On the mechanistic underpinning of discrete-time population models with complex dynamics

We present a mechanistic underpinning for various discrete-time population models that can produce limit cycles and chaotic dynamics. Specific examples include the discrete-time logistic model and the Hassell model, which for a long time eluded convincing mechanistic interpretations, and also the Ricker- and Beverton-Holt models. We first formulate a continuous-time resource consumption model for the dynamics within a year, and from that we derive a discrete-time model for the between-year dynamics. Without influx of resources from the outside into the system, the resulting between-year dynamics is always overcompensating and hence may produce complex dynamics as well as extinction in finite time. We recover a connection between various standard types of continuous-time models for the resource dynamics within a year on the one hand and various standard types of discrete-time models for the population dynamics between years on the other. The model readily generalizes to several resource and consumer species as well as to more than two trophic levels for the within-year dynamics.     

Helminth infections in Apodemus sylvaticus in southern England: interactive effects of host age, sex and year on the prevalence and abundance of infections.

Helminth parasites were studied in the wood mouse, Apodemus sylvaticus, in southern England in September of each of four successive years (1994-1997). Nine species of helminths were recorded: five nematodes (Heligmosomoides polygyrus, Syphacia stroma, Pelodera strongyloides, Trichuris muris, Capillaria murissylvatici), two cestodes (Microsomacanthus crenata, Taenia taeniaeformis) and two trematodes (Corrigia vitta, Brachylaemus recurvum). In total, 134 mice were examined and 91.8% carried at least one species of helminth. The majority of mice carried two to three species (60.5%) and the highest combination was six of the nine species recorded in the study. The patterns of between-year variations in the prevalence and abundance of infection were different for each of the six species for which sufficient quantitative data were available to enable statistical analysis. For H. polygyrus, the most important source of variation arose from between-year differences, host age and the interaction of these factors: abundance increased with host age but in 1995 the age pattern was markedly different from that in the remaining years. The abundance of C. vitta also varied significantly between years but additionally there was a strong independent age effect. For M. crenata, the year x age interaction was significant, indicating that abundance among different age cohorts varied from year to year but there was also a weak significant main effect of age arising from the youngest age cohort carrying no parasites and the oldest age cohort the heaviest infections. For P. strongyloides the only significant factor was between-year variation with 1995 being a year of exceptionally low prevalence and abundance of infection. No significant between-year variation was detected for S. stroma but there was a strong sex effect (males carrying heavier infections) and an age effect (older mice of both sexes carrying heavier infections). The abundance of Trichuris muris varied only in relation to host age, worm burdens growing in intensity with increasing age, but there was also a significant interaction between year and host sex with respect to prevalence. For the remaining three species, the prevalence of infections was too low (< 8.2%) to enable any meaningful interpretation. This analysis emphasizes the need for carefully controlled statistical procedures in aiding the interpretation and the prioritization of the factors affecting worm burdens in wild rodents.     

Fruit production in Sorbus aucuparia L. (Rosaceae) and pre-dispersal seed predation by the apple fruit moth (Argyresthia conjugella Zell.)

Variation in fruit production and pre-dispersal seed predation by Argyresthia conjugella was studied in␣four populations of Sorbus aucuparia in northern Sweden.␣The number of infructescences, fruits per infructescence, consumed seeds and developed unattacked seeds per fruit were scored in marked trees from 1984 to 1990. The results showed that the number of fruits produced in each population determined the number of seed predators occurring in the host population, as the yearly number of seed predators was significantly and positively correlated with yearly number of fruits, in all but one population. The seed predators showed a delay in response to variation in number of fruits produced. This lag in response resulted in a large proportion of fruits being attacked and seeds consumed in a bad fruiting year that followed a good fruiting year, and vice versa. The proportion of fruits attacked and seeds consumed was largest in the population showing the greatest between-year variation in fruit production and lowest in the population showing the lowest between-year variation in fruit production. Furthermore, the individuals within the former population were synchronised, while they were not in the latter population. These results contradict one of the possible explanations of mast-seeding, where large synchronised between-year variation is supposed to reduce pre-dispersal seed predation. Instead, differences in attraction of the seed predator to differences in fruit crop size could explain the observed difference in seed predation between the two populations with opposite fruiting patterns. Within each population, irrespective of year, the proportion of fruits attacked and seeds consumed was independent of a tree's fruiting display. Therefore, trees with high fruit production, despite harbouring the largest number of seed predators, produced the largest number of developed seeds in absolute numbers, compared to trees that produced few fruits. Received: 25 February 1996 / Accepted: 30 November 1996     

