Diffusion analysis and stationary distribution of the two-species lottery competition model

The lottery model is a stochastic population model in which juveniles compete for space. Examples include sedentary organisms such as trees in a forest and members of marine benthic communities. The behavior of this model appears to be characteristic of that found in other sorts of stochastic competition models. In a community with two species, it was previously demonstrated that coexistence of the species is possible if adult death rates are small and environmental variation is large. Environmental variation is incorporated by assuming that the birth rates and death rates are random variables. Complicated conditions for coexistence and competitive exclusion have been derived elsewhere. In this paper, simple and easily interpreted conditions are found by using the technique of diffusion approximation. Formulae are given for the stationary distribution and means and variances of population fluctuations. The shape of the stationary distribution allows the stability of the coexistence to be evaluated.     

Coexistence of competitive species with a stage-structured life cycle

Ecological theory provides explanations for exclusion or coexistence of competing species. Most theoretical works on competition dynamics that have shaped current perspectives on coexistence assume a simple life cycle. This simplification, however, may omit important realities. We present a simple two-stage structured competition model to investigate the effects of life-history characteristics on coexistence. The achievement and the stability of coexistence depend not only on competition coefficients but also on a set of life-history parameters that reflect the viability of an individual, namely, adult death rate, maturation rate, and birth rate. High individual viability is necessary for a species to persist, but it does not necessarily facilitate coexistence. Intense competition at the juvenile or adult stage may require higher or lower viability, respectively, for stable coexistence to be possible. The stability mechanism can be explained by the refuge effect of the less competitive stage, and the birth performance, which preserves the less competitive stage as a refuge. Coexistence might readily collapse if the life-history characteristics, which together constitute individual viability, change, even though two species have an inherent competitive relation conducive to stable coexistence.     

The stabilizing effect of a random environment

It is shown that the lottery competition model permits coexistence in a stochastic environment, but not in a constant environment. Conditions for coexistence and competitive exclusion are determined. Analysis of these conditions shows that the essential requirements for coexistence are overlapping generations and fluctuating birth rates which ensure that each species has periods when it is increasing. It is found that a species may persist provided only that it is favored sufficiently by the environment during favorable periods independently of the extent to which the other species is favored during its favorable periods.Coexistence is defined in terms of the stochastic boundedness criterion for species persistence. Using the lottery model as an example this criterion is justified and compared with other persistence criteria. Properties of the stationary distribution of population density are determined for an interesting limiting case of the lottery model and these are related to stochastic boundedness. An attempt is then made to relate stochastic boundedness for infinite population models to the behavior of finite population models.     

Short-term instabilities and long-term community dynamics

Competition in a temporally variable environment leads to sequences of short-term instabilities that in some cases are the mechanism of long-term coexistence; in other cases they promote long-term instability. Recent work associates long-term stability with a positive relationship between environmental and competitive effects and with population growth rates that are buffered against jointly unfavorable environmental and competitive events. Buffered growth rates arise from population subdivision over life-history stages, microenvironments or phenotypes. A distinct but related mechanism of long-term stability relies on population growth rates that are nonlinear functions of competition. New ways of understanding and investigating species diversity follow from these results.     

Coexistence of predator and prey in intraguild predation systems with ontogenetic niche shifts

In basic intraguild predation (IGP) systems, predators and prey also compete for a shared resource. Theory predicts that persistence of these systems is possible when intraguild prey is superior in competition and productivity is not too high. IGP often results from ontogenetic niche shifts, in which the diet of intraguild predators changes as a result of growth in body size (life-history omnivory). As a juvenile, a life-history omnivore competes with the species that becomes its prey later in life. Competition can hence limit growth of young predators, while adult predators can suppress consumers and therewith neutralize negative effects of competition. We formulate and analyze a stage-structured model that captures both basic IGP and life-history omnivory. The model predicts increasing coexistence of predators and consumers when resource use of stage-structured predators becomes more stage specific. This coexistence depends on adult predators requiring consumer biomass for reproduction and is less likely when consumers outcompete juvenile predators, in contrast to basic IGP. Therefore, coexistence occurs when predation structures the community and competition is negligible. Consequently, equilibrium patterns over productivity resemble those of three-species food chains. Life-history omnivory thus provides a mechanism that allows intraguild predators and prey to coexist over a wide range of resource productivity.     

