When can higher-order interactions produce stable coexistence?

Most ecological models are based on the assumption that species interact in pairs. Diverse communities, however, can have higher-order interactions, in which two or more species jointly impact the growth of a third species. A pitfall of the common pairwise approach is that it misses the higher-order interactions potentially responsible for maintaining natural diversity. Here, we explore the stability properties of systems where higher-order interactions guarantee that a specified set of abundances is a feasible equilibrium of the dynamics. Even these higher-order interactions which lead to equilibria do not necessarily produce stable coexistence. Instead, these systems are more likely to be stable when the pairwise interactions are weak or facilitative. Correlations between the pairwise and higher-order interactions, however, do permit robust coexistence even in diverse systems. Our work not only reveals the challenges in generating stable coexistence through higher-order interactions but also uncovers interaction patterns that can enable diversity.

Landscape fragmentation overturns classical metapopulation thinking.

Habitat loss and isolation caused by landscape fragmentation represent a growing threat to global biodiversity. Existing theory suggests that the process will lead to a decline in metapopulation viability. However, since most metapopulation models are restricted to simple networks of discrete habitat patches, the effects of real landscape fragmentation, particularly in stochastic environments, are not well understood. To close this major gap in ecological theory, we developed a spatially explicit, individual-based model applicable to realistic landscape structures, bridging metapopulation ecology and landscape ecology. This model reproduced classical metapopulation dynamics under conventional model assumptions, but on fragmented landscapes, it uncovered general dynamics that are in stark contradiction to the prevailing views in the ecological and conservation literature. Notably, fragmentation can give rise to a series of dualities: a) positive and negative responses to environmental noise, b) relative slowdown and acceleration in density decline, and c) synchronization and desynchronization of local population dynamics. Furthermore, counter to common intuition, species that interact locally ("residents") were often more resilient to fragmentation than long-ranging "migrants." This set of findings signals a need to fundamentally reconsider our approach to ecosystem management in a noisy and fragmented world.

Effect of migrations on synchrony in host-parasitoid system.

Insect outbreaks can cause large scale defoliation of forest trees or destruction of crops, leading to ecosystem degradation and economic losses. Some outbreaks occur simultaneously across large geographic scales and some outbreaks occur periodically every few years across space. Parasitoids are a natural enemy of these defoliators and could help mitigate these pest outbreaks. A holistic understanding of the host-parasitoid interactions in a spatial context would thus enhance our ability to understand, predict and prevent these outbreaks. We use a discrete time deterministic model of the host parasitoid system with populations migrating between 2 patches to elucidate features of spatial host outbreaks. We show that whenever populations persist indefinitely, host outbreaks in both patches can occur alternatively (out of phase) at low migration between patches whereas host outbreaks always occur simultaneously (in phase) in both patches at high migration between patches. We show that our results are robust across a large range of parameters across different modelling approaches used typically to model intraspecific competition among hosts and parasitism, in the host-parasitoid literature. We give an analytical expression for the period of oscillations when the migration is low i.e., when host outbreaks in both patches are out of phase, show it is in agreement with numerical results. We end our paper by showing that we get the same results whether we include the biologically rooted formulations from May et al. (1981) or a general cellular automata model with qualitative rules.

A model free method of predicting transient dynamics in anaerobic digestion.

Transient dynamics pose unique challenges when dealing with predictions and management of ecological systems yet little headway has been made on understanding when an ecological system might be in a transient state. As a start we consider a specific model, here focusing on a canonical model for anaerobic digestion. Through a series of simplifications, we analyse the potential of the model for transient dynamics, and the driving mechanisms. Using a stochastic analogue of this model, we create synthetic ecological data. Thus, combining our understanding of the deterministic transient dynamics with the use of empirical dynamical modelling, we propose several new metrics to indicate when the synthetic time series is leaving a transient state.

Variation in Avian Predation Pressure as a Driver for the Diversification of Periodical Cicada Broods.

