Variability in resource consumption rates and the coexistence of competing species

Towards a heuristic understanding of the storage effect

The storage effect is a general explanation for coexistence in a variable environment. Unfortunately, the storage effect is poorly understood, in part because the generality of the storage effect precludes an interpretation that is simultaneously simple, intuitive and correct. Here, we explicate the storage effect by dividing one of its key conditions-covariance between environment and competition-into two pieces, namely that there must be a strong causal relationship between environment and competition, and that the effects of the environment do not change too quickly. This finer-grained definition can explain a number of previous results, including (1) that the storage effect promotes annual plant coexistence when the germination rate fluctuates, but not when the seed yield fluctuates, (2) that the storage effect is more likely to be induced by resource competition than the apparent competition, and (3) why the storage effect arises readily in models with either stage structure or environmental autocorrelation. Additionally, our expanded definition suggests two novel mechanisms by which the temporal storage effect can arise-transgenerational plasticity and causal chains of environmental variables-thus suggesting that the storage effect is a more common phenomenon than previously thought.

Temporal variation may have diverse impacts on range limits

Environmental fluctuations are pervasive in nature, but the influence of non-directional temporal variation on range limits has received scant attention. We synthesize insights from the literature and use simple models to make conceptual points about the potentially wide range of ecological and evolutionary effects of temporal variation on range limits. Because organisms respond nonlinearly to environmental conditions, temporal variation can directionally alter long-term growth rates, either to shrink or to expand ranges. We illustrate this diversity of outcomes with a model of competition along a mortality gradient. Temporal variation can permit transitions between alternative states, potentially facilitating range expansion. We show this for variation in dispersal, using simple source-sink population models (with strong Allee effects, or with gene flow hampering local adaptation). Temporal variation enhances extinction risk owing to demographic stochasticity, rare events, and loss of genetic variation, all tending to shrink ranges. However, specific adaptations to exploit variation (including dispersal) may permit larger ranges than in similar but constant environments. Grappling with temporal variation is essential both to understand eco-evolutionary dynamics at range limits and to guide conservation and management strategies. This article is part of the theme issue 'Species' ranges in the face of changing environments (Part II)'.

Mixed interactions among life history stages of two harvested related species

Climate change and harvesting can affect the ecosystems' functioning by altering the population dynamics and interactions among species. Knowing how species interact is essential for better understanding potentially unintended consequences of harvest on multiple species in ecosystems. I analyzed how stage‐specific interactions between two harvested competitors, the haddock (Melanogrammus aeglefinus) and Atlantic cod (Gadus morhua), living in the Barents Sea affect the outcome of changes in the harvest of the two species. Using state‐space models that account for observation errors and stochasticity in the population dynamics, I run different harvesting scenarios and track population‐level responses of both species. The increasing temperature elevated the number of larvae of haddock but did not significantly influence the older age‐classes. The nature of the interactions between both species shifted from predator‐prey to competition around age‐2 to ‐3. Increased cod fishing mortality, which led to decreasing abundance of cod, was associated with an increasing overall abundance of haddock, which suggests compensatory dynamics of both species. From a stage‐specific approach, I show that a change in the abundance in one species may propagate to other species, threatening the exploited species' recovery. Thus, this study demonstrates that considering interactions among life history stages of harvested species is essential to enhance species' co‐existence in harvested ecosystems. The approach developed in this study steps forward the analyses of effects of harvest and climate in multi‐species systems by considering the comprehension of complex ecological processes to facilitate the sustainable use of natural resources.

Seed banks alter metacommunity diversity: The interactive effects of competition, dispersal and dormancy

Dispersal and dormancy are two common strategies allowing for species persistence and the maintenance of biodiversity in variable environments. However, theory and empirical tests of spatial diversity patterns tend to examine either mechanism in isolation. Here, we developed a stochastic, spatially explicit metacommunity model incorporating seed banks with varying germination and survival rates. We found that dormancy and dispersal had interactive, nonlinear effects on the maintenance and distribution of metacommunity diversity. Seed banks promoted local diversity when seed survival was high and maintained regional diversity through interactions with dispersal. The benefits of seed banks for regional diversity were largest when dispersal was high or intermediate, depending on whether local competition was equal or stabilising. Our study shows that classic predictions for how dispersal affects metacommunity diversity can be strongly influenced by dormancy. Together, these results emphasise the need to consider both temporal and spatial processes when predicting multi-scale patterns of diversity.

