Phase- and density-dependent population dynamics in Norwegian lemmings: interaction between deterministic and stochastic processes

Temporal Dynamics of the Blueberry Maggot Fly (Diptera: Tephritidae) and its Primary Parasitoid, Biosteres melleus (Hymenoptera: Braconidae), in the Maine Wild Blueberry System

Between 1998 and 2017, we conducted studies in wild blueberry, Vaccinium angustifolium Aiton (Ericales: Ericaceae), to elucidate the temporal dynamics of the blueberry maggot fly, Rhagoletis mendax Curran, and its parasitoid, Biosteres melleus (Gahan). A predictive model for the emergence of R. mendax was validated at two sites over 3 yr. A second predictive model for the major parasitoid, B. melleus, of R. mendax was constructed and suggests that the delay in emergence of the parasitoid relative to its host provides a period or 'biological window' of 9 d where insecticide sprays can be applied to manage R. mendax with a limited impact on the parasitoid. A 20-yr study on the parasitoid/host dynamics showed parasitism rates ranging from 0.5 to 28.2%. It appears that R. mendax populations in Maine wild blueberry are characterized by stable equilibrium dynamics, significantly affected by stochastic processes. There was a weak, but significant relationship between B. melleus density and R. mendax intrinsic rates of growth. Our data suggest that R. mendax population dynamics in wild blueberry is characterized by an unstable equilibrium tipping point of 7.9 maggots per liter of blueberries or an average of 10 flies per trap.

An Inverse Problem: Trappers Drove Hares to Eat Lynx

The Canadian lynx and snowshoe hare pelt data by the Hudson Bay Company did not fit the classical predator-prey theory. Rather than following the peak density of the hare, that of the lynx leads it, creating the hares-eat-lynx (HEL) paradox. Although trappers were suspected to play a role, no mathematical model has ever demonstrated the HEL effect. Here we show that the long-held assumption that the pelt number is a proxy of the wild populations is false and that when the data are modeled by the harvest rates by the trappers, the problem is finally resolved: both the HEL paradox and the classical theory are unified in our mechanistic hare-lynx-competitor-trapper (HLCT) model where competitor stands for all predators of the hares other than the lynx. The result is obtained by systematically fitting the data to various models using Newton's inverse problem method. Main findings of this study include: the prey-eats-predator paradox in kills by an intraguild top-predator can occur if the top-predator prefers the predator to the prey; the benchmark HLCT model is more sensitive to all lynx-trapper interactions than to the respective hare-trapper interactions; the Hudson Bay Company's hare pelt number maybe under-reported; and, the most intriguing of all, the trappers did not interfere in each other's trapping activities.

Graph-facilitated resonant mode counting in stochastic interaction networks

Oscillations in dynamical systems are widely reported in multiple branches of applied mathematics. Critically, even a non-oscillatory deterministic system can produce cyclic trajectories when it is in a low copy number, stochastic regime. Common methods of finding parameter ranges for stochastically driven resonances, such as direct calculation, are cumbersome for any but the smallest networks. In this paper, we provide a systematic framework to efficiently determine the number of resonant modes and parameter ranges for stochastic oscillations relying on real root counting algorithms and graph theoretic methods. We argue that stochastic resonance is a network property by showing that resonant modes only depend on the squared Jacobian matrix J2, unlike deterministic oscillations which are determined by J By using graph theoretic tools, analysis of stochastic behaviour for larger interaction networks is simplified and stochastic dynamical systems with multiple resonant modes can be identified easily.

Variation in stability of elk and red deer populations with abiotic and biotic factors at the species-distribution scale

