Moving forward in circles: challenges and opportunities in modelling population cycles

Temporal variability in mortality and recruitment jointly influence the periodic fluctuations in Antarctic krill populations

Antarctic krill (Euphausia superba) is a key part of the food web in the Southern Ocean ecosystem. Significant inter-annual fluctuations in population dynamics make stock assessment and management of its population a significant challenge. To better understand the population dynamics and fluctuation of krill, a survey-based age-structured catch-at-length model (ACL) is used to estimate the periodic fluctuations, based on length data collected from scientific surveys under the US Antarctic Marine Living Resources (AMLR) Program between 1992 and 2011. Spectral analysis of the model estimates revealed periodic fluctuations of 5-6 years in the recruitment, total abundance, and total biomass of krill in the Antarctic Peninsula, while spawning stock biomass exhibited periodic fluctuations of both 5-6 years and 2-3 years. The variations in krill total abundance and total biomass were mainly driven by recruitment, but the variation in spawning stock biomass was likely driven by both recruitment and time-varying, age-specific mortality. Our study contributes to a better understanding of the patterns and drivers of the periodic dynamics of Antarctic krill, which may help lead to improved assessments and fishery management for this important stock.

Systematic shifts in the variation among host individuals must be considered in climate-disease theory

To make more informed predictions of host-pathogen interactions under climate change, studies have incorporated the thermal performance of host, vector and pathogen traits into disease models to quantify effects on average transmission rates. However, this body of work has omitted the fact that variation in susceptibility among individual hosts affects disease spread and long-term patterns of host population dynamics. Furthermore, and especially for ectothermic host species, variation in susceptibility is likely to be plastic, influenced by variables such as environmental temperature. For example, as host individuals respond idiosyncratically to temperature, this could affect the population-level variation in susceptibility, such that there may be predictable functional relationships between variation in susceptibility and temperature. Quantifying the relationship between temperature and among-host trait variation will therefore be critical for predicting how climate change and disease will interact to influence host-pathogen population dynamics. Here, we use a model to demonstrate how short-term effects of temperature on the distribution of host susceptibility can drive epidemic characteristics, fluctuations in host population sizes and probabilities of host extinction. Our results emphasize that more research is needed in disease ecology and climate biology to understand the mechanisms that shape individual trait variation, not just trait averages.

Accumulation of pathogens in soil microbiome can explain long-term fluctuations of legumes in a grassland community

All plant populations fluctuate in time. Apart from the dynamics imposed by external forces such as climate, these fluctuations can be driven by endogenous processes taking place within the community. In this study, we aimed to identify potential role of soil-borne microbial communities in driving endogenous fluctuations of plant populations. We combined a unique, 35-yr long abundance data of 11 common plant species from a species-rich mountain meadow with development of their soil microbiome (pathogenic fungi, arbuscular mycorrhizal fungi and oomycetes) observed during 4 yr of experimental cultivation in monocultures. Plant species which abundance fluctuated highly in the field (particularly legumes) accumulated plant pathogens in their soil mycobiome. We also identified increasing proportion of mycoparasitic fungi under highly fluctuating legume species, which may indicate an adaptation of these species to mitigate the detrimental effects of pathogens. Our study documented that long-term fluctuations in the abundance of plant species in grassland communities can be explained by the accumulation of plant pathogens in plant-soil microbiome. By contrast, we found little evidence of the role of mutualists in plant population fluctuations. These findings offer new insights for understanding mechanisms driving both long-term vegetation dynamics and patterns of species coexistence and richness.

Small rodent population cycles and plants - after 70 years, where do we go?

Small rodent population cycles characterise northern ecosystems, and the cause of these cycles has been a long-lasting central topic in ecology, with trophic interactions currently considered the most plausible cause. While some researchers have rejected plant-herbivore interactions as a cause of rodent cycles, others have continued to research their potential roles. Here, we present an overview of whether plants can cause rodent population cycles, dividing this idea into four different hypotheses with different pathways of plant impacts and related assumptions. Our systematic review of the existing literature identified 238 studies from 150 publications. This evidence base covered studies from the temperate biome to the tundra, but the studies were scattered across study systems and only a few specific topics were addressed in a replicated manner. Quantitative effects of rodents on vegetation was the best studied topic, and our evidence base suggests such that such effects may be most pronounced in winter. However, the regrowth of vegetation appears to take place too rapidly to maintain low rodent population densities over several years. The lack of studies prevented assessment of time lags in the qualitative responses of vegetation to rodent herbivory. We conclude that the literature is currently insufficient to discard with confidence any of the four potential hypotheses for plant-rodent cycles discussed herein. While new methods allow analyses of plant quality across more herbivore-relevant spatial scales than previously possible, we argue that the best way forward to rejecting any of the rodent-plant hypotheses is testing specific predictions of dietary variation. Indeed, all identified hypotheses make explicit assumptions on how rodent diet taxonomic composition and quality will change across the cycle. Passing this bottleneck could help pinpoint where, when, and how plant-herbivore interactions have - or do not have - plausible effects on rodent population dynamics.

Five fundamental ways in which complex food webs may spiral out of control

Theory suggests that increasingly long, negative feedback loops of many interacting species may destabilize food webs as complexity increases. Less attention has, however, been paid to the specific ways in which these 'delayed negative feedbacks' may affect the response of complex ecosystems to global environmental change. Here, we describe five fundamental ways in which these feedbacks might pave the way for abrupt, large-scale transitions and species losses. By combining topological and bioenergetic models, we then proceed by showing that the likelihood of such transitions increases with the number of interacting species and/or when the combined effects of stabilizing network patterns approach the minimum required for stable coexistence. Our findings thus shift the question from the classical question of what makes complex, unaltered ecosystems stable to whether the effects of, known and unknown, stabilizing food-web patterns are sufficient to prevent abrupt, large-scale transitions under global environmental change.

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.

