Sensitivity analysis of coexistence in ecological communities: theory and application

Sensitivity analysis, the study of how ecological variables of interest respond to changes in external conditions, is a theoretically well‐developed and widely applied approach in population ecology. Though the application of sensitivity analysis to predicting the response of species‐rich communities to disturbances also has a long history, derivation of a mathematical framework for understanding the factors leading to robust coexistence has only been a recent undertaking. Here we suggest that this development opens up a new perspective, providing advances ranging from the applied to the theoretical. First, it yields a framework to be applied in specific cases for assessing the extinction risk of community modules in the face of environmental change. Second, it can be used to determine trait combinations allowing for coexistence that is robust to environmental variation, and limits to diversity in the presence of environmental variation, for specific community types. Third, it offers general insights into the nature of communities that are robust to environmental variation. We apply recent community‐level extensions of mathematical sensitivity analysis to example models for illustration. We discuss the advantages and limitations of the method, and some of the empirical questions the theoretical framework could help answer.     

Periodic coexistence of four species competing for three essential resources

We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.     

Population biology of multispecies helminth infection: competition and coexistence

The role that interspecific interactions play in shaping parasite communities is uncertain. To date, models of competition between helminth species have assumed that interaction occurs through parasite-induced host death. To our knowledge, there has been no theoretical exploration of other forms of competition. We examine models in which competition acts at the point of establishment within the host, and at the time of egg production by the adult worm. The models used are stochastic and we allow hosts to vary in their rate of exposure to infective larvae. We derive the Lotka-Volterra model of competition when exposure is homogenous and thus demonstrate that two helminth species cannot coexist on a single limiting resource. We show that coexistence of species is promoted by heterogeneity in host exposure provided that the rates of exposure to the two species are not perfectly correlated, and, if they are positively correlated, provided that the degree of heterogeneity in host exposure is similar for the two competing helminth species. These results are robust to the mechanism of competition.     

Temperature fluctuation facilitates coexistence of competing species in experimental microbial communities

1. Temperature fluctuation is a general phenomenon affecting many, if not all, species in nature. While a few studies have shown that temperature fluctuation can promote species coexistence, little is known about the effects of different regimes of temperature fluctuation on coexistence. 2. We experimentally investigated how temperature fluctuation and different regimes of temperature fluctuation ('red' environments in which temperature series exhibited positive temporal autocorrelation vs. 'white' environments in which temperature series showed little autocorrelation) affected the coexistence of two ciliated protists, Colpidium striatum Stein and Paramecium tetraurelia Sonneborn, which competed for bacterial resources. 3. We have previously shown that the two species differed in their growth responses to changes in temperature and in their resource utilization patterns. The two species were not always able to coexist at constant temperatures (22, 24, 26, 28 and 30 degrees C), with Paramecium being competitively excluded at 26 and 28 degrees C. This indicated that resource partitioning was insufficient to maintain coexistence at these temperatures. 4. Here we show that in both red and white environments in which temperature varied between 22 and 32 degrees C, Paramecium coexisted with Colpidium. Consistent with the differential effects of temperature on their intrinsic growth rates, Paramecium population dynamics were largely unaffected by temperature regimes, and Colpidium showed more variable population dynamics in the red environments. 5. Temperature-dependent competitive effects of Colpidium on Paramecium, together with resource partitioning, appeared to be responsible for the coexistence in the white environments; resource partitioning and the storage effect appeared to account for the coexistence in the red environments. 6. These results suggest that temperature fluctuation may play important roles in regulating species coexistence and diversity in ecological communities.     

How to quantify the temporal storage effect using simulations instead of math

The storage effect has become a core concept in community ecology, explaining how environmental fluctuations can promote coexistence and maintain biodiversity. However, limitations of existing theory have hindered empirical applications: the need for detailed mathematical analysis whenever the study system requires a new model, and restricted theory for structured populations. We present a new approach that overcomes both these limitations. We show how temporal storage effect can be quantified by Monte Carlo simulations in a wide range of models for competing species. We use the lottery model and a generic integral projection model (IPM) to introduce ideas, and present two empirical applications: (1) algal species in a chemostat with variable temperature, showing that the storage effect can operate without a long‐lived life stage and (2) a sagebrush steppe community IPM. Our results highlight the need for careful modelling of nonlinearities so that conclusions are not driven by unrecognised model constraints.     

