Successional state dynamics: a novel approach to modeling nonequilibrium foodweb dynamics

Periodic temporal environmental variations induce coexistence in resource competition models

Natural ecosystems, in particular on the microbial scale, are inhabited by a large number of species. The population size of each species is affected by interactions of individuals with each other and by spatial and temporal changes in environmental conditions, such as resource abundance. Here, we use a generic population dynamics model to study how, and under what conditions, a periodic temporal environmental variation can alter an ecosystem's composition and biodiversity. We demonstrate that using timescale separation allows one to qualitatively predict the long-term population dynamics of interacting species in varying environments. We show that the notion of Tilman's R* rule, a well-known principle that applies for constant environments, can be extended to periodically varying environments if the timescale of environmental changes (e.g., seasonal variations) is much faster than the timescale of population growth (doubling time in bacteria). When these timescales are similar, our analysis shows that a varying environment deters the system from reaching a steady state, and stable coexistence between multiple species becomes possible. Our results posit that biodiversity can in part be attributed to natural environmental variations.

Global stability of a tridiagonal competition model with seasonal succession

In this paper, we investigate a tridiagonal three-species competition model with seasonal succession. The Floquet multipliers of all nonnegative periodic solutions of such a time-periodic system are estimated via the stability analysis of equilibria. Together with the Brouwer degree theory, sufficient conditions for existence and uniqueness of the positive periodic solution are given. We further obtain the global dynamics of coexistence and extinction for three competing species in this periodically forced environment. Finally, some numerical examples are presented to illustrate the effectiveness of our theoretical results.

A West Nile virus nonlocal model with free boundaries and seasonal succession

The paper deals with a West Nile virus (WNv) model, in which the nonlocal diffusion characterizes the long-range movement of birds and mosquitoes, the free boundaries describe their spreading fronts, and the seasonal succession accounts for the effect of the warm and cold seasons. The well-posedness of the mathematical model is established, and its long-term dynamical behaviours, which depend upon the generalized eigenvalues of the corresponding linearized differential operator, are investigated. For both spatially independent and nonlocal WNv models with seasonal successions, the generalized eigenvalues are studied and applied to determine whether the spreading or vanishing occurs. Our results extend those for the case with nonlocal diffusion but no free boundary and those for the case with free boundary but local diffusion, respectively. The generalized eigenvalues reveal that there exists positive correlation between the duration of the warm season and the risk of infection. Moreover, the initial infection length, the initial infection scale and the spreading ability to new areas all play important roles for the long time behaviors of the time dependent solutions.

Extending the gleaner-opportunist trade-off

Species exhibit various trade-offs that can result in stable coexistence of competitors. The gleaner-opportunist trade-off to fluctuations in resource abundance is one of the most intuitive, yet also misunderstood, coexistence-promoting trade-offs. Here, we review its history as an ecological concept, discuss extensions to the classical theory and outline opportunities to advance its understanding. The mechanism of coexistence between species that grow relatively faster than their competitors in a low-resource environment (i.e. a gleaner) versus a high-resource environment (i.e. an opportunist) was first proposed in the 1970s. Stable coexistence could emerge between gleaners and opportunists if the opportunist species (dominant in unstable environments) dampens resource fluctuations via relatively convex functional responses, while the gleaner species (dominant in stable environments) promotes fluctuations, or diminishes them less than the opportunist does, via relatively saturating functional responses. This fluctuation-dependent coexistence mechanism has since been referred to by various names, including the Armstrong-McGehee mechanism and relative nonlinearity of competition. Several researchers have argued this mechanism likely plays a relatively minor role in species coexistence owing in part to the restricted range of conditions that allow it to operate. More recent theoretical research, however, suggests that relative nonlinearity can operate over wider conditions than previously thought. Here, we identify several novel, or little explored, extensions to the gleaner-opportunist trade-off that can yield species coexistence under phenomena as diverse as fluctuations in predation/pathogen pressure, multiple resources, phenotypic plasticity and rapid evolution, amongst other phenomena. While the original definition of the gleaner-opportunist trade-off may be imperfect as a collective for these extensions, we argue that a subtle reframing of the trade-off focusing on species' performance in equilibrium versus fluctuating conditions (irrespective of preferences for high or low resources, predation pressure or other competitive factors) reveals their fundamental commonality in stable coexistence via relative nonlinearity. An extended framing shines a light on the potential ubiquity of this canonical trade-off in nature and on the breadth of theoretical and empirical terrain that remains to be trodden.