Stability of ecosystems under oscillatory driving with frequency modulation

Consumer-resource cycles are widespread in ecosystems, and seasonal forcing is known to influence them profoundly. Typically, seasonal forcing perturbs an ecosystem with time-varying frequency; however, previous studies have explored the dynamics of such systems under oscillatory forcing with constant frequency. Studies of the effect of time-varying frequency on ecosystem stability are lacking. Here we investigate isolated and network models of a cyclic consumer-resource ecosystem with oscillatory driving subjected to frequency modulation. We show that frequency modulation can induce stability in the system in the form of stable synchronized solutions, depending on intrinsic model parameters and extrinsic modulation strength. The stability of synchronous solutions is determined by calculating the maximal Lyapunov exponent, which determines that the fraction of stable synchronous solution increases with an increase in the modulation strength. We also uncover intermittent synchronization when synchronous dynamics are intermingled with episodes of asynchronous dynamics. Using the phase-reduction method for the network model, we reduce the system into a phase equation that clearly distinguishes synchronous, intermittently synchronous, and asynchronous solutions. While investigating the role of network topology, we find that variation in rewiring probability has a negligible effect on the stability of synchronous solutions. This study deepens our understanding of ecosystems under seasonal perturbations.

Heterogeneity in Behaviour and Movement can Influence the Stability of Predator-Prey Periodic Travelling Waves

Cyclic predator-prey systems are often observed in nature. In a spatial setting, these can manifest as periodic traveling waves (PTW). Environmental change and direct human activity are known to, among other effects, increase the heterogeneity of the physical environment, which prey and predator inhabit. Aiming to understand the effects of heterogeneity on predator-prey PTWs, we consider a one-dimensional infinite landscape Rosenzweig-MacArthur reaction-diffusion model, with alternating patch types, and study the PTWs in this system. Applying the method of homogenisation, we show how heterogeneity can affect the stability of PTW solutions. We illustrate how the effects of heterogeneity can be understood and interpreted using Turchin's concept of residence index (encapsuling diffusion rate and patch preference). In particular, our results show that prey heterogeneity acts to modulate the effects of predator heterogeneity, by this we mean that as prey increasingly spend more time in one patch type over another the stability of the PTWs becomes more sensitive to heterogeneity in predator movement and behaviour.

Intrinsic nonlinear dynamics drive single-species systems

The importance of oscillations and deterministic chaos in natural biological systems has been discussed for several decades and was originally based on discrete-time population growth models (May 1974). Recently, all types of nonlinear dynamics were shown for experimental communities where several species interact. Yet, there are no data exhibiting the whole range of nonlinear dynamics for single-species systems without trophic interactions. Up until now, ecological experiments and models ignored the intracellular dimension, which includes multiple nonlinear processes even within one cell type. Here, we show that dynamics of single-species systems of protists in continuous experimental chemostat systems and corresponding continuous-time models reveal typical characteristics of nonlinear dynamics and even deterministic chaos, a very rare discovery. An automatic cell registration enabled a continuous and undisturbed analysis of dynamic behavior with a high temporal resolution. Our simple and general model considering the cell cycle exhibits a remarkable spectrum of dynamic behavior. Chaos-like dynamics were shown in continuous single-species populations in experimental and modeling data on the level of a single type of cells without any external forcing. This study demonstrates how complex processes occurring in single cells influence dynamics on the population level. Nonlinearity should be considered as an important phenomenon in cell biology and single-species dynamics and also, for the maintenance of high biodiversity in nature, a prerequisite for nature conservation.

Broad-scale climate variation drives the dynamics of animal populations: a global multi-taxa analysis

Climate is a major extrinsic factor affecting the population dynamics of many organisms. The Broad-Scale Climate Hypothesis (BSCH) was proposed by Elton to explain the large-scale synchronous population cycles of animals, but the extent of support and whether it differs among taxa and geographical regions is unclear. We reviewed publications examining the relationship between the population dynamics of multiple taxa worldwide and the two most commonly used broad-scale climate indices, El Niño-Southern Oscillation (ENSO) and North Atlantic Oscillation (NAO). Our review and synthesis (based on 561 species from 221 papers) reveals that population changes of mammals, birds and insects are strongly affected by major oceanic shifts or irregular oceanic changes, particularly in ENSO- and NAO-influenced regions (Pacific and Atlantic, respectively), providing clear evidence supporting Elton's BSCH. Mammal and insect populations tended to increase during positive ENSO phases. Bird populations tended to increase in positive NAO phases. Some species showed dual associations with both positive and negative phases of the same climate index (ENSO or NAO). These findings indicate that some taxa or regions are more or less vulnerable to climate fluctuations and that some geographical areas show multiple weather effects related to ENSO or NAO phases. Beyond confirming that animal populations are influenced by broad-scale climate variation, we document extensive patterns of variation among taxa and observe that the direct biotic and abiotic mechanisms for these broad-scale climate factors affecting animal populations are very poorly understood. A practical implication of our research is that changes in ENSO or NAO can be used as early signals for pest management and wildlife conservation. We advocate integrative studies at both broad and local scales to unravel the omnipresent effects of climate on animal populations to help address the challenge of conserving biodiversity in this era of accelerated climate change.

Evolution of class-structured populations in periodic environments

What is the influence of periodic environmental fluctuations on life-history evolution? We present a general theoretical framework to understand and predict the long-term evolution of life-history traits under a broad range of ecological scenarios. Specifically, we investigate how periodic fluctuations affect selection when the population is also structured in distinct classes. This analysis yields time-varying selection gradients that clarify the influence of the fluctuations of the environment on the competitive ability of a specific life-history mutation. We use this framework to analyse the evolution of key life-history traits of pathogens. We examine three different epidemiological scenarios and we show how periodic fluctuations of the environment can affect the evolution of virulence and transmission as well as the preference for different hosts. These examples yield new and testable predictions on pathogen evolution, and illustrate how our approach can provide a better understanding of the evolutionary consequences of time-varying environmental fluctuations in a broad range of scenarios.