Can a More Variable Species Win Interspecific Competition?

An individual-based approach is used to describe population dynamics. Two kinds of models have been constructed with different distributions illustrating individual variability. In both models, the growth rate of an individual and its final body weight at the end of the growth period, which determines the number of offspring, are functions of the amount of resources assimilated by an individual. In the model with a symmetric distribution, the half saturation constant in the Michaelis-Menten function describing the relationship between the growth of individuals and the amount of resources has a normal distribution. In the model with an asymmetric distribution, resources are not equally partitioned among individuals. The individual who acquired more resources in the past, will acquire more resources in the future. A single population comprising identical individuals has a very short extinction time. If individuals differ in the amount of food assimilated, this time significantly increases irrespectively of the type of model describing population dynamics. Individuals of two populations of competing species use common resources. For larger differences in individual variability, the more variable species will have a longer extinction time and will exclude less variable species. Both populations can also coexist when their variabilities are equal or even when they are slightly different, in the latter case under the condition of high variability of both species. These conclusions have a deterministic nature in the case of the model with the asymmetric distribution-repeated simulations give the same results. In the case of the model with the symmetric distribution, these conclusions are of a statistical nature-if we repeat the simulation many times, then the more variable species will have a longer extinction time more frequently, but some results will happen (although less often) when the less variable species has a longer extinction time. Additionally, in the model with the asymmetric distribution, the result of competition will depend on the way of the introduction of variability into the model. If the higher variability is due to an increase in the proportion of individuals with a low assimilation of resources, it can produce a longer extinction time of the less variable species.

Coexistence of “Cream Skimmer” and “Crumb Picker” Phenotypes in Nature and in Cancer

Over 40 years ago, seminal papers by Armstrong and McGehee and by Levins showed that temporal fluctuations in resource availability could permit coexistence of two species on a single resource. Such coexistence results from non-linearities or non-additivities in the way resource supply translates into fitness. These reflect trade-offs where one species benefits more than the other during good periods and suffers more (or does less well) than the other during less good periods, be the periods stochastic, unstable population dynamics, or seasonal. Since, coexistence based on fluctuating conditions has been explored under the guises of “grazers” and “diggers,” variance partitioning, relative non-linearity, “opportunists” and “gleaners,” and as the storage effect. Here we focus on two phenotypes, “cream skimmers” and “crumb pickers,” the former having the advantage in richer times and the latter in less rich times. In nature, richer and poorer times, with regular or stochastic appearances, are the norm and occur on many time scales. Fluctuations among richer and poorer times also appear to be the norm in cancer ecosystems. Within tumors, nutrient availability, oxygen, and pH can fluctuate stochastically or periodically, with swings occurring over seconds to minutes to hours. Despite interest in tumor heterogeneity and how it promotes the coexistence of different cancer cell types, the effects of fluctuating resource availability have not been explored for cancer. Here, in the context of pulsed resources, we (1) develop models of foraging consumers who experience pulsed resources to examine four types of trade-offs that can promote coexistence of phenotypes that do relatively better in richer versus in poorer times, (2) establish that conditions in tumors are conducive for this mechanism, (3) propose and empirically explore biomarkers indicative of the two phenotypes (HIF-1, GLUT-1, CA IX, CA XII), and (4) and compare cream skimmer and crumb picker biology and ecology in nature and cancer to provide cross-disciplinary insights into this interesting, and, we argue, likely very common, mechanism of coexistence.

Competitive coexistence of seasonal breeders

Many species are annual breeders who, between reproductive events, consume resources and may die. Their resource often reproduces continuously or has short, overlapping generations. An accurate model for such life cycles needs to represent both, the discrete- and the continuous-time processes in the community. The dynamics of a single discrete breeder and its resource can differ significantly from that of a fully continuous consumer-resource community (e.g., Lotka-Volterra) and that of a fully discrete one (e.g., Nicholson-Bailey). We study the dynamics of multiple discrete breeders on a single resource and identify a number of coexistence mechanisms and complex dynamics. The resource grows logistically, resource consumption is linear and consumer reproduction can be linear or nonlinear. We derive explicit conditions for the positive equilibrium state to exist and for mutual invasion to occur at that equilibrium. Stable equilibrium coexistence of more than one consumer is possible only when reproduction is nonlinear. Higher resource growth rate generally allows more consumers to stably coexist. Our explicit formulas allow us to generate communities of many coexisting consumers. Total biomass in the system seems to increase with the number of coexisting consumers. Complex patterns of coexistence arise, including bistability of equilibrium and non-equilibrium coexistence. The mixed continuous-discrete modeling approach can easily be adapted to study how certain aspects of global change affect discrete breeder communities.