Disturbances allow coexistence of competing species

We give the first mathematically rigorous proof that disturbances allow competing species to coexist. This work provides a mathematical framework to explain the existence of fugitive species and the role played by disturbances in increasing or decreasing the biodiversity of ecosystems. We study modifications of the metapopulation model for patchy environments proposed by Caswell and Cohen (1990, 1991). For the one- and two-species models we give necessary and sufficient conditions on the parameters for the existence of a non-trivial equilibrium solution, which is shown to be always globally stable.     

An interdisciplinary conception of human-wildlife coexistence

Human-wildlife coexistence is a pressing worldwide concern with significant impacts on biodiversity conservation and sustainable development. Despite the growing interest in coexistence research and practice, recent analytical attention remains fragmented, inconsistent, and of questionable practical use. To address this, we draw on the integrative framework of policy sciences to clarify coexistence concepts. The term “coexistence” is currently used with various interpretations, such as a desirable state, a factual occurrence, or a conservation approach. We adopt the classical ecological definition of coexistence and view human-wildlife coexistence as a multispecies social process that involves diverse interactions between human and non-human actors. We propose a typology of coexistence that can be applied to human-wildlife and human-human interactions across different contexts. We categorize the factors that condition coexistence into two groups: the behavioral predispositions of actors, which we conceptualize as a physical-physiological-psychological complex, and the environmental circumstances of interactions, which we conceptualize as a natural-social-cultural nexus. These interactions involved in coexistence are characterized by impermanence and inter-determination. We use the example of human-snow leopard interactions on the Tibetan Plateau in China to illustrate our points. To better understand and address coexistence challenges, we recommend that conservationists take an integrative approach that is problem-oriented, contextual, self-reflective, and multi-method.     

Stabilizing factors interact in promoting host-parasite coexistence

Understanding the mechanisms that promote coexistence among species is a fundamental problem in evolutionary ecology. Such mechanisms include environmental noise, spatial population structure, density dependence, and genetic variation. In natural populations such factors may exert combined effects on coexistence. Thus, to disentangle the contribution of several factors to coexistence, their effects have to be considered simultaneously. Here we investigate the effects of Ricker-type density dependence, genetic variation, and the frequency of sex on host-parasite coexistence, using Nicholson-Bailey models with and without host density dependence. Interestingly, a low frequency of sex (and the genetic variation induced by sex) is the most important factor in explaining the stability of the host-parasite interaction. However, the carrying capacity K and the frequency of sex interact in affecting coexistence. If K is low (strong density regulation), coexistence is easily attained in the density-dependent model, independently of the frequency of sex. In contrast, for high values of K (weak density regulation), low frequencies of sex considerably improve coexistence. Thus, our results suggest that coexistence among species may strongly depend on interactions among several stabilizing factors. These results seem to be robust since they remain qualitatively unchanged if one assumes (1) Beverton-Holt-type or genotype-specific rather than Ricker-type density dependence in the host, or (2) different genotype-specific susceptibilities of hosts to their parasites, or if one adds (3) moderate levels of environmental stochasticity.     

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.     

A sampling theory for asymmetric communities

We introduce the first analytical model of asymmetric community dynamics to yield Hubbell's neutral theory in the limit of functional equivalence among all species. Our focus centers on an asymmetric extension of Hubbell's local community dynamics, while an analogous extension of Hubbell's metacommunity dynamics is deferred to an appendix. We find that mass-effects may facilitate coexistence in asymmetric local communities and generate unimodal species abundance distributions indistinguishable from those of symmetric communities. Multiple modes, however, only arise from asymmetric processes and provide a strong indication of non-neutral dynamics. Although the exact stationary distributions of fully asymmetric communities must be calculated numerically, we derive approximate sampling distributions for the general case and for nearly neutral communities where symmetry is broken by a single species distinct from all others in ecological fitness and dispersal ability. In the latter case, our approximate distributions are fully normalized, and novel asymptotic expansions of the required hypergeometric functions are provided to make evaluations tractable for large communities. Employing these results in a Bayesian analysis may provide a novel statistical test to assess the consistency of species abundance data with the neutral hypothesis.     