Climate Change-Driven Regime Shifts in a Planktonic Food Web

AbstractPredicting how food webs will respond to global environmental change is difficult because of the complex interplay between the abiotic forcing and biotic interactions. Mechanistic models of species interactions in seasonal environments can help understand the effects of global change in different ecosystems. Seasonally ice-covered lakes are warming faster than many other ecosystems and undergoing pronounced food web changes, making the need to forecast those changes especially urgent. Using a seasonally forced food web model with a generalist zooplankton grazer and competing cold-adapted winter and warm-adapted summer phytoplankton, we show that with declining ice cover, the food web moves through different dynamic regimes, from annual to biennial cycles, with decreasing and then disappearing winter phytoplankton blooms and a shift of maximum biomass to summer season. Interestingly, when predator-prey interactions were not included, a declining ice cover did not cause regime shifts, suggesting that both are needed for regime transitions. A cluster analysis of long-term data from Lake Baikal, Siberia, supports the model results, revealing a change from regularly occurring winter blooms of endemic diatoms to less frequent winter bloom years with decreasing ice cover. Together, the results show that even gradual environmental change, such as declining ice cover duration, may cause discontinuous or abrupt transitions between dynamic regimes in food webs.

Seasonal food webs with migrations: multi-season models reveal indirect species interactions in the Canadian Arctic tundra

Models incorporating seasonality are necessary to fully assess the impact of global warming on Arctic communities. Seasonal migrations are a key component of Arctic food webs that still elude current theories predicting a single community equilibrium. We develop a multi-season model of predator-prey dynamics using a hybrid dynamical systems framework applied to a simplified tundra food web (lemming-fox-goose-owl). Hybrid systems models can accommodate multiple equilibria, which is a basic requirement for modelling food webs whose topology changes with season. We demonstrate that our model can generate multi-annual cycling in lemming dynamics, solely from a combined effect of seasonality and state-dependent behaviour. We compare our multi-season model to a static model of the predator-prey community dynamics and study the interactions between species. Interestingly, including seasonality reveals indirect interactions between migrants and residents not captured by the static model. Further, we find that the direction and magnitude of interactions between two species are not necessarily accurate using only summer time-series. Our study demonstrates the need for the development of multi-season models and provides the tools to analyse them. Integrating seasonality in food web modelling is a vital step to improve predictions about the impacts of climate change on ecosystem functioning. This article is part of the theme issue 'The changing Arctic Ocean: consequences for biological communities, biogeochemical processes and ecosystem functioning'.

The competitive exclusion principle in stochastic environments

In its simplest form, the competitive exclusion principle states that a number of species competing for a smaller number of resources cannot coexist. However, it has been observed empirically that in some settings it is possible to have coexistence. One example is Hutchinson's 'paradox of the plankton'. This is an instance where a large number of phytoplankton species coexist while competing for a very limited number of resources. Both experimental and theoretical studies have shown that temporal fluctuations of the environment can facilitate coexistence for competing species. Hutchinson conjectured that one can get coexistence because nonequilibrium conditions would make it possible for different species to be favored by the environment at different times. In this paper we show in various settings how a variable (stochastic) environment enables a set of competing species limited by a smaller number of resources or other density dependent factors to coexist. If the environmental fluctuations are modeled by white noise, and the per-capita growth rates of the competitors depend linearly on the resources, we prove that there is competitive exclusion. However, if either the dependence between the growth rates and the resources is not linear or the white noise term is nonlinear we show that coexistence on fewer resources than species is possible. Even more surprisingly, if the temporal environmental variation comes from switching the environment at random times between a finite number of possible states, it is possible for all species to coexist even if the growth rates depend linearly on the resources. We show in an example (a variant of which first appeared in Benaim and Lobry '16) that, contrary to Hutchinson's explanation, one can switch between two environments in which the same species is favored and still get coexistence.

Contributions to nonstationary community theory

The study of the role of environmental variation in community dynamics has traditionally assumed that the environment is a stationary stochastic process or a periodic deterministic process. However, the physical environment in nature is nonstationary. Moreover, anthropogenically driven climate change provides a new challenge emphasizing a persistent but frequently ignored problem: how to make predictions about the dynamics of communities when the nonstationarity of the physical environment is recognized. Recent work is providing a path to conclusions with none of the traditional assumptions of environmental stationarity or periodicity. Traditional assumptions about convergence of long-term averages of functions of environmental states can be replaced by assumptions about temporal sums, allowing convergence and persistence of population processes to be demonstrated in general nonstationary environments. These tools are further developed and illustrated here with some simple models of nonstationary community dynamics, including the Beverton-Holt model, the threshold exponential and the lottery model.

Synchronization and entrainment of metapopulations: A trade-off among time-induced heterogeneity, dispersal, and seasonal force

Demographic and environmental heterogeneities are prevalent across many natural systems. Earlier studies on metapopulation models have mostly considered heterogeneities either in the demographic parameters or in the interaction strength and topology between the spatially separated patches. In contrast, here we study the dynamics of a metapopulation model where each of the uncoupled patches has different periods of oscillations (period mismatch). We show different synchronization dynamics governed by both period mismatch and dispersal in neighboring patches. Indeed, we find both appearance and disappearance of phase synchronization, quasiperiodic oscillations, and period doubling of limit cycle. We also quantify the effect of seasonal variation (entrainment) and dispersal on species synchrony using phase-response curve and a synchrony measure, which thereof identify the influence of stochasticity on species persistence through trade-off mechanisms. Our results show that trade-offs among period mismatch, dispersal, and external force can drive entrained oscillations as well as asynchronous population dynamics that structure ecological communities.

