Invasibility and stochastic boundedness in monotonic competition models

We give necessary and sufficient conditions for stochastically bounded coexistence in a class of models for two species competing in a randomly varying environment. Coexistence is implied by mutual invasibility, as conjectured by Turelli. In the absence of invasibility, a species converges to extinction with large probability if its initial population is small, and extinction of one species must occur with probability one regardless of the initial population sizes. These results are applied to a general symmetric competition model to find conditions under which environmental fluctuations imply coexistence or competitive exclusion.     

Relative nonlinearity and permanence

We modify the commonly used invasibility concept for coexistence of species to the stronger concept of uniform invasibility. For two-species discrete-time competition and predator-prey models, we use this concept to find broad easily checked sufficient conditions for the rigorous concept of permanent coexistence. With these results, permanent coexistence becomes a tractable concept for many discrete-time population models. To understand how these conditions apply to nonpoint attractors, we generalize the concept of relative nonlinearity and use it to show how population fluctuations affect the long-term low-density growth rate (“the invasion rate”) of a species when it is invading the system consisting of the other species (“the resident”) at a single-species attractor. The concept of relative nonlinearity defines circumstances when this invasion rate is increased or decreased by resident population fluctuations arising from a nonpoint attractor. The presence and sign of relative nonlinearity is easily checked in models of interacting species. When relative nonlinearity is zero or positive, fluctuations cannot decrease the invasion rate. It follows that permanence is then determined by invasibility of the resident’s fixed points. However, when relative nonlinearity is negative, invasibility, and hence permanent coexistence, can be undermined by resident population fluctuations. These results are illustrated with specific two-species competition and predator-prey models of generic forms.     

Spatial heterogeneity of plant–soil feedbacks increases per capita reproductive biomass of species at an establishment disadvantage

Plant–soil feedbacks have been widely implicated as a driver of plant community diversity, and the coexistence prediction generated by a negative plant–soil feedback can be tested using the mutual invasibility criterion: if two populations are able to invade one another, this result is consistent with stable coexistence. We previously showed that two co-occurring Rumex species exhibit negative pairwise plant–soil feedbacks, predicting that plant–soil feedbacks could lead to their coexistence. However, whether plants are able to reproduce when at an establishment disadvantage (“invasibility”), or what drivers in the soil might correlate with this pattern, are unknown. To address these questions, we created experimental plots with heterogeneous and homogeneous soils using field-collected conditioned soils from each of these Rumex species. We then allowed resident plants of each species to establish and added invader seeds of the congener to evaluate invasibility. Rumex congeners were mutually invasible, in that both species were able to establish and reproduce in the other’s resident population. Invaders of both species had twice as much reproduction in heterogeneous compared to homogeneous soils; thus the spatial arrangement of plant–soil feedbacks may influence coexistence. Soil mixing had a non-additive effect on the soil bacterial and fungal communities, soil moisture, and phosphorous availability, suggesting that disturbance could dramatically alter soil legacy effects. Because the spatial arrangement of soil patches has coexistence implications, plant–soil feedback studies should move beyond studies of mean effects of single patch types, to consider how the spatial arrangement of patches in the field influences plant communities.     

The age-structured lottery model

The lottery model of competition between species in a variable environmental has been influential in understanding how coexistence may result from interactions between fluctuating environmental and competitive factors. Of most importance, it has led to the concept of the storage effect as a mechanism of species coexistence. Interactions between environment and competition in the lottery model stem from the life-history assumption that environmental variation and competition affect recruitment to the adult population, but not adult survival. The strong role of life-history attributes in this coexistence mechanism implies that its robustness should be checked for a variety of life-history scenarios. Here, age structure is added to the adult population, and the results are compared with the original lottery model. This investigation uses recently developed shape characteristics for mortality and fecundity schedules to quantify the effects of age structure on the long-term low-density growth rate of a species in competition with its competitor when applying the standard invasibility coexistence criterion. Coexistence conditions are found to be affected to a small degree by the presence of age structure in the adult population: Type III mortality broadens coexistence conditions, and type I mortality makes them narrower. The rates of recovery from low density for coexisting species, and the rates of competitive exclusion in other cases, are modified to a greater degree by age structure. The absolute rates of recovery or decline of a species from low density are increased by type I mortality or early peak reproduction, but reduced by type III mortality or late peak reproduction. Analytical approximations show how the most important effects can be considered as simple modifications of the long-term low-density growth rates for the original lottery model.     