Stability in population dynamics is an emergent property of the interaction between direct and delayed density dependence, the strengths of which vary with environmental covariates. Analysis of variation across populations in the strength of direct and delayed density dependence can reveal variation in stability properties of populations at the species level. We examined the stability properties of 22 elk/red deer populations in a two-stage analysis. First, we estimated direct and delayed density dependence applying an AR(2) model in a Bayesian hierarchical framework. Second, we plotted the coefficients of direct and delayed density dependence in the Royama parameter plane. We then used a hierarchical approach to test the significance of environmental covariates of direct and delayed density dependence. Three populations exhibited highly stable and convergent dynamics with strong direct, and weak delayed, density dependence. The remaining 19 populations exhibited more complex dynamics characterized by multi-annual fluctuations. Most (15 of 19) of these exhibited a combination of weak to moderate direct and delayed density dependence. Best-fit models included environmental covariates in 17 populations (77% of the total). Of these, interannual variation in growing-season primary productivity and interannual variation in winter temperature were the most common, performing as the best-fit covariate in six and five populations, respectively. Interannual variation in growing-season primary productivity was associated with the weakest combination of direct and delayed density dependence, while interannual variation in winter temperature was associated with the strongest combination of direct and delayed density dependence. These results accord with a classic theoretical prediction that environmental variability should weaken population stability. They furthermore suggest that two forms of environmental variability, one related to forage resources and the other related to abiotic conditions, both reduce stability, but in opposing fashion: one through weakened direct density dependence and the other through strengthened delayed density dependence. Importantly, however, no single abiotic or biotic environmental factor emerged as generally predictive of the strengths of direct or delayed density dependence, nor of the stability properties emerging from their interaction. Our results emphasize the challenges inherent to ascribing primacy to drivers of such parameters at the species level and distribution scale.

Overcompensation and phase effects in a cyclic common vole population: between first and second-order cycles

Population cycles in voles are often thought to be generated by one-year delayed density dependence on the annual population growth rate. In common voles, however, it has been suggested by Turchin (2003) that some populations exhibit first-order cycles, resulting from strong overcompensation (i.e. carrying capacity overshoots in peak years, with only an effect of the current year abundance on annual growth rates). We focus on a common vole (Microtus arvalis) population from western France that exhibits 3-year cycles. Several overcompensating nonlinear models for populations dynamics are fitted to the data, notably those of Hassell, and Maynard-Smith and Slatkin. Overcompensating direct density dependence (DD) provides a satisfactory description of winter crashes, and one-year delayed density dependence is not responsible for the crashes, thus these are not classical second-order cycles. A phase-driven modulation of direct density dependence maintains a low-phase, explaining why the cycles last three years instead of two. Our analyses suggest that some of this phase dependence can be expressed as one-year delayed DD, but phase dependence provides a better description. Hence, modelling suggests that cycles in this population are first-order cycles with a low phase after peaks, rather than fully second-order cycles. However, based on the popular log-linear second-order autoregressive model, we would conclude only that negative delayed density dependence exists. The additive structure of this model cannot show when delayed DD occurs (here, during lows rather than peaks). Our analyses thus call into question the automated use of second-order log-linear models, and suggests that more attention should be given to non-(log)linear models when studying cyclic populations. From a biological viewpoint, the fast crashes through overcompensation that we found suggest they might be caused by parasites or food rather than predators, though predators might have a role in maintaining the low phase and spatial synchrony.

Specific non-monotonous interactions increase persistence of ecological networks

The relationship between stability and biodiversity has long been debated in ecology due to opposing empirical observations and theoretical predictions. Species interaction strength is often assumed to be monotonically related to population density, but the effects on stability of ecological networks of non-monotonous interactions that change signs have not been investigated previously. We demonstrate that for four kinds of non-monotonous interactions, shifting signs to negative or neutral interactions at high population density increases persistence (a measure of stability) of ecological networks, while for the other two kinds of non-monotonous interactions shifting signs to positive interactions at high population density decreases persistence of networks. Our results reveal a novel mechanism of network stabilization caused by specific non-monotonous interaction types through either increasing stable equilibrium points or reducing unstable equilibrium points (or both). These specific non-monotonous interactions may be important in maintaining stable and complex ecological networks, as well as other networks such as genes, neurons, the internet and human societies.

Stochastic formulation of ecological models and their applications

The increasing use of computer simulation by theoretical ecologists started a move away from models formulated at the population level towards individual-based models. However, many of the models studied at the individual level are not analysed mathematically and remain defined in terms of a computer algorithm. This is not surprising, given that they are intrinsically stochastic and require tools and techniques for their study that may be unfamiliar to ecologists. Here, we argue that the construction of ecological models at the individual level and their subsequent analysis is, in many cases, straightforward and leads to important insights. We discuss recent work that highlights the importance of stochastic effects for parameter ranges and systems where it was previously thought that such effects would be negligible.

