The dynamics of natural populations : feedback structures in fluctuating environments

The fluctuations exhibited by natural populations have fascinated ecologists for the last eighty years. However, a vigorous debate between different schools of population ecologists has hampered reaching a consensus about the causes of such numerical fluctuations. Recent findings and a more synthetic view of population change espoused by ecologists, statisticians, and mathematicians have integrated the role of nonlinear feedback (deterministic) and external environmental (deterministic or stochastic) processes in the dynamics of natural populations. The new challenge for population ecologists is to understand how these two different forces interact in nature. In this commentary, I review some of the basic principles of population analysis during the last 50 years. Finally, this commentary emphasize that one of the most promising approaches in population ecology will be the analysis and interpretation of time series data from several species in the same place, and the integration of demographic analysis and mathematical modeling. In both cases we need long-term data of biological populations and the factors that effect them. The potential insights gained from such an approach will help ecologists to understand better the dynamics of natural populations and will have large implications for applied issues such as conservation, management, and control of natural populations.


INTRODUCTION
Understanding the dynamic of natural populations is one of the central problems in population ecology (Royama 1977, 1992, Berryman 1981, 1999).Population ecologists have long been troubled by what factors regulate population dynamics, which lead to the hotly debated controversy on the role of density-dependent (intrinsic) vs. density-independent (extrinsic) factors in determining population dynamics (Nicholson 1933, Bodenheimer 1938, Andrewartha & Birch 1954, Royama 1977, 1992, Turchin 1995).Today, most population ecologists agree in that endogenous (density-dependent) and exogenous (density-independent) factors operate simultaneously in natural populations (Murdoch 1994, Turchin 1995, Huffaker et al. 1999) and that population regulation has to be understood within the context of stochastic influences (see Royama 1977, 1992, Turchin 1995 for a detailed discussion).Therefore, demonstrating density-dependence is no longer a key issue in population ecology (Royama 1977, Berryman 1991, Turchin 1995), and has been replaced by the problem of deducing the basic properties and structure of population processes (Royama 1977, 1992, Berryman 1981, 1999, Berryman et al. 1987).
On the other hand, since May (1974) showed that very simple nonlinear deterministic population models can have very complex dynamics (but see Ricker 1954 for a earlier reference), population ecologists have been looking for the signature of nonlinear interactions in nature (Schaffer & Kot 1985, Hastings et al. 1993).Although, recent studies have clearly demonstrated how nonlinear interactions between individuals can lead population fluctuations toward complex dynamics (Constantino et al. 1995, 1997, Dennis et al. 1995, 1997, Cushing et al. 1998).1t is still not clear whether determining chaos in natural populations will help us to understand the underlying ecological processes (Stenseth 1999).Most population ecologists are aware that both nonlinear feedback (deterministic) forces and external ( abiotic) environmental variability are operating together in natural populations.The basic problems to be faced are; (l) determining the feedback structure and deducing the dominant structure of the food web, (2) determining the nature of the stochastic forces and (3) understand how these two different forces interact to give rise to the observed dynamic patterns.
In this essay, a broad and general overview of some statistical, mathematical and ecological aspects of classical population analysis is presented.I review the development of population analysis, and show the main similarities and differences in philosophy and approach among the different authors.Although (nowadays) ecology seems to be more focused toward theoretical modeling and field experiments, long-term monitoring and time series analysis of population data are (and have been) very important for understanding natural systems.Thus, I highlight some of the advancements made in population dynamics, most of them due to the efforts of collecting population data during long time periods.In fact, the collecting tradition of Eltonian (Crowcroft 1991) and Nordic ecologists (Stenseth 1999) and entomologists (Huffaker et al. 1999) has inspired and supported population dynamic studies for decades.This essay represents only a general comment, especially directed to ecologists interested in population dynamics and those who think that population dynamics studies have a central role in ecology.