How spatial structure and neighbor uncertainty promote mutualists and weaken black queen effects

The ubiquity of cooperative cross-feeding (a resource-exchange mutualism) raises two related questions: Why is cross-feeding favored over self-sufficiency, and how are cross-feeders protected from non-producing cheaters? The Black Queen Hypothesis suggests that if leaky resources are costly, then there should be selection for either gene loss or self-sufficiency, but selection against mutualistic inter-dependency. Localized interactions have been shown to protect mutualists against cheaters, though their effects in the presence of self-sufficient organisms are not well understood. Here we develop a stochastic spatial model to examine how spatial effects alter the predictions of the Black Queen Hypothesis. Microbes need two essential resources to reproduce, which they can produce themselves (at a cost) or take up from neighbors. Additionally, microbes need empty sites to give birth into. Under well mixed mean-field conditions, the cross-feeders will always be displaced by a non-producer and a self-sufficient microbe. However, localized interactions have two effects that favor production. First, a microbe that interacts with a small number of neighbors will not always receive the essential resources it needs; this effect slightly harms cross-feeders but greatly harms non-producers. Second, microbes tend to displace other microbes that produce resources they need; this effect also slightly harms cross-feeders but greatly harms non-producers. Our work therefore suggests localized interactions produce an accelerating cost of non-production. Thus, the right trade-off between the cost of producing resources and the cost of sometimes being resource-limited can favor mutualistic inter-dependence over both self-sufficiency and non-production.

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.

Functional traits explain phytoplankton responses to environmental gradients across lakes of the United States

Ecological communities exhibit regular shifts in structure along environmental gradients, but it has proved difficult to dissect the mechanisms by which environmental conditions determine the relative success of species. Functional traits may provide a link between environmental drivers and mechanisms of community membership, but this has not been well tested for phytoplankton, which dominate primary production in many aquatic ecosystems. Here we test whether functional traits of phytoplankton can explain how species respond to gradients of light and phosphorus across U.S. lakes. We find that traits related to light utilization and maximum growth rate can predict species' differential responses to the relative availability of these resources. These results show that laboratory-measured traits are predictive of species' performance under natural conditions, that functional traits provide a mechanistic foundation for community ecology, and that variation in community structure is predictable in spite of the complexity of ecological communities.

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.

Successional dynamics in the seasonally forced diamond food web

Plankton seasonal succession is a classic example of nonequilibrium community dynamics. Despite the fact that it has been well studied empirically, it lacks a general quantitative theory. Here we investigate a food web model that includes a resource, two phytoplankton, and a shared grazer-the diamond food web-in a seasonal environment. The model produces a number of successional trajectories that have been widely discussed in the context of the verbal Plankton Ecology Group model of succession, such as a spring bloom of a good competitor followed by a grazer-induced clear-water phase, setting the stage for the late-season dominance of a grazer-resistant species. It also predicts a novel, counterintuitive trajectory where the grazer-resistant species has both early- and late-season blooms. The model often generates regular annual cycles but sometimes produces multiyear cycles or chaos, even with identical forcing each year. Parameterizing the model, we show how the successional trajectory depends on nutrient supply and the length of the growing season, two key parameters that vary among water bodies. This model extends nonequilibrium theory to food webs and is a first step toward a quantitative theory of plankton seasonal succession.

Large-scale biodiversity patterns in freshwater phytoplankton

Our planet shows striking gradients in the species richness of plants and animals, from high biodiversity in the tropics to low biodiversity in polar and high-mountain regions. Recently, similar patterns have been described for some groups of microorganisms, but the large-scale biogeographical distribution of freshwater phytoplankton diversity is still largely unknown. We examined the species diversity of freshwater phytoplankton sampled from 540 lakes and reservoirs distributed across the continental United States and found strong latitudinal, longitudinal, and altitudinal gradients in phytoplankton biodiversity, demonstrating that microorganisms can show substantial geographic variation in biodiversity. Detailed analysis using structural equation models indicated that these large-scale biodiversity gradients in freshwater phytoplankton diversity were mainly driven by local environmental factors, although there were residual direct effects of latitude, longitude, and altitude as well. Specifically, we found that phytoplankton species richness was an increasing saturating function of lake chlorophyll a concentration, increased with lake surface area and possibly increased with water temperature, resembling effects of productivity, habitat area, and temperature on diversity patterns commonly observed for macroorganisms. In turn, these local environmental factors varied along latitudinal, longitudinal, and altitudinal gradients. These results imply that changes in land use or climate that affect these local environmental factors are likely to have major impacts on large-scale biodiversity patterns of freshwater phytoplankton.

A Lotka-Volterra competition model with seasonal succession

A complete classification for the global dynamics of a Lotka-Volterra two species competition model with seasonal succession is obtained via the stability analysis of equilibria and the theory of monotone dynamical systems. The effects of two death rates in the bad season and the proportion of the good season on the competition outcomes are also discussed.