The duality of stability: towards a stochastic theory of species interactions

Stability of consumer-resource interactions in periodic environments

Periodic fluctuations in abiotic conditions are ubiquitous across a range of temporal scales and regulate the structure and function of ecosystems through dynamic biotic responses that are adapted to these external forces. Research has suggested that certain environmental signatures may play a crucial role in the maintenance of biodiversity and the stability of food webs, while others argue that coupled oscillators ought to promote chaos. As such, numerous uncertainties remain regarding the intersection of temporal environmental patterns and biological responses, and we lack a general understanding of the implications for food web stability. Alarmingly, global change is altering the nature of both environmental rhythms and biological rates. Here, we develop a general theory for how continuous periodic variation in productivity, across temporal scales, influences the stability of consumer-resource interactions: a fundamental building block of food webs. Our results suggest that consumer-resource dynamics under environmental forcing are highly complex and depend on asymmetries in both the speed of forcing relative to underlying dynamics and in local stability properties. These asymmetries allow for environmentally driven stabilization under fast forcing, relative to underlying dynamics, as well as extremely complex and unstable dynamics at slower periodicities. Our results also suggest that changes in naturally occurring periodicities from climate change may lead to precipitous shifts in dynamics and stability.

No sensitivity to functional forms in the Rosenzweig-MacArthur model with strong environmental stochasticity

The classic Rosenzweig-MacArthur predator-prey model has been shown to exhibit, like other coupled nonlinear ordinary differential equations (ODEs) from ecology, worrying sensitivity to model structure. This sensitivity manifests as markedly different community dynamics arising from saturating functional responses with nearly identical shapes but different mathematical expressions. Using a stochastic differential equation (SDE) version of the Rosenzweig-MacArthur model with the three functional responses considered by Fussmann & Blasius (2005), I show that such sensitivity seems to be solely a property of ODEs or stochastic systems with weak noise. SDEs with strong environmental noise have by contrast very similar fluctuations patterns, irrespective of the mathematical formula used. Although eigenvalues of linearized predator-prey models have been used as an argument for structural sensitivity, they can also be an argument against structural sensitivity. While the sign of the eigenvalues' real part is sensitive to model structure, its magnitude and the presence of imaginary parts are not, which suggests noise-driven oscillations for a broad range of carrying capacities. I then discuss multiple other ways to evaluate structural sensitivity in a stochastic setting, for predator-prey or other ecological systems.

Managing aquatic habitat structure for resilient trophic interactions

Structural habitat (the three-dimensional arrangement of physical matter, abiotic and biotic, at a location) is a foundational element for the resilience and maintenance of biodiversity, yet anthropogenic development is driving the global simplification of aquatic environments. Resource managers regularly seek to conserve aquatic food webs by increasing structural habitat complexity with expected benefits to fisheries; however, the global effectiveness of such actions is unclear. Our synthesis and theoretical analyses found that the response of a consumer-resource interaction (predatory sportfish and forage fish prey) to the addition of prey refuge habitat differed among systems with low and high rates of biomass transfer from resource to consumer (i.e., biomass potential); stabilization was not the rule. Greater prey refuge habitat availability tended to stabilize systems characterized by high biomass potential while simultaneously increasing consumer densities. In contrast, increasing prey refuge habitat availability in systems characterized by low biomass potential tended to mute energy transfer and moved consumer densities toward local extinction. Importantly, biomass potential and prey refuge can have antagonistic effects on stability and relative consumer densities, and it is therefore important to consider the local conditions of a system when using habitat manipulation as a management measure. Further development of our context-dependent perspective to whole food webs, and across different environments, may help to guide structural habitat management to better restore and protect aquatic ecosystems.

