutualist-free and mutualist-rich regions. We apply our general findings to two different biological scenarios: invasion of a non-responsive plant into an established microbe-responsive native population, and successional replacement of non-responders by microbe-responsive species. We find that resistance to invasion is greatest when seed dispersal by the native plant is modest and dispersal by the invader is greater. Nonetheless, a native plant that relies on microbial mutualists for competitive dominance may be particularly vulnerable to invasion because any disturbance that temporarily reduces its density or that of the mutualist creates a window for a non-responsive invader to establish dominance. We further find that the positive feedbacks from associations with beneficial soil microbes create resistance to successional turnover. Our theoretical results constitute an important first step toward developing a general understanding of the interplay between mutualism and competition in patchy landscapes, and generate qualitative predictions that may be tested in future empirical studies.

Cooperation and cheating in Pseudomonas aeruginosa: the roles of the las, rhl and pqs quorum-sensing systems

Pseudomonas aeruginosa coordinates the transcription of hundreds of genes, including many virulence genes, through three hierarchically arranged quorum-sensing (QS) systems, namely las, rhl and pqs. Each system consists of genes involved in autoinducer synthesis, lasI, rhlI and pqsABCDH, as well as cognate-regulatory genes, lasR, rhlR and pqsR. In this study, we analyzed the social behavior of signal-blind (ΔlasR, ΔrhlR, ΔpqsR) and signal-negative (ΔlasI, ΔrhlI, ΔpqsA) mutants from each QS system. As each system controls extracellular common goods but differs in the extent of regulatory control, we hypothesized that all signal-blind mutants can behave as cheaters that vary in their ability to invade a QS-proficient population. We found that lasR and pqsR, but not rhlR, mutants evolve from a wild-type ancestor in vitro under conditions that favor QS. Accordingly, defined lasR and pqsR mutants enriched in wild-type co-culture, whereas rhlR and all signal-negative mutants did not. Both lasR and pqsR mutants enriched with negative frequency dependence, suggesting social interactions with the wild type, although the pqsR mutant also grew well on its own. Taken together, the lasR mutant behaved as a typical cheater, as reported previously. However, the pqsR and rhlR mutants exhibited more complex behaviors, which can be sufficiently explained by positive and negative pleiotropic effects through differential regulation of pqs gene expression in the interconnected QS network. The evolutionary approach adopted here may account for the prevalence of naturally occurring QS mutants.

Differential response to frequency-dependent interactions: an experimental test using genotypes of an invasive grass

Positive feedbacks have been suggested as a means for non-indigenous species to successfully invade novel environments. Frequency-dependent feedbacks refer to a species performance being dependent on its local abundance in the population; however, frequency dependence is often described as a monolithic trait of a species rather than examining the variation in response for individual genotypes and fitness traits. Here, we investigate frequency-dependent outcomes for individual genotypes and fitness-related traits for the invasive grass Phalaris arundinacea. We tested for competition-mediated frequency dependence by establishing hexagonal arrays with the center target plant surrounded by either same, different or no genotype neighbors to determine how changing the small-scale frequency neighborhood-influenced invasion success. We used a Bayesian ANOVA approach which allowed us to easily accommodate our non-normal dataset and found that same neighbor plots had greater biomass production than different neighbor plots. Target plants also had greater stem height and aboveground biomass when surrounded by same genotype neighbors. A greenhouse experiment did not support the hypothesis that increased mycorrhizal associations were the cause of positive frequency dependence. We devised a frequency-dependent metric to quantify the extent of fitness-related differences for individual genotypes and found that individual genotypes showed a range of both positive and negative responses to different frequency treatments; however, only positive responses were statistically significant. The small-scale genotypic neighborhood had no effect for the fitness-related traits of leaf number, belowground biomass and total biomass. We demonstrate that individual invasive genotypes respond differently to changing frequency neighborhoods and that growth responses do not respond with the same direction and magnitude. A range of frequency-dependent responses may allow genotypes to invade a wide range of environments.