Strategy Space and the Disturbance Spectrum: A Life-History Model for Tree Species Coexistence

The disturbance spectrum consists of disturbance patterns differing in type, size, intensity, and frequency. It is proposed that tree life-history traits are adaptations to particular disturbance regimes. Four independent axes are proposed to define the dominant dimensions of tree strategy space: shade tolerance, tree height, capacity for vegetative reproduction, and seed dispersal distance. A fitness model was developed to elucidate interactions between the proposed life-history traits. The model shows how alternate life-history sets can coexist when disturbance patterns fluctuate in space and time. Variable disturbance regimes were shown, based on data and simulation results, to enhance species coexistence, as predicted. The strategy space model accurately predicts the number of common tree species for the eastern United States, boreal Canada, and southwestern pi?on-juniper woodlands. The model also provides an explanation for latitudinal gradients in tree species richness in North America and Europe. The proposed model predicts a relationship between disturbance characteristics and the species composition of a forest that allows for the coexistence of large numbers of species. The life-history traits of size, growth rate, life span, shade tolerance, age of reproduction, seed dispersal distance, and vegetative reproduction are all incorporated into the model.     

Synchronized reproduction promotes species coexistence through reproductive facilitation

Theories for species coexistence often emphasize niche differentiation and temporal segregation of recruitment to avoid competition. Recent work on mutualism suggested that plant species sharing pollinators provide mutual facilitation when exhibit synchronized reproduction. The facilitation on reproduction may enhance species persistence and coexistence. Theoretical ecologists paid little attention to such indirect mutualistic systems by far. We propose a new model for a two-species system using difference equations. The model focuses on adult plants and assumes no resource competition between these well-established individuals. Our formulas include demographic parameters, such as mortality and recruitment rates, and functions of reproductive facilitation. Both recruitment and facilitation effects reach saturation levels when flower production is at high levels. We conduct mathematical analyses to assess conditions of coexistence. We establish demographical conditions permitting species coexistence. Our analyses suggest a “rescue” effect from a “superior” species to a “weaker” species under strong recruitment enhancement effect when the later is not self-sustainable. The facilitation on rare species may help to overcome Allee effect.     

Untangling the mechanisms of cryptic species coexistence in a nematode community through individual-based modelling

Cryptic species are morphologically identical but genetically distinct, and are prominent across numerous phyla. The coexistence of such closely related species on local scales would seem to run counter to traditional coexistence and competition theory; it has been hypothesized as a consequence of differences in their resource use or tolerances to environmental conditions. We developed an individual-based model of a community of three cryptic Litoditis marina (nematode) species, to understand how individual-level interspecific and intraspecific interactions might explain the coexistence of these closely related species. The model incorporates individuals' reproduction, competition, dispersal and resource use. Data characterizing the cryptic species (growth rates, dispersal ability, competitive interactions and responses to changing environmental conditions) were obtained from laboratory experiments involving both mono- and multispecific nematode cultures, and are used to parameterize the model. Simulation studies are used to investigate which individual-level mechanisms of dispersal and interaction lead to the characteristic population-level patterns observed experimentally. Our results highlight the key role of intraspecific competition in mediating dispersal and therefore co-occurrence of the cryptic species. The differences in dispersal also influence the response of the cryptic species to competition, a combination of factors that provides an explanation for their co-occurrence. These results provide insights into how changes in individual-level processes can be amplified to affect population-level co-occurrence.     

Life-history trade-offs and ecological dynamics in the evolution of longevity

Longevity is a life-history trait that is shaped by natural selection. An unexplored consequence is how selection on this trait affects diversity and diversification in species assemblages. Motivated by the diverse rockfish (Sebastes) assemblage in the North Pacific, the effects of trade-offs in longevity against competitive ability are explored. A competition model is developed and used to explore the potential for species diversification and coexistence. Invasion analyses highlight that life-history trait trade-offs in longevity can mitigate the effects of competitive ability and favour the coexistence of a finite number of species. Our results have implications for niche differentiation, limiting similarity and assembly dynamics in multispecies interactions.     