A general theory of coexistence and extinction for stochastic ecological communities

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393-431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893-1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.

Polyrhythmic foraging and competitive coexistence

The current ecological understanding still does not fully explain how biodiversity is maintained. One strategy to address this issue is to contrast theoretical prediction with real competitive communities where diverse species share limited resources. I present, in this study, a new competitive coexistence theory-diversity of biological rhythms. I show that diversity in activity cycles plays a key role in coexistence of competing species, using a two predator-one prey system with diel, monthly, and annual cycles for predator foraging. Competitive exclusion always occurs without activity cycles. Activity cycles do, however, allow for coexistence. Furthermore, each activity cycle plays a different role in coexistence, and coupling of activity cycles can synergistically broaden the coexistence region. Thus, with all activity cycles, the coexistence region is maximal. The present results suggest that polyrhythmic changes in biological activity in response to the earth's rotation and revolution are key to competitive coexistence. Also, temporal niche shifts caused by environmental changes can easily eliminate competitive coexistence.

Time is honey: circadian clocks of bees and flowers and how their interactions may influence ecological communities

The interactions between flowering plants and insect pollinators shape ecological communities and provide one of the best examples of coevolution. Although these interactions have received much attention in both ecology and evolution, their temporal aspects are little explored. Here we review studies on the circadian organization of pollination-related traits in bees and flowers. Research, mostly with the honeybee, Apis mellifera, has implicated the circadian clock in key aspects of their foraging for flower rewards. These include anticipation, timing of visits to flowers at specified locations and time-compensated sun-compass orientation. Floral rhythms in traits such as petal opening, scent release and reward availability also show robust daily rhythms. However, in only few studies was it possible to adequately determine whether these oscillations are driven by external time givers such as light and temperature cycles, or endogenous circadian clocks. The interplay between the timing of flower and pollinator rhythms may be ecologically significant. Circadian regulation of pollination-related traits in only few species may influence the entire pollination network and thus affect community structure and local biodiversity. We speculate that these intricate chronobiological interactions may be vulnerable to anthropogenic effects such as the introduction of alien invasive species, pesticides or environmental pollutants.This article is part of the themed issue 'Wild clocks: integrating chronobiology and ecology to understand timekeeping in free-living animals'.

Interactions among competing nematode species affect population growth rates

Investigations of the interplay of organisms in an ecological community are a prerequisite to understanding the processes that shape the structures of those communities. Among several types of interactions, interest in the positive interactions of species that compete for the same resource has grown, as they may provide a mechanism enabling coexistence. In the laboratory experiment described herein, the effects of interspecific interaction on the population growth of two bacterial-feeding nematode species, Panagrolaimus cf. thienemanni and Poikilolaimus cf. regenfussi, were investigated. Specifically, we asked: (1) whether there is an interspecific interaction between organisms competing for a mutual resource and (2) whether these interactions are altered by the competitors' initial densities and (3) their variable growth rates (induced by different food supplies). Each treatment initially contained 48 nematode individuals, but at different species ratios (48:0; 32:16; 24:24; 16:32; 0:48). The populations were provided with three different bacterial densities (10⁸, 10⁹, and 10¹⁰ cells ml^-1) as food. The data were analyzed using a generalized linear mixed model. The best-fitting model revealed a significant decline in population growth rates with an increasing species ratio, but depending on the food density and species. These results provide strong evidence for positive interspecific interactions that vary with both species density and food-supply level. They also suggest important roles for positive interspecific interactions in habitat colonization and in maintaining the coexistence of species in the same trophic group.