MODELLING POPULATION DYNAMICS IN A STOCHAS-TIC ENVIRONMENT
Let N, be the abundance (or a proxy thereof) of a population at time t.furthermore, let εt be a noise term represented by some randomly distributed quantity (see Engen & Lande 1996, Bjørnstad et al. 1998, 1999a, for a detailed discussion about the error structure of the stochastic forces influencing population dynamics in natural and laboratory systems).A population dynamics model incorporating both system-intrinsic processes (both within the population and between various trophic levels) and stochastic influences may be depicted as: where p denotes the number of lags included in the autoregressive process (i.e., the order of the autoregressive process, see Berryman et al. 1987, Royama 1992, Turchin 1995).In the absence of stochastic noise (εt = 0, for all t) equation 1 is just a difference equation (linear or non-linear) capable of several dynamical behaviors, ranging from stability (exponential or oscillatory), through limit cycles and quasiperiodicity to chaos (May 1974, Schaffer & Kot 1985, Turchin & Taylor 1992).This noise-free case of equation 1 can be referred to as a skeleton (Tong 1993).The skeleton (function F) represents the individual interactions (within and between populations) or the feedback structure (Berryman 1981, 1999, Royama 1977, 1992).An alternative way to express equation 1 is in terms of the realized per capita population growth rates that represent the processes of individual survival and reproduction (Berryman 1999).Defining Rt = log(N,) -log(N,_ 1 ), equation 1 can be expressed as a R-function (sensu Berryman 1999): This model represents the basic feedback structure and integrates the stochastic forces that drive population dynamics in nature.The basic idea for population analysis is to choose a family of functional forms for f to fit time series data.In the following I will review some of different approaches followed by different authors.
Probably the first time series analysis of a population dynamic process was made by Moran ( 1953a) who focused on the lynx fur-returns data.Moran (1953aMoran ( , 1953b) fitted a linear secondorder autoregressive model to the log-transformed lynx data, which provides an important insight on two essential aspects of the lynx dynamics.The first, was the order of the feedback structure and the second was the cause of the synchrony of the cycles among regional populations.The order of the feedback structure represents an appropriate and useful way to deduce the food web structure (see Stenseth et al. 1997, Berryman 1999).The second insight of Moran's study was related to the effect of a correlated density-independent stochastic factor on the synchrony of regional populations ("the Moran's effect", see Royama 1992).Despite of the implications of Moran's studies they were largely ignored in the ecological literature for decades.One year later, Ricker ( 1954) introduced the concept of "reproduction curves" as the relationship " ... between the existing stock and the future stock which the existing stock produces".The author describes the different types of reproduction curves (discrete maps) that may be found in nature, and develop two important issues for population dynamics.One was how the shape of the reproduction curve determines the kind of deterministic dynamics of a population.In fact, Ricker (1954) was the first ecologist who showed that negative feedback can give rise to regular and irregular population oscillations (cycles and chaos).The second insight of Ricker's monographs was his intention to link the type of reproduction curves (and the subsequent population dynamic patterns) to some explicit biological mechanisms, such as, predation, competitive interactions and cannibalism.In addition, Ricker was very interested in the interaction between (stochastic) disturbances and reproduction curves, and how the relative importance of both factors influences population fluctuations.
Although, Moran ( 1953a, 1953b), Ricker ( 1954) and also Morris ( 1959) described the basic structures of the population dynamics processes, that is the order and the shape of the feedback structure.These insights were ignored in most of the ecological literature (but many entomologists were involved in these issues in the 60's and 70's, see Berryman 1978 for details) for more than twenty years.However, they were clearly expressed in the Royama (1977) influential work on population dynamics and the notions of density-dependence and basic population structures (see Royama 1977 pp.6-7).These basic structures are the components of the factors that determine R (per capita population growth rate), that is the Rfunction of equation ( 2), which represent a general form of the "reproduction curves" of Ricker (1954).Following Royama's (1977) notations, the factors that determine R can be classified as density uninfluenced or "exogenous" (Z) and density influenced or "endogenous" (F).Royama's scheme represents the basic structures of population dynamic processes and they can be linked to ecological time series data using equation ( 2).The factors belonging to the F category are those which conform the feedback structure of the population processes.The F factor involves the idea of first-order and second-order negative feedback (Morris 1959, Varley et al. 1973, Berryman 1981, 1999).Determining the structure of the system (F factors), that is the sign(+ or-) and order of the feedback structure, is the key issue in population dynamic studies, and in linking such studies to food web structure (Berryman 1999), a fact not widely appreciated by ecologists.
The approaches followed by the different authors to choose the type off models to represent the structure of population dynamics have been varied.Some authors have used log-linear models to describe population dynamics (Royama 1992, B j ø r n s t a d et al. 1995, Stenseth et al. 1996a, 1996b, 1997, Lima & Jaksic 1999).These kinds of models are one of the simplest representations of population dynamic processes.Log-linear autoregressive models are very useful to determine the density-dependent structure (the order of the density-dependent feedback), particularly in short time series (15-20 points).In addition, the feedback coefficients (first-and second-order) can be used for comparisons among populations or species ( B j ø r n s t a d et al. 1995et al. , Stenseth et al. 1996a) and to link time series and mathematical models (Stenseth et al. 1996b(Stenseth et al. , 1997)).However, linear models have an unrealistic ecological property, the per capita population growth rate is unbounded and also cannot exhibit complex dynamics (Royama 1992).Another option is to use theoretically-based models, such as the discrete non-linear logistic model (Berryman 1991, 1992, Royama 1992), or the two-lag linear modification of the discrete logistic model (Turchin 1990).Theoretically based time series models have the advantage that parameters have a direct biological interpretation, fulfill some basic biological and logical properties, have a parsimonious structure and also enable extrapolation beyond the data (Royama 1992).The cost comes from assuming a specific functional form for f.However, one can fit several alternative theoretical models and use model selection tools, such as cross validation or Akaike Information Criteria (AIC) (see Burnahm & Anderson 1998) to choose the best model according to the available data.
A more complex approach is to choose the functional form of f using the response surface methodology (Turchin & Taylor 1992) and neural nets (Ellner & Turchin 1995).Recently, nonparametric autoregressive models with non-Gaussian error have been used to determine the feedback structure ( B j ø r n s t a d et al. 1998).This kind of model may be estimated using the framework of Generalized Additive Models (GAM) (see Stenseth et al. 1997, B j ø r n s t a d et al. 1998, 1999a for applications to population dynamics): The basis functions f of the additive terms are d natural cubic-B splines smoothers ( B j ø r n s t a d et al. 1998).However, there are some problems, one is overfitting when there are not enough data points, and the other is the biological interpretation of the model parameters and extrapolation (see Royama 1992).Taking in consideration that population data are mostly short (15-30 points), and that some essential aspect of population dynamic structures and processes in natural systems are unknown, the use of simple theoretical models has clear advantages and merits.