AbstractPeriodical cicadas live 13 or 17 years underground as nymphs, then emerge in synchrony as adults to reproduce. Developmentally synchronized populations called broods rarely coexist, with one dominant brood locally excluding those that emerge in off years. Twelve modern 17-year cicada broods are believed to have descended from only three ancestral broods following the last glaciation. The mechanisms by which these daughter broods overcame exclusion by the ancestral brood to synchronously emerge in a different year, however, are elusive. Here, we demonstrate that temporal variation in the population density of generalist predators can allow intermittent opportunities for new broods to invade, even though a single brood remains dominant most of the time. We show that this mechanism is consistent, in terms of the type and frequency of brood replacements, with the distribution of periodical cicada broods throughout North America today. Although we investigate one particularly charismatic case study, the mechanisms involved (competitive exclusion, Allee effects, trait variation, predation, and temporal variability) are ubiquitous and could contribute to patterns of species diversity in a range of systems.

Making Your Own Luck: Weak Vertical Swimming Improves Dispersal Success for Coastal Marine Larvae.

Dispersive early life stages are common in nature. Although many dispersing organisms ("propagules") are passively moved by outside forces, some improve their chances of successful dispersal through weak movements that exploit the structure of the environment to great effect. The larvae of many coastal marine invertebrates, for instance, swim vertically through the water column to exploit depth-varying currents, food abundance, and predation risk. Several swimming behaviors and their effects on dispersal between habitats are characterized in the literature, yet it remains unclear when and why these behaviors are advantageous. We addressed this gap using a mathematical model of larval dispersal that scored how well behaviors allowed larvae to simultaneously locate habitats, avoid predators, and gather energy. We computed optimal larval behaviors through dynamic programming, and compared those optima against passive floating and three well documented behaviors from the literature. Optimal behaviors often (but not always) resembled the documented ones. However, our model predicted that the behaviors from the literature performed robustly well, if not optimally, across many conditions. Our results shed light on why some larval behaviors are widespread geographically and across species, and underscore the importance of carefully considering the weak movements of otherwise passive propagules when studying dispersal.

Rate-induced tipping in complex high-dimensional ecological networks.

In an ecosystem, environmental changes as a result of natural and human processes can cause some key parameters of the system to change with time. Depending on how fast such a parameter changes, a tipping point can occur. Existing works on rate-induced tipping, or R-tipping, offered a theoretical way to study this phenomenon but from a local dynamical point of view, revealing, e.g., the existence of a critical rate for some specific initial condition above which a tipping point will occur. As ecosystems are subject to constant disturbances and can drift away from their equilibrium point, it is necessary to study R-tipping from a global perspective in terms of the initial conditions in the entire relevant phase space region. In particular, we introduce the notion of the probability of R-tipping defined for initial conditions taken from the whole relevant phase space. Using a number of real-world, complex mutualistic networks as a paradigm, we find a scaling law between this probability and the rate of parameter change and provide a geometric theory to explain the law. The real-world implication is that even a slow parameter change can lead to a system collapse with catastrophic consequences. In fact, to mitigate the environmental changes by merely slowing down the parameter drift may not always be effective: Only when the rate of parameter change is reduced to practically zero would the tipping be avoided. Our global dynamics approach offers a more complete and physically meaningful way to understand the important phenomenon of R-tipping.

The comparison of dispersal rate between invasive and native species varied by plant life form and functional traits.