Life-history speed, population disappearances and noise-induced ratchet effects

Nature is replete with variation in the body sizes, reproductive output and generation times of species that produce life-history responses known to vary from small and fast to large and slow. Although researchers recognize that life-history speed likely dictates fundamental processes in consumer-resource interactions like productivity and stability, theoretical work remains incomplete in this critical area. Here, we examine the role of life-history speed on consumer-resource interactions by using a well-used mathematical approach that manipulates the speed of the consumer's growth rate in a consumer-resource interaction. Importantly, this approach holds the isocline geometry intact, allowing us to assess the impacts of altered life-history speed on stability (coefficient of variation, CV) without changing the underlying qualitative dynamics. Although slowing life history can be initially stabilizing, we find that in stochastic settings slowing ultimately drives highly destabilizing population disappearances, especially under reddened noise. Our results suggest that human-driven reddening of noise may decrease species stability because the autocorrelation of red noise enlarges the period and magnitude of perturbations, overwhelming a species' natural compensatory responses via a ratchet-like effect. This ratchet-like effect then pushes species' population dynamics far away from equilibria, which can lead to precipitous local extinction.

Cascading effects of predators on algal size structure

The presence of edible and inedible prey species in a food web can influence the strength that nutrients (bottom-up) or herbivores (top-down) have on primary production. In boreal peatlands, wetter more nutrient-rich conditions associated with ongoing climate change are expanding consumer access to aquatic habitat and promoting sources of primary production (i.e., algae) that are susceptible to trophic regulation. Here, we used an in situ mesocosm experiment to evaluate the consequences of enhanced nutrient availability and food-web manipulation (herbivore and predator exclusion) on algal assemblage structure in an Alaskan fen. Owing to the potential for herbivores to selectively consume edible algae (small cells) in favor of more resistant forms, we predicted that the proportion of less-edible algae (large cells) would determine the strength of top-down or bottom-up effects. Consistent with these expectations, we observed an increase in algal-cell size in the presence of herbivores (2-tiered food web) that was absent in the presence of a trophic cascade (3-tiered food web), suggesting that predators indirectly prevented morphological changes in the algal assemblage by limiting herbivory. Increases in algal-cell size with herbivory were driven by a greater proportion of filamentous green algae and nitrogen-fixing cyanobacteria, whose size and morphological characteristics mechanically minimize consumption. While consumer-driven shifts in algal assemblage structure were significant, they did not prevent top-down regulation of biofilm development by herbivores. Our findings show that increasing wet periods in northern peatlands will provide new avenues for trophic regulation of algal production, including directly through consumption and indirectly via a trophic cascade.

Coupled phase-amplitude dynamics in heterogeneous metacommunities

Spatial synchrony of population fluctuations is an important tool for predicting regional stability. Its application to natural systems is still limited by the complexity of ecological time series displaying great variation in the frequency and amplitude of their fluctuations, which are not fully resolved by current ecological theories of spatial synchrony. In particular, while environmental fluctuations and limited dispersal can each control the dynamics of frequency and amplitude of population fluctuations, ecological theories of spatial synchrony still need to resolve their role on synchrony and stability in heterogeneous metacommunities. Here, we adopt a heterogeneous predator-prey metacommunity model and study the response of dispersal-driven phase locking and frequency modulation to among-patch heterogeneity in carrying capacity. We find that frequency modulation occurs at intermediate values of dispersal and habitat heterogeneity. We also show how frequency modulation can emerge in metacommunities of autonomously oscillating populations as well as through the forcing of local communities at equilibrium. Frequency modulation was further found to produce temporal variation in population amplitudes, promoting local and regional stability through cyclic patterns of local and regional variability. Our results highlight the importance of approaching spatial synchrony as a non-stationary phenomenon, with implications for the assessment and interpretation of spatial synchrony observed in experimental and natural systems.

Ecological Character Displacement Destabilizes Food Webs

AbstractEcological character displacement is an adaptive process that generally increases phenotypic diversity. Despite the fact that this diversification is due to an eco-evolutionary feedback between consumers competing for shared resources, its consequences for food-web dynamics have received little attention. Here, I study a model of two consumers competing for two shared resources to examine how character displacement in consumer attack rates affects resource abundances and the resilience of food webs to perturbations. I found that character displacement always strengthened consumer-resource interactions whenever consumers competed for resources that occurred in different habitats. This increase in interaction strength resulted in lower resource abundances and less resilient food webs. This occurred under different evolutionary trade-offs and in both simple and more realistic foraging scenarios. Taken together, my results show that the adaptive process of character displacement may come with the ecological cost of decreasing food-web resilience.