Cycles, stochasticity and density dependence in pink salmon population dynamics

Complex dynamics of animal populations often involve deterministic and stochastic components. A fascinating example is the variation in magnitude of 2-year cycles in abundances of pink salmon (Oncorhynchus gorbuscha) stocks along the North Pacific rim. Pink salmon have a 2-year anadromous and semelparous life cycle, resulting in odd- and even-year lineages that occupy the same habitats but are reproductively isolated in time. One lineage is often much more abundant than the other in a given river, and there are phase switches in dominance between odd- and even-year lines. In some regions, the weak line is absent and in others both lines are abundant. Our analysis of 33 stocks indicates that these patterns probably result from stochastic perturbations of damped oscillations owing to density-dependent mortality caused by interactions between lineages. Possible mechanisms are cannibalism, disease transmission, food depletion and habitat degradation by which one lineage affects the other, although no mechanism has been well-studied. Our results provide comprehensive empirical estimates of lagged density-dependent mortality in salmon populations and suggest that a combination of stochasticity and density dependence drives cyclical dynamics of pink salmon stocks.

Linking climate change to lemming cycles

The population cycles of rodents at northern latitudes have puzzled people for centuries, and their impact is manifest throughout the alpine ecosystem. Climate change is known to be able to drive animal population dynamics between stable and cyclic phases, and has been suggested to cause the recent changes in cyclic dynamics of rodents and their predators. But although predator-rodent interactions are commonly argued to be the cause of the Fennoscandian rodent cycles, the role of the environment in the modulation of such dynamics is often poorly understood in natural systems. Hence, quantitative links between climate-driven processes and rodent dynamics have so far been lacking. Here we show that winter weather and snow conditions, together with density dependence in the net population growth rate, account for the observed population dynamics of the rodent community dominated by lemmings (Lemmus lemmus) in an alpine Norwegian core habitat between 1970 and 1997, and predict the observed absence of rodent peak years after 1994. These local rodent dynamics are coherent with alpine bird dynamics both locally and over all of southern Norway, consistent with the influence of large-scale fluctuations in winter conditions. The relationship between commonly available meteorological data and snow conditions indicates that changes in temperature and humidity, and thus conditions in the subnivean space, seem to markedly affect the dynamics of alpine rodents and their linked groups. The pattern of less regular rodent peaks, and corresponding changes in the overall dynamics of the alpine ecosystem, thus seems likely to prevail over a growing area under projected climate change.

Modelling non-additive and nonlinear signals from climatic noise in ecological time series: Soay sheep as an example

Understanding how climate can interact with other factors in determining patterns of species abundance is a persistent challenge in ecology. Recent research has suggested that the dynamics exhibited by some populations may be a non-additive function of climate, with climate affecting population growth more strongly at high density than at low density. However, we lack methodologies to adequately explain patterns in population growth generated as a result of interactions between intrinsic factors and extrinsic climatic variation in non-linear systems. We present a novel method (the Functional Coefficient Threshold Auto-Regressive (FCTAR) method) that can identify interacting influences of climate and density on population dynamics from time-series data. We demonstrate its use on count data on the size of the Soay sheep population, which is known to exhibit dynamics generated by nonlinear and non-additive interactions between density and climate, living on Hirta in the St Kilda archipelago. The FCTAR method suggests that climate fluctuations can drive the Soay sheep population between different dynamical regimes--from stable population size through limit cycles and non-periodic fluctuations.

Discrete and continuous state population models in a noisy world

Simple ecological models operate mostly with population densities using continuous variables. However, in reality densities could not change continuously, since the population itself consists of integer numbers of individuals. At first sight this discrepancy appears to be irrelevant, nevertheless, it can cause large deviations between the actual statistical behaviour of biological populations and that predicted by the corresponding models. We investigate the conditions under which simple models, operating with continuous numbers of individuals can be used to approximate the dynamics of populations consisting of integer numbers of individuals. Based on our definition for the (statistical) distance between the two models we show that the continuous approach is acceptable as long as sufficiently high biological noise is present, or, the dynamical behaviour is regular (non-chaotic). The concepts are illustrated with the Ricker model and tested on the Tribolium castaneum data series. Further, we demonstrate with the help of T. castaneum's model that if time series are not much larger than the possible population states (as in this practical case) the noisy discrete and continuous models can behave temporarily differently, almost independently of the noise level. In this case the noisy, discrete model is more accurate [OR has to be applied].