A long dispersal distance is widely used to indicate high invasiveness, but it ignores the temporal dimensions of plant invasion. Faster dispersal rates (= distance/time) of invasive species than native ones have been widely used in modeling species invasion and planning control management. However, the comparison of dispersal rate between invasive and native plants, particularly for dispersal on a local or landscape scale, has not been tested with a comprehensive dataset. Moreover, both the effects of plant functional traits on the dispersal rate and variation in the functional-trait effects between invasive and native plants remain elusive. Compiling studies from 30 countries globally, we compared seed dispersal rates (km/year) on a local or landscape scale between 64 observations of invasive and 78 observations of native plants given effects of plant life forms, disturbance levels, and measurement methods. Furthermore, we compared the effects of functional traits on dispersal rate between invasive and native species. We found that: (1) Trait values were similar between the invasive and native plants except for the greater height of woody native plants than woody invasive ones; (2) Compared within the same plant life form, the faster dispersal rates of invasive species were found in herbaceous plants, not in woody plants, and disturbance level and measurement methods did not affect the rate comparison; (3) Plant height and seed length had significant effects on dispersal rates of both invasive and native plants, but the effect of leaf dry matter content (LDMC) was only significant on herbaceous invasive plants. The comparison of dispersal rate between invasive and native plants varied by plant life form. The convergent values but divergent dispersal effects of plant traits between invasive and native species suggest that the trait effects on invasiveness could be better understood by trait association with key factors in invasiveness, e.g., dispersal rate, than the direct trait comparison between invasive and native plants.

Stable diverse food webs become more common when interactions are more biologically constrained.

Ecologists have long sought to understand how diversity and structure mediate the stability of whole ecosystems. For high-diversity food webs, the interactions between species are typically represented using matrices with randomly chosen interaction strengths. Unfortunately, this procedure tends to produce ecological systems with no underlying equilibrium solution, and so ecological inferences from this approach may be biased by nonbiological outcomes. Using recent computationally efficient methodological advances from metabolic networks, we employ for the first time an inverse approach to diversity-stability research. We compare classical random interaction matrices of realistic food web topology (hereafter the classical model) to feasible, biologically constrained, webs produced using the inverse approach. We show that an energetically constrained feasible model yields a far higher proportion of stable high-diversity webs than the classical random matrix approach. When we examine the energetically constrained interaction strength distributions of these matrix models, we find that although these diverse webs have consistent negative self-regulation, they do not require strong self-regulation to persist. These energetically constrained diverse webs instead show an increasing preponderance of weak interactions that are known to increase local stability. Further examination shows that some of these weak interactions naturally appear to arise in the model food webs from a constraint-generated realistic generalist-specialist trade-off, whereby generalist predators have weaker interactions than more specialized species. Additionally, the inverse technique we present here has enormous promise for understanding the role of the biological structure behind stable high-diversity webs and for linking empirical data to the theory.

Modeling and prediction of phase shifts in noisy two-cycle oscillations.

Understanding and predicting ecological dynamics in the presence of noise remains a substantial and important challenge. This is particularly true in light of the poor quality of much ecological data and the imprecision of many ecological models. As a first approach to this problem, we focus here on a simple system expressed as a discrete time model with 2-cycle behavior, reflecting alternating high and low population sizes. Such dynamics naturally arise in ecological systems with overcompensatory density dependence. We ask how the amount of detail included in the population estimates affects the ability to forecast the likelihood of changes in the phase of oscillation, meaning whether high populations occur in odd or in even years. We adjust the level of detail by converting continuous population levels to simple, coarse-grained descriptions using two-state and four-state models. We also consider a cubic noisy over-compensatory model with three parameters. The focus on phase changes is what distinguishes the question we are asking and the methods we use from more standard time series approaches. Obviously, adding observation states improves the ability to forecast phase shifts. In particular, the four-state model and cubic model outperform the two-state model because they include a transition state, through which the dynamics typically pass during a phase change. Nonetheless, at high noise levels the improvement in forecast skill is relatively modest. Additionally, the frequency of phase changes depends strongly on the noise level, and is much less affected by the parameter determining amplitude in the population model, so phase shift frequencies could possibly be used to infer noise levels.

Social copying drives a tipping point for nonlinear population collapse.

