Harmless and Profitless Delays in Discrete Competitive Lotka–Volterra Systems

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

Density Dependence in Voles and Mice: A Comparative Study

DENSITY DEPENDENCE IN VOLES AND MICE: A COMPARATIVE STUDY

In a recent paper, Woiwod & Hanski (1992) presented the analysis of 5715 time series from aphid and moth populations for density dependence. In the longer (>20 generation) series that lacked temporal trends in abundances they found non-delayed density dependence in 67-88% of moth species and 87-91% of aphids (depending on the choice of statistical test). The high frequency with which density dependence was found was taken to indicate that density dependence is pervasive and common in the year-to-year population changes of moths and aphids (Woiwod & Hanski 1992; Godfray & Hassell 1992). Surprisingly, delayed density dependence was found less frequently than would be expected by chance alone (from < 5% of series) (Hanski & Woiwod 1991; Woiwod & Hanski 1992). This is in stark contrast to Turchin (1990) who found delayed (lag 2) density dependence in eight of 14 (57%) time series from forest insects using identical statistical techniques to those employed by Woiwod & Hanski (1992). Turchin (1990) found non-delayed density dependence in only a single species (7%). Woiwod & Hanski explain this difference in the frequency of delayed density dependence as being due to Turchin's choice of forest insect pests that often show cyclical dynamics, whereas their study used a more balanced sample of species. However, it seems somewhat unlikely that virtually all of the 456 species considered by Woiwod & Hanski would lack delayed density dependence. A possible alternative explanation is considered here, which stems from the temporal scale at which the species are sampled. The possibility explored here is not a sole explanation but is a contribution to explanation of Woiwod and Hanski's results. There are many studies looking at the effects of measuring ecological processes at different spatial scales (e.g. Greig-Smith 1954; Heads & Lawton 1983; Menge & Olson 1990; Rose & Leggett 1990; Yamamura 1990; Holling 1992; Doak, Marino & Kareiva 1992; Holt 1992). A lucid review of studies of spatial scale and patterning is provided by Levin (1992). There are fewer studies that consider both spatial and temporal scales (e.g. McArdle, Gaston & Lawton 1990; Thomas 1991; Fahrig 1992; Johnson, Milne & Wiens 1992; Loreau 1992) and studies of the effects of temporal scales are even less common (e.g. Pimm & Redfearn 1988). If we turn to the question of scale and detecting density dependence there are very few studies. In a field study of the viburnum whitefly (Aleurotrachelus jelinekii Frauenf.) (Southwood & Reader 1976), key factor analysis failed to reveal any source of densitydependent mortality in 16 generations of data; however, more detailed within-generation studies on a per leaf basis revealed that egg mortality was densitydependent (Hassell, Southwood & Reader 1987). Additionally, Hassell (1985, 1986) used simulations to demonstrate that for both parasitoids and herbivorous insect species, density dependence within generations did not necessarily lead to density dependence showing up between generations (in time series). In parasitoids, analysis at inappropriate spatial scales could lead to direct density dependence being incorrectly identified as inverse density dependence (Walde & Murdoch 1988). Furthermore, subsampling a population (a form of sampling at an inappropriate spatial scale) could mask within-generation densitydependent processes (Hassell. 1987). It appears that sampling scale in both time and space is important for detecting density dependence. Since Woiwod & Hanski (1992) used only a single mean annual abundance for each species and some species have more than a single generation per year the results of their analysis for these species might be biased in some way. The purpose of this note is to demonstrate that sampling at different temporal scales has consequences for detecting delayed density dependence, but not for identifying some form of density dependence. I concentrate on the effects of taking the mean of more than a single generation per year on detecting density dependence. In Woiwod & Hanski's (1992) study only annual abundances of moths and aphids were used, whereas there may have been two or more generations per year in some species. The consequence of taking the mean of more than one consecutive generation is that time series are smoothed and the amount of smoothing increases with the number of generations per year that are sampled. Aphids, in particular, typically have several generations per year (the exact number depending on temperature and nutritional state). The identity for 263 moth species is published (Taylor & Woiwod 1980) and Ian Woiwod kindly provided the names of the other species. The numbers of generations per year were obtained from Skinner (1984) 479

Based on recent advances in time‐series analyses of ecological dynamics using statistical and mathematical models, we summarise our recent results on the seasonal processes in the annual population dynamics of the grey‐sided vole Clethrionomys rufocanus (Sundevall, 1846) in Hokkaido, Japan, and report additional analyses on annual and seasonal density dependence. Annual direct density dependence was strong in almost all populations. In contrast, delayed density dependence was generally weak, although clear delayed density dependence was detected in some of the studied populations. Although seasonal density dependence was observed both in winter and summer, direct density dependence was much more profound during winter; thus, winter density dependence contributed most to the overall annual direct density dependence. We found no correlation between the seasonal components of annual direct density dependence; however, the corresponding seasonal components for annual delayed density dependence were positively correlated. We conclude that winter conditions influence the strength of annual direct density dependence most profoundly. Moreover, we conclude that direct density dependence during summer and winter may be generated by different mechanisms, whereas delayed density dependence seems to be generated by a common mechanism. Candidate mechanisms are discussed in relation to general knowledge of northern rodent populations and to specific insights provided by earlier studies of grey‐sided voles in Hokkaido.