Competition and coexistence with multiple life-history stages

Do complex life histories affect the conditions under which competitors can coexist? We investigated this using a two-species, two-stage Ricker model. With complex life cycles, the competition coefficients associated with each life-history stage suggest one of three competitive outcomes-coexistence, alternate stable states, or competitive exclusion-that depend on the relative magnitudes of intraspecific and interspecific competition. When the two stages suggest the same outcome, only that outcome can occur. When the stages suggest different outcomes, either one may prevail. It is also possible to have emergent outcomes, in which the outcome is not suggested by either stage. This can occur when the two stages suggest competitive exclusion by opposite species or when one stage suggests alternate stable states and the other suggests coexistence. Therefore, determining the mechanisms of coexistence in species with complex life histories may require consideration of competitive interactions within all life-history stages.     

Life-cycle switching and coexistence of species with no niche differentiation

The increasing evidence of coexistence of cryptic species with no recognized niche differentiation has called attention to mechanisms reducing competition that are not based on niche-differentiation. Only sex-based mechanisms have been shown to create the negative feedback needed for stable coexistence of competitors with completely overlapping niches. Here we show that density-dependent sexual and diapause investment can mediate coexistence of facultative sexual species having identical niches. We modelled the dynamics of two competing cyclical parthenogens with species-specific density-dependent sexual and diapause investment and either equal or different competitive abilities. We show that investment in sexual reproduction creates an opportunity for other species to invade and become established. This may happen even if the invading species is an inferior competitor. Our results suggests a previously unnoticed mechanism for species coexistence and can be extended to other facultative sexual species and species investing in diapause where similar density-dependent life-history switches could act to promote coexistence.     

Evolution of life-history traits collapses competitive coexistence

Trade-offs between competitive ability and the other life-history traits are considered to be a major mechanism of competitive coexistence. Many theoretical studies have demonstrated the robustness of such a coexistence mechanism ecologically; however, it is unknown whether the coexistence is robust evolutionarily. Here, we report that evolution of life-history traits not directly related to competition, such as longevity, and predator avoidance, easily collapses competitive coexistence in several competition systems: spatially structured, and predator-mediated two-species competition systems. In addition, we found that a superior competitor can be excluded by an inferior one by common mechanisms among the models. Our results suggest that ecological competitive coexistence due to a life-history trait trade-off balance may not be balanced on an evolutionary timescale, that is, it may be evolutionarily fragile.     

Modelling the Coexistence of Annual and Perennial Plants in Temporally Varying Environments*

Abstract The long-term growth and coexistence of species with large mixed populations in varying environments were modelled for representative environments and life-history characteristics of annual and perennial plants. The effects of the relationships between the means, variances, and covariances of seed yield, establishment, and survival, were explored by Taylor's expansion. The main findings are: 1. Individual variation in reproductive success within generations has no effect on long-term growth, which is determined only by the mean growth rate of the individuals of the species. 2. In annual species with nonoverlapping generations and without seed banks, the species with the largest mean log of the annual growth rate Y, that is the product of the average seed yield per plant and the establishment probability per seed, will win in competition with other species, independent of the correlations between the growth rates of the different species. In this case there is a negative tradeoff between the mean and the variance. 3. In perennial species with a lottery type of equal access to vacant sites, a high annual survival probability allows stable coexistence between perennial species with independent or negatively correlated variance in their mean annual product of seed production and establishment Y. 4. The coexistence range and the likely number of coexisting perennial species increase as a function of the variance of the common species, and is decreased by the variance of the rare species. The coexistence range is decreased by the covariance between the growth rates of the species, and between the survival of the rare species and its growth rate. 5. If mortality in the community of long-lived perennials is synchronized, the generations become nonoverlapping, and the competitive dynamics become similar to that of annuals. 6. Coexistence between annual and perennial species is promoted if the covariances between the annual survival and the relative yield of perennials, and between the yields of perennials and annuals, decrease and become more negative. 7. Selection for seed yield and establishment in different conditions in annuals favours a generalist strategy with low variance between years which provides a moderate yield and establishment over a wide range of environmental conditions. In perennial plants, long-term growth rate is determined by the lifetime seed yield and establishment. Because of strong competition with annuals in the more common conditions, selection in perennials favours instead a specialist strategy, with a high seed yield and establishment at relatively rare occasions in space and time, in which there is only weak competition with annuals. 8. Coexistence of annual species with a long-lived seed bank in the soil is also made possible by independent variation in different years of the germination, seed yield and establishment of different species, analogous to the situation of perennial plants.     