Autocorrelated environmental variation and the establishment of invasive species

Environmental parameters such as temperature and rainfall have a positively autocorrelated variance structure which makes it likely that runs of good or bad conditions will occur. It has previously been demonstrated that such autocorrelated environmental variance can increase the probability of extinction in small populations, in much the same way that increased variance without autocorrelation can increase extinction risk. As a result, it has also been suggested that positive autocorrelation will decrease the probability that a species will establish in a novel location. We suggest that describing the probability of invasion success as the probability of indefinite persistence may be an inappropriate definition of risk. Economic or ecological damage may be associated with a population that initially reaches high densities before going extinct in the new location. In addition, such populations may spread to new locations before extirpation. We use a modeling approach to examine the effect of positively autocorrelated conditions on the probability that small populations will reach large size before extinction. We find that where variance is high and the geometric mean of the population growth rate is low, autocorrelation increases the risk that a population will pass a an upper threshold density, even when extinction probability is unaffected. Therefore species classified as having low probability of invasion risk on the basis of population growth rates measured in low variance environments may actually have quite a substantial probability of establishing a large population for a period of time. The mechanism behind the effect is the disproportionate influence of short runs of good conditions initially following introduction.     

Coexistence and Spread of Competitors in Heterogeneous Landscapes

Competition between species is ubiquitous in nature and therefore widely studied in ecology through experiment and theory. One of the central questions is under which conditions a (rare) invader can establish itself in a landscape dominated by a resident species at carrying capacity. Applying the same question with the roles of the invader and resident reversed leads to the principle that “mutual invasibility implies coexistence.” A related but different question is how fast a locally introduced invader spreads into a landscape (with or without competing resident), provided it can invade. We explore some aspects of these questions in a deterministic, spatially explicit model for two competing species with discrete non-overlapping generations in a patchy periodic environment. We obtain threshold values for fragmentation levels and dispersal distances that allow for mutual invasion and coexistence even if the non-spatial competition model predicts competitive exclusion. We obtain exact results when dispersal is governed by a Laplace kernel. Using the average dispersal success, we develop a mathematical framework to obtain approximate results that are independent of the exact dispersal patterns, and we show numerically that these approximations are very accurate.     

Thermal Carrying Capacity for a Thermally-Sensitive Species at the Warmest Edge of Its Range

Anthropogenic environmental change is causing unprecedented rates of population extirpation and altering the setting of range limits for many species. Significant population declines may occur however before any reduction in range is observed. Determining and modelling the factors driving population size and trends is consequently critical to predict trajectories of change and future extinction risk. We tracked during 12 years 51 populations of a cold-water fish species (brown trout Salmo trutta) living along a temperature gradient at the warmest thermal edge of its range. We developed a carrying capacity model in which maximum population size is limited by physical habitat conditions and regulated through territoriality. We first tested whether population numbers were driven by carrying capacity dynamics and then targeted on establishing (1) the temperature thresholds beyond which population numbers switch from being physical habitat- to temperature-limited; and (2) the rate at which carrying capacity declines with temperature within limiting thermal ranges. Carrying capacity along with emergent density-dependent responses explained up to 76% of spatio-temporal density variability of juveniles and adults but only 50% of young-of-the-year''s. By contrast, young-of-the-year trout were highly sensitive to thermal conditions, their performance declining with temperature at a higher rate than older life stages, and disruptions being triggered at lower temperature thresholds. Results suggest that limiting temperature effects were progressively stronger with increasing anthropogenic disturbance. There was however a critical threshold, matching the incipient thermal limit for survival, beyond which realized density was always below potential numbers irrespective of disturbance intensity. We additionally found a lower threshold, matching the thermal limit for feeding, beyond which even unaltered populations declined. We predict that most of our study populations may become extinct by 2100, depicting the gloomy fate of thermally-sensitive species occurring at thermal range margins under limited potential for adaptation and dispersal.     