Sudden changes in populations are ubiquitous in ecological systems, especially under perturbations. The agents of global change may increase the frequency and severity of anthropogenic perturbations, but complex populations' responses hamper our understanding of their dynamics and resilience. Furthermore, the long-term environmental and demographic data required to study those sudden changes are rare. Fitting dynamical models with an artificial intelligence algorithm to population fluctuations over 40 y in a social bird reveals that feedback in dispersal after a cumulative perturbation drives a population collapse. The collapse is well described by a nonlinear function mimicking social copying, whereby dispersal made by a few individuals induces others to leave the patch in a behavioral cascade for decision-making to disperse. Once a threshold for deterioration of the quality of the patch is crossed, there is a tipping point for a social response of runaway dispersal corresponding to social copying feedback. Finally, dispersal decreases at low population densities, which is likely due to the unwillingness of the more philopatric individuals to disperse. In providing the evidence of copying for the emergence of feedback in dispersal in a social organism, our results suggest a broader impact of self-organized collective dispersal in complex population dynamics. This has implications for the theoretical study of population and metapopulation nonlinear dynamics, including population extinction, and managing of endangered and harvested populations of social animals subjected to behavioral feedback loops.

Pool dynamics of time-dependent compartmental systems with application to the terrestrial carbon cycle.

Compartmental models play an important role to describe the dynamics of systems that involve mass movements between different types of pools. We develop a theory to analyse the average ages of mass in different pools in a linear compartmental system with time-dependent (i.e. non-autonomous) transfer rates, which involves transit times that characterize the average time a particle has spent in a particular pool. We apply our theoretical results to investigate a nine-dimensional compartmental system with time-dependent fluxes between pools modelling the carbon cycle which is a modification of the Carnegie-Ames-Stanford approach model. Knowledge of transit time and mean age allows calculation of carbon storage in a pool as a function of time. The general result that has important implications for understanding and managing carbon storage is that the change in storage in different pools does not change monotonically through time: as rates change monotonically a pool which initially shows a decrease may then show an increase in storage or vice versa. Thus caution is needed in extrapolating even the direction of future changes in storage in carbon storage in different pools with global change.

How far to build it before they come? Analyzing the use of the Field of Dreams hypothesis in bull kelp restoration.

In restoration ecology, the Field of Dreams hypothesis posits that restoration efforts that create a suitable environment could lead to the eventual recovery of the remaining aspects of the ecosystem through natural processes. Natural processes following partial restoration has led to ecosystem recovery in both terrestrial and aquatic systems. However, understanding the efficacy of a "Field of Dreams" approach requires a comparison of different approaches to partial restoration in terms of spatial, temporal, and ecological scale with what would happen given more comprehensive restoration efforts. We explore the relative effect of partial restoration and ongoing recovery on restoration efficacy with a dynamical model based on temperate rocky reefs in Northern California. We analyze our model for both the ability and rate of bull kelp forest recovery under different restoration strategies. We compare the efficacy of a partial restoration approach with a more comprehensive restoration effort by exploring how kelp recovery likelihood and rate change with varying intensities of urchin removal and kelp outplanting over different time periods and spatial scales. We find that, in the case of bull kelp forests, setting more favorable initial conditions for kelp recovery by implementing both urchin harvesting and kelp outplanting at the start of the restoration project has a bigger impact on the kelp recovery rate than applying restoration efforts through a longer period of time. Therefore, partial restoration efforts, in terms of spatial and temporal scale, can be significantly more effective when applied across multiple ecological scales in terms of both the capacity and rate for achieving the target outcomes.

An explanation for unexpected population crashes in a constant environment.