Time lags and the balance of positive and negative interactions in driving grassland community dynamics

Both facilitative and competitive interactions occur simultaneously among plants, and the net balance between them can vary over time. Despite this, recent model-fitting studies have found that negative interactions predominate. This suggests that more complex models may be necessary to uncover facilitation. Here we fitted models including seasonality, interannual variation, and time lags to survey data to test for patterns in positive and negative interactions among plants in a Michigan dry sand prairie. We hypothesized that interactions would be generally facilitative in this dry environment. Results indicate that most immediate (direct) interactions among dominant species are actually competitive, although interactions were more facilitative over the drier summer season. Interestingly, lagged density dependence was strong for all species in both seasons; it was positive for conspecific interactions and both positive and negative for heterospecific interactions. Observed lagged density dependence is likely due to effects from litter and/or past storage in rhizomes. Conspecific immediate and lagged interactions tended to be stronger than heterospecific interactions, suggesting that population dynamics in this community are driven mostly by conspecifics. Overall, the presence of strong lagged density dependence in this system suggests that it may be more widespread in plants than previously thought.

Intergenomic epistasis and coevolutionary constraint in plants and rhizobia

Studying how the fitness benefits of mutualism differ among a wide range of partner genotypes, and at multiple spatial scales, can shed light on the processes that maintain mutualism and structure coevolutionary interactions. Using legumes and rhizobia from three natural populations, I studied the symbiotic fitness benefits for both partners in 108 plant maternal family by rhizobium strain combinations. Genotype-by-genotype (G x G) interactions among local genotypes and among partner populations determined, in part, the benefits of mutualism for both partners; for example, the fitness effects of particular rhizobium strains ranged from uncooperative to mutualistic depending on the plant family. Correlations between plant and rhizobium fitness benefits suggest a trade off, and therefore a potential conflict, between the interests of the two partners. These results suggest that legume-rhizobium mutualisms are dynamic at multiple spatial scales, and that strictly additive models of mutualism benefits may ignore dynamics potentially important to both the maintenance of genetic variation and the generation of geographic patterns in coevolutionary interactions.

Positive feedbacks promote power-law clustering of Kalahari vegetation

The concept of local-scale interactions driving large-scale pattern formation has been supported by numerical simulations, which have demonstrated that simple rules of interaction are capable of reproducing patterns observed in nature. These models of self-organization suggest that characteristic patterns should exist across a broad range of environmental conditions provided that local interactions do indeed dominate the development of community structure. Readily available observations that could be used to support these theoretical expectations, however, have lacked sufficient spatial extent or the necessary diversity of environmental conditions to confirm the model predictions. We use high-resolution satellite imagery to document the prevalence of self-organized vegetation patterns across a regional rainfall gradient in southern Africa, where percent tree cover ranges from 65% to 4%. Through the application of a cellular automata model, we find that the observed power-law distributions of tree canopy cluster sizes can arise from the interacting effects of global-scale resource constraints (that is, water availability) and local-scale facilitation. Positive local feedbacks result in power-law distributions without entailing threshold behaviour commonly associated with criticality. Our observations provide a framework for integrating a diverse suite of previous studies that have addressed either mean wet season rainfall or landscape-scale soil moisture variability as controls on the structural dynamics of arid and semi-arid ecosystems.

Context dependence in the coevolution of plant and rhizobial mutualists

Several mechanisms are expected to rapidly rid mutualisms of genetic variation in partner quality. Variation for mutualist quality, however, appears to be widespread. We used a model legume-rhizobium mutualism to test for evidence that context-dependent selection may maintain variation in partner quality. In a greenhouse experiment using 10 natural populations of Medicago truncatula and two strains of Sinorhizobium medicae, we detected significant genotype x genotype (G x G) interactions for plant fitness, indicating that the most beneficial rhizobium strain depends on the host genotype. In a second experiment using a subset of the plant populations used in the first experiment, we detected significant G x G interactions for both plant and rhizobium fitness. Moreover, the plant population with which rhizobium strains gained the greatest benefit depended on the nitrogen environment. Finally, we found that in a high nitrogen environment, all plant populations had lower fitness when inoculated with a 1:1 mixture of strains than with the worse single strain alone, suggesting that nitrogen shifts the exchange of benefits in favour of rhizobia. Our data suggest that genotype, nitrogen and biotic dependency might contribute to the maintenance of genetic variation in mutualist quality when coupled with spatial or temporal heterogeneity in the environment.