ESS emergence pattern of male butterflies in stochastic environments

Summary The evolutionarily stable (or ESS) emergence schedule for males of univoltine butterflies is analysed in an environment in which the female emergence schedule fluctuates stochastically between years. The ESS emergence curve, computed using the mutant invadability criterion, is shown to be the one that maximizes mean logarithmic lifetime mating success in the population in which it dominates. If males have accurate information about the female emergence schedule within each year, their emergence curve would evolve to the one predicted by a deterministic game model. The male emergence curve would then shift between years, closely following year to year changes in the female emergence pattern. If, instead, males have uncertainty about the female emergence schedule, the ESS male emergence curve becomes broader than the one predicted by the deterministic game model and will not track the between-year fluctuation of female emergence well. In a special case, we show how the between-year variation of mean emergence date, the variance of emergence date, the sexual difference in mean emergence dates (protandry) and the between-year correlation of mean emergence dates of both sexes should change with the degree of accuracy of information available to males.     

Modelling the effect of temperature variation on the seasonal dynamics of Ixodes ricinus tick populations

Seasonal variation in temperature is known to drive annual patterns of tick activity and can influence the dynamics of tick-borne diseases. An age-structured model of the dynamics of Ixodes ricinus populations was developed to explore how changes in average temperature and different levels of temperature variability affect seasonal patterns of tick activity and the transmission of tick-borne diseases. The model produced seasonal patterns of tick emergence that are consistent with those observed throughout Great Britain. Varying average temperature across a continuous spectrum produced a systematic pattern in the times of peak emergence of questing ticks which depends on cumulative temperature over the year. Examination of the effects of between-year stochastic temperature variation on this pattern indicated that peak emergence times are more strongly affected by temperature stochasticity at certain levels of average temperature. Finally the model was extended to give a simple representation of the dynamics of a tick-borne disease. A threshold level of annual cumulative temperature was identified at which disease persistence is sensitive to stochastic temperature variation. In conclusion, the effect of changing patterns of temperature variation on the dynamics of I. ricinus ticks and the diseases they transmit may depend on the cumulative temperature over the year and will therefore vary across different locations. The results also indicate that diapause mechanisms have an important influence on seasonal patterns of tick activity and require further study.     

Year-to-year Oscillations in Demography of the Strictly Biennial <Emphasis Type="Italic">Pedicularis sylvatica</Emphasis> and Effects of Experimental Disturbances

In 1998–2001, I studied disturbance effects on the population structure and dynamics of a grassland strict biennial Pedicularis sylvatica, and on the species demography (monthly dynamics of seedling recruitment in 1998 and within- and between-year survival in 1998–2000). In two Czech populations, I established three experimental disturbance regimes: (1) a gap treatment, that simulated grazing by clipping vegetation and creating small gaps, (2) a mowing treatment, where I clipped the vegetation, and (3) a no management treatment, where I left the vegetation untreated. The number of recruiting seedlings varied greatly by year, and demographic structure of populations showed significant year-to-year oscillations in mean seedling numbers, from low (3 ± 0.7 s.e. per 0.25 m² plot) to high (103 ± 20). Inversely in the same years and plots, mean adult numbers in populations oscillated from high (12 ± 2) to low (0.7 ± 0.3). Disturbance effects were only important for seedling recruitment in early census dates in all years. In 1998, most seedlings recruited in April–May in gaps in both sites, but most died before winter. Within- and between-year survival was not affected by disturbance regimes but fluctuated significantly among years. Between-year survival increased with increasing size of the overwintering bud and was higher in disturbance treatments. Since the oscillations in population structure did not significantly vary in response to experimental disturbances, population dynamics may be driven endogenously rather than by disturbance events. The weak disturbance effects on species demography may also indicate population resilience to changes in habitat quality. However, since disturbances promoted seedling recruitment, grazing or mowing regimes are strongly recommended, as they create regeneration opportunities and maintain habitat quality, meeting the species long-term conservation goals.     