Testing for common structures in a panel of threshold models

We consider the problem of examining the extent of (partial) similarity in the dynamics of a panel of independent threshold autoregressive processes. We develop some tests for common structure via Wald's approach and by checking whether the parameter estimates of the unconstrained threshold models satisfy the constraints defining the common structure. One test concerns the equality of independent ratios of normal means, which is shown to have nonstandard asymptotic null distribution. These tests are illustrated with a modern panel of Canadian lynx data; our analysis suggests that the lynx data over Canada share similar dynamics in the decrease phase, but they appear to be different in the increase phase.

Smoothing for spatiotemporal models and its application to modeling muskrat-mink interaction

For a set of spatially dependent dynamical models, we propose a method for estimating parameters that control temporal dynamics by spatial smoothing. The new approach is particularly relevant for analyzing spatially distributed panels of short time series. The asymptotic results show that spatial smoothing will improve the estimation in the presence of nugget effect, even when the sample size in each location is large. The proposed methodology is used to analyze the annual mink and muskrat data collected in a period of 25 years in 81 Canadian locations. Based on the proposed method, we are able to model the temporal dynamics which reflects the food chain interaction of the two species.

Quasicycles revisited: apparent sensitivity to initial conditions

Environmental noise is known to sustain cycles by perturbing a deterministic approach to equilibrium that is itself oscillatory. Quasicycles produced in this way display a regular period but varied amplitude. They were proposed by Nisbet and Gurney (Nature 263 (1976) 319) as one possible explanation for population fluctuations in nature. Here, we revisit quasicyclic dynamics from the perspective of nonlinear time series analysis. Time series are generated with a predator-prey model whose prey's growth rate is driven by environmental noise. A method for the analysis of short and noisy data provides evidence for sensitivity to initial conditions, with a global Lyapunov exponent often close to zero characteristic of populations 'at the edge of chaos'. Results with methods restricted to long time series are consistent with a finite-dimensional attractor on which dynamics are sensitive to initial conditions. These results are compared with those previously obtained for quasicycles in an individual-based model with heterogeneous spatial distributions. Patterns of sensitivity to initial conditions are shown to differentiate phase-forgetting from phase-remembering quasicycles involving a periodic driver. The previously reported mode at zero of Lyapunov exponents in field and laboratory populations may reflect, in part, quasicyclic dynamics.

Rates of increase in age-structured populations: a lesson from the European harbour seals

Behavioural differences among population segments coupled with the transient dynamics of perturbed population structures lead to severely biased estimates of the intrinsic rates of increase in natural populations. This phenomenon is expected to occur in most populations that are structured by age, sex, state, or rank. The 1988 epizootic in European harbour seals (Phoca vitulina) perturbed the population composition radically. Detailed documentation of mass mortality, 20 years of population surveys, and data on age- and sex-specific behaviour were used to quantify biases in the observed rate of increase (λobs.), which in many areas substantially exceeded the "maximum rate of increase". This is serious, since λobs. is a key parameter, for example, in estimating potential biological removal or modelling population dynamics. For populations where the underlying age and sex composition is unknown, we suggest that data on fecundity and survival rates be used to find the upper theoretical rate of population increase. We found that the intrinsic rates of increase (λ1) in populations of true seals with even sex ratios and stable age structures cannot exceed 13% per year (λ1max. = 1.13). Frequently reported larger values are indicative of nonstable population structures or populations affected by migrations.

Spatial structure of lemming populations (Dicrostonyx groenlandicus) fluctuating in density

The pattern and scale of the genetic structure of populations provides valuable information for the understanding of the spatial ecology of populations, including the spatial aspects of density fluctuations. In the present paper, the genetic structure of periodically fluctuating lemmings (Dicrostonyx groenlandicus) in the Canadian Arctic was analysed using mitochondrial DNA (mtDNA) control region sequences and four nuclear microsatellite loci. Low genetic variability was found in mtDNA, while microsatellite loci were highly variable in all localities, including localities on isolated small islands. For both genetic markers the genetic differentiation was clear among geographical regions but weaker among localities within regions. Such a pattern implies gene flow within regions. Based on theoretical calculations and population census data from a snap-trapping survey, we argue that the observed genetic variability on small islands and the low level of differentiation among these islands cannot be explained without invoking long distance dispersal of lemmings over the sea ice. Such dispersal is unlikely to occur only during population density peaks.

Common structure in panels of short ecological time-series

Typically, in many studies in ecology, epidemiology, biomedicine and others, we are confronted with panels of short time-series of which we are interested in obtaining a biologically meaningful grouping. Here, we propose a bootstrap approach to test whether the regression functions or the variances of the error terms in a family of stochastic regression models are the same. Our general setting includes panels of time-series models as a special case. We rigorously justify the use of the test by investigating its asymptotic properties, both theoretically and through simulations. The latter confirm that for finite sample size, bootstrap provides a better approximation than classical asymptotic theory. We then apply the proposed tests to the mink-muskrat data across 81 trapping regions in Canada. Ecologically interpretable groupings are obtained, which serve as a necessary first step before a fuller biological and statistical analysis of the food chain interaction.

Geographic variation in population cycles of Canadian muskrats (Ondatra zibethicus)

We investigated the dynamic properties of population cycles in Canadian muskrats (Ondatra zibethicus). Ninety-one historic time series of muskrat-harvest data obtained from the Hudson's Bay Company Archives were analyzed. Most series were 25 years in length (1925–1949) and were distributed primarily throughout five ecozones. For each series, we estimated period length and coefficients for a second-order autoregressive model. Estimated period length varied between 3 and 13 years, with 3- to 5-year periods located in Subarctic-Arctic ecozones. We hypothesize that the 4-year cycles are largely a result of predation by red fox (Vulpes vulpes), which exhibit 4-year cycles in Arctic regions. The remaining ecozones generally averaged 8–9 years in period length. However, the relative contributions of direct and delayed density dependence varied along a latitudinal gradient. We hypothesize that both social and trophic interactions are necessary to produce the observed dynamics, but that shifts in the nature of mink predation were responsible for the changes in the relative contribution of direct and delayed density dependence. Essentially, there is a tension between population-intrinsic and trophic interactions that may bound the length of the cycle.

Are lemmings prey or predators?

Large oscillations in the populations of Norwegian lemmings have mystified both professional ecologists and lay public. Ecologists suspect that these oscillations are driven by a trophic mechanism: either an interaction between lemmings and their food supply, or an interaction between lemmings and their predators. If lemming cycles are indeed driven by a trophic interaction, can we tell whether lemmings act as the resource ('prey') or the consumer ('predator')? In trophic interaction models, peaks of resource density generally have a blunt, rounded shape, whereas peaks of consumer density are sharp and angular. Here we have applied several statistical tests to three lemming datasets and contrasted them with comparable data for cyclic voles. We find that vole peaks are blunt, consistent with their cycles being driven by the interaction with predators. In contrast, the shape of lemming peaks is consistent with the hypothesis that lemmings are functional predators, that is, their cycles are driven by their interaction with food plants. Our findings suggest that a single mechanism, such as interaction between rodents and predators, is unlikely to provide the 'universal' explanation of all cyclic rodent dynamics.

Common dynamic structure of canada lynx populations within three climatic regions

Across the boreal forest of Canada, lynx populations undergo regular density cycles. Analysis of 21 time series from 1821 onward demonstrated structural similarity in these cycles within large regions of Canada. The observed population dynamics are consistent with a regional structure caused by climatic features, resulting in a grouping of lynx population dynamics into three types (corresponding to three climatic-based geographic regions): Pacific-maritime, Continental, and Atlantic-maritime. A possible link with the North Atlantic Oscillation is suggested.

Multiannual Vole Cycles and Population Regulation during Long Winters: An Analysis of Seasonal Density Dependence

The small-rodent community in northern Fennoscandia shows regular multiannual fluctuations in abundance. We analyze time series of spring and fall abundances covering 44 yr from the vole community at Kilpisjärvi, northern Finnish Lapland, which is dominated by the gray-sided vole (Clethrionomys rufocanus). Treating spring and fall abundances as variables in a bivariate time-series model, we study seasonal aspects of population regulation and thereby utilize more of the available data than in most previous analyses of rodent fluctuations. The analysis demonstrates substantial density dependence of population growth during the winter. This includes a strong direct effect of previous fall density and a strong 2-yr delayed effect of spring density. Population growth during the summer shows evidence of delayed density dependence but is generally weaker than that during the winter season. Winter regulation appears to be a crucial element in the production of multiannual cycles in this community, and we suggest that long and severe winters coupled with delayed density dependence may be a direct cause of the vole fluctuations in northern and high-elevation areas.