Extreme Selection Unifies Evolutionary Game Dynamics in Finite and Infinite Populations

We show that when selection is extreme—the fittest strategy always reproduces or is imitated—the unequivalence between the possible evolutionary game scenarios in finite and infinite populations resolves, in the sense that the three generic outcomes—dominance, coexistence, and mutual exclusion—emerge in well-mixed populations of any size. We consider the simplest setting of a 2-player-2-strategy symmetric game and the two most common microscopic definitions of strategy spreading—the frequency-dependent Moran process and the imitation process by pairwise comparison—both in the case allowing any intensity of selection. We show that of the seven different invasion and fixation scenarios that are generically possible in finite populations—fixation being more or less likely to occur and rapid compared to the neutral game—the three that are possible in large populations are the same three that occur for sufficiently strong selection: (1) invasion and fast fixation of one strategy; (2) mutual invasion and slow fixation of one strategy; (3) no invasion and no fixation. Moreover (and interestingly), in the limit of extreme selection 2 becomes mutual invasion and no fixation, a case not possible for finite intensity of selection that better corresponds to the deterministic case of coexistence. In the extreme selection limit, we also derive the large population deterministic limit of the two considered stochastic processes.     

Characterizing population vulnerability for 758 species

We investigate relationships between life history traits and the character of population dynamics as revealed by time series data. Our classification of time series is according to 'extinction category,' where we identify three classes of populations: (i) weakly varying populations with such high growth rates that long-term persistence is likely (unless some extreme catastrophe occurs); (ii) populations with such low growth rates that average population size must be large to buffer them against extinction in a variable environment; and (iii) highly variable populations that fluctuate so dramatically that dispersal or some other refuge mechanism is likely to be key to their avoidance of extinction. Using 1941 time series representing 758 species from the Global Population Dynamics Database, we find that, depending on the form of density dependence one assumes, between 46 and 90% of species exhibit dynamics that are so variable that even large carrying capacities could not buffer them against extinction on a 100-year time horizon. The fact that such a large proportion of population dynamics are so locally variable vindicates the growing realization that dispersal, habitat connectedness, and large-scale processes are key to local persistence. Furthermore, for mammals, simply by knowing body size, age at first reproduction, and average number of offspring we could correctly predict extinction categories for 83% of species (60 of 72).     

Environmental stochasticity and extinction risk in a population of a small songbird, the great tit

Using a long-term demographic data set, we estimated the separate effects of demographic and environmental stochasticity in the growth rate of the great tit population in Wytham Wood, United Kingdom. Assuming logistic density regulation, both the demographic (sigma2d = 0.569) and environmental (sigma2e = 0.0793) variance, with interactions included, were significantly greater than zero. The estimates of the demographic variance seemed to be relatively insensitive to the length of the study period, whereas reliable estimates of the environmental variance required long time series (at least 15 yr of data). The demographic variance decreased significantly with increasing population density. These estimates are used in a quantitative analysis of the demographic factors affecting the risk of extinction of this population. The very long expected time to extinction of this population (approximately 10(19) yr) was related to a relatively large population size (>/=120 pairs during the study period). However, for a given population size, the expected time to extinction was sensitive to both variation in population growth rate and environmental stochasticity. Furthermore, the form of the density regulation strongly affected the expected time to extinction. Time to extinction decreased when the maximum density regulation approached K. This suggests that estimates of viability of small populations should be given both with and without inclusion of density dependence.     

