Effectively vertical transmission of a Drosophila‐parasitic nematode: mechanism and consequences

1. Long‐term control of insects by parasites is possible only if the parasite populations persist. Because parasite transmission rate depends on host density, parasite populations may go extinct during periods of low host density. Vertical transmission of parasites, however, is independent of host density and may therefore provide a demographic bridge through times when their insect hosts are rare. 2. The nematode Howardula aoronymphium, which parasitises mycophagous species of Drosophila, can experience both horizontal and effectively vertical transmission, relative rates of which depend, in theory at least, on the density of hosts at breeding sites. 3. A nine‐generation experiment was carried out in which nematodes were transmitted either exclusively vertically or primarily horizontally. This experiment revealed that these parasites can persist and exhibit positive population growth even when there is only vertical transmission. 4. Assays at the end of the experiment revealed that the vertically transmitted nematodes had suffered no inbreeding depression and that they were similar to the horizontally transmitted nematodes in terms of virulence, infectivity, within‐host growth rate, and fecundity. Thus, vertical transmission of H. aoronymphium did not appear to compromise the ability of these parasites to control Drosophila populations.     

Rapid evolution of host specificity in a parasitic nematode

Summary In eastern North America, the nematodeHowardula aoronymphium parasitizes four species of mushroom-breedingDrosophila:D. falleni andD. recens of the quinaria species group, andD. putrida andD. testacea of the testacea group. One strain ofH. aoronymphium, designated Mendon-87, was initially capable of infecting all four of these host species. After less than 3 years in laboratory culture usingD. falleni as the sole host, this strain had completely lost the ability to infectD. putrida. Two other nematode strains parasitizedD. falleni andD. putrida at equal rates. These results demonstrate the existence of genetic variation for host specificity within this nematode species. More importantly, they show that host specificity can evolve rapidly when only one host is available for parasitization. Ecological conditions are such that natural populations ofH. aoronymphium may comprise numerous host races, lineages incapable of parasitizing the full range of host species. However, I argue that such host races are probably ephemeral and thus unlikely to persist long enough to undergo speciation.     

The expression of virulence during double infections by different parasites with conflicting host exploitation and transmission strategies

In many natural populations, hosts are found to be infected by more than one parasite species. When these parasites have different host exploitation strategies and transmission modes, a conflict among them may arise. Such a conflict may reduce the success of both parasites, but could work to the benefit of the host. For example, the less‐virulent parasite may protect the host against the more‐virulent competitor. We examine this conflict using the waterflea Daphnia magna and two of its sympatric parasites: the blood‐infecting bacterium Pasteuria ramosa that transmits horizontally and the intracellular microsporidium Octosporea bayeri that can concurrently transmit horizontally and vertically after infecting ovaries and fat tissues of the host. We quantified host and parasite fitness after exposing Daphnia to one or both parasites, both simultaneously and sequentially. Under conditions of strict horizontal transmission, Pasteuria competitively excluded Octosporea in both simultaneous and sequential double infections, regardless of the order of exposure. Host lifespan, host reproduction and parasite spore production in double infections resembled those of single infection by Pasteuria. When hosts became first vertically (transovarilly) infected with O. bayeri, Octosporea was able to withstand competition with P. ramosa to some degree, but both parasites produced less transmission stages than they did in single infections. At the same time, the host suffered from reduced fecundity and longevity. Our study demonstrates that even when competing parasite species utilize different host tissues to proliferate, double infections lead to the expression of higher virulence and ultimately may select for higher virulence. Furthermore, we found no evidence that the less‐virulent and vertically transmitting O. bayeri protects its host against the highly virulent P. ramosa.     