Mechanisms of coexistence in competitive metacommunities

Although there is a large body of theory on spatial competitive coexistence, very little of it involves comparative analyses of alternative mechanisms. We thus have limited knowledge of the conditions under which multiple spatial mechanisms can operate or of emergent properties arising from interactions between mechanisms. Here we present a mathematical framework that allows for comparative analysis of spatial coexistence mechanisms. The basis for comparison is mechanisms operating in spatially homogeneous competitive environments (e.g., life-history trade-offs) versus mechanisms operating in spatially heterogeneous competitive environments (e.g., source-sink dynamics). Our comparative approach leads to several new insights about spatial coexistence. First, we show that spatial variation in the expression of a life-history trade-off leads to a unique regional pattern that cannot be predicted by considering trade-offs or source-sink dynamics alone. This result represents an instance where spatial heterogeneity constrains rather than promotes coexistence, and it illustrates the kind of counterintuitive emergent properties that arise due to interactions between different classes of mechanisms. Second, we clarify the role of dispersal mortality in spatial coexistence. Previous studies have shown that coexistence can be constrained or facilitated by dispersal mortality. Our broader analysis distinguishes situations where dispersal mortality is not necessary for coexistence from those where such mortality is essential for coexistence because it preserves spatial variation in the strength of competition. These results form the basis for two important future directions: evolution of life-history traits in spatially heterogeneous environments and elucidation of the cause and effect relationship(s) between biodiversity and ecosystem functioning.     

Evolutionarily stable age at first reproduction in a density-dependent model.

We develop a new model of life history evolution to investigate the evolution of age at first reproduction. Density dependence is taken into account. For a given "species", age of maturity, offspring survival, immature survival, adult survival, fecundity, immature age-classes entering in competition with adults and immature competitive ability are traits adjustable by natural selection, and constitute a particular strategy. On the contrary, the type of intraspecific competition (scramble or contest), strength of competition and inherent net reproductive rate Ro(inh) are fixed (specific) characteristics. As a consequence of fixing Ro(inh), the evolution of any trait will affect trade-offs between others. Evolutionarily stable strategies are determined numerically by using the mathematical concept of Lyapunov exponents. Altogether, we consider 960 different hypothetical "species" (i.e. different combinations of fixed traits). Corresponding ESSs are analyzed with respect to their age at first reproduction, adult survival and immature competitive ability components. They appear to be gathered in three groups. One is intuitive and characterized by a reduction of immature competitive ability and a correlation of age of maturity with adult survival; populations reach mainly equilibria. The two other groups respectively include "species" with low age of maturity but high adult survival, and "species" close to semelparity with delayed maturity; immature competitive ability may not be minimized, and populations possibly exhibit complex dynamics. In conclusion, the hypothesis that the evolution of a demographic parameter modifies trade-offs between others turns out to have important consequences. We argue that life history theory cannot ignore the source and mode-of-operation of density dependence and must regard potential short-term instability as essential.     

Spatial variation and density-dependent dispersal in competitive coexistence

It is well known that dispersal from localities favourable to a species' growth and reproduction (sources) can prevent competitive exclusion in unfavourable localities (sinks). What is perhaps less well known is that too much emigration can undermine the viability of sources and cause regional competitive exclusion. Here, I investigate two biological mechanisms that reduce the cost of dispersal to source communities. The first involves increasing the spatial variation in the strength of competition such that sources can withstand high rates of emigration; the second involves reducing emigration from sources via density-dependent dispersal. I compare how different forms of spatial variation and modes of dispersal influence source viability, and hence source-sink coexistence, under dominance and pre-emptive competition. A key finding is that, while spatial variation substantially reduces dispersal costs under both types of competition, density-dependent dispersal does so only under dominance competition. For instance, when spatial variation in the strength of competition is high, coexistence is possible (regardless of the type of competition) even when sources experience high emigration rates; when spatial variation is low, coexistence is restricted even under low emigration rates. Under dominance competition, density-dependent dispersal has a strong effect on coexistence. For instance, when the emigration rate increases with density at an accelerating rate (Type III density-dependent dispersal), coexistence is possible even when spatial variation is quite low; when the emigration rate increases with density at a decelerating rate (Type II density-dependent dispersal), coexistence is restricted even when spatial variation is quite high. Under pre-emptive competition, density-dependent dispersal has only a marginal effect on coexistence. Thus, the diversity-reducing effects of high dispersal rates persist under pre-emptive competition even when dispersal is density dependent, but can be significantly mitigated under dominance competition if density-dependent dispersal is Type III rather than Type II. These results lead to testable predictions about source-sink coexistence under different regimes of competition, spatial variation and dispersal. They identify situations in which density-independent dispersal provides a reasonable approximation to species' dispersal patterns, and those under which consideration of density-dependent dispersal is crucial to predicting long-term coexistence.     