Population size dependence, competitive coexistence and habitat destruction

1. Spatial dynamics can lead to coexistence of competing species even with strong asymmetric competition under the assumption that the inferior competitor is a better colonizer given equal rates of extinction. Patterns of habitat fragmentation may alter competitive coexistence under this assumption.
2. Numerical models were developed to test for the previously ignored effect of population size on competitive exclusion and on extinction rates for coexistence of competing species. These models neglect spatial arrangement.
3. Cellular automata were developed to test the effect of population size on competitive coexistence of two species, given that the inferior competitor is a better colonizer. The cellular automata in the present study were stochastic in that they were based upon colonization and extinction probabilities rather than deterministic rules.
4. The effect of population size on competitive exclusion at the local scale was found to have little consequence for the coexistence of competitors at the metapopulation (or landscape) scale. In contrast, population size effects on extinction at the local scale led to much reduced landscape scale coexistence compared to simulations not including localized population size effects on extinction, especially in the cellular automata models. Spatially explicit dynamics of the cellular automata vs. deterministic rates of the numerical model resulted in decreased survival of both species. One important finding is that superior competitors that are widespread can become extinct before less common inferior competitors because of limited colonization.
5. These results suggest that population size–extinction relationships may play a large role in competitive coexistence. These results and differences are used in a model structure to help reconcile previous spatially explicit studies which provided apparently different results concerning coexistence of competing species.     

Building modern coexistence theory from the ground up: The role of community assembly

Modern coexistence theory (MCT) is one of the leading methods to understand species coexistence. It uses invasion growth rates—the average, per-capita growth rate of a rare species—to identify when and why species coexist. Despite significant advances in dissecting coexistence mechanisms when coexistence occurs, MCT relies on a ‘mutual invasibility’ condition designed for two-species communities but poorly defined for species-rich communities. Here, we review well-known issues with this component of MCT and propose a solution based on recent mathematical advances. We propose a clear framework for expanding MCT to species-rich communities and for understanding invasion resistance as well as coexistence, especially for communities that could not be analysed with MCT so far. Using two data-driven community models from the literature, we illustrate the utility of our framework and highlight the opportunities for bridging the fields of community assembly and species coexistence.     

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.     

Bayesian Inference on the Effect of Density Dependence and Weather on a Guanaco Population from Chile

Understanding the mechanisms that drive population dynamics is fundamental for management of wild populations. The guanaco (Lama guanicoe) is one of two wild camelid species in South America. We evaluated the effects of density dependence and weather variables on population regulation based on a time series of 36 years of population sampling of guanacos in Tierra del Fuego, Chile. The population density varied between 2.7 and 30.7 guanaco/km², with an apparent monotonic growth during the first 25 years; however, in the last 10 years the population has shown large fluctuations, suggesting that it might have reached its carrying capacity. We used a Bayesian state-space framework and model selection to determine the effect of density and environmental variables on guanaco population dynamics. Our results show that the population is under density dependent regulation and that it is currently fluctuating around an average carrying capacity of 45,000 guanacos. We also found a significant positive effect of previous winter temperature while sheep density has a strong negative effect on the guanaco population growth. We conclude that there are significant density dependent processes and that climate as well as competition with domestic species have important effects determining the population size of guanacos, with important implications for management and conservation.     

Invasibility in a spatiotemporally fluctuating environment is determined by the periodicity of fluctuations and resident turnover rates

The ability of a species to invade a community is influenced by the traits of the invader, the resident community and the environment. However, qualitative generalizations are possible. Using a model of perennial plants in a spatiotemporally fluctuating environment, we find that fluctuating environments may be more or less invasible than static environments. Invasibility is strongly dependent on the interaction of the difference in turnover rates of resident and invader populations and the rate of temporal change of the environment. If resident population turnover is faster than the invader's, then invasibility is an initially positive, decreasing function of the period temporal variation, such that invasibility is increased by rapid temporal fluctuations but slightly reduced in slowly fluctuating environments. If resident turnover is slower than the invader's, then invasibility is an initially negative, increasing function of temporal period, such that invasibility is reduced in rapidly changing environment facilitated by slow temporal fluctuations. These results are explained by the relative abilities of resident and invader populations to successfully respond to environmental variation at different temporal scales.     