Effect of domain-shape on coexistence problems in a competition-diffusion system

We discuss a competition-diffusion system to study coexistence problems of two competing species in a homogeneous environment. In particular, by using invariant manifold theory, effects of domain-shape are considered on this problem.     

Spatial and temporal patterns of coexistence between competing <Emphasis Type="Italic">Aedes</Emphasis> mosquitoes in urban Florida

Understanding mechanisms fostering coexistence between invasive and resident species is important in predicting ecological, economic, or health impacts of invasive species. The mosquito Aedes aegypti coexists at some urban sites in southeastern United States with invasive Aedes albopictus, which is often superior in interspecific competition. We tested predictions for three hypotheses of species coexistence: seasonal condition-specific competition, aggregation among individual water-filled containers, and colonization–competition tradeoff across spatially partitioned habitat patches (cemeteries) that have high densities of containers. We measured spatial and temporal patterns of abundance for both species among water-filled resident cemetery vases and experimentally positioned standard cemetery vases and ovitraps in metropolitan Tampa, Florida. Consistent with the seasonal condition-specific competition hypothesis, abundances of both species in resident and standard cemetery vases were higher early in the wet season (June) versus late in the wet season (September), but the proportional increase of A. albopictus was greater than that of A. aegypti, presumably due to higher dry-season egg mortality and strong wet-season competitive superiority of larval A. albopictus. Spatial partitioning was not evident among cemeteries, a result inconsistent with the colonization-competition tradeoff hypothesis, but both species were highly independently aggregated among standard cemetery vases and ovitraps, which is consistent with the aggregation hypothesis. Densities of A. aegypti but not A. albopictus differed among land use categories, with A. aegypti more abundant in ovitraps in residential areas compared to industrial and commercial areas. Spatial partitioning among land use types probably results from effects of land use on conditions in both terrestrial and aquatic-container environments. These results suggest that both temporal and spatial variation may contribute to local coexistence between these Aedes in urban areas.     

Interspecific competition and coexistence of the two chrysomelids,Gastrophysa atrocyanea motschulsky andGalerucella vittaticollis baly (Coleoptera: Chrysomelidae), under limited food resource conditions

Field experiments were conducted in order to investigate the mode of exploitation of food resources and the mechanism of coexistence of mixed larval populations of the two chrysomelids,Gastrophysa atrocyanea andGalerucella vittaticollis, under limited food resource conditions. The larval survival rates seemed high enough to assure coexistence when hatchlings of the two species were released in 1∶1 and 1∶3 ratios on a host plant. However, the survival rate became almost nil for both species when a 3∶1 ratio was employed, suggesting asymmetrical interspecific competition. Wasted food consumption was much higher inG. atrocyanea larvae. The population ofG. atrocyanea seemed to be regulated more by intraspecific competition, while on the other hand, the population ofG. vittaticollis was considered to be more likely affected by the interspecific competition withG. atrocyanea, depending on the initial ratio and density of the two species.     

Spatial patterns and coexistence mechanisms in systems with unidirectional flow

River ecosystems are the prime example of environments where unidirectional flow influences the dispersal of individuals. Spatial patterns of community composition and species replacement emerge from complex interplays of hydrological, geochemical, biological, and ecological factors. Local processes affecting algal dynamics are well understood, but a mechanistic basis for large scale emerging patterns is lacking. To understand how these patterns could emerge in rivers, we analyze a reaction-advection-diffusion model for two competitors in heterogeneous environments. The model supports waves that invade upstream up to a well-defined "upstream invasion limit". We discuss how these waves are produced and present their key properties. We suggest that patterns of species replacement and coexistence along spatial axes reflect stalled waves, produced from diffusion, advection, and species interactions. Emergent spatial scales are plausible given parameter estimates for periphyton. Our results apply to other systems with unidirectional flow such as prevailing winds or climate-change scenarios.     

Heteromyopia and the spatial coexistence of similar competitors

Most spatial models of competing species assume symmetries in the spatial scales of dispersal and interactions. This makes analysis tractable, and has led to the conclusion that segregation of species in space does not promote coexistence. However, these symmetries leave parts of the parameter space uninvestigated. Using a moment‐approximation method, we present a spatial version of the Lotka–Volterra competition equations to investigate effects of removing symmetries in the distances over which individuals disperse and interact. Some spatial segregation of the species always comes about due to competition, and such segregation does not necessarily lead to coexistence. But, if interspecific competition occurs over shorter distances than intraspecific competition (heteromyopia), spatial segregation becomes strong enough to promote coexistence. Such coexistence is most likely when the species have similar dynamics, in contrast to the competition–colonization trade‐off that requires large competitive differences between species.     