Density-dependent demography and movements in a cyclic brown lemming population

Theoretical modeling predicts that both direct and delayed density-dependence are key factors to generate population cycles. Deciphering density-dependent processes that lead to variable population growth characterizing different phases of the cycles remains challenging. This is particularly the case for the period of prolonged low densities, which is inherently data deficient. However, demographic analyses based on long-term capture-mark-recapture datasets can help resolve this question. We relied on a 16-year (2004-2019) live-trapping program to analyze the summer demography and movements of a cyclic brown lemming population in the Canadian Arctic. More specifically, we examined if inversely density-dependent processes could explain why population growth can remain low during the prolonged low phase. We found that the proportion of females in the population was inversely density-dependent with a strong male-biased sex ratio at low densities but not at high densities. However, survival of adult females was higher than adult males, but both had lower survival at low densities than at high ones. Distances moved by both adult males and females were density-dependent, and proportion of females in reproductive condition was weakly density-dependent as it tended to increase at low density. Individual body condition, measured as monthly change in body mass, was not density-dependent. Overall, the strong male-biased sex ratio at very low densities suggests a loss of reproductive potential due to the rarity of females and appears to be the most susceptible demographic factor that could contribute to the prolonged low phase in cyclic brown lemmings. What leads to this sex-bias in the first place is still unclear, potentially owing to our trapping period limited to the summer, but we suggest that it could be due to high predation rate on breeding females in winter.

Stochasticity-induced persistence in coupled social-ecological systems

Stochasticity is often associated with negative consequences for population dynamics since a population may die out due to random chance during periods when population size is very low (stochastic fade-out). Here we develop a coupled social-ecological model based on stochastic differential equations that includes natural expansion and harvesting of a forest ecosystem, and dynamics of conservation opinions, social norms and social learning in a human population. Our objective was to identify mechanisms that influence long-term persistence of the forest ecosystem in the presence of noise. We found that most of the model parameters had a significant influence on the time to extinction of the forest ecosystem. Increasing the social learning rate and the net benefits of conservation significantly increased the time to extinction, for instance. Most interestingly, we found a parameter regime where an increase in the amount of system stochasticity caused an increase in the mean time to extinction, instead of causing stochastic fade-out. This effect occurs for a subset of realizations, but the effect is large enough to increase the mean time to extinction across all realizations. Such "stochasticity-induced persistence" occurs when stochastic dynamics in the social system generates benefits in the forest system at crucial points in its temporal dynamics. We conclude that studying relatively simple social-ecological models has the benefit of facilitating characterization of dynamical states and thereby enabling us to formulate new hypothesis about mechanisms that could be operating in empirical social-ecological systems.

Tradeoffs of managing cod as a sustainable resource in fluctuating environments

Sustainable human exploitation of living marine resources stems from a delicate balance between yield stability and population persistence to achieve socioeconomic and conservation goals. But our imperfect knowledge of how oceanic oscillations regulate temporal variation in an exploited species can obscure the risk of missing management targets. We illustrate how applying a management policy to suppress fluctuations in fishery yield in variable environments (prey density and regional climate) can present unintended outcomes in harvested predators and the sustainability of harvesting. Using Atlantic cod (Gadus morhua, an apex predatory fish) in the Barents Sea as a case study we simulate age-structured population and harvest dynamics through time-varying, density-dependent and density-independent processes with a stochastic, process-based model informed by 27-year monitoring data. In this model, capelin (Mallotus villosus, a pelagic forage fish), a primary prey of cod, fluctuations modulate the strength of density-dependent regulation primarily through cannibalistic pressure on juvenile cod survival; sea temperature fluctuations modulate thermal regulation of cod feeding, growth, maturation, and reproduction. We first explore how capelin and temperature fluctuations filtered through cod intrinsic dynamics modify catch stability and then evaluate how management to suppress short-term variability in catch targets alters overharvest risk. Analyses revealed that suppressing year-to-year catch variability impedes management responses to adjust fishing pressure, which becomes progressively out of sync with variations in cod abundance. This asynchrony becomes amplified in fluctuating environments, magnifying the amplitudes of both fishing pressure and cod abundance and then intensifying the density-dependent regulation of juvenile survival through cannibalism. Although these transient dynamics theoretically give higher average catches, emergent, quasicyclic behaviors of the population would increase long-term yield variability and elevate overharvest risk. Management strategies that overlook the interplay of extrinsic (fishing and environment) and intrinsic (life history and demography) fluctuations thus can inadvertently destabilize fish stocks, thereby jeopardizing the sustainability of harvesting. These policy implications underscore the value of ecosystem approaches to designing management measures to sustainably harvest ecologically connected resources while achieving socioeconomic security.

A synthetic review of terrestrial biological research from the Alberta oil sands region: 10 years of published literature

In the past decade, a large volume of peer-reviewed papers has examined the potential impacts of oil and gas resource extraction in the Canadian oil sands (OS). A large proportion focuses on terrestrial biology: wildlife, birds, and vegetation. We provide a qualitative synthesis of the condition of the environment in the oil sands region (OSR) from 2009 to 2020 to identify gaps and progress cumulative effects assessments. Our objectives were to (1) qualitatively synthesize and critically review knowledge from the OSR; (2) identify consistent trends and generalizable conclusions; and (3) pinpoint gaps in need of greater monitoring or research effort. We visualize knowledge and terrestrial monitoring foci by allocating papers to a conceptual model for the OS. Despite a recent increase in publications, focus has remained concentrated on a few key stressors, especially landscape disturbance, and a few taxa of interest. Stressor and response monitoring is well represented, but direct monitoring of pathways (linkages between stressors and responses) is limited. Important knowledge gaps include understanding effects at multiple spatial scales, mammal health effects monitoring, focused monitoring of local resources important to Indigenous communities, and geospatial coverage and availability, including higher attribute resolution in human footprint, comprehensive land cover mapping, and up-to-date LiDAR coverage. Causal attribution based on spatial proximity to operations or spatial orientation of monitoring in the region is common but may be limited in the strength of inference that it provides. Integr Environ Assess Manag 2022;18:388-406. © 2021 The Authors. Integrated Environmental Assessment and Management published by Wiley Periodicals LLC on behalf of Society of Environmental Toxicology & Chemistry (SETAC).