The immunoepidemiology of nematode parasites of farmed animals: a mathematical approach.

The population dynamics of farmed animals are controlled by humans, and often involve high host densities, which encourage higher parasite burdens than would be usual in wild animals. As a result, the immunity to reinfection acquired by the host is an important determinant of parasite population dynamics. For example, lambs are highly susceptible to gastrointestinal nematodes as they begin to graze, but develop an immunity that accounts for the observed within-year variation in parasite load and pasture contamination. In the longer term, control measures are compromised by the development of parasite strains resistant to chemotherapy, focusing attention on the development of 'natural' measures, including the selection for resistant hosts and the development of antiparasite vaccines. Mick Roberts here considers the immunoepidemiology of parasites of farmed animals on three levels: the interaction between the parasite and the host's immune system determining the individual's level of protection; the development of acquired immunity determining the within-year parasite population dynamics; and the long-term effects of control measures on the between-year parasite population dynamics.     

On the mechanistic underpinning of discrete-time population models with Allee effect

The Allee effect means reduction in individual fitness at low population densities. There are many discrete-time population models with an Allee effect in the literature, but most of them are phenomenological. Recently, Geritz and Kisdi [2004. On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol. 228, 261-269] presented a mechanistic underpinning of various discrete-time population models without an Allee effect. Their work was based on a continuous-time resource-consumer model for the dynamics within a year, from which they derived a discrete-time model for the between-year dynamics. In this article, we obtain the Allee effect by adding different mate finding mechanisms to the within-year dynamics. Further, by adding cannibalism we obtain a higher variety of models. We thus present a generator of relatively realistic, discrete-time Allee effect models that also covers some currently used phenomenological models driven more by mathematical convenience.     

Diversity and Coexistence of Sonoran Desert Winter Annuals

Abstract Annual plants make up ca. 50% of local floras in the Sonoran Desert. As with most plant communities, there is no shortage of potential coexistence generating mechanisms, and several mechanisms are likely contributors to coexistence at different spatial scales in the Sonoran Desert, e.g. spatial heterogeneity and the behaviors of predators and grazers. We explore one mechanism of likely importance for desert annuals: temporal environmental variation. It is widely recognized that coexistence is promoted by temporal variation if species such as desert annuals have "temporal niches" in the sense that each has years in which it out-performs the others. It is usually suggested that some resistent life-history stage, such as a seed bank, is also necessary to buffer each species from the negative population dynamic impact of unfavorable years. Using ten years of demographic data, we document the large year-to-year variation in population dynamics of desert annuals and show that ten species respond differently to temporal variation. Competition experiments document reversals in competitive superiority. Also, all species have a between-year seed bank, such that only a proportion of the seed bank germinates in any given year. Thus this system meets our intuitive requirements for variance-based coexistence. Dynamic models of this system demonstrate that subtle aspects of the species biology determine whether coexistence criteria are actually met. Specifically, variable germination fractions are required and coexistence is most readily favored with "predictive" germination. Germination fractions in this system do vary among years in a species specific fashion. Also, for the three years of available data, germination was predictive, in that each species had greater germination fractions in year of greater demographic success. Thus all of the population dynamic elements necessary for temporal variance mediated coexistence seem to be present in this system.     

Between-year variation in community structure and biomass–size distributions of benthic lake fish communities

The benthic fish communities of 26 Swedish lakes were monitored annually with multi-mesh gillnets in a standardized way in the years 1994–1997. No major environmental changes (e.g. in nutrient level or acidity) occurred within the lakes during the 4 years of study. In most of the lakes, the between-year variation of biomass (per unit of effort) was close to the sampling precision, and biomass was usually less variable than abundance. Median values of lakespecific indices of variance compensation indicated no general covariance of biomass between species, but a tendency for positive covariance between size classes. This indicates that the benthic fish communities of small to intermediate-sized Swedish lakes most often show low between-year variation in total biomass of fish older than 0+, and that their biomass–size distributions are not, in general, shaped by periodic strong year-classes of keystone species. In a few individual lakes, density compensation was indicated as complementary proportions of a pair of species or size classes that made major contributions to the total biomass.     