The Effects of Dynamical Rates on Species Coexistence in a Variable Environment: The Paradox of the Plankton Revisited

Hutchinson's famous hypothesis for the "paradox of the plankton" has been widely accepted, but critical aspects have remained unchallenged. Hutchinson argued that environmental fluctuations would promote coexistence when the timescale for environmental change is comparable to the timescale for competitive exclusion. Using a consumer-resource model, we do find that timescales of processes are important. However, it is not the time to exclusion that must be compared with the time for environmental change but the time for resource depletion. Fast resource depletion, when resource consumption is favored for different species at different times, strongly promotes coexistence. The time for exclusion is independent of the rate of resource depletion. Therefore, the widely believed predictions of Hutchinson are misleading. Fast resource depletion, as determined by environmental conditions, ensures strong coupling of environmental processes and competition, which leads to enhancement over time of intraspecific competition relative to interspecific competition as environmental shifts favor different species at different times. This critical coupling is measured by the covariance between environment and competition. Changes in this quantity as densities change determine the stability of coexistence and provide the key to rigorous analysis, both theoretically and empirically, of coexistence in a variable environment. These ideas apply broadly to diversity maintenance in variable environments whether the issue is species diversity or genetic diversity and competition or apparent competition.

Predicting the outcome of competition when fitness inequality is variable

Traditional niche theory predicts that when species compete for one limiting resource in simple ecological settings the more fit competitor should exclude the less fit competitor. Since the advent of neutral theory ecologists have increasingly become interested both in how the magnitude of fitness inequality between competitors and stochasticity may affect this prediction. We used numerical simulations to investigate the outcome of two-species resource competition along gradients of fitness inequality (inequality in R*) and initial population size in the presence of demographic stochasticity. We found that the deterministic prediction of more fit competitors excluding less fit competitors was often unobserved when fitness inequalities were low or stochasticity was strong, and unexpected outcomes such as dominance by the less fit competitor, long-term co-persistence of both competitors or the extinction of both competitors could be common. By examining the interaction between fitness inequality and stochasticity our results mark the range of parameter space in which the predictions of niche theory break down most severely, and suggest that questions about whether competitive dynamics are driven by neutral or niche processes may be locally contingent.

Coexistence in a variable environment: eco-evolutionary perspectives

A central question in community ecology is the means by which species coexist. Models of coexistence often assume that species have fixed trait values and consider questions such as how tradeoffs and environmental variation influence coexistence and diversity. However, species traits can be dynamic, varying between populations and individuals and changing over time as species adapt and evolve, at rates that are relevant to ecological processes. Consequently, adding evolution to ecological coexistence models may modify their predictions and stability in complex or unexpected ways. We extend a well-studied coexistence mechanism depending on resource fluctuations by allowing evolution along a tradeoff between maximum growth rate and competitive ability. Interactions between favorable season length and the period of fluctuations constrain coexistence, with two species coexistence favored by intermediate season length and arising through evolutionary branching or non-local invasion. However, these results depend on the relative rates of ecological and evolutionary processes: rapid evolution leads to a complete breakdown of otherwise stable coexistence. Other coexistence mechanisms should be evaluated from an evolutionary perspective to examine how evolutionary forces may alter predicted ecological dynamics.

ZOOPLANKTON BIODIVERSITY AND LAKE TROPHIC STATE: EXPLANATIONS INVOKING RESOURCE ABUNDANCE AND DISTRIBUTION

While empirical studies linking biodiversity to local environmental gradients have emphasized the importance of lake trophic status (related to primary productivity), theoretical studies have implicated resource spatial heterogeneity and resource relative ratios as mechanisms behind these biodiversity patterns. To test the feasibility of these mechanisms in natural aquatic systems, the biodiversity of crustacean zooplankton communities along gradients of total phosphorus (TP) as well as the vertical heterogeneity and relative abundance of their phytoplankton resources were assessed in 18 lakes in Quebec, Canada. Zooplankton community richness was regressed against TP, the spatial distribution of phytoplankton spectral groups, and the relative biomass of spectral groups. Since species richness does not adequately capture ecological function and life history of different taxa, features which are important for mechanistic theories, relationships between zooplankton functional diversity (FD) and resource conditions were examined. Zooplankton species richness showed the previously established tendency to a unimodal relationship with TP, but functional diversity declined linearly over the same gradient. Changes in zooplankton functional diversity could be attributed to changes in both the spatial distribution and type of phytoplankton resource. In the studied lakes, spatial heterogeneity of phytoplankton groups declined with TP, even while biomass of all groups increased. Zooplankton functional diversity was positively related to increased heterogeneity in cyanobacteria spatial distribution. However, a smaller amount of variation in functional diversity was also positively related to the ratio of biomass in diatoms/chrysophytes to cyanobacteria. In all observed relationships, a greater variation of functional diversity than species richness measures was explained by measured factors, suggesting that functional measures of zooplankton communities will benefit ecological research attempting to identify mechanisms behind environmental gradients affecting diversity.