Spatial mosaic formation through frequency-dependent selection in Müllerian mimicry complexes

Although contemporary models of Müllerian mimicry have considered the movement of interfacial boundaries between two distinct mimetic forms, and even the possibility of polymorphisms in two patch systems, no model has considered how multiple forms of Müllerian mimics might evolve and be maintained over large geographical areas. A spatially explicit individual-based model for the evolution of Müllerian mimicry is presented, in which two unpalatable species are distributed over discrete cells within a regular lattice. Populations in each cell are capable of genetic drift and experience localized dispersal as well as frequency-dependent selection by predators. When each unpalatable prey species was introduced into a random cell and allowed to spread, then mimicry evolved throughout the system in the form of a spatial mosaic of phenotypes, separated by narrow "hybrid zones". The primary mechanism generating phenotypic diversity was the occasional establishment of new mutant forms in unoccupied cells and their subsequent maintenance (and spread) through frequency-dependent selection. The mean number of discrete clusters of the same morph that formed in the lattice was higher the higher the intensity of predation, and higher the lower the dispersal rate of unpalatable prey. Under certain conditions the hybrid zones moved, in a direction dependent on the curvature of their interfacial boundaries. However, the mimetic mosaics were highly stable when the intensity of predation was high and the rate of prey dispersal was low. Overall, this model highlights how a stable mosaic of different mimetic forms can evolve from a range of starting conditions through a combination of chance effects and localized frequency-dependent selection.

The evolution of a Müllerian mimic in a spatially distributed community

Strong positive density-dependence should lead to a loss of diversity, but warning-colour and Müllerian mimicry systems show extraordinary levels of diversity. Here, we propose an analytical model to explore the dynamics of two forms of a Müllerian mimic in a heterogeneous environment with two alternative model species. Two connected populations of a dimorphic, chemically defended mimic are allowed to evolve and disperse. The proportions of the respective model species vary spatially. We use a nonlinear approximation of Müller's number-dependent equations to model a situation where the mortality for either form of the mimic decreases hyberbolically when its local density increases. A first non-spatial analysis confirms that the positive density-dependence makes coexistence of mimetic forms unstable in a single isolated patch, but shows that mimicry of the rarer model can be stable once established. The two-patch analysis shows that when models have different abundance in different places, local mimetic diversity in the mimic is maintained only if spatial heterogeneity is strong, or, more interestingly, if the mimic is not too strongly distasteful. Therefore, mildly toxic species can become polymorphic in a wider range of ecological settings. Spatial dynamics thus reveal a region of Müllerian polymorphism separating classical Batesian polymorphism and Müllerian monomorphism along the mimic's palatability spectrum. Such polymorphism-palatability relationship in a spatial environment provides a parsimonious hypothesis accounting for the observed Müllerian polymorphism that does not require quasi-Batesian dynamics. While the stability of coexistence depends on all factors, only the migration rate and strength of selection appear to affect the level of diversity at the polymorphic equilibrium. Local adaptation is predicted in most polymorphic cases. These results are in very good accordance with recent empirical findings on the polymorphic butterflies Heliconius numata and H. cydno.

To succeed globally, disperse locally: effects of local pollen and seed dispersal on tetraploid establishment

Newly formed tetraploid plants in sympatry with their diploid progenitors should face significant obstacles to persistence and population establishment because of low-fitness triploids formed by cross-ploidy pollinations. Prior models have found restrictive conditions for a minority tetraploid subpopulation to persist. A stochastic spatial model, parameterized using snow buttercups (Ranunculus adoneus), was used to examine the influence of limited seed and pollen dispersal distances on the success of minority tetraploids and the interaction of these factors with different rates of self-pollination and tetraploid advantage. Higher rates of self-pollination and increased tetraploid advantage increase the probability of tetraploid persistence. Limiting the dispersal of seeds and pollen further increases the positive impact of any given level of self-pollination and tetraploid advantage. Taxa with short-distance seed and pollen dispersal should face much less stringent barriers to sympatric polyploid speciation than taxa with long-distance dispersal patterns. With short-distance seed and pollen dispersal, polyploid speciation should be possible in the absence of ecological differentiation or recurrent polyploid formation through unreduced gametes.