Noise-induced versus intrinsic oscillation in ecological systems

Studies of oscillatory populations have a long history in ecology. A first-principles understanding of these dynamics can provide insights into causes of population regulation and help with selecting detailed predictive models. A particularly difficult challenge is determining the relative role of deterministic versus stochastic forces in producing oscillations. We employ statistical physics concepts, including measures of spatial synchrony, that incorporate patterns at all scales and are novel to ecology, to show that spatial patterns can, under broad and well-defined circumstances, elucidate drivers of population dynamics. We find that when neighbours are coupled (e.g. by dispersal), noisy intrinsic oscillations become distinguishable from noise-induced oscillations at a transition point related to synchronisation that is distinct from the deterministic bifurcation point. We derive this transition point and show that it diverges from the deterministic bifurcation point as stochasticity increases. The concept of universality suggests that the results are robust and widely applicable.

Bifurcation analysis of the predator-prey model with the Allee effect in the predator

The use of predator–prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka–Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator–prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator–prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

Climate variability and density-dependent population dynamics: Lessons from a simple High Arctic ecosystem

Whether the renowned population cycles of small mammals in northern food webs are driven by bottom-up (plant–herbivore) or top-down (predator–prey) interactions is still a debated question but crucial to our understanding of their ecological functions and response to climate change. A long-term study of a graminivorous vole population in an exceptionally simple High Arctic food web allowed us to identify which population dynamics features are present without top-down regulation. Unique features were high-amplitude, noncyclic population fluctuations driven by a combination of stochastic weather events and season-specific density dependence likely arising from plant–herbivore interactions. That such features are not present in more complex food webs points to the importance of top-down regulation in small mammal populations.

Ecologists are still puzzled by the diverse population dynamics of herbivorous small mammals that range from high-amplitude, multiannual cycles to stable dynamics. Theory predicts that this diversity results from combinations of climatic seasonality, weather stochasticity, and density-dependent food web interactions. The almost ubiquitous 3- to 5-y cycles in boreal and arctic climates may theoretically result from bottom-up (plant–herbivore) and top-down (predator–prey) interactions. Assessing, empirically, the roles of such interactions and how they are influenced by environmental stochasticity has been hampered by food web complexity. Here, we take advantage of a uniquely simple High Arctic food web, which allowed us to analyze the dynamics of a graminivorous vole population not subjected to top-down regulation. This population exhibited high-amplitude, noncyclic fluctuations—partly driven by weather stochasticity. However, the predominant driver of the dynamics was overcompensatory density dependence in winter that caused the population to frequently crash. Model simulations showed that the seasonal pattern of density dependence would yield regular 2-y cycles in the absence of stochasticity. While such short cycles have not yet been observed in mammals, they are theoretically plausible if graminivorous vole populations are deterministically bottom-up regulated. When incorporating weather stochasticity in the model simulations, cyclicity became disrupted and the amplitude was increased—akin to the observed dynamics. Our findings contrast with the 3- to 5-y population cycles that are typical of graminivorous small mammals in more complex food webs, suggesting that top-down regulation is normally an important component of such dynamics.

Altered precipitation dynamics lead to a shift in herbivore dynamical regime

The interaction between endogenous dynamics and exogenous environmental variation is central to population dynamics. Although investigations into the effects of changing mean climate are widespread, changing patterns of variation in environmental forcing also affect dynamics in complex ways. Using wavelet and time series analyses, we identify a regime shift in the dynamics of a moth species in California from shorter to longer period oscillations over a 34-year census, and contemporaneous changes in regional precipitation dynamics. Simulations support the hypothesis that shifting precipitation dynamics drove changes in moth dynamics, possibly due to stochastic resonance with delayed density-dependence. The observed shift in climate dynamics and the interaction with endogenous dynamics mean that predicting future population dynamics will require information on both climatic shifts and their interaction with endogenous density-dependence, a combination that is rarely available. Consequently, models based on historical data may be unable to predict future population dynamics.

Congruent Genetic and Demographic Dispersal Rates in a Natural Metapopulation at Equilibrium

Understanding the functioning of natural metapopulations at relevant spatial and temporal scales is necessary to accurately feed both theoretical eco-evolutionary models and conservation plans. One key metric to describe the dynamics of metapopulations is dispersal rate. It can be estimated with either direct field estimates of individual movements or with indirect molecular methods, but the two approaches do not necessarily match. We present a field study in a large natural metapopulation of the butterfly Boloria eunomia in Belgium surveyed over three generations using synchronized demographic and genetic datasets with the aim to characterize its genetic structure, its dispersal dynamics, and its demographic stability. By comparing the census and effective population sizes, and the estimates of dispersal rates, we found evidence of stability at several levels: constant inter-generational ranking of population sizes without drastic historical changes, stable genetic structure and geographically-influenced dispersal movements. Interestingly, contemporary dispersal estimates matched between direct field and indirect genetic assessments. We discuss the eco-evolutionary mechanisms that could explain the described stability of the metapopulation, and suggest that destabilizing agents like inter-generational fluctuations in population sizes could be controlled by a long adaptive history of the species to its dynamic local environment. We finally propose methodological avenues to further improve the match between demographic and genetic estimates of dispersal.