Spatiotemporal population distributions and their implications for species coexistence in a variable environment

A population experiences environmental variation both directly, through effects on life history parameters such as fecundity, and indirectly, through effects on the population distributions of competitors and thus on the distribution of competition. Which spatial and temporal scales of environmental variation most influence the coexistence of two species thus depends in part on the degree to which the resident population responds to different scales of variation. In this paper, I calculate an approximation for a spatiotemporal population distribution as the result of a filter function convolved with the environmental variation. I find that there is no straightforward connection between spatial or temporal scales inherent to an organism's life history, such as mean lifetime or dispersal distance, and the population's sensitivity to variation at different scales. Rather, life history traits interact sensitively with the way environmental variation affects the organism. I comment on the implications for variation-mediated coexistence.

On the impossibility of coexistence of infinitely many strategies

We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the "interaction operator" which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of "limiting similarity" of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.

Evolution restricts the coexistence of specialists and generalists: the role of trade-off structure

Environmental variability and adaptive foraging behavior have been shown to favor coexistence of specialists and generalists on an ecological timescale. This leaves unaddressed the question of whether such coexistence can also be expected on an evolutionary timescale. In this article, we study the attainability, through gradual evolution, of specialist-generalist coexistence, as well as the evolutionary stability of such communities when allowing for immigration. Our analysis shows that the potential for specialist-generalist coexistence is much more restricted than originally thought and strongly depends on the trade-off structure assumed. We establish that ecological coexistence is less likely for species facing a trade-off between per capita reproduction in different habitats than when the trade-off acts on carrying capacities alone. We also demonstrate that coexistence is evolutionarily stable whenever it is ecologically stable but that in most cases, such coexistence cannot be reached through gradual evolution. We conclude that an evolutionarily stable community of specialists and generalists may be created only through immigration from elsewhere or through mutations of large effect. Our results highlight that trade-offs in fitness-determining traits can have counterintuitive effects on the evolution of specialization.

Food web dynamics in correlated and autocorrelated environments

The densities of populations in a community or food web vary as a consequence of both population interactions and environmental (e.g. weather) fluctuations. Populations often respond to the same kinds of environmental fluctuations, and therefore experience correlated environments. Furthermore, some environmental factors change slowly over time, thereby producing positive environmental autocorrelation. We show that the effects of environmental correlation and autocorrelation on the dynamics of the populations in a food web can be large and unintuitive, but can be understood by analyzing the eigenvectors of the community (system) matrix of interactions among populations. For example, environmental correlation and autocorrelation may either obscure or enhance the cyclic dynamics that generally characterize predator-prey interactions even when there is no direct effect of the environment on how species interact. Thus, understanding the population dynamics of species in a food web requires explicit attention to the correlation structure of environmental factors affecting all species.

The impact of consumer-resource cycles on the coexistence of competing consumers

This article seeks to determine the extent to which endogenous consumer-resource cycles can contribute to the coexistence of competing consumer species. It begins with a numerical analysis of a simple model proposed by Armstrong and McGehee. This model has a single resource and two consumers, one with a linear functional response and one with a saturating response. Coexistence of the two consumer species can occur when the species with a saturating response generates population cycles of the resource, and also has a lower resource requirement for zero population growth. Coexistence can be achieved over a wide range of relative efficiencies of the two consumers provided that the functional response of the saturating consumer reaches its half-saturation value when the resource population is a small fraction of its carrying capacity. In this case, the range of efficiencies allowing coexistence is comparable to that when two competitors have stable dynamics and a high degree of resource partitioning. A variety of modifications of this basic model are analyzed to investigate the consequences for coexistence of different resource growth equations, different functional and numerical response shapes, and other factors. Large differences in functional response shape appear to be the most important factor in producing robust coexistence via resource cycles. If the unstable species has a concave numerical response, this greatly expands the conditions allowing coexistence. If the stable consumer species has a convex (accelerating) functional and/or numerical response, the range of conditions allowing coexistence is also expanded. We argue that large between-species differences in functional response form can often be produced by between-consumer differences in the adaptive adjustments of foraging effort to food density. Consumer-resource cycles can also expand the conditions allowing coexistence when there is resource partitioning, but do so primarily when resource partitioning is relatively slight; this makes the ease of coexistence relatively independent of consumer similarity.