Variation in life-history traits of male Japanese fluvial sculpin Cottus pollux in relation to nest abundance along a stream course

Life-history theory predicts the occurrence of variation in the life-history traits of fish populations under different environmental conditions; however, most studies have focused on such variation between geographically separated populations. We compared breeding characteristics and life-history traits of the Japanese fluvial sculpin (Cottus pollux), a bottom-dwelling nest-holding fish, between two adjacent sites sub-divided by a weir along a stream course in central Japan. Males in the area with a lower abundance of nest sites reached sexual maturity at an earlier age and had a shorter life span than males in the area with sufficient nest abundance. Size-dependent male reproduction was found only in areas with a shortage of nest sites, supporting the assumption of competitive exclusion among males for nests. Females matured at the same age in both sites with no differences in age-specific growth rates and mortality. Our results provide evidence for life-history variation in age and size at maturity and age-specific mortality schedule of males in nest-holding fishes in a single stream population via different sexual selection regimes related to differences in nest abundance between sites.     

LIFE-HISTORY EVOLUTION IN GUPPIES (POECILIA RETICULATA) 6. DIFFERENTIAL MORTALITY AS A MECHANISM FOR NATURAL SELECTION

We have previously reported a correlation between the life-history patterns of guppies and the types of predators with which they coexist. Guppies from localities with an abundance of large predators (high predation localities) mature at an earlier age and devote more resources to reproduction than those found in localities with only a single, small species of predator (low predation localities). We also found that when guppies were introduced from a high to low predation locality, the guppy life history evolved to resemble what was normally found in this low predation locality. The presumed mechanism of natural selection is differences among localities in age/size-specific mortality (the age/size-specific mortality hypothesis); in high predation localities we assumed that guppies experienced high adult mortality rates while in the low predation localities we assumed that guppies experienced high juvenile mortality rates. These assumptions were based on stomach content analyses of wild-caught predators and on laboratory experiments. Here, we evaluate these assumptions by directly estimating the mortality rates of guppies in natural populations. We found that guppies from high predation localities experience significantly higher mortality rates than their counterparts from low predation localities, but that these higher mortality rates are uniformly distributed across all size classes, rather than being concentrated in the larger size classes. This result appears to contradict the predictions of the age/size-specific predation hypothesis. However, we argue, using additional data on growth rates and the probabilities of survival to maturity in each type of locality, that the age-specific mortality hypothesis remains plausible. This is because the probability of survival to first reproduction is very similar in each type of locality, but the guppies from high predation localities have a much lower probability of survival per unit time after maturity. We also argue for the plausibility of two other mechanisms of natural selection. These results thus reveal mortality patterns that provide a potential cause of natural selection, but expand, rather than narrow, the number of possible mechanisms responsible for life-history evolution in guppies.     

Dynamics of species coexistence: maintenance of a plant-ant competitive metacommunity

The metacommunity approach is an adequate framework to study coexistence between interacting species at different spatial scales. However, empirical evidence from natural metacommunities necessary to evaluate the predictive power of theoretical models of species coexistence remains sparse. We use two African ant species, Cataulacus mckeyi and Petalomyrmex phylax , symbiotically associated with the myrmecophyte Leonardoxa africana africana , to examine spatio-temporal dynamics of species coexistence and to investigate which environmental and life-history parameters may contribute to the maintenance of species diversity in this guild of symbiotic ants. Using environmental niche partitioning as a conceptual framework, we combined data on habitat variation, social structure of colonies, and population genetics with data from a colonisation experiment and from observation of temporal dynamics. We propose that the dynamics of ant species colonisation and replacement at local and regional scales can be explained by a set of life history traits for which the two ants exhibit hierarchies, coupled with strong environmental differences between the different patches in the level of environmental disturbances. The role of the competition–colonisation tradeoff is discussed and we propose that interspecific tradeoffs for traits related to dispersal and to reproduction are also determinant for species coexistence. We therefore suggest that species-sorting mechanisms are predominant in the dynamics of this metacommunity, but we also emphasise that there may be many ways for two symbionts in competition for the same host to coexist. The results speak in favour of a more complete integration of the various metacommunity models in a single theoretical framework.