Mathematical Modeling of Population Dynamics Based on Recurrent Equations: Results and Prospects. Part I

Abstract

Approaches to modeling population dynamics using discrete-time models are described in this two-part review. The development of the scientific ideas of discrete time models, from the Malthus model to modern population models that take into account many factors affecting the structure and dynamics, is presented. The most important and interesting results of recurrent equation application to biological system analysis obtained by the authors are given. In the first part of this review, the population dynamic effects that result from density-dependent regulation of population, the age and sex structures, and the influence of external factors are considered.

Fitting stochastic predator-prey models using both population density and kill rate data

Most mechanistic predator-prey modelling has involved either parameterization from process rate data or inverse modelling. Here, we take a median road: we aim at identifying the potential benefits of combining datasets, when both population growth and predation processes are viewed as stochastic. We fit a discrete-time, stochastic predator-prey model of the Leslie type to simulated time series of densities and kill rate data. Our model has both environmental stochasticity in the growth rates and interaction stochasticity, i.e., a stochastic functional response. We examine what the kill rate data brings to the quality of the estimates, and whether estimation is possible (for various time series lengths) solely with time series of population counts or biomass data. Both Bayesian and frequentist estimation are performed, providing multiple ways to check model identifiability. The Fisher Information Matrix suggests that models with and without kill rate data are all identifiable, although correlations remain between parameters that belong to the same functional form. However, our results show that if the attractor is a fixed point in the absence of stochasticity, identifying parameters in practice requires kill rate data as a complement to the time series of population densities, due to the relatively flat likelihood. Only noisy limit cycle attractors can be identified directly from population count data (as in inverse modelling), although even in this case, adding kill rate data - including in small amounts - can make the estimates much more precise. Overall, we show that under process stochasticity in interaction rates, interaction data might be essential to obtain identifiable dynamical models for multiple species. These results may extend to other biotic interactions than predation, for which similar models combining interaction rates and population counts could be developed.

Combined Impacts of Predation, Mutualism and Dispersal on the Dynamics of a Four-Species Ecological System

Multi-species and ecosystem models have provided ecologist with an excellent opportunity to study the effects of multiple biotic interactions in an ecological system. Predation and mutualism are among the most prevalent biotic interactions in the multi-species system. Several ecological studies exist, but they are based on one-or two-species interactions, and in real life, multiple interactions are natural characteristics of a multi-species community. Here, we use a system of partial differential equations to study the combined effects of predation, mutualism and dispersal on the multi-species coexistence and community stability in the ecological system. Our results show that predation provided a defensive mechanism against the negative consequences of the multiple species interactions by reducing the net effect of competition. Predation is critical in the stability and coexistence of the multi-species community. The combined effects of predation and dispersal enhance the multiple species coexistence and persistence. Dispersal exerts a positive effect on the system by supporting multiple species coexistence and stability of community structures. Dispersal process also reduces the adverse effects associated with multiple species interactions. Additionally, mutualism induces oscillatory behaviour on the system through Hopf bifurcation. The roles of mutualism also support multiple species coexistence mechanisms (for some threshold values) by increasing the stable coexistence and the stable limit cycle regions. We discover that the stability and coexistence mechanisms are controlled by the transcritical and Hopf bifurcation that occurs in this system. Most importantly, our results show the important influences of predation, mutualism and dispersal in the stability and coexistence of the multi-species communities

Quasi-stationary states of game-driven systems: A dynamical approach

Evolutionary game theory is a framework to formalize the evolution of collectives ("populations") of competing agents that are playing a game and, after every round, update their strategies to maximize individual payoffs. There are two complementary approaches to modeling evolution of player populations. The first addresses essentially finite populations by implementing the apparatus of Markov chains. The second assumes that the populations are infinite and operates with a system of mean-field deterministic differential equations. By using a model of two antagonistic populations, which are playing a game with stationary or periodically varying payoffs, we demonstrate that it exhibits metastable dynamics that is reducible neither to an immediate transition to a fixation (extinction of all but one strategy in a finite-size population) nor to the mean-field picture. In the case of stationary payoffs, this dynamics can be captured with a system of stochastic differential equations and interpreted as a stochastic Hopf bifurcation. In the case of varying payoffs, the metastable dynamics is much more complex than the dynamics of the means.

Threshold harvesting as a conservation or exploitation strategy in population management

Threshold harvesting removes the surplus of a population above a set threshold and takes no harvest below the threshold. This harvesting strategy is known to prevent overexploitation while obtaining higher yields than other harvesting strategies. However, the harvest taken can vary over time, including seasons of no harvest at all. While this is undesirable in fisheries or other exploitation activities, it can be an attractive feature of management strategies where removal interventions are costly and desirable only occasionally. In the presence of population fluctuations, the issue of variable harvests and population sizes becomes even more notorious. Here, we investigate the impact of threshold harvesting on the dynamics of both population size and harvests, especially in the presence of population cycles. We take into account semelparous and iteroparous life cycles, Allee effects, observation uncertainty, and demographic as well as environmental stochasticity, using generic mathematical models in discrete time. Our results show that threshold harvesting enhances multiple forms of population stability, namely persistence, constancy, resilience, and dynamic stability. We discuss plausible choices of threshold values, depending on whether the aim is resource exploitation, pest control, or the stabilization of fluctuations.

Resource-harvester cycles caused by delayed knowledge of the harvested population state can be dampened by harvester forecasting

The monitoring of ecosystems and the spread of information concerning their state among human stakeholders is often a lengthy process. The importance of mutual feedbacks between socioeconomic and ecological dynamics is being increasingly recognised in recent studies, but it is generally assumed that the feedback from the environment is instantaneous, ignoring any delay in the spread of ecosystem knowledge and the resulting potential for system stability loss. On the other hand, human actors rarely make purely myopic socioeconomic decisions as is often assumed. Rather, they show a degree of foresight for future utility which may have an opposing, stabilising effect to any delay in knowledge. In this paper, we consider a generic resource-harvester model with delayed ecosystem knowledge and predictive behaviour by the harvesters. We show that delays in the spread of information about the resource level can destabilise the bioeconomic equilibrium in the system and induce harvesting cycles or the collapse of the resource. Sufficiently farsighted prediction by the harvesters can stabilise the system, provided the delay is not too long. However, if the time horizon of prediction is too long relative to the timescale of resource growth, prediction can be destabilising even in the absence of delay. The results imply that effective monitoring of ecosystems and fast dissemination of the results are necessary for their sustainable use and that efforts to promote appropriate foresight among ecosystem users on the personal and institutional level would be beneficial to the stability of coupled socioeconomic-ecological systems.