Notes on Alexandrium population dynamics

We review within-year and between-year survival strategies ofthe meroplanktonic dinoflagellate Alexandrium, with specialattention to the role of cyst beds and extended dormancy. Someof the constraints on the evolution of cyst bed dynamics arediscussed in the framework of a model borrowed from desert seedecology, in which Q, the annual germination rate, is selectedby p, the probability that the vegetative phase will be successfulon decadal time scales. Since Alexandrium, and the closely relatedPyrodinium, undergo gametogenesis at relatively low cell concentrations,specialized traits must have evolved to achieve syngamy. Itis suggested that motility and the use of chemical signals promotemating, and that the toxins act as pheromones. It is also proposedthat toxins in cysts are used as signals to influence planozygotesettlement so as to control dispersal of this stage, and ensurethat cyst beds are sufficiently stocked to inoculate the watercolumn adequately at the appro priate time of year.     

Population dynamics of diploid and hexaploid populations of a perennial herb

BACKGROUND AND AIMS: Despite the recent enormous increase in the number of studies on polyploid species, no studies to date have explored the population dynamics of these taxa. It is thus not known whether the commonly reported differences in single life-history traits between taxa of different ploidy levels result in differences in population dynamics. METHODS: This study explores differences in single life-history traits and in the complete life cycle between populations of different ploidy levels and compares these differences with differences observed between different habitat types and years. Diploid and hexaploid populations of a perennial herb, Aster amellus, are used as the study system. Transition matrix models were used to describe the dynamics of the populations, and population growth rates, elasticity values and life-table response experiments were used to compare the dynamics between populations and years. KEY RESULTS: The results indicate that between-year variation in population dynamics is much larger than variation between different ploidy levels and different habitat conditions. Significant differences exist, however, in the structure of the transition matrices, indicating that the dynamics of the different ploidy levels are different. Strong differences in probability of extinction of local populations were also found, with hexaploid populations having higher probability than diploid populations, indicating strong potential differences in persistence of these populations. CONCLUSIONS: This is the first study on complete population dynamics of plants of different ploidy levels. This knowledge will help to understand the ability of new ploidy levels to spread into new areas and persist there, and the interactions of different ploidy levels in secondary contact zones. This knowledge will also contribute to understanding of interactions of different ploidy levels with other plant species or other interacting organisms such as pollinators or herbivores.     

Climate-driven changes on phytoplankton–zooplankton coupling and nutrient availability in high mountain lakes of Southern Europe

1. The effect of climate variability on phytoplankton and zooplankton dynamics and nutrient availability was studied in two high mountain fishless lakes (La Caldera and Río Seco) of contrasting morphology, hydrology and dissolved inorganic nitrogen : soluble reactive phosphate (DIN : SRP) ratios during 1986 and after a 10‐year‐long drought in 1996 and 1997. 2. Thaw was delayed and water temperatures were lower in both lakes in 1996 than in 1986 and 1997. However, the lake‐specific DIN : SRP ratio was maintained in the 3 years studied, reflecting its local control. 3. On other hand, the presumptive limiting nutrient in each lake, P in La Caldera and N in Río Seco, showed higher concentrations in 1996 versus 1986 and 1997. Significant positive correlations between temperature and chlorophyll a were found in both lakes in 1996 but these relationships were negative or not significant in 1986 and 1997. Zooplankton biomass showed lower values in 1996 than in 1986 or 1997. 4. These findings can be explained by a decoupling of the phytoplankton–zooplankton interaction because of a constraint on zooplankton growth by low temperatures in the coldest year studied. This observation furnishes evidence that regional climatic control on the phytoplankton–zooplankton link can modulate the overall demand for nutrients.     