Polymorphic mimicry, microhabitat use, and sex-specific behaviour

In order to assess the adaptive importance of microhabitat segregation for the maintenance of mimetic diversity, I explore how flight height varies between the sympatric forms of the polymorphic butterfly Heliconius numata and their respective models in the genus Melinaea. There is no evidence for vertical stratification of mimicry rings in these tiger-patterned butterflies, but males of H. numata tend to fly significantly higher than females and the Melinaea models. This difference in microhabitat preference likely results from females searching for host plants whereas males are patrolling for mates. I then present an extension of Muller's mimicry model for the case of partial behavioural or spatial segregation of sexes. The analysis suggests that sex-specific behaviours can make mimicry more beneficial, simply by reducing the effective population size participating in mimicry. The interaction between mimicry and sex-specific behaviours may therefore facilitate the evolution of polymorphism via enhanced, fine-scale local adaptation.

Spatial dynamics in model plant communities: what do we really know?

A variety of models have shown that spatial dynamics and small-scale endogenous heterogeneity (e.g., forest gaps or local resource depletion zones) can change the rate and outcome of competition in communities of plants or other sessile organisms. However, the theory appears complicated and hard to connect to real systems. We synthesize results from three different kinds of models: interacting particle systems, moment equations for spatial point processes, and metapopulation or patch models. Studies using all three frameworks agree that spatial dynamics need not enhance coexistence nor slow down dynamics; their effects depend on the underlying competitive interactions in the community. When similar species would coexist in a nonspatial habitat, endogenous spatial structure inhibits coexistence and slows dynamics. When a dominant species disperses poorly and the weaker species has higher fecundity or better dispersal, competition-colonization trade-offs enhance coexistence. Even when species have equal dispersal and per-generation fecundity, spatial successional niches where the weaker and faster-growing species can rapidly exploit ephemeral local resources can enhance coexistence. When interspecific competition is strong, spatial dynamics reduce founder control at large scales and short dispersal becomes advantageous. We describe a series of empirical tests to detect and distinguish among the suggested scenarios.

Soil community feedback and the coexistence of competitors: conceptual frameworks and empirical tests

A growing body of empirical work suggests that soil organisms can exert a strong role in plant community dynamics and may contribute to the coexistence of plant species. Some of this evidence comes from examining the feedback on plant growth through changes in the composition of the soil community. Host specific changes in soil community composition can generate feedback on plant growth and this feedback can be positive or negative. Previous work has demonstrated that negative soil community feedback can contribute to the coexistence of equivalent competitors. In this paper, I show that negative soil community feedback can also contribute to the coexistence of strong competitors, maintaining plant species that would not coexist in the absence of soil community dynamics. I review the evidence for soil community feedback and find accumulating evidence that soil community feedback can be common, strongly negative, and generated by a variety of complementary soil microbial mechanisms, including host-specific changes in the composition of the rhizosphere bacteria, nematodes, pathogenic fungi, and mycorrhizal fungi. Finally, I suggest topics needing further examination.

A novel theory to explain species diversity in landscapes: positive frequency dependence and habitat suitability

Theories to explain the diversity of species have required that individual species occupy unique niches and/or vary in their response to environmental factors. Positive interactions within a species, although common in communities, have not been thought to maintain species diversity because in non-spatial models the more abundant species always outcompetes the rarer species. Here, we show, using a stochastic spatial model, that positive intraspecific interactions such as those caused by positive frequency dependence and/or priority effects, can maintain species diversity if interactions between individuals are primarily local and the habitat contains areas that cannot be colonized by any species, such as boulders or other physical obstructions. When intraspecific interactions are primarily neutral, species diversity will eventually erode to a single species. When the landscape is homogeneous (i.e. does not contain areas that cannot be colonized by any species), the presence of strong intraspecific interactions will not maintain diversity.

Allee effect, spatial structure and species coexistence

Both positive and negative interactions among species are common in communities. Until recently, attention has focused on negative interactions such as competition. However, the importance of positive interactions such as the Allee effect has recently been recognized. We construct a single-patch model that incorporates both an Allee effect and competition between two species. A species that experiences an Allee effect cannot establish in a patch which is already occupied by a competitor unless its density is over a critical value. This effect, when translated into a metapopulation, makes migrants of a species unable to colonize patches where another species has established. This interaction between the Allee effect and inter-specific competition creates and stabilizes spatial segregation of species. Therefore, under circumstances in which competition would preclude local coexistence, the presence of an Allee effect can allow coexistence at a metapopulation scale. Furthermore, we found that a species can resist displacement if stronger competitors experience an Allee effect.