The effect of seasonal strength and abruptness on predator-prey dynamics

Coupled dynamical systems in ecology are known to respond to the seasonal forcing of their parameters with multiple dynamical behaviours, ranging from seasonal cycles to chaos. Seasonal forcing is predominantly modelled as a sine wave. However, the transition between seasons is often more sudden as illustrated by the effect of snow cover on predation success. A handful of studies have mentioned the robustness of their results to the shape of the forcing signal but did not report any detailed analyses. Therefore, whether and how the shape of seasonal forcing could affect the dynamics of coupled dynamical systems remains unclear, while abrupt seasonal transitions are widespread in ecological systems. To provide some answers, we conduct a numerical analysis of the dynamical response of predator-prey communities to the shape of the forcing signal by exploring the joint effect of two features of seasonal forcing: the magnitude of the signal, which is classically the only one studied, and the shape of the signal, abrupt or sinusoidal. We consider both linear and saturating functional responses, and focus on seasonal forcing of the predator's discovery rate, which fluctuates with changing environmental conditions and prey's ability to escape predation. Our numerical results highlight that a more abrupt seasonal forcing mostly alters the magnitude of population fluctuations and triggers period-doubling bifurcations, as well as the emergence of chaos, at lower forcing strength than for sine waves. Controlling the variance of the forcing signal mitigates this trend but does not fully suppress it, which suggests that the variance is not the only feature of the shape of seasonal forcing that acts on community dynamics. Although theoretical studies may predict correctly the sequence of bifurcations using sine waves as a representation of seasonality, there is a rationale for applied studies to implement as realistic seasonal forcing as possible to make precise predictions of community dynamics.

Temporal patterns of dispersal-induced synchronization in population dynamics

The mechanisms and properties of synchronization of oscillating ecological populations attract attention because it is a fairly common phenomenon and because spatial synchrony may elevate a risk of extinction and may lead to other environmental impacts. Conditions for stable synchronization in a system of linearly coupled predator-prey oscillators have been considered in the past. However, the spatial dispersal coupling may be relatively weak and may not necessarily lead to a stable, complete synchrony. If the coupling between oscillators is too weak to induce a stable synchrony, oscillators may be engaged into intermittent synchrony, when episodes of synchronized dynamics are interspersed with the episodes of desynchronized dynamics. In the present study we consider the temporal patterning of this kind of intermittent synchronized dynamics in a system of two dispersal-coupled Rosenzweig-MacArthur predator-prey oscillators. We consider the properties of the distributions of durations of desynchronized intervals and their dependence on the model parameters. We show that the temporal patterning of synchronous dynamics (an ecological network phenomenon) may depend on the properties of individual predator-prey patch (individual oscillator) and may vary independently of the strength of dispersal. We also show that if the dynamics of predator is slow relative to the dynamics of the prey (a situation that may promote brief but large outbreaks), dispersal-coupled predator-prey oscillating populations exhibit numerous short desynchronizations (as opposed to few long desynchronizations) and may require weaker dispersal in order to reach strong synchrony.

Circumpolar status of Arctic ptarmigan: Population dynamics and trends

Rock ptarmigan (Lagopus muta) and willow ptarmigan (L. lagopus) are Arctic birds with a circumpolar distribution but there is limited knowledge about their status and trends across their circumpolar distribution. Here, we compiled information from 90 ptarmigan study sites from 7 Arctic countries, where almost half of the sites are still monitored. Rock ptarmigan showed an overall negative trend on Iceland and Greenland, while Svalbard and Newfoundland had positive trends, and no significant trends in Alaska. For willow ptarmigan, there was a negative trend in mid-Sweden and eastern Russia, while northern Fennoscandia, North America and Newfoundland had no significant trends. Both species displayed some periods with population cycles (short 3-6 years and long 9-12 years), but cyclicity changed through time for both species. We propose that simple, cost-efficient systematic surveys that capture the main feature of ptarmigan population dynamics can form the basis for citizen science efforts in order to fill knowledge gaps for the many regions that lack systematic ptarmigan monitoring programs.

Demography of snowshoe hare population cycles

Cyclic fluctuations in abundance exhibited by some mammalian populations in northern habitats ("population cycles") are key processes in the functioning of many boreal and tundra ecosystems. Understanding population cycles, essentially demographic processes, necessitates discerning the demographic mechanisms that underlie numerical changes. Using mark-recapture data spanning five population cycles (1977-2017), we examined demographic mechanisms underlying the 9-10-yr cycles exhibited by snowshoe hares (Lepus americanus Erxleben) in southwestern Yukon, Canada. Snowshoe hare populations always decreased during winter and increased during summer; the balance between winter declines and summer increases characterized the four, multiyear cyclic phases: increase, peak, decline, and low. Little or no recruitment occurred during winter, but summer recruitment varied markedly across the four phases with the highest and lowest recruitment observed during the increase and decline phase, respectively. Population crashes during the decline were triggered by a substantial decline in winter survival and by a lack of subsequent summer recruitment. In contrast, initiation of the increase phase was triggered by a twofold increase in summer recruitment abetted secondarily by improvements in subsequent winter survival. We show that differences in peak density across cycles are explained by differences in overall population growth rate, amount of time available for population growth to occur, and starting population density. Demographic mechanisms underlying snowshoe hare population cycles were consistent across cycles in our study site but we do not yet know if similar demographic processes underlie population cycles in other northern snowshoe hare populations.