Species coexistence in a variable world

The contribution of deterministic and stochastic processes to species coexistence is widely debated. With the introduction of powerful statistical techniques, we can now better characterise different sources of uncertainty when quantifying niche differentiation. The theoretical literature on the effect of stochasticity on coexistence, however, is often ignored by field ecologists because of its technical nature and difficulties in its application. In this review, we examine how different sources of variability in population dynamics contribute to coexistence. Unfortunately, few general rules emerge among the different models that have been studied to date. Nonetheless, we believe that a greater understanding is possible, based on the integration of coexistence and population extinction risk theories. There are two conditions for coexistence in the presence of environmental and demographic variability: (1) the average per capita growth rates of all coexisting species must be positive when at low densities, and (2) these growth rates must be strong enough to overcome negative random events potentially pushing densities to extinction. We propose that critical tests for species coexistence must account for niche differentiation arising from this variability and should be based explicitly on notions of stability and ecological drift.     

Factors and mechanisms that explain coexistence in a Mediterranean carnivore assemblage: An integrated study based on camera trapping and diet

To promote management and conservation, it is useful to identify the factors that determine species distribution and to understand the mechanisms that regulate the organization of species assemblages or influence the dynamics of communities. Using information provided by 842 camera-trap photos and 8175 scats, we studied the factors that favour the coexistence of European badgers (Meles meles), red foxes (Vulpes vulpes) and stone martens (Martes foina) in a Mediterranean, agroforestry environment in the Iberian Peninsula. With extensive, simultaneous occupation of the space, and simultaneous activity during a broad time period (basically nocturnal and crepuscular activity patterns), the carnivores displayed different strategies depending on the availability of resources. In summer when plant resources were abundant and easy to access, there was a high overlap in patterns of diet and activity, and the temporal avoidance of the superior competitor allowed joint use of the same plots. In autumn, when there were fewer resources (although still sufficient) that were harder to access, the maintenance of food overlap was compensated for by avoidance and a reduction in overlapping activity. In winter and spring, the differentiation in response behaviour was evident in the partial substitution of plant resources.Differentiation in niche dimensions has been linked to complementarity, the differential needs and capacities of each species, and their biology. Differentiation in response behaviour was compatible with the hierarchical structure of the carnivores: European badger; red fox; stone marten. Knowledge of these factors and mechanisms increases our understanding and can help in the prediction of responses to disturbances. Consequently, it helps to improve management and conservation actions.     

Niche construction in the light of niche theory

Ecological niche construction, the process whereby an organism improves its environment to enhance its growth and persistence, is an important missing element of niche theory. Niche theory has mainly focused on niche-deteriorating processes, such as resource consumption, predation and competition, which have negative effects on population growth. Here, we integrate niche construction explicitly into modern niche theory. We use a graphical approach to analyse how a species' niche-improving impacts interplay with niche-deteriorating impacts to modify its response to the environment. In a model of two consumers that compete for one limiting resource and one predator, we show how niche construction modifies the traditional niche-deteriorating impacts of its agent or of competing species, and hence the potential for species coexistence. By altering the balance between intraspecific and interspecific competitive effects, niche construction can either generate net interspecific facilitation or strengthen interspecific competition. The adaptive benefit derived from niche construction also strongly affects the realized niche of a niche-constructing species.     

Spatial mechanisms for coexistence of species sharing a common natural enemy

We examine the conditions under which spatial structure can mediate coexistence of apparent competitors. We use a spatially explicit, host-parasitoid metapopulation model incorporating local dynamics of Nicholson-Bailey type and global dispersal. Depending on the model parameters, the resulting system displays a plethora of asynchronous dynamical behaviors for which permanent or transient coexistence is observed. We identify a number of spatially mediated tradeoffs which apparent competitors can utilize and demonstrate that the dynamics of spatial coexistence can typically be understood from consideration of two and three patch systems. The phase relationships of species abundances are different for our model than for some other mechanisms of spatial coexistence. We discuss the implications of our findings relative to issues of community organization and biological conservation.