Between-year variation in population sex ratio increases with complexity of the breeding system in Hymenoptera

While adaptive adjustment of sex ratio in the function of colony kin structure and food availability commonly occurs in social Hymenoptera, long-term studies have revealed substantial unexplained between-year variation in sex ratio at the population level. In order to identify factors that contribute to increased between-year variation in population sex ratio, we conducted a comparative analysis across 47 Hymenoptera species differing in their breeding system. We found that between-year variation in population sex ratio steadily increased as one moved from solitary species, to primitively eusocial species, to single-queen eusocial species, to multiple-queen eusocial species. Specifically, between-year variation in population sex ratio was low (6.6% of total possible variation) in solitary species, which is consistent with the view that in solitary species, sex ratio can vary only in response to fluctuations in ecological factors such as food availability. In contrast, we found significantly higher (19.5%) between-year variation in population sex ratio in multiple-queen eusocial species, which supports the view that in these species, sex ratio can also fluctuate in response to temporal changes in social factors such as queen number and queen-worker control over sex ratio, as well as factors influencing caste determination. The simultaneous adjustment of sex ratio in response to temporal fluctuations in ecological and social factors seems to preclude the existence of a single sex ratio optimum. The absence of such an optimum may reflect an additional cost associated with the evolution of complex breeding systems in Hymenoptera societies.     

Demography and population dynamics of the mouse opossum (Thylamys elegans) in semi-arid Chile: seasonality, feedback structure and climate

Here we present, to the authors' knowledge for the very first time for a small marsupial, a thorough analysis of the demography and population dynamics of the mouse opossum (Thylamys elegans) in western South America. We test the relative importance of feedback structure and climatic factors (rainfall and the Southern Oscillation Index) in explaining the temporal variation in the demography of the mouse opossum. The demographic information was incorporated into a stage-structured population dynamics model and the model's predictions were compared with observed patterns. The mouse opossum's capture rates showed seasonal (within-year) and between-year variability, with individuals having higher capture rates during late summer and autumn and lower capture rates during winter and spring. There was also a strong between-year effect on capture probabilities. The reproductive (the fraction of reproductively active individuals) and recruitment rates showed a clear seasonal and a between-year pattern of variation with the peak of reproductive activity occuring during winter and early spring. In addition, the fraction of reproductive individuals was positively related to annual rainfall, while population density and annual rainfall positively influenced the recruitment rate. The survival rates were negatively related to annual rainfall. The average finite population growth rate during the study period was estimated to be 1.011 +/- 0.0019 from capture-recapture estimates. While the annual growth rate estimated from the seasonal linear matrix models was 1.026, the subadult and adult survival and maturation rates represent between 54% (winter) and 81% (summer) of the impact on the annual growth rate.     

Role of a parasitoid with a low fecundity,Praestochrysis shanghaiensis (Hymenoptera: Chrysididae), in the population dynamics of its host

1. Analysis of life tables of the oriental moth, Monema flavescens, obtained for 8 generations over 4 years, disclosed that the cocoon parasitoid, Praestochrysis shanghaiensis, acted as a density-disruptive factor.
2. The density of the host cocoon remained stable (max./min.=3.2), whereas that of the host adult varied (max./min.=14.3) although both showed similar fluctation patterns.
3. Stability of the host population was associated with the density-dependence in the ratio of first generation cocoons to overwintered generation moths, which was the key factor for the rate of change throughout the year. Chrysidid parasitism among the first generation cocoons ranged from 37.7 to 70.1%, and that among the second generation cocoons from 16.7 to 63.2%, each showing an inverse density-dependence and acting as the main determinant (key-factor) of the between-year variation in the density of the adult moths.
4. The density-dependence of the rate of change from overwintered generation adults to first generation cocoons was so strong that the parasitism on the second generation hosts had not effect on the cocoon density of the first generation. On the other hand, the density-dependence of the rate of change from first generation adults to second generation cocoons was weak, and the parasitism on the first generation hosts became the key factor for the between-year variation of the second generation cocoons.
5. It is suggested that the stability of the parasitoid-host system will be disrupted without three parasitism-restricting factors: asynchrony in the parasitoid attack on the second generation hosts, high mortality among parasitoid larvae of the second generation, and the high proportion of those first generation parasitoids that enter diapause. These factors are considered to be effective only in cooler parts of the distribution of the parasitoid.