Unexpected population crashes are an important feature of natural systems, yet many observed crashes have not been explained. Two difficulties in explaining population crashes are their relative rarity and the multi-causal nature of ecological systems. We approach this issue with experimental microcosms, with large numbers of replicates of red flour beetle populations (Tribolium castaneum). We determined that population crashes are caused by an interaction between stochasticity and successive episodes of density dependence: demographic stochasticity in oviposition rates occasionally produces a high density of eggs; so high that there are insufficient flour resources for subsequent larvae. This mechanism can explain unexpected population crashes in more general settings: stochasticity 'pushes' population into a regime where density dependence is severely overcompensatory. The interaction between nonlinearity and stochasticity also produces chaotic population dynamics and a double-humped one-generation population map, suggesting further possibilities for unexpected behaviour in a range of systems. We discuss the generality of our proposed mechanism, which could potentially account for previously inexplicable population crashes.

Pulse disturbances in age-structured populations: Life history predicts initial impact and recovery time.

Understanding population responses to discrete 'pulsed' environmental disturbances is essential to conservation and adaptive management. Populations of concern can be driven to low levels by disturbance, and understanding interspecific differences in recovery trajectories is necessary for evaluating management options. We analysed single-species models to investigate the demographic and management factors determining the two components of population 'resilience': the magnitude of initial impact on population abundance, and duration of the recovery time. We simulated age-structured populations with density-dependent recruitment, subjected to a pulse disturbance consisting of a period of increased mortality of either the juvenile age class or all age classes, and calculated both impact and return time. For illustration, we used demographic parameters from a suite of 16 fish species. We formulated the model as a renewal equation, allowing us to describe disturbance impacts mathematically as a convolution. We also included nonlinear dynamics, representing populations that recover to a steady state; this is more realistic (in most cases) than prior analyses of resilience in linear models without density-dependence. When the disturbance affected only one or a few young age-classes, longevity was the major life-history determinant of impact and recovery time. Shorter-lived species endured greater impacts when disturbed because each age class is a greater proportion of the population. However, shorter-lived species also had faster recovery times, for the same reason. When disturbance affected adult age-classes, the impact was more immediate and no longer affected by species' longevity, though the effect of longevity on recovery time remained. These results improve our understanding of interspecific differences in resilience and increase our ability to make predictions for adaptive management. Additionally, formulating the problem as a renewal equation and using mathematical convolutions allows us to quantify how disturbances with different time courses (not just an immediate, constant level of disturbance but gradually increasing or decreasing levels of disturbance) would have different effects on population resilience: delayed responses for species in which biomass is concentrated in older age classes, and for disturbances that become progressively more severe.

Entender las respuestas de la población a perturbaciones ambientales, específicamente a pulsadas individuales, es esencial para la conservación y la gestión adaptativa. Las poblaciones de interés pueden reducirse a niveles bajas debido a la perturbación, y es necesario entender las diferencias interespecíficas en las trayectorias de recuperación para evaluar las opciones de gestión. Analizamos modelos para especies individuales para investigar los factores demográficos y de gestión que determinan los dos componentes de la 'resiliencia' de la población: la magnitud del impacto inicial sobre la abundancia de la población y la duración del tiempo de recuperación. Simulamos poblaciones estructuradas por edad con reclutamiento que depende de la densidad, las sometimos a una perturbación pulsada que consiste en un período de mayor mortalidad del grupo etário juvenil o de todos los grupos etários, y calculamos tanto el impacto como el tiempo de retorno. A modo de ilustración, utilizamos parámetros demográficos de un conjunto de 16 especies de peces. Formulamos el modelo como una ecuación de renovación, lo que nos permite describir matemáticamente los impactos de las perturbaciones como una convolución. También incluimos dinámicas no lineales que representan poblaciones que se recuperan hacia un estado estable; esto es más realista (en la mayoría de los casos) que los análisis previos de resiliencia en modelos lineales sin la dependencia de la densidad. Cuando la perturbación ha afectado a uno o a algunos pocos grupos etários jóvenes, la longevidad fue el principal determinante de la historia de vida del impacto y el tiempo de recuperación. Las especies de vida más corta sufrieron mayores impactos cuando fueron perturbadas porque cada grupo etáreo representa una mayor proporción de la población. Sin embargo, las especies con vidas más cortas también tuvieron tiempos de recuperación más rápidos, por la misma razón. Cuando la perturbación afectó a los grupos etários adultos, el impacto fue más inmediato y ya no se vio afectado por la longevidad de las especies, aunque se mantuvo el efecto de la longevidad sobre el tiempo de recuperación. Estos resultados mejoran nuestra comprensión de las diferencias interespecíficas de la resiliencia y aumentan nuestra capacidad para hacer predicciones con fin a la gestión adaptativa. Además, formular el problema como una ecuación de renovación y usar convoluciones matemáticas nos permite cuantificar cómo las perturbaciones con distintos lapsos de tiempo (no solo un nivel de perturbación constante e inmediato, sino niveles de perturbación que aumentan o disminuyen gradualmente) tendrían diferentes efectos sobre la resiliencia de la población: respuestas tardías para especies en las que la biomasa se concentra en grupos etários de mayor edad y para perturbaciones que se vuelven progresivamente más severas.