General theory of competitive coexistence in spatially-varying environments

A general model of competitive and apparent competitive interactions in a spatially-variable environment is developed and analyzed to extend findings on coexistence in a temporally-variable environment to the spatial case and to elucidate new principles. In particular, coexistence mechanisms are divided into variation-dependent and variation-independent mechanisms with variation-dependent mechanisms including spatial generalizations of relative nonlinearity and the storage effect. Although directly analogous to the corresponding temporal mechanisms, these spatial mechanisms involve different life history traits which suggest that the spatial storage effect should arise more commonly than the temporal storage effect and spatial relative nonlinearity should arise less commonly than temporal relative nonlinearity. Additional mechanisms occur in the spatial case due to spatial covariance between the finite rate of increase of a local population and its local abundance, which has no clear temporal analogue. A limited analysis of these additional mechanisms shows that they have similar properties to the storage effect and relative nonlinearity and potentially may be considered as enlargements of the earlier mechanisms. The rate of increase of a species perturbed to low density is used to quantify coexistence. A general quadratic approximation, which is exact in some important cases, divides this rate of increase into contributions from the various mechanisms above and admits no other mechanisms, suggesting that opportunities for coexistence in a spatially-variable environment are fully characterized by these mechanisms within this general model. Three spatially-implicit models are analyzed as illustrations of the general findings and of techniques using small variance approximations. The contributions to coexistence of the various mechanisms are expressed in terms of simple interpretable formulae. These spatially-implicit models include a model of an annual plant community, a spatial multispecies version of the lottery model, and a multispecies model of an insect community competing for spatially-patchy and ephemeral food.

Fluctuating Environments and Phytoplankton Community Structure: A Stochastic Model

Spatial heterogeneity in organism and resource distributions can generate temporal heterogeneity in resource access for simple organisms like phytoplankton. The role of temporal heterogeneity as a structuring force for simple communities is investigated via models of phytoplankton with contrasting life histories competing for a single fluctuating resource. A stochastic model in which environmental and demographic stochasticity are treated separately is compared with a model with deterministic resource variation to assess the importance of stochasticity. When compared with the deterministic model, the stochastic model allows for coexistence over a wider range of parameter values (or life-history types). The model suggests that demographic stochasticity alone is far more important in increasing the possibility of coexistence than environmental stochasticity alone. However, the combined effects of both types of stochasticity produce the largest likelihood of coexistence. Finally, the influence of relative nutrient levels and nutrient pulse frequency on these results is addressed. We relate our findings to variable environment theory with evidence for both relative nonlinearity and the storage effect acting in this model. We show for the first time that temporal dynamics generated by demographic stochasticity may operate like the storage effect at particular spatial scales.

Grazers and Diggers: Exploitation Competition and Coexistence among Foragers with Different Feeding Strategies on a Single Resource

A mathematical model is presented that describes a system where two consumer species compete exploitatively for a single renewable resource. The resource is distributed in a patchy but homogeneous environment; that is, all patches are intrinsically identical. The two consumer species are referred to as diggers and grazers, where diggers deplete the resource within a patch to lower densities than grazers. We show that the two distinct feeding strategies can produce a heterogeneous resource distribution that enables their coexistence. Coexistence requires that grazers must either move faster than diggers between patches or convert the resources to population growth much more efficiently than diggers. The model shows that the functional form of resource renewal within a patch is also important for coexistence. These results contrast with theory that considers exploitation competition for a single resource when the resource is assumed to be well mixed throughout the system.

Geometry, heterogeneity and competition in variable environments

The effects of environmental fluctuations on coexistence of competing species can be understood by a new geometric analysis. This analysis shows how a species at low density gains an average growth rate advantage when the environm ent fluctuates and all species have growth rates of the particular geometric form called subadditive. This low density advantage opposes competitive exclusion. Additive growth rates confer no such low density advantage, while superadditive growth rates promote competitive exclusion. Growth-rate geometry can be understood in terms of heterogeneity within populations. Total population growth is divided into different components, such as may be contributed by different lifehistory stages, phenotypes, or subpopulations in different microhabitats. The relevant aspects of such within-population heterogeneity can be displayed as a scatter plot of sensitivities of different components of population growth to environm ental and competitive factors, and can be measured quantitatively as a covariance. A three-factor model aids the conceptual division of population growth into suitable components