Metapopulation extinction risk: Dispersal’s duplicity

Metapopulation extinction risk is the probability that all local populations are simultaneously extinct during a fixed time frame. Dispersal may reduce a metapopulation’s extinction risk by raising its average per-capita growth rate. By contrast, dispersal may raise a metapopulation’s extinction risk by reducing its average population density. Which effect prevails is controlled by habitat fragmentation. Dispersal in mildly fragmented habitat reduces a metapopulation’s extinction risk by raising its average per-capita growth rate without causing any appreciable drop in its average population density. By contrast, dispersal in severely fragmented habitat raises a metapopulation’s extinction risk because the rise in its average per-capita growth rate is more than offset by the decline in its average population density. The metapopulation model used here shows several other interesting phenomena. Dispersal in sufficiently fragmented habitat reduces a metapopulation’s extinction risk to that of a constant environment. Dispersal between habitat fragments reduces a metapopulation’s extinction risk insofar as local environments are asynchronous. Grouped dispersal raises the effective habitat fragmentation level. Dispersal search barriers raise metapopulation extinction risk. Nonuniform dispersal may reduce the effective fraction of suitable habitat fragments below the extinction threshold. Nonuniform dispersal may make demographic stochasticity a more potent metapopulation extinction force than environmental stochasticity.     

Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.     

Caught in the middle: Asymmetric competition causes high variance in intermediate trait abundances

In asymmetric competition between two individuals of the same or different species, one individual has a distinct advantage over the other due to a particular beneficial trait. An important trait that induces asymmetric competition is size (body size in animals, height in plants). There is usually a trade-off between fecundity and the trait that leads to competitive superiority (e.g. seed number vs seed size), enabling coexistence of populations with different trait values. These predictions on coexistence are based on classic deterministic models. Here, we explore the behaviour of a stochastic model of asymmetric competition where stochasticity is assumed to be demographic. We derive approximations for the temporal variance and covariance of the population sizes of the coexisting species. The derivations highlight that the variability of the population size of a species strongly depends on the stochastic fluctuations of species with higher trait values, while they are less influenced by species with lower trait values. Particularly, species with intermediate trait values are strongly affected resulting in relatively high variability. As a result these species have a relative high probability of extinction even though they have a larger population size than species with high trait values. We confirm these approximations with individual-based simulations. Thus, our analysis can explain gaps in size distributions as an emergent property of systems with a fecundity–competition trade-off.     

Does habitat specialization shape the evolutionary potential of wild bird populations?

Because specialist species evolved in more temporally and spatially homogeneous environments than generalist species, they are supposed to experience less fluctuating selection. For this reason, we expect specialists to show lower overall genetic variation as compared to generalists. We also expect populations from specialist species to be smaller and more fragmented, with lower neutral genetic diversity. We tested these hypotheses by investigating patterns of genetic diversity along a habitat specialization gradient in wild birds, based on estimates of heritability, coefficients of variation of additive genetic variance, and heterozygosity available in the literature. We found no significant effect of habitat specialization on any of the quantitative genetic estimators but generalists had higher heterozygosity. This effect was mainly a consequence of the larger population size of generalists. Our results suggest that evolutionary potential does not differ at the population level between generalist and specialist species, but the trend observed in heterozygosity levels and population sizes may explain their difference in susceptibility to extinction.     

Coevolution of competing Callosobruchus species does not stabilize coexistence

Interspecific resource competition is expected to select for divergence in resource use, weakening interspecific relative to intraspecific competition, thus promoting stable coexistence. More broadly, because interspecific competition reduces fitness, any mechanism of interspecific competition should generate selection favoring traits that weaken interspecific competition. However, species also can adapt to competition by increasing their competitive ability, potentially destabilizing coexistence. We reared two species of bean beetles, the specialist Callosobruchus maculatus and the generalist C. chinensis, in allopatry and sympatry on a mixture of adzuki beans and lentils, and assayed mutual invasibility after four, eight, and twelve generations of evolution. Contrary to the expectation that coevolution of competitors will weaken interspecific competition, the rate of mutual invasibility did not differ between sympatry and allopatry. Rather, the invasion rate of C. chinensis, but not C. maculatus, increased with duration of evolution, as C. chinensis adapted to lentils without experiencing reduced adaptation to adzuki beans, and regardless of the presence or absence of C. maculatus. Our results highlight that evolutionary responses to interspecific competition promote stable coexistence only under specific conditions that can be difficult to produce in practice.