INTERACTIONS BETWEEN FEEDBACK STRUCTURES AND STOCHASTICITY
A very interesting type of interaction between deterministic and stochastic processes was proposed by B j ø r n s t a d et al. (1999b) and Stenseth et al. (1999a) for explaining the two-peaked spectrum (cycles and trends) observed in the recruitment time series of a cod (Gadus morhua) population.These authors argued that the short-term cycles and long-term trends are the results of stochastic (white noise) recruitment echoed through the age structure of the population.In this case stochastic recruitment appears to interact with population-intrinsic density dependent processes (inter-cohort competitive interactions) to produce a dynamical pattern that is very different to that predicted by the purely deterministic or stochastic models.In ecological terms, the effect is caused by the inter-cohort interaction operating as a "resonator" for the variability in recruitment.As a consequence, the dynamics of this cod population would be the result of the combined age-structured deterministic processes and stochastic forces.In fact, other marine populations appear to have the potential to give rise to this type of complex dynamics when inter-cohort interactions are strong enough (Lima et al. 2000).
The recruitment dynamics of the yellow clam (Mesodesma mactroides) in Uruguayan coasts is an interesting example of how density-dependent structure and stochasticity can interact to give rise to complex dynamics for determined values of inter-cohort interactions.
On the other hand, recent studies have shown the importance of global and local climatic variables in determining the dynamics of natural populations.Large northern ungulates (Post et al. 1997, Forchhammer et al. 1998, Post & Stenseth 1999), small mammals (Merrit et al. 2001) and song birds (Saether et al. 2000) populations are influenced by a large-scale climatic variable such as the Northern Atlantic Oscillation (NAO).In the same vein, small mammal populations in semiarid regions of western South America are strongly influenced the El Nifio Southern Oscillation (ENSO) phenomena (Jaksic et al. 1997, Lima et al. 1999a, 1999b).However, understand and predicting the dynamical consequences of these climatic influences on population change is not an easy task.First, because population responses to environmental forces depend of the type of feedback structure (Royama 1992).Also, the relation between demographic responses and climate variability can be highly nonlinear implying that population change may show large responses to small climatic changes and vice versa (Saether et al. 2000).On the other hand, global climate has its own complex temporal dynamics that can be the result of a nonlinear dynamical system driven by random fluctuations (Stone et al. 1998).For example, the strong aperiodicity (every 3-7 years) observed in the El Nifio Southern Oscillation ENSO (Allen et al. 1996) may result from stochastic resonance via chaotic dynamics induced by external noise (Stone et al. 1998).Otherwise, the NAO is closely associated with global fluctuations in temperature and an increasing trend of this index has been observed during the last 30 years (Hurrel 1995).This trend in climate change will have positive effects on ungulate (Forchhammer eta!. 1998, Post & Stenseth 1999) and bird (Saether eta!. 2000) populations in northern Europe, but negative consequences for small mammal populations in eastern North America (Merrit et a!. 2001 ).