Landscape modification and nutrient-driven instability at a distance

Almost 50 years ago, Michael Rosenzweig pointed out that nutrient addition can destabilise food webs, leading to loss of species and reduced ecosystem function through the paradox of enrichment. Around the same time, David Tilman demonstrated that increased nutrient loading would also be expected to cause competitive exclusion leading to deleterious changes in food web diversity. While both concepts have greatly illuminated general diversity-stability theory, we currently lack a coherent framework to predict how nutrients influence food web stability across a landscape. This is a vitally important gap in our understanding, given mounting evidence of serious ecological disruption arising from anthropogenic displacement of resources and organisms. Here, we combine contemporary theory on food webs and meta-ecosystems to show that nutrient additions are indeed expected to drive loss in stability and function in human-impacted regions. Our models suggest that destabilisation is more likely to be caused by the complete loss of an equilibrium due to edible plant species being competitively excluded. In highly modified landscapes, spatial nutrient transport theory suggests that such instabilities can be amplified over vast distances from the sites of nutrient addition. Consistent with this theoretical synthesis, the empirical frequency of these distant propagating ecosystem imbalances appears to be growing. This synthesis of theory and empirical data suggests that human modification of the Earth is strongly connecting distantly separated ecosystems, causing rapid, expansive and costly nutrient-driven instabilities over vast areas of the planet. Similar to existing food web theory, the corollary to this spatial nutrient theory is that slowing down spatial nutrient pathways can be a potent means of stabilising degraded ecosystems.

Stochastic epidemic metapopulation models on networks: SIS dynamics and control strategies

While deterministic metapopulation models for the spread of epidemics between populations have been well-studied in the literature, variability in disease transmission rates and interaction rates between individual agents or populations suggests the need to consider stochastic fluctuations in model parameters in order to more fully represent realistic epidemics. In the present paper, we have extended a stochastic SIS epidemic model - which introduces stochastic perturbations in the form of white noise to the force of infection (the rate of disease transmission from classes of infected to susceptible populations) - to spatial networks, thereby obtaining a stochastic epidemic metapopulation model. We solved the stochastic model numerically and found that white noise terms do not drastically change the overall long-term dynamics of the system (for sufficiently small variance of the noise) relative to the dynamics of a corresponding deterministic system. The primary difference between the stochastic and deterministic metapopulation models is that for large time, solutions tend to quasi-stationary distributions in the stochastic setting, rather than to constant steady states in the deterministic setting. We then considered different approaches to controlling the spread of a stochastic SIS epidemic over spatial networks, comparing results for a spectrum of controls utilizing local to global information about the state of the epidemic. Variation in white noise was shown to be able to counteract the treatment rate (treated curing rate) of the epidemic, requiring greater treatment rates on the part of the control and suggesting that in real-life epidemics one should be mindful of such random variations in order for a treatment to be effective. Additionally, we point out some problems using white noise perturbations as a model, but show that a truncated noise process gives qualitatively comparable behaviors without these issues.

On the prevalence and dynamics of inverted trophic pyramids and otherwise top-heavy communities

Classically, biomass partitioning across trophic levels was thought to add up to a pyramidal distribution. Numerous exceptions have, however, been noted including complete pyramidal inversions. Elevated levels of biomass top-heaviness (i.e. high consumer/resource biomass ratios) have been reported from Arctic tundra communities to Brazilian phytotelmata, and in species assemblages as diverse as those dominated by sharks and ants. We highlight two major pathways for creating top-heaviness, via: (1) endogenous channels that enhance energy transfer across trophic boundaries within a community and (2) exogenous pathways that transfer energy into communities from across spatial and temporal boundaries. Consumer-resource models and allometric trophic network models combined with niche models reveal the nature of core mechanisms for promoting top-heaviness. Outputs from these models suggest that top-heavy communities can be stable, but they also reveal sources of instability. Humans are both increasing and decreasing top-heaviness in nature with ecological consequences. Current and future research on the drivers of top-heaviness can help elucidate fundamental mechanisms that shape the architecture of ecological communities and govern energy flux within and between communities. Questions emerging from the study of top-heaviness also usefully draw attention to the incompleteness and inconsistency by which ecologists often establish definitional boundaries for communities.