     

Short- and long-term effects of winter and spring weather on growth and survival of red deer in Norway

Populations of red deer (Cervus elaphus) in Norway have increased continuously over the last decades. We tested the possible effects of climate and increase in population size on the survival rates and body condition of individuals in one of the northernmost populations of red deer in Europe. Based on 678 individuals of known age marked between 1977 and 1995, we estimated annual survival rates, the probabilities of being harvested and the recapture probability according to sex, age, year, winter and spring weather, population size, and, body weight and body condition, using capture-mark-recapture models. Winter harshness negatively influenced the body weight of yearlings and the survival of calves of both sexes. Spring weather influenced the survival of males in all age classes. A negative trend during the study period was detected in body weight and condition of calves and yearlings, but not in any age- or sex- specific survival rates. No significant gender differences in mean survival were shown in any age class. Moreover, there was little (male) or no (female) detectable between-year variation in survival rates for yearlings and adults. Winter weather acts as a limiting factor on population growth through a short-term effect on first-year survival and a long-term effect on body weight. We discuss the surprising low sex differences in natural survival rates and the differential effects of winter harshness on body weight, body condition and survival in relation to life history characteristics of red deer. Received: 10 November 1997 / Accepted: 2 June 1998     

Plant community changes induced by experimental climate change: Seedling and adult species composition

Experimental manipulation of climate provides a powerful tool for studying plant community dynamics with respect to current climate change. We experimentally investigated the vegetation dynamics of a Mediterranean shrubland under directional climate change by manipulating rain and temperature at stand level throughout 7 years. We focused on seedling establishment in relation to the between-year variability of drought conditions. We also compared seedling dynamics to changes in the established adult vegetation to assess the coupling between both dynamics. We used multivariate techniques (principal response curves (PRC) and redundancy analysis (RDA)) to explore changes in the whole community, and Generalized Linear Model (GLZM) to analyse the influence of drought on the abundance and survival of the most abundant species.Drought treatment induced significant changes in the species composition of the seedlings, via a differential decrease in the seedling density of most species. No species was particularly favoured in terms of seedling abundance under water-deficit conditions. Warming only explained a low percentage of the variability in seedling species composition. The emergence of seedlings in control plots – which may be considered an estimation of the between-year variability in the conditions for seedling establishment – was a better predictor of seedling emergence in experimental plots than climate manipulation treatments. The PRC analysis of the adults showed dynamics that were different from those recorded for seedlings, and it also showed that drought treatment significantly explained species composition. This result is reinforced by the change in the relative abundance of seedling and adults of the more common species in the drought and warming treatments, supporting the hypothesis that climatic directional change heightens discrepancies between recruitment and the adult performance. The RDA analysis applied to species composition at the end of the experiment failed, however, to attain any statistical significance. The warming treatment did not produce any significant shifts in adult vegetation.In conclusion, directional climate change – particularly drier conditions in Mediterranean shrublands – would result in a change in the recruitment of the plant community. This change in seedling recruitment tends to be different from the dynamics of adults, suggesting that potential adult mortality would not be compensated by actual seedling recruitment, thus enhancing shifts in community composition.     

Temporal change in the herbaceous understory community of an old-growth forest: from seasons to decades

Studies focusing on the herbaceous groundlayer of forest communities have become increasingly common in the literature, yet rarely do these studies adequately capture the temporal dynamics that typify this stratum. Using long-term data from an old-growth remnant stand in southeastern Ohio, we compare seasonal, annual, and decadal changes in herb composition and abundance on a north- and south-facing slope. Herbs were sampled within 70, 2-m² circular plots at three times throughout the growing season in 1997, 1998, 2011, and 2012. Slope, season, and year all had significant effects on species composition, as did their interactions. Species richness and diversity varied considerably throughout seasons, yet remained consistent over the fifteen-year study interval. A dramatic reduction in overall abundance, from an average of 63 % in 1997 to just 12 % in 2012, was observed. This decline was most severe from 1997 to 1998. A nonmetric multidimensional scaling ordination for all 4 years showed a collapse in phenological distinction among seasonal-sampling units. Finally, we looked at year-to-year changes in individual species abundances by comparing occurrence frequencies. The frequency of a species in year i was usually a good indicator of its frequency in year j; however, a few species showed considerable increases or decreases, such as the invasive Alliaria petiolata, which increased from zero plots in 1997 to 15 plots in 2012. These results suggest the importance of adequately and simultaneously documenting vegetation changes at short, intermediate, and long-term timescales—especially for understanding the dynamics of temperate deciduous forest herb communities.     

Two SIS epidemiologic models with delays

 The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values. Received: 10 January 1997 / 18 November 1997