CHANGES OF THE FEEDBACK STRUCTURE
Natural populations are potentially embedded in a web of interaction with other species so that many feedback loops can be created, and a complex feedback structure is expected.However, the empirical evidence from time series analysis suggests that natural populations have feedback structures characterized by dimension one or two.This is an indication that population dynamics are controlled by simple feedback structures and that only certain feedbacks in the food web dominate the dynamics of natural populations (see Berryman 1993, 1999 for details).This phenomenon is related to the concept of limiting factors or the "law of the minimum" (Liebig 1840), and it has been integrated within a population dynamic framework by Berryman (1993Berryman ( , 1999)).However, although only one feedback is able to dominate population dynamics close to equilibrium, this can change from time to time, given some external or internal factors.Consequently it is possible to find complex regulatory structures and alternative stable states (meta-stable dynamics, see Berryman 1999) in natural populations.
For example, the feedback structures may change according to some population density threshold.This phenomenon has been clearly represented by a kind of modeling approach using Threshold Autoregressive models (TAR, see Tong 1993 ).These models are piecewise Taylor approximations to more general nonlinear models (Stenseth 1999), the fitting procedures are well developed (Tong 1993), and they have been applied recently in an ecological setting (Framstad et a!. 1997, Grenfell et a!. 1998, Stenseth et a!. 1998).The model depicted by equation ( 1) may be simplified in several ways.The function F( •) may be represented by an exponential function (e.g., Stenseth et a!.1996a, 1996b; i.e., F(•) = exp(f(•)), where f is some function of the same variables as F. It may further be argued that it is appropriate to consider the f-function to be represented primarily as a function of Xt-1 =loge (Nt-1); Xt-2 = loge (Nt-2), etc. (Berryman 1992, Royama 1992, Turchin 1995, Stenseth eta!. 1996a).A loglinear TAR model can be expressed as (see Stenseth 1999): where B;, j (i = 1 and 2) are the autoregressive parameters for the two domains.Notice that εt represent a noise term normally distributed and σ 1 is a positive parameter representing the degree of stochasticity in the two domains.The assumption about the error structure is based on the fact that natural populations are generally lognormal or gamma distributed (Engen & Lande 1996) and thus the variance is stabilized by a log-transformation (Sen & Sivrastava 1990).The biological meaning of this model lies in that a different feedback structure exists for each domain (θ), and that the threshold represents some critical density level (Fig. 1 ).For instance, aT AR model with population density as the threshold to separate the two domains has been used to describe population dynamics of the lynx (Stenseth et a!.1998a) and sheep (Grenfell et a!.1998).In this case, there are two different density-dependent structures depending on the actual or some lagged population density.If these population systems are faced with some external environmental perturbations (be it deterministic or stochastic), the density-dependent structure can switch from one state to the other depending on the magnitude of the perturbation.For example, Grenfell et a!.
( 1998) showed that the feedback structure of soay sheep populations can be represented by a firstorder feedback above the population threshold, while below the threshold the realized population per capita growth rate is constant and positive (exponential growth) (Fig. 1 A).On the other hand, the observed population dynamics of the lynx across northern Canada appears to be the result of a common TAR structure (first-and second-order feedback), but second-order feedback are stronger above the population threshold (Stenseth et a!.1998) (Fig. 1B).
Another approach to the problem of feedback dominance, thresholds and shift in the feedback structures is based on the interaction between negative and positive feedbacks (Berryman 1999).Because the opposed forces of negative (e.g., competition) and positive (co-operation) feedbacks complex R-functions can be produced with three (or more) equilibrium points (Fig. 2A).The population dynamics of such feedback structure can shift from one to other stable state given the existence of an unstable intermediate equilibrium point or escape threshold where positive feedbacks dominate (Fig. 2A).This kind of dynamics is called meta-stable (Berryman 1999) and can be approached by using two different R-functions