The Dynamic Shift Detector: An algorithm to identify changes in parameter values governing populations

Environmental factors interact with internal rules of population regulation, sometimes perturbing systems to alternate dynamics though changes in parameter values. Yet, pinpointing when such changes occur in naturally fluctuating populations is difficult. An algorithmic approach that can identify the timing and magnitude of parameter shifts would facilitate understanding of abrupt ecological transitions with potential to inform conservation and management of species. The “Dynamic Shift Detector” is an algorithm to identify changes in parameter values governing temporal fluctuations in populations with nonlinear dynamics. The algorithm examines population time series data for the presence, location, and magnitude of parameter shifts. It uses an iterative approach to fitting subsets of time series data, then ranks the fit of break point combinations using model selection, assigning a relative weight to each break. We examined the performance of the Dynamic Shift Detector with simulations and two case studies. Under low environmental/sampling noise, the break point sets selected by the Dynamic Shift Detector contained the true simulated breaks with 70–100% accuracy. The weighting tool generally assigned breaks intentionally placed in simulated data (i.e., true breaks) with weights averaging >0.8 and those due to sampling error (i.e., erroneous breaks) with weights averaging <0.2. In our case study examining an invasion process, the algorithm identified shifts in population cycling associated with variations in resource availability. The shifts identified for the conservation case study highlight a decline process that generally coincided with changing management practices affecting the availability of hostplant resources. When interpreted in the context of species biology, the Dynamic Shift Detector algorithm can aid management decisions and identify critical time periods related to species’ dynamics. In an era of rapid global change, such tools can provide key insights into the conditions under which population parameters, and their corresponding dynamics, can shift.

Populations naturally fluctuate in abundance, and the rules governing these fluctuations are a result of both internal (density dependent) and external (environmental) processes. For these reasons, pinpointing when changes in populations occur is difficult. In this study, we develop a novel break-point analysis tool for population time series data. Using a density dependent model to describe a population’s underlying dynamic process, our tool iterates through all possible break point combinations (i.e., abrupt changes in parameter values) and applies information-theoretic decision tools (i.e. Akaike's Information Criterion corrected for small sample sizes) to determine best fits. Here, we develop the approach, simulate data under a variety of conditions to demonstrate its utility, and apply the tool to two case studies: an invasion of multicolored Asian ladybeetle and declining monarch butterflies. The Dynamic Shift Detector algorithm identified parameter changes that correspond to known environmental change events in both case studies.

Chaos theory discloses triggers and drivers of plankton dynamics in stable environment

Despite the enticing discoveries of chaos in nature, triggers and drivers of this phenomenon remain a classical enigma which needs irrefutable empirical evidence. Here we analyze results of the yearlong replicated mesocosm experiment with multi-species plankton community that allowed revealing signs of chaos at different trophic levels in strictly controlled abiotic environment. In mesocosms without external stressors, we observed the “paradox of chaos” when biotic interactions (internal drivers) were acting as generators of internal abiotic triggers of complex plankton dynamics. Chaos was registered as episodes that vanished unpredictably or were substituted by complex behaviour of other candidates when longer time series were considered. Remarkably, episodes of chaos were detected even in the most abiotically stable conditions. We developed the Integral Chaos Indicator to validate the results of the Lyapunov exponent analysis. These findings are essential for modelling and forecasting behaviour of a variety of natural and other global systems.

Enhancing population stability with combined adaptive limiter control and finding the optimal harvesting-restocking balance

Fluctuations in population size may have negative consequences (e.g., an increased risk of extinction or the occurrence of repeated outbreaks), and many management strategies are aimed at avoiding them by either only restocking or only harvesting the population. Two of these strategies are adaptive limiter control (ALC) and adaptive threshold harvesting (ATH). With ALC the population is controlled by only restocking and with ATH by only harvesting. We propose the strategy of combined adaptive limiter control (CALC) as the combination of ALC and ATH and study the potential advantages of CALC over ALC and ATH. We consider two different population models, namely a stochastic overcompensatory model and a host-pathogen-predator model. For the first model, our results show that the combination of restocking and harvesting under CALC improves the constancy stability of the managed populations when the harvesting and restocking intensities are high enough. Otherwise the effect is marginal or in rare cases negative. For the second model, we show that combining harvesting with restocking reduces the outbreak risk only if the harvesting intensity is low. For medium harvesting intensities the effect is marginal and for high harvesting intensities the risk of outbreaks is increased. In addition, we study the optimal harvesting-restocking balance by considering a proxy of the benefit obtained in terms of the reduction in the outbreak risk and the harvesting and restocking costs.

CHAOTIC COEXISTENCE IN A RESOURCE–CONSUMER MODEL

We study the interspecies competition in a simple resource–consumer model which includes the resource supply as a dynamic variable. In the model, an organism of each species needs to consume a certain minimum rate of resource (food) to survive; a higher rate of consumption is required for reproduction. We analyze the orbit diagrams and Lyapunov exponents in order to characterize the system dynamical behavior. We observe that under a periodic food supply, the system can exhibit coexistence in the form of limit cycle oscillations. For a certain parameter range, we observe chaotic behavior emerging from successive period doublings and quasi-periodicity.

Population cycles: generalities, exceptions and remaining mysteries

Population cycles are one of nature's great mysteries. For almost a hundred years, innumerable studies have probed the causes of cyclic dynamics in snowshoe hares, voles and lemmings, forest Lepidoptera and grouse. Even though cyclic species have very different life histories, similarities in mechanisms related to their dynamics are apparent. In addition to high reproductive rates and density-related mortality from predators, pathogens or parasitoids, other characteristics include transgenerational reduced reproduction and dispersal with increasing-peak densities, and genetic similarity among populations. Experiments to stop cyclic dynamics and comparisons of cyclic and noncyclic populations provide some understanding but both reproduction and mortality must be considered. What determines variation in amplitude and periodicity of population outbreaks remains a mystery.