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.

Characterizing Long Transients in Consumer-Resource Systems With Group Defense and Discrete Reproductive Pulses.

During recent years, the study of long transients has been expanded in ecological theory to account for shifts in long-term behavior of ecological systems. These long transients may lead to regime shifts between alternative states that resemble the dynamics of alternative stable states for a prolonged period of time. One dynamic that potentially leads to long transients is the group defense of a resource in a consumer-resource interaction. Furthermore, time lags in the population caused by discrete reproductive pulses have the potential to produce long transients, either independently or in conjunction to the transients caused by the group defense. In this work, we analyze the potential for long transients in a model for a consumer-resource system in which the resource exhibits group defense and reproduces in discrete reproductive pulses. This system exhibits crawl-by transients near the extinction and carrying capacity states of resource, and a transcritical bifurcation, under which a ghost limit cycle appears. We estimate the transient time of our system from these transients using perturbation theory. This work advances an understanding of how systems shift between alternate states and their duration of staying in a given regime and what ecological dynamics may lead to long transients.

Optimal management of stochastic invasion in a metapopulation with Allee effects.

Invasive species account for incalculable damages worldwide, in both ecological and bioeconomic terms. The question of how a network of invasive populations can be optimally managed is one that deserves further exploration. A study accounting for partial observability and imperfect detection, in particular, could yield useful insights into species eradication efforts. Here, we generalized a simple model system that we developed in previous work. This model consists of three interacting populations with underlying strong Allee effects and stochastic dynamics, inhabiting distinct locations connected by dispersal, which can generate bistability. To explore the stochastic dynamics, we formulated an individual-based modeling approach. Next, using the theory of continuous-time Markov chains, we approximated the original high-dimensional model by a Markov chain with eight states, with each state corresponding to a combination of population thresholds. We then used the reduced model as the core for a powerful decision-making tool, referred to as a Partially Observable Markov Decision Process (POMDP). Analysis of this POMDP indicates when the system results in optimal management outcomes.

Occupancy times for time-dependent stage-structured models.

During their lifetimes, individuals in populations pass through different states, and the notion of an occupancy time describes the amount of time an individual spends in a given set of states. Questions related to this idea were studied in a recent paper by Roth and Caswell for cases where the environmental conditions are constant. However, it is truly important to consider the case where environments are changing randomly or in directional way through time, so the transition probabilities between different states change over time, motivating the use of time-dependent stage-structured models. Using absorbing inhomogenous Markov chains and the discrete-time McKendrick-von Foerster equation, we derive explicit formulas for the occupancy time, its expectation, and its higher-order moments for stage-structured models with time-dependent transition rates. The results provide insights into the dynamics of long lived plant or animal populations where individuals transition in both directions between reproductive and non reproductive stages. We apply our approach to study a specific time-dependent model of the Southern Fulmar, and obtain insights into how the number of breeding attempts depends on external conditions that vary through time.