Below threshold Nt
Above threshold Nt A

Population density
Fig. 1: Changes in the feedback structure according to population density.(A) The population rate of change (R-function) is constant (exponential growth) below the population threshold and there is a negative feedback above the population threshold (see Grenfell et al. 1998 for details).(B) Below the population threshold (left inset) the realized per capita growth rate showed first-order (Nt-1) and weak second-order (N ) feedback, while above the population threshold (right inset) the second-order feedback is stronger (see Stenseth et al. 1998 for details).
for each domain (equation 4) because natural populations are almost never observed near unstable equilibrium (Fig. 2B).Population dynamics regulated by two different first-order feedbacks exhibit sudden transitions from low-to high-density dynamics, while catastrophic "boom and bust" dynamics are produced if second-order feedback dominate at high-densities (Berryman 1999).One of the best examples of a meta-stable system is population dynamics of bark beetles and how they escape tree defenses (by positive feedback) producing massive tree mortality in forest ecosystems (Berryman et al. 1984 ).
A somewhat different kind of change in the feedback structure has been recently proposed for explaining rodent outbreaks in semiarid Chile.It has been proposed that the frequent (but aperiodic) rodent outbreaks ("ratadas") are the outcome of shift in feedback dominance mediated by El Niño disturbances (Lima, Stenseth, Crespin, Yoccoz & Jaksic unplublished results).A strong first-order feedback will dominate population dynamics during low-rainfall periods (competition for food limitation) producing low population density and saw-toothed oscillations (Fig. 3A), while second-order feedback, suggestive of

Population density
Fig. 2: (A) Example of a complex R-function, which is characterized by negative feedback at low densities, positive feedback at intermediate densities and at high-density population is limited again by a negative feedback.This R-function is characterized by showing two stable (1 and 3) and one unstable (2) equilibrium.Predators that switch or aggregate in response to prey densities are able to create this feedback structure in prey populations (see Berryman 1999 pp.75 for details).(B) Approximation to this complex R-function by two first-order stabilizing R-functions because natural populations are almost never observed near unstable equilibrium.
trophic interactions (predators or plants), dominates during high-density and high-rainfall years and causes low frequency dynamics (Fig. 3B).Consequently, rodent outbreaks ("ratadas") in semiarid Chile may be caused by the interaction between climatic variability (ENSO), intra-specific interactions and food web structure.

SEASONAL DENSITY-DEPENDENT STRUCTURE
Recent studies on cyclic species of voles show an important consequence of seasonality; namely, density-dependent regulation that differs between seasons (Mallorie & Flowerdew 1994, Hansen et al. 1999).In northern Fennoscandia winter regulation appears to be the crucial factor in the production of multi-annual cycles of the vole community (Hansen et al. 1999).Also, many small rodent populations living in seasonal environ-ments experience different sequential density dependent processes within a year (Nelson et al. 1991, Hornfeldt 1994, Morris 1996).Hansen et al. (1999) represented the seasonal feedback structure using an autoregressive model that considers spring and autumn abundances as separate variables.Let St and Ft be the log abundances in the year t at spring and fall respectively, the population growth rates of winter and summer can be written as a linear function of their seasonal densities: Rwt and Rst represent the population growth rates during winter and summer respectively.The ai parameters represent the feedback acting during the winter and the bi parameters the feedback Cambios de Ia estructura de retroalimentaci6n en respuesta a una perturbaci6n ex6gena (e.g., El Niño) en Ia dinamica poblacional dellauch6n orejudo (Phyllotis darwini).La tasa de cambio poblacional (funci6n-R) modi fica su estructura de retroalimentaci6n con el nivel de precipitaciones.Durante años de bajas precipitaciones (La Niña) Ia funci6n-R y Ia dinamica poblacional son dominadas por una retroalimentaci6n de primer orden (N,_ 1 ), mientras que durante los años lluviosos (e.g., EL Niño) una retroalimentaci6n de segundo orden (N,_ 2 ) se vuelve dominante.
during the summer.The parameter a 1 , a 2 , b 1 and b 2 measure first-order feedback, while second-order feedback is represented by the other parameters and the noise terms are represented by εwt and εst, One important result is that winter regulation appears to be the crucial element in producing multi-annual cycles in gray-sided vole populations (Hansen et al. 1999) (Fig. 4).The comparison of this result to the seasonal dynamics of southern red-backed voles in eastern North America showed some interesting insights on seasonal regulation in vole population dynamics Estructura de retroalimentaci6n estacional del roedor (Clethrionomys rufocanus) en el norte de Finlandia (Hansen eta!. 1999).(A) Regulaci6n durante el invierno caracterizada por una fuerte retroalimentaci6n de primer y segundo orden (Rwt = tasa de cambio poblacional durante los inviernos; Ft-1 = densidad durante el otoiio previo; St-2 = densidad durante la primavera dos aiios antes).(B).Regulaci6n durante el verano caracterizada por una débil retroalimentaci6n de primer y segundo orden (Rst = tasa de cambio poblacional durante los veranos; Ft-1 = densidad durante el otoiio previo; Ft-2 = densidad durante el otoiio dos aiios antes).(Merrit et al. 2001) (Fig. 5).Hansen et al. (1999) found that one of the most important factors that may explain cyclic dynamics in gray-sided voles is a seasonal change in the strength of regulation.They found that the gray-sided vole seasonal dynamics at Fennoscandia was characterised by a stronger first-and second-order negative feedback structure in winters than in summers, implying that a major part of the regulation occurs during winter (Fig. 4 ).However, seasonal regulation in southern red-backed voles switches from firstand second-order feedback in winter to only first- order feedback in summer (Fig. 5).This dominance of first-order feedback along the year may explain why southern red-backed voles did not exhibit cyclic dynamics as gray-sided voles in Fennoscandia.On the other hand, this kind of approach makes insights about the food web structure of these two ecosystems.Despite of the apparent complexity of the food webs in both systems (boreal and deciduous forests), the realised dynamics of gray-sided voles in Fennoscandia is of order two (due to trophic interactions during winters), while the dynamics of southern red-backed voles is of order one.The paradox is that despite being embedded in a richer small mammal and predator community, the southern red-backed vole dynamics suggest a simpler food web structure, mostly dominated by first order feedbacks.