Co-adaptation impacts the robustness of predator-prey dynamics against perturbations

Global change threatens the maintenance of ecosystem functions that are shaped by the persistence and dynamics of populations. It has been shown that the persistence of species increases if they possess larger trait adaptability. Here, we investigate whether trait adaptability also affects the robustness of population dynamics of interacting species and thereby shapes the reliability of ecosystem functions that are driven by these dynamics. We model co-adaptation in a predator-prey system as changes to predator offense and prey defense due to evolution or phenotypic plasticity. We investigate how trait adaptation affects the robustness of population dynamics against press perturbations to environmental parameters and against pulse perturbations targeting species abundances and their trait values. Robustness of population dynamics is characterized by resilience, elasticity, and resistance. In addition to employing established measures for resilience and elasticity against pulse perturbations (extinction probability and return time), we propose the warping distance as a new measure for resistance against press perturbations, which compares the shapes and amplitudes of pre- and post-perturbation population dynamics. As expected, we find that the robustness of population dynamics depends on the speed of adaptation, but in nontrivial ways. Elasticity increases with speed of adaptation as the system returns more rapidly to the pre-perturbation state. Resilience, in turn, is enhanced by intermediate speeds of adaptation, as here trait adaptation dampens biomass oscillations. The resistance of population dynamics strongly depends on the target of the press perturbation, preventing a simple relationship with the adaptation speed. In general, we find that low robustness often coincides with high amplitudes of population dynamics. Hence, amplitudes may indicate the robustness against perturbations also in other natural systems with similar dynamics. Our findings show that besides counteracting extinctions, trait adaptation indeed strongly affects the robustness of population dynamics against press and pulse perturbations.

Virulence-driven trade-offs in disease transmission: A meta-analysis

The virulence-transmission trade-off hypothesis proposed more than 30 years ago is the cornerstone in the study of host-parasite co-evolution. This hypothesis rests on the premise that virulence is an unavoidable and increasing cost because the parasite uses host resources to replicate. This cost associated with replication ultimately results in a deceleration in transmission rate because increasing within-host replication increases host mortality. Empirical tests of predictions of the hypothesis have found mixed support, which cast doubt about its overall generalizability. To quantitatively address this issue, we conducted a meta-analysis of 29 empirical studies, after reviewing over 6000 published papers, addressing the four core relationships between (1) virulence and recovery rate, (2) within-host replication rate and virulence, (3) within-host replication and transmission rate, and (4) virulence and transmission rate. We found strong support for an increasing relationship between replication and virulence, and replication and transmission. Yet, it is still uncertain if these relationships generally decelerate due to high within-study variability. There was insufficient data to quantitatively test the other two core relationships predicted by the theory. Overall, the results suggest that the current empirical evidence provides partial support for the trade-off hypothesis, but more work remains to be done.

Predator-prey feedback in a gyrfalcon-ptarmigan system?

Specialist predators with oscillating dynamics are often strongly affected by the population dynamics of their prey, yet they are not always the cause of prey cycling. Only those that exert strong (delayed) regulation of their prey can be. Inferring predator-prey coupling from time series therefore requires contrasting models with top-down versus bottom-up predator-prey dynamics. We study here the joint dynamics of population densities of the Icelandic gyrfalcon Falco rusticolus, and its prey, the rock ptarmigan Lagopus muta. The dynamics of both species are likely not only linked to each other but also to stochastic weather variables acting as confounding factors. We infer the degree of coupling between populations, as well as forcing by abiotic variables, using multivariate autoregressive models MAR(p), with p = 1 and 2 time lags. MAR(2) models, allowing for species to cycle independently from each other, further suggest alternative scenarios where a cyclic prey influences its predator but not the other way around (i.e., bottom-up scenarios). The classical MAR(1) model predicts that the time series exhibit predator-prey feedback (i.e., reciprocal dynamic influence between prey and predator), and that weather effects are weak and only affecting the gyrfalcon population. Bottom-up MAR(2) models produced a better fit but less realistic cross-correlation patterns. Simulations of MAR(1) and MAR(2) models further demonstrate that the top-down MAR(1) models are more likely to be misidentified as bottom-up dynamics than vice versa. We therefore conclude that predator-prey feedback in the gyrfalcon-ptarmigan system is likely the main cause of observed oscillations, though bottom-up dynamics cannot yet be excluded with certainty. Overall, we showed how to make more out of ecological time series by using simulations to gauge the quality of model identification, and paved the way for more mechanistic modeling of this system by narrowing the set of important biotic and abiotic drivers.

Transferability of biotic interactions: Temporal consistency of arctic plant-rodent relationships is poor

Variability in biotic interaction strength is an integral part of food web functioning. However, the consequences of the spatial and temporal variability of biotic interactions are poorly known, in particular for predicting species abundance and distribution. The amplitude of rodent population cycles (i.e., peak-phase abundances) has been hypothesized to be determined by vegetation properties in tundra ecosystems. We assessed the spatial and temporal predictability of food and shelter plants effects on peak-phase small rodent abundance during two consecutive rodent population peaks. Rodent abundance was related to both food and shelter biomass during the first peak, and spatial transferability was mostly good. Yet, the temporal transferability of our models to the next population peak was poorer. Plant-rodent interactions are thus temporally variable and likely more complex than simple one-directional (bottom-up) relationships or variably overruled by other biotic interactions and abiotic factors. We propose that parametrizing a more complete set of functional links within food webs across abiotic and biotic contexts would improve transferability of biotic interaction models. Such attempts are currently constrained by the lack of data with replicated estimates of key players in food webs. Enhanced collaboration between researchers whose main research interests lay in different parts of the food web could ameliorate this.