Scaling of the mean and variance of population dynamics under fluctuating regimes.

Theoretical ecologists have long sought to understand how the persistence of populations depends on the interactions between exogenous (biotic and abiotic) and endogenous (e.g., demographic and genetic) drivers of population dynamics. Recent work focuses on the autocorrelation structure of environmental perturbations and its effects on the persistence of populations. Accurate estimation of extinction times and especially determination of the mechanisms affecting extinction times is important for biodiversity conservation. Here we examine the interaction between environmental fluctuations and the scaling effect of the mean population size with its variance. We investigate how interactions between environmental and demographic stochasticity can affect the mean time to extinction, change optimal patch size dynamics, and how it can alter the often-assumed linear relationship between the census size and the effective population size. The importance of the correlation between environmental and demographic variation depends on the relative importance of the two types of variation. We found the correlation to be important when the two types of variation were approximately equal; however, the importance of the correlation diminishes as one source of variation dominates. The implications of these findings are discussed from a conservation and eco-evolutionary point of view.

Environmental fluctuations and level of density-compensation strongly affects the probability of fixation and fixation times.

The probability of, and time to, fixation of a mutation in a population has traditionally been studied by the classic Wright-Fisher model where population size is constant. Recent theoretical expansions have covered fluctuating populations in various ways but have not incorporated models of how the environment fluctuates in combination with different levels of density-compensation affecting fecundity. We tested the hypothesis that the probability of, and time to, fixation of neutral, advantageous and deleterious mutations is dependent on how the environment fluctuates over time, and on the level of density-compensation. We found that fixation probabilities and times were dependent on the pattern of autocorrelation of carrying capacity over time and interacted with density-compensation. The pattern found was most pronounced at small population sizes. The patterns differed greatly depending on whether the mutation was neutral, advantageous, or disadvantageous. The results indicate that the degree of mismatch between carrying capacity and population size is a key factor, rather than population size per se, and that effective population sizes can be very low also when the census population size is far above the carrying capacity. This study highlights the need for explicit population dynamic models and models for environmental fluctuations for the understanding of the dynamics of genes in populations.

Divergent timing of egg-laying may maintain life history polymorphism in potentially multivoltine insects in seasonal environments.

The length of the favourable season determines voltinism in insect populations. In some insects, there is variation in fecundity and timing of reproduction among females. If the length of the favourable season does not allow all offspring to develop into adults without diapause, the benefits of high early fecundity may outweigh the associated cost of low lifetime fecundity. We tested this by exploring mating frequencies of Pieris napi females along a latitudinal gradient in different generations. Pieris napi is a bivoltine butterfly, and genetically polyandrous females enjoy higher lifetime fecundity than monandrous ones. Polyandry is, however, coupled with a relatively low early fecundity. We found that monandrous females are more likely to produce an additional generation than polyandrous ones under conditions that allow production of only a partial summer generation. Monandrous females were also the first to emerge and slightly over-represented in the summer generation under conditions that allow the development of a complete summer generation. Further, a stochastic model shows that variation in the timing of reproduction between strategies is sufficient to explain the observed patterns. Thus, seasonality may counter-select against polyandry, or more generally against low early reproductive rate, and promote maintenance of polymorphism in life history strategies.

Predominance of outcrossing in Lymnaea stagnalis despite low apparent fitness costs of self-fertilization.

We have quantified the natural mating system in eight populations of the simultaneously hermaphroditic aquatic snail Lymnaea stagnalis, and studied the ecological and genetic forces that may be directing mating system evolution in this species. We investigated whether the natural mating system can be explained by the availability of mates, by the differential survival of self- and cross-fertilized snails in nature, and by the effects of mating system on parental fecundity and early survival. The natural mating system of L. stagnalis was found to be predominantly cross-fertilizing. Density of snails in the populations had no relationship with the mating system, suggesting that outcrossing rates are not limited by mate availability at the population densities observed. Contrary to expectations for outcrossing species, we detected no evidence for inbreeding depression in survival in nature with inferential population genetic methods. Further, experimental manipulations of mating system in the laboratory revealed that self-fertilization had no effect on parental fecundity, and only minor effects on offspring survival. Predominance of cross-fertilization despite low apparent fitness costs of self-fertilization is at odds with the paradigm that high self-fertilization depression is necessary for maintenance of cross-fertilization in self-compatible hermaphrodites.

Self-organized dynamics in spatially structured populations.

Self-organization and pattern formation represent the emergence of order in temporal and spatial processes. Self-organization in population ecology is gaining attention due to the recent advances concerning temporal fluctuations in the population size of dispersal-linked subunits. We shall report that spatially structured models of population renewal promote the emergence of a complex power law order in spatial population dynamics. We analyse a variety of population models showing that self-organization can be identified as a temporal match in population dynamics among local units, and how the synchrony changes in time. Our theoretical results are concordant with analyses of population data on the Canada lynx.

Is the impact of environmental noise visible in the dynamics of age-structured populations?

Climate change has ignited lively research into its impact on various population-level processes. The research agenda in ecology says that some of the fluctuations in population size are accountable for by the external noise (e.g. weather) modulating the dynamics of populations. We obeyed the agenda by assuming population growth after a resource-limited Leslie matrix model in an age-structured population. The renewal process was disturbed by superimposing noise on the development of numbers in one or several age groups. We constructed models for iteroparous and semelparous breeders so that, for both categories, the population growth rate was matching. We analysed how the modulated population dynamics correlates with the noise signal with different time-lags. No significant correlations were observed for semelparous breeders, whereas for iteroparous breeders high correlations were frequently observed with time-lags of 71 year or longer. However, the latter occurs under red-coloured noise and for low growth rates when the disturbance is on the youngest age group only. It is laborious to find any clear signs of the (red) noise- and age group-specific fluctuations if the disturbance influences older age groups only. These results cast doubts on the possibility of detecting the signature of external disturbance after it has modulated temporal fluctuations in age-structured populations.

From arctic lemmings to adaptive dynamics: Charles Elton's legacy in population ecology.

We shall examine the impact of Charles S. Elton's 1924 article on periodic fluctuations in animal populations on the development of modern population ecology. We argue that his impact has been substantial and that during the past 75 years of research on multi-annual periodic fluctuations in numbers of voles, lemmings, hares, lynx and game animals he has contributed much to the contemporary understanding of the causes and consequences of population regulation. Elton was convinced that the cause of the regular fluctuations was climatic variation. To support this conclusion, he examined long-term population data then available. Despite his firm belief in a climatic cause of the self-repeating periodic dynamics which many species display, Elton was insightful and far-sighted enough to outline many of the other hypotheses since put forward as an explanation for the enigmatic long-term dynamics of some animal populations. An interesting, but largely neglected aspect in Elton's paper is that it ends with speculation regarding the evolutionary consequences of periodic population fluctuations. The modern understanding of these issues will also be scrutinised here. In population ecology, Elton's 1924 paper has spawned a whole industry of research on populations displaying multi-annual periodicity. Despite the efforts of numerous research teams and individuals focusing on the origins of multi-annual population cycles, and despite the early availability of different explanatory hypotheses, we are still lacking rigorous tests of some of these hypotheses and, consequently, a consensus of the causes of periodic fluctuations in animal populations. Although Elton would have been happy to see so much effort spent on cyclic populations, we also argue that it is unfortunate if this focus on a special case of population dynamics should distract our attention from more general problems in population and community dynamics.

Population variability in space and time.

One of the most ubiquitous phenomena of all natural populations is their variability in numbers in space and time. However, there are notable differences among populations in the way the population size fluctuates. One of the major challenges in population and community ecology is to explain and understand this variety and to find possible underlying rules that might be modified from case-to-case. Population variability also has a spatial component because fluctuations are often synchronized over relatively large distances. Recently, this has led to growing interest in how 'internal' (density-dependent) processes interact with 'external' factors such as environmental variability.

Visibility of the environmental noise modulating population dynamics.

Characterizing population fluctuations and their causes is a major theme in population ecology. The debate is on the relative merits of density-dependent and density-independent effects. One paradigm (revived by the research on global warming and its relation to long-term population data) states that fluctuations in population densities can often be accounted for by external noise. Several empirical models have been suggested to support this view. We followed this by assuming a given population skeleton dynamics (Ricker dynamics and second-order autoregressive dynamics) topped off with noise composed of low- and high-frequency components. Our aim was to determine to what extent the modulated population dynamics correlate with the noise signal. High correlations (with time-lag -1) were observed with both model categories in the region of stable dynamics, but not in the region of periodic or complex dynamics. This finding is not very sensitive to low-frequency noise. High correlations throughout the entire range of dynamics are only achievable when the impact of the noise is very high. Fitted parameter values of skeleton dynamics modulated with noise are prone to err substantially. This casts doubt as to what degree the underlying dynamics are any more recognizable after being modulated by the external noise.

Does evolution of iteroparous and semelparous reproduction call for spatially structured systems?

A persistent question in the evolution of life histories is the fitness trade-off between reproducing only once (semelparity) in a lifetime or reproducing repeated times in different seasons (iteroparity). The problem can be formulated into a research agenda by assuming that one reproductive strategy is resident (has already evolved) and by asking whether invasion (evolution) of an alternative reproductive strategy is possible. For a spatially nonstructured system, Bulmer (1994) derived the relationship v + PA < 1 (PA is adult survival; vbS and bS are offspring numbers for iteroparous and semelparous breeding strategies, respectively) at which semelparous population cannot be invaded by an iteroparous mutant. When the inequality is changed to v + PA > 1, invasion of a semelparous mutant is not possible. From the inequalities, it is easy to see that possibilities for evolutionary establishment of a novel reproductive strategy are rather narrow. We extended the evolutionary scenario into a spatially structured system with dispersal linkage among the subunits. In this domain, a rare reproductive strategy can easily invade a population dominated by a resident reproductive strategy. The parameter space enabling invasion is far more generous with spatially structured evolutionary scenarios than in a spatially nonstructured system.

Dynamic complexities in host-parasitoid interaction

In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, the investigations of complex population dynamics have mainly concentrated on single populations and not on higher dimensional ecological systems. Here we report a detailed study of the complicated dynamics occurring in a basic discrete-time model of host-parasitoid interaction. The complexities include (a) non-unique dynamics, meaning that several attractors coexist, (b) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties (pattern of self-similarity and fractal basin boundaries), (c) intermittency, (d) supertransients, (e) chaotic attractors, and (f) "transient chaos". Because of these complexities minor changes in parameter or initial values may strikingly change the dynamic behavior of the system. All the phenomena presented in this paper should be kept in mind when examining and interpreting the dynamics of ecological systems. Copyright 1999 Academic Press.

A general theory of environmental noise in ecological food webs.

We examine the effects of environmental noise on populations that are parts of simple two-species food webs. We assume that the species are strongly interacting and that one or the other population is affected by the noise signal. Further assuming that a stable equilibrium with positive population densities exists, we are able to perform a complete frequency analysis of the system. If only one of the populations is subject to noise, the relative noise response by both populations is fully determined by the sign of a single element of the Jacobian matrix. The analysis is readily extended to cases when both species are affected by noise or when the food web has more than two species. The general conclusion about relative responses to noise is then less unambiguous, but the power spectra describing the frequency composition of the population variabilities are nevertheless completely determined. These results are entirely independent on the exact nature of the interaction (i.e., predation, competition, mutualism) between the populations. The results show that the interpretation of the "color" of ecological time series (i.e., the frequency composition of population variability over time) may be complicated by species interactions. The propagation of noise signals through food webs and the importance of web structure for the expected response of all parts of the web to such signals is a challenging field for future studies.

The enigma of frequency-dependent selection.

Frequency-dependent selection is so fundamental to modern evolutionary thinking that everyone `knows' the concept. Yet the term is used to refer to different types of selection. The concept is well defined in the original context of population genetics theory, which focuses on short-term evolutionary change. The original concept becomes ambiguous, however, when used in the context of long-term evolution, where density dependence becomes essential. Weak and strong frequency dependence, as distinguished in this article, refer to two very different forms of selection.

The spatial dimension in population fluctuations

Theoretical research into the dynamics of coupled populations has suggested a rich ensemble of spatial structures that are created and maintained either by external disturbances or self-reinforcing interactions among the populations. Long-term data of the Canadian lynx from eight Canadian provinces display large-scale spatial synchrony in population fluctuations. The synchronous dynamics are not time-invariant, however, as pairs of populations that are initially in step may drift out of phase and back into phase. These observations are in agreement with predictions of a spatially-linked population model and support contemporary population ecology theory.

Population dynamics and the colour of environmental noise.

The effect of red, white and blue environmental noise on discrete-time population dynamics is analyzed. The coloured noise is superimposed on Moran-Ricker and Maynard Smith dynamics, the resulting power spectra are less than examined. Time series dominated by short- and long-term fluctuations are said to be blue and red, respectively. In the stable range of the Moran-Ricker dynamics, environmental noise of any colour will make population dynamics red or blue depending the intrinsic growth rate. Thus, telling apart the colour of the noise from the colour of the population dynamics may not be possible. Population dynamics subjected to red and blue environmental noises show, respectively, more red or blue power spectra than those subjected to white noise. The sensitivity to differences in the noise colours decreases with increasing complexity and ultimately disappears in the chaotic range of the population dynamics. These findings are duplicated with the Maynard Smith model for high growth rates when the strength of density dependence changes. However, for low growth rates the power spectra of the population dynamics with noise are red in stable, periodic and aperiodic ranges irrespective of the noise colour. Since chaotic population fluctuations may show blue spectra in the deterministic case, this implies that blue deterministic chaos may become red under any colour of the noise.

Host-parasite dynamics and the evolution of host immunity and parasite fecundity strategies.

We explore evolutionarily stable co-evolution of host-macroparasite++ interactions in a discrete-time two-species population dynamics model, in which the dynamics may be stable, cyclic or chaotic. The macroparasites are assumed to harm host individuals through decreased reproductive output. Hosts may develop costly immune responses to defend themselves against parasites. Parasites compete with conspecifics by adjusting their fecundities. Overall, the presence of both parasites and the immune response in hosts produces more stable dynamics and lower host population sizes than that observed in the absence of the parasites. In our evolutionary analyses, we show that maximum parasite fecundity is always an evolutionarily stable strategy (ESS), irrespective of the type of population interaction, and that maximum parasite fecundity generally induces a minimum parasite population size through over-exploitation of the host. Phenotypic polymorphisms with respect to immunity in the host species are common and expected in ESS host strategies: the benefits of immunication depend on the frequency of the immune hosts in the population. In particular, the steady-state proportions of immune hosts depend, in addition to all the parameters of the parasite dynamics only on the cost of immunity and on the virulence of parasites in susceptible hosts. The implicit ecological dynamics of the host-parasite interaction affect the proportion of immune host individuals in the population. Furthermore, when changes in certain population parameters cause the dynamics of the host-parasite interaction to move from stability to cyclicity and then to chaos, the proportion of immune hosts tend to decrease; however, we also detected counter-examples to this result. As a whole, incorporating immunological and genetic aspects, as well as life-history trade-offs, into host-macroparasite dynamics produces a rich extension to the patterns observed in the models of ecological interactions and epidemics, and deserves more attention than is currently the case.