CONCLUSIONS
The issues discussed above emphasize the importance of determining the feedback structure and the exogenous factors for understanding population dynamics in nature (Royama 1977, 1992, Berryman 1981, 1999, Berryman & Millstein 1994, Turchin 1995, Stenseth et al. 1996b, 1998, Stenseth 1999).It is clear that the underlying mechanisms of the interaction between environmental variation and regulatory structure have not been fully explored in population ecology.There is much work to be done for ecologists, statisticians, and mathematicians in order to develop new conceptual frameworks, statistical methods and mathematical tools to deal with the questions raised by this theme.Although these new questions render population dynamics and demographic studies a little bit more complex, it is certainly true that they also bring new challenges and open new and exciting research opportunities.
I believe that the major advances in the understanding of population dynamics had come from the analysis and interpretations of ecological time series.Of course that mathematical modeling and experimentation are important elements in population ecology, but without time series data and analysis the crucial point in population ecology is lost; that is, the temporal pattern of fluctuation (the population phenomena).Thus, without a clear description and interpretation of the population dynamic pattern, it is very difficult to see how plausible biological hypothesis can be constructed (Berryman 1999).I think that the most promising approach in population ecology will be the analysis and interpretation of time series data from several species in the same place (Berryman 1999) and the integration of demographic analysis and mathematical modeling (Stenseth 1999).In both cases we need long-term data of biological populations and the factors that effect them.The potential insights gained from such interactions will help ecologists to understand better the dynamics of natural populations and will have large implications for applied issues such as conservation, management, and control of natural populations.

Fig. 3 :
Fig.3: Changes of feedback structure according to some exogenous perturbation (e.g., El Niño) for leafeared mice (Phyllotis darwini) population dynamics.The population rate of change (R-function) changes the feedback structure in relation with the rainfall level.Note that during low-rainfall years (La Niña drought-years) a first-order feedback (Nt-1) dominates the R-function and population dynamics, while during high-rainfall years a second-order feedback (Nt-2) become dominant.

Fig. 4 :
Fig. 4: Seasonal feedback structure of gray-sided voles (Clethrionomys rufocanus) in Fennoscandia (from Hansen et al. 1999).(A) Winter regulation characterized by strong first-and second-order negative feedback (Rwt =population growth rates during winters; Ft-1 =previous autumn density; st-2 = spring density two years before).(B) Summer regulation characterized by weak first-and second-order negative feedback (Rst =population growth rates during summers; Ft-1 =previous autumn density; Ft-2 =autumn density two years before).