1. The presence of direct and delayed (1 year) density dependence in fluctuating grey-sided vole populations (Clethrionomys rufocanus) in Hokkaido, Japan, were tested using 90 time series spanning 31 years (1962 92). 2. The autocorrelation test of Bulmer, the randomization test of Pollard. Lakhani & Rothery, and the bootstrap test of Dennis & Taper using two different models (Ricker and Gompertz) detected direct density dependence in most of the time series (90.0-100%) at the 5% level (one-tailed test). 3. Plotting population growth rates as functions of abundance suggested that the Gompertz model is more appropriate than the Ricker model for the studied populations. The tests for direct density dependence using the Gompertz model also rejected the null hypothesis of density-independent growth more frequently than the tests using the Ricker model. 4. We extended the randomization and the bootstrap methods to investigate delayed density dependence by using partial correlation and partial regression, respectively. The extended tests detected delayed density dependence in 8-15 time series (8.9%-16.7% at the 5% level (two-tailed test). 5. The high incidence of direct density dependence and the low, but significant, incidence of delayed density dependence in these vole populations are discussed. Delayed maturation at high densities by social interference and a rich generalist fauna are thought to represent plausible mechanisms generating direct density dependence. As a candidate mechanism for delayed density dependence, specialist predator and climate conditions (affecting the exposure to predators) are discussed.

BackgroundStatistical autoregressive analyses of direct and delayed density dependence are widespread in ecological research. The models suggest that changes in ecological factors affecting density dependence, like predation and landscape heterogeneity are directly portrayed in the first and second order autoregressive parameters, and the models are therefore used to decipher complex biological patterns. However, independent tests of model predictions are complicated by the inherent variability of natural populations, where differences in landscape structure, climate or species composition prevent controlled repeated analyses. To circumvent this problem, we applied second-order autoregressive time series analyses to data generated by a realistic agent-based computer model. The model simulated life history decisions of individual field voles under controlled variations in predator pressure and landscape fragmentation. Analyses were made on three levels: comparisons between predated and non-predated populations, between populations exposed to different types of predators and between populations experiencing different degrees of habitat fragmentation.ResultsThe results are unambiguous: Changes in landscape fragmentation and the numerical response of predators are clearly portrayed in the statistical time series structure as predicted by the autoregressive model. Populations without predators displayed significantly stronger negative direct density dependence than did those exposed to predators, where direct density dependence was only moderately negative. The effects of predation versus no predation had an even stronger effect on the delayed density dependence of the simulated prey populations. In non-predated prey populations, the coefficients of delayed density dependence were distinctly positive, whereas they were negative in predated populations. Similarly, increasing the degree of fragmentation of optimal habitat available to the prey was accompanied with a shift in the delayed density dependence, from strongly negative to gradually becoming less negative.ConclusionWe conclude that statistical second-order autoregressive time series analyses are capable of deciphering interactions within and across trophic levels and their effect on direct and delayed density dependence.

Animal populations vary in response to a combination of density-dependent and density-independent forces, which interact to drive their population dynamics. Understanding how abiotic forces mediate the form and strength of density-dependent processes remains a central goal of ecology, and is of increasing urgency in a rapidly changing world. Here, we report for the first time that industrial pollution determines the relative strength of rapid and delayed density dependence operating on an animal population. We explored the impacts of pollution and climate on the population dynamics of an eruptive leafmining moth, Phyllonorycter strigulatella, around a coal-fired power plant near Apatity, north-western Russia. Populations were monitored at 14 sites over 26years. The relative strengths of rapid and delayed density dependence varied with distance from the power plant. Specifically, the strength of rapid density dependence increased while the strength of delayed density dependence decreased with increasing distance from the pollution source. Paralleling the increasing strength of rapid density dependence, we observed declines in the densities of P.strigulatella, increases in predation pressure from birds and ants, and declines in an unknown source of mortality (perhaps plant antibiosis) with increasing distance from the power plant. In contrast to the associations with pollution, associations between climate change and leafminer population densities were negligible. Our results may help to explain the outbreaks of insect herbivores that are frequently observed in polluted environments. We show that they can result from the weakening of rapid (stabilizing) density dependence relative to the effects of destabilizing delayed density dependence. Moreover, our results may explain some of the variation reported in published studies of animal populations in polluted habitats. Variable results may emerge in part because of the location of the study sites on different parts of pollution gradients. Finally, in a rapidly changing world, effects of anthropogenic pollution may be as, or more, important than are effects of climate change on the future dynamics of animal populations.

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

The effect of refuge used by prey has a stabilizing impact on population dynamics and the effect of time delay has its destabilizing influences. Little attention has been paid to the combined effects of prey refuge and time delay on the dynamic consequences of the predator-prey interaction. Here, a predator-prey model with a class of functional responses was studied by using the analytical approach. The refuge is considered as protecting a constant proportion of prey and the discrete time delay is the gestation period. We evaluated both effects with regard to the local stability of the interior equilibrium point of the considered model. The results showed that the effect of prey refuge has stronger influences than that of time delay on the considered model when the time lag is smaller than the threshold. However, if the time lag is larger than the threshold, the effect of time delay has stronger influences than that of refuge used by prey.

A variety of techniques were used to test for density dependence in 32 time series from bracken-feeding insects. Seventeen taxa (primarily species, but including some pooled data from two or more closely related species whose larvae could not be distinguished in frond surveys) occurred on an open site; a woodland site held 15 taxa. For series of 12 years, collected on the open habitat, direct density dependence was detected by one or more of the techniques in 10 (58.8%) of 17 taxa, compared to only 5 (33.3%) of 15 taxa with time series of 8 years in length from the woodland habitat. Delayed density dependence was detected in 6 cases for the open site and in no cases at the woodland site. Either direct or delayed density dependence was found in 13 (76.5%) of 17 taxa for the open site and 13 (86.7%) of the 15 taxa which occurred on both sites. Although these results suggest a high frequency of density dependence in the species making up the bracken insect community, results from individual tests were extremely variable. Density dependence was detected least often by Vickery and Nudds' (1984) test, and most frequently by Varley and Gradwell's (1960) test, although the latter is prone to high rates of detecting spurious density dependence. Direct density dependence was detected most frequently in taxa that were univoltine and did not have delayed diapause, i.e. in those taxa whose life-histories conform most closely to the assumptions of the models underlying the analyses. Delayed density dependence occurred more frequently in species with more complex life-histories at the open site (taxa that were either bivoltine or multivoltine, or had delayed diapause). The results are consistent with the view that that the bracken herbivore assemblage consists of populations which are independently regulated by density dependent processes, although the present analyses suggest that we cannot rely on these tests to firmly show whether density dependence is present or not in an individual time series of the lengths considered here.

The predator-prey systems with harvesting have received a great deal of attentions for last few decades. Incorporating discrete time delays into predator-prey models could induce instability and bifurcation. In this paper we are interested in studying the combined effects of harvesting and discrete time delay on the dynamics of a predator-prey model. A comparative analysis is provided for stability behaviour in absence as well as in presence of time delay. The length of discrete time delay to preserve stability of the model system is obtained. Existence of Hopf-bifurcating small amplitude periodic solutions is derived by taking discrete time delay as a bifurcation parameter.

AUTOCORRELATED EXOGENOUS FACTORS AND THE DETECTION OF DELAYED DENSITY DEPENDENCE

Temporal variation in population size is regulated by both exogenous forces and density‐dependent feedbacks. Furthermore, accumulating evidence indicates that temporal and spatial variation in climate and resources can modify the strength of density dependence in animal populations. We analyzed six long‐term time series estimates of Peromyscus leucopus (white‐footed mouse) abundance from Kansas, Ohio, Pennsylvania, Virginia, Vermont, and Maine, USA, using the Kalman filter. Model‐averaged estimates of the strength of delayed density dependence increased from west to east and from south to north. The strength of direct and delayed density dependence was positively related to the annual number of days with minimum temperature below −17.8°C. Annual population growth rates of P. leucopus at the Maine site were positively related to acorn abundance and P. leucopus populations tracked the changes in red‐oak acorn abundance. The populations of P. leucopus living in northern latitudes might be more dependent on northern red oak (Quercus rubra) acorns for winter food than P. leucopus in southern latitudes. Furthermore, northern red oak trees mast every 4–5 years. Thus, longer, colder winters in northerly latitudes might result in stronger delayed density dependence in mouse populations with a shortage of winter food. Mice might simply track the acorn fluctuations in a delayed autocorrelated manner; however, delayed density dependence remained in our models for the Maine mouse populations after accounting for acorns, suggesting additional sources for delayed density dependence. Our results suggest that, in seed‐eating Peromyscus, cyclicity may be regulated, in part, from low to high trophic levels.

SummaryWe performed an extensive statistical modelling study on the population fluctuations and population growth rates of 15 raptor species in the Kalahari desert in South Africa.The correlation pattern between rainfall and population abundance changed systematically with raptor body weight and diet type. The abundance of heavier raptors feeding on larger prey‐items had lower correlations with rainfall than lighter raptors feeding on small prey‐items. Whereas raptor species feeding on small prey‐items were more affected by immediate rainfall, species feeding on large prey‐items were more affected by rainfall in the previous year.Population abundances were explained most parsimoniously by direct and delayed density dependence and rainfall during the current and previous breeding season. Interspecific competition was never a predictor variable. Population abundances of species best described by rainfall fed on larger prey‐items than population abundances of species best described by density dependence.Population growth rates were always best described by direct density dependence. The strength of density dependence was positively correlated with reproduction rate, due mainly to Falconiform species having higher reproduction rates than Accipitrid species.Shifting from the species to the guild level, we found that abundance and biomass shares of feeding guilds did not vary significantly over time, supporting the hypothesis of guild constancy.