No evidence for behavioural adaptations to nematode parasitism by the fly Drosophila putrida

Behavioural adaptations of hosts to their parasites form an important component of the evolutionary dynamics of host–parasite interactions. As mushroom‐feeding Drosophila can tolerate deadly mycotoxins, but their Howardula nematode parasites cannot, we asked how consuming the potent mycotoxin α‐amanitin has affected this host–parasite interaction. We used the fly D. putrida and its parasite H. aoronymphium, which is both highly virulent and at high prevalence in some populations, and investigated whether adult flies utilize food with toxin to prevent infection in the next generation or consume the toxin to reduce the virulence of an already established infection. First, we found that uninfected females did not prefer to eat or lay their eggs on toxic food, indicating that selection has not acted on the flies to alter their behaviour towards α‐amanitin to prevent their offspring from becoming infected by Howardula. However, we cannot rule out that flies use an alternate cue that is associated with toxin presence in the wild. Second, we found that infected females did not prefer to eat food with α‐amanitin and that consuming α‐amanitin did not cure or reduce the virulence of the parasite in adults that were already infected. In sum, our results indicate there are no direct effects of eating α‐amanitin on this host–parasite interaction, and we suggest that toxin tolerance is more likely maintained by selection due to competition for resources than as a mechanism to avoid parasite infection or to reduce the virulence of infection.     

HOST GROWTH CONDITIONS INFLUENCE EXPERIMENTAL EVOLUTION OF LIFE HISTORY AND VIRULENCE OF A PARASITE WITH VERTICAL AND HORIZONTAL TRANSMISSION

In parasites with mixed modes of transmission, ecological conditions may determine the relative importance of vertical and horizontal transmission for parasite fitness. This may lead to differential selection pressure on the efficiency of the two modes of transmission and on parasite virulence. In populations with high birth rates, increased opportunities for vertical transmission may select for higher vertical transmissibility and possibly lower virulence. We tested this idea in experimental populations of the protozoan Paramecium caudatum and its bacterial parasite Holospora undulata. Serial dilution produced constant host population growth and frequent vertical transmission. Consistent with predictions, evolved parasites from this “high‐growth” treatment had higher fidelity of vertical transmission and lower virulence than parasites from host populations constantly kept near their carrying capacity (“low‐growth treatment”). High‐growth parasites also produced fewer, but more infectious horizontal transmission stages, suggesting the compensation of trade‐offs between vertical and horizontal transmission components in this treatment. These results illustrate how environmentally driven changes in host demography can promote evolutionary divergence of parasite life history and transmission strategies.     

Virulence and competitive ability in an obligately killing parasite

Mixed infections are thought to have a major influence on the evolution of parasite virulence. During a mixed infection, higher within‐host parasite growth is favored under the assumption that it is critical to the competitive success of the parasite. As within‐host parasite growth may also increase damage to the host, a positive correlation is predicted between virulence and competitive success. However, when parasites must kill their hosts in order be transmitted, parasites may spend energy on directly attacking their host, even at the cost of their within‐host growth. In such systems, a negative correlation between virulence and competitive success may arise. We examined virulence and competitive ability in three sympatric species of obligately killing nematode parasites in the genus Steinernema. These nematodes exist in a mutualistic symbiosis with bacteria in the genus Xenorhabdus. Together the nematodes and their bacteria kill the insect host soon after infection, with reproduction of both species occurring mainly after host death. We found significant differences among the three nematode species in the speed of host killing. The nematode species with the lowest and highest levels of virulence were associated with the same species of Xenorhabdus, indicating that nematode traits, rather than the bacterial symbionts, may be responsible for the differences in virulence. In mixed infections, host mortality rate closely matched that associated with the more virulent species, and the more virulent species was found to be exclusively transmitted from the majority of coinfected hosts. Thus, despite the requirement of rapid host death, virulence appears to be positively correlated with competitive success in this system. These findings support a mechanistic link between parasite growth and both anti‐competitor and anti‐host factors.     

Life history and virulence are linked in the ectoparasitic salmon louse Lepeophtheirus salmonis

Models of virulence evolution for horizontally transmitted parasites often assume that transmission rate (the probability that an infected host infects a susceptible host) and virulence (the increase in host mortality due to infection) are positively correlated, because higher rates of production of propagules may cause more damages to the host. However, empirical support for this assumption is scant and limited to microparasites. To fill this gap, we explored the relationships between parasite life history and virulence in the salmon louse, Lepeophtheirus salmonis, a horizontally transmitted copepod ectoparasite on Atlantic salmon Salmo salar. In the laboratory, we infected juvenile salmon hosts with equal doses of infective L. salmonis larvae and monitored parasite age at first reproduction, parasite fecundity, area of damage caused on the skin of the host, and host weight and length gain. We found that earlier onset of parasite reproduction was associated with higher parasite fecundity. Moreover, higher parasite fecundity (a proxy for transmission rate, as infection probability increases with higher numbers of parasite larvae released to the water) was associated with lower host weight gain (correlated with lower survival in juvenile salmon), supporting the presence of a virulence–transmission trade‐off. Our results are relevant in the context of increasing intensive farming, where frequent anti‐parasite drug use and increased host density may have selected for faster production of parasite transmission stages, via earlier reproduction and increased early fecundity. Our study highlights that salmon lice, therefore, are a good model for studying how human activity may affect the evolution of parasite virulence.     

Evolution of virulence under intensive farming: salmon lice increase skin lesions and reduce host growth in salmon farms

Parasites rely on resources from a host and are selected to achieve an optimal combination of transmission and virulence. Human‐induced changes in parasite ecology, such as intensive farming of hosts, might not only favour increased parasite abundances, but also alter the selection acting on parasites and lead to life‐history evolution. The trade‐off between transmission and virulence could be affected by intensive farming practices such as high host density and the use of antiparasitic drugs, which might lead to increased virulence in some host–parasite systems. To test this, we therefore infected Atlantic salmon (Salmo salar) smolts with salmon lice (Lepeophtheirus salmonis) sampled either from wild or farmed hosts in a laboratory experiment. We compared growth and skin damage (i.e. proxies for virulence) of hosts infected with either wild or farmed lice and found that, compared to lice sampled from wild hosts in unfarmed areas, those originating from farmed fish were more harmful; they inflicted more skin damage to their hosts and reduced relative host weight gain to a greater extent. We advocate that more evolutionary studies should be carried out using farmed animals as study species, given the current increase in intensive food production practices that might be compared to a global experiment in parasite evolution.     

EVOLUTIONARY ASSOCIATIONS OF BROOD PARASITIC FINCHES (VIDUA) AND THEIR HOST SPECIES: ANALYSES OF MITOCHONDRIAL DNA RESTRICTION SITES

The species-specific associations of the African brood parasitic finches Vidua with their estrildid finch host species may have originated by cospeciation with the host species or by later colonizations of new hosts. Predictions of these alternative models were tested in two species groups of brood parasites (indigobirds, paradise whydahs) and their hosts. Phylogenetic analyses suggested that the brood parasites and their hosts did not speciate in parallel. The parasitic indigobirds share mitochondrial haplotypes with each other, and species limits in both indigobirds and paradise whydahs do not correspond with their gene trees. Different parasite species within a region are more closely related to each other than any is to parasites that are associated with its same host species in other regions of Africa. There is little genetic difference between parasite species D?_i,j < 0.001 in the indigobirds, D?_i,j = 0.01 in the whydahs). Genetic distances D?_i,j between the parasite species are less than the genetic distances between their corresponding host species in all parasite-host comparisons, and average only 7.2% as large in the indigobirds as in their hosts and 42% as large in the paradise whydahs as in their hosts. A phylogenetic model that allows ancestral haplotype polymorphisms to be retained in descendant species was compared to a constraint model of species monophyly requiring all but the one ancestral haplotype to be independently derived within each species. The constraint model increases the length of the indigobird tree by 50% over that of the model of retained ancestral polymorphisms; the difference is statistically significant. Both phylogenetic and distance analyses indicate that the brood parasites have become associated with their host species through host switches and independent colonizations of the hosts, rather than through parallel cospeciation with them. The molecular genetic results are supported by recent discoveries of additional host species that are associated with the indigobirds in the field and by variation in the species-specific song behaviors of the brood parasites.     

The Ratchet and the Red Queen: the maintenance of sex in parasites

The adaptive significance of sexual reproduction remains as an unsolved problem in evolutionary biology. One promising hypothesis is that frequency‐dependent selection by parasites selects for sexual reproduction in hosts, but it is unclear whether such selection on hosts would feed back to select for sexual reproduction in parasites. Here we used individual‐based computer simulations to explore this possibility. Specifically, we tracked the dynamics of asexual parasites following their introduction into sexual parasite populations for different combinations of parasite virulence and transmission. Our results suggest that coevolutionary interactions with hosts would generally lead to a stable coexistence between sexual parasites and a single parasite clone. However, if multiple mutations to asexual reproduction were allowed, we found that the interaction led to the accumulation of clonal diversity in the asexual parasite population, which led to the eventual extinction of the sexual parasites. Thus, coevolution with sexual hosts may not be generally sufficient to select for sex in parasites. We then allowed for the stochastic accumulation of mutations in the finite parasite populations (Muller's Ratchet). We found that, for higher levels of parasite virulence and transmission, the population bottlenecks resulting from host–parasite coevolution led to the rapid accumulation of mutations in the clonal parasites and their elimination from the population. This result may explain the observation that sexual reproduction is more common in parasitic animals than in their free‐living relatives.     

HOST–PARASITE GENETIC INTERACTIONS AND VIRULENCE-TRANSMISSION RELATIONSHIPS IN NATURAL POPULATIONS OF MONARCH BUTTERFLIES

Evolutionary models predict that parasite virulence (parasite-induced host mortality) can evolve as a consequence of natural selection operating on between-host parasite transmission. Two major assumptions are that virulence and transmission are genetically related and that the relative virulence and transmission of parasite genotypes remain similar across host genotypes. We conducted a cross-infection experiment using monarch butterflies and their protozoan parasites from two populations in eastern and western North America. We tested each of 10 host family lines against each of 18 parasite genotypes and measured virulence (host life span) and parasite transmission potential (spore load). Consistent with virulence evolution theory, we found a positive relationship between virulence and transmission across parasite genotypes. However, the absolute values of virulence and transmission differed among host family lines, as did the rank order of parasite clones along the virulence-transmission relationship. Population-level analyses showed that parasites from western North America caused higher infection levels and virulence, but there was no evidence of local adaptation of parasites on sympatric hosts. Collectively, our results suggest that host genotypes can affect the strength and direction of selection on virulence in natural populations, and that predicting virulence evolution may require building genotype-specific interactions into simpler trade-off models.     

To eject or to abandon? Life history traits of hosts and parasites interact to influence the fitness payoffs of alternative anti‐parasite strategies

Hosts either tolerate avian brood parasitism or reject it by ejecting parasitic eggs, as seen in most rejecter hosts of common cuckoos, Cuculus canorus, or by abandoning parasitized clutches, as seen in most rejecter hosts of brown‐headed cowbirds, Molothrus ater. What explains consistent variation between alternative rejection behaviours of hosts within the same species and across species when exposed to different types of parasites? Life history theory predicts that when parasites decrease the fitness of host offspring, but not the future reproductive success of host adults, optimal clutch size should decrease. Consistent with this prediction, evolutionarily old cowbird hosts, but not cuckoo hosts, have lower clutch sizes than related rarely‐ or newly parasitized species. We constructed a mathematical model to calculate the fitness payoffs of egg ejector vs. nest abandoner hosts to determine if various aspects of host life history traits and brood parasites’ virulence on adult and young host fitness differentially influence the payoffs of alternative host defences. These calculations showed that in general egg ejection was a superior anti‐parasite strategy to nest abandonment. Yet, increasing parasitism rates and increasing fitness values of hosts’ eggs in both currently parasitized and future replacement nests led to switch points in fitness payoffs in favour of nest abandonment. Nonetheless, nest abandonment became selectively more favourable only at lower clutch sizes and only when hosts faced parasitism by a cowbird‐ rather than a cuckoo‐type brood parasite. We suggest that, in addition to evolutionary lag and gape‐size limitation, our estimated fitness differences based on life history trait variation provide new insights for the consistent differences observed in the anti‐parasite rejection strategies between many cuckoo‐ and cowbird‐hosts.     

Local adaptation and enhanced virulence of Nosema granulosis artificially introduced into novel populations of its crustacean host, Gammarus duebeni

Local adaptation theory predicts that, on average, most parasite species should be locally adapted to their hosts (more suited to hosts from local than distant populations). Local adaptation has been studied for many horizontally transmitted parasites, however, vertically transmitted parasites have received little attention. Here we present the first study of local adaptation in an animal/parasite system where the parasite is vertically transmitted. We investigate local adaptation and patterns of virulence in a crustacean host infected with the vertically transmitted microsporidian Nosema granulosis. Nosema granulosis is vertically transmitted to successive generations of its crustacean host, Gammarus duebeni and infects up to 46% of adult females in natural populations. We investigate local adaptation using artificial horizontal infection of different host populations in the UK. Parasites were artificially inoculated from a donor population into recipient hosts from the sympatric population and into hosts from three allopatric populations in the UK. The parasite was successfully established in hosts from all populations regardless of location, infecting 45% of the recipients. Nosema granulosis was vertically (transovarially) transmitted to 39% of the offspring of artificially infected females. Parasite burden (intensity of infection) in developing embryos differed significantly between host populations and was an order of magnitude higher in the sympatric population, suggesting some degree of host population specificity with the parasite adapted to its local host population. In contrast with natural infections, artificial infection with the parasite resulted in substantial virulence, with reduced host fecundity (24%) and survival (44%) of infected hosts from all the populations regardless of location. We discuss our findings in relation to theories of local adaptation and parasite-host coevolution.     

The virulence–transmission relationship in an obligate killer holds under diverse epidemiological and ecological conditions,but where is the tradeoff?

Parasite virulence is a leading theme in evolutionary biology. Modeling the course of virulence evolution holds the promise of providing practical insights into the management of infectious diseases and the implementation of vaccination strategies. A key element of virulence modeling is a tradeoff between parasite transmission rate and host lifespan. This assumption is crucial for predicting the level of optimal virulence. Here, I test this assumption using the water flea Daphnia magna and its castrating and obligate‐killing bacterium Pasteuria ramosa. I found that the virulence–transmission relationship holds under diverse epidemiological and ecological conditions. In particular, parasite genotype, absolute and relative parasite dose, and within‐host competition in multiple infections did not significantly affect the observed trend. Interestingly, the relationship between virulence and parasite transmission in this system is best explained by a model that includes a cubic term. Under this relationship, parasite transmission initially peaks and saturates at an intermediate level of virulence, but then it further increases as virulence decreases, surpassing the previous peak. My findings also highlight the problem of using parasite‐induced host mortality as a “one‐size‐fits‐all” measure of virulence for horizontally transmitted parasites, without considering the onset and duration of parasite transmission as well as other equally virulent effects of parasites (e.g., host castration). Therefore, mathematical models may be required to predict whether these particular characteristics of horizontally transmitted parasites can direct virulence evolution into directions not envisaged by existing models.     

Parasite and host assemblages: embracing the reality will improve our knowledge of parasite transmission and virulence

Interactions involving several parasite species (multi-parasitized hosts) or several host species (multi-host parasites) are the rule in nature. Only a few studies have investigated these realistic, but complex, situations from an evolutionary perspective. Consequently, their impact on the evolution of parasite virulence and transmission remains poorly understood. The mechanisms by which multiple infections may influence virulence and transmission include the dynamics of intrahost competition, mediation by the host immune system and an increase in parasite genetic recombination. Theoretical investigations have yet to be conducted to determine which of these mechanisms are likely to be key factors in the evolution of virulence and transmission. In contrast, the relationship between multi-host parasites and parasite virulence and transmission has seen some theoretical investigation. The key factors in these models are the trade-off between virulence across different host species, variation in host species quality and patterns of transmission. The empirical studies on multi-host parasites suggest that interspecies transmission plays a central role in the evolution of virulence, but as yet no complete picture of the phenomena involved is available. Ultimately, determining how complex host–parasite interactions impact the evolution of host–parasite relationships will require the development of cross-disciplinary studies linking the ecology of quantitative networks with the evolution of virulence.     

Microsporidian life cycles and diversity: the relationship between virulence and transmission

The microsporidia are obligate intracellular parasites which have diverse life cycles involving both horizontal and vertical transmission and parasitise a wide range of vertebrate and invertebrate hosts. In this paper we consider the life cycles and diversity of the microsporidia. We focus in particular on the relationship between parasite transmission and virulence and its implications for host-parasite coevolution. The use of horizontal and vertical routes of transmission varies between species and there is a strong link between transmission and virulence. Horizontal transmission is characterised by a high parasite burden and associated pathogenicity. In contrast, vertical transmission is characterised by low virulence, which has led to under-reporting of this important transmission route. Vertically transmitted microsporidia may also cause male killing or feminisation of their host, with implications for host population sex ratio and stability. Phylogenetic analysis shows that vertical transmission occurs in diverse branches of the Microspora. We find that there is evidence for vertical transmission in both vertebrate and invertebrate hosts and conclude that it is a common or possibly even ubiquitous transmission route within this phylum.     

Host condition as a constraint for parasite reproduction

Environmental stress has been suggested to increase host susceptibility to infections and reduce host ability to resist parasite growth and reproduction, thus benefiting parasites. This prediction stems from expected costs of immune defence; hosts in poor condition should have less resources to be allocated to immune function. However, the alternative hypothesis for response to environmental stress is that hosts in poor condition provide less resources for parasites and/or suffer higher mortality, leading to reduced parasite growth, reproduction and survival. We contrasted these alternative hypotheses in a trematode–snail ( Diplostomum spathaceum – Lymnaea stagnalis ) system by asking: (1) how host condition affects parasite reproduction (amount and quality of produced transmission stages) and (2) how host condition affects the survival of infected host individuals. We experimentally manipulated host condition by starving the snails, and found that parasites produced fewer and poorer quality transmission stages in stressed hosts. Furthermore, starvation increased snail mortality. These findings indicate that in well-established trematode infections, reduced ability of immune allocation has no effect on host exploitation by parasites. Instead, deteriorating resources for the snail host can directly limit the amount of resources available for the parasite. This, together with increased host mortality, may have negative effects on parasite populations in the wild.     

Host patch selection induced by parasitism: basic reproduction ratio r(0) and optimal virulence

The effects of parasites on the behavior of their hosts are well documented. For example, parasites may affect the habitat selection of the host individual. We used variables aggregation methods to investigate the way in which parasites affect the spatial pattern of susceptible hosts. We developed a simple epidemiological model, taking into account both the reproduction processes of hosts (density-dependent birth and death) and infection, considered separately on two different patches, and the migration of susceptible hosts between these two patches. We used the complete model of three equations to generate an aggregated model describing the dynamics of the combined susceptible and infected host populations. We obtained the basic reproduction ratio (R(0)) from the aggregated model, and then studied the effect of the migratory behavior of susceptible hosts on the ability of the parasite to invade the system. We also used the basic reproduction ratio to investigate the evolution of parasite virulence in relation to the migration decisions of susceptible hosts. We found that host investment in avoidance of the infected patch leads to an increase in optimal virulence if host investment is costly.     

Species-specific prevalence of mermithid parasites in populations of six congeneric host caddisflies of Ecnomus McLachlan, 1864 (Trichoptera: Ecnomidae)

Mermithid nematodes are entomophagous parasites and, despite being present in diverse aquatic insects, studies of caddisflies acting as definitive hosts are few and the ecological impacts on host populations are largely unknown. During a four-year study in a stream in southeastern Australia, parasitic mermithid worms were commonly found inside adult caddisflies, but only species of the genus Ecnomus McLachlan, 1864 (Ecnomidae). Ecnomus were the definitive host and parasite prevalence ranged from <1% to nearly 50% across six species. Species-specific prevalence varied little between years and was typically higher in males than in females. Parasite intensity ranged from 1 to 6 (median = 1), but did not vary between species or sexes. Infected hosts could fly, but were castrated and died when worms emerged. High prevalence and virulence (reproductive failure and death of the host) suggests that these parasites could potentially play a role in the population dynamics of some Ecnomus spp.     

Interspecific competition among macroparasites in a density-dependent host population

 We analyze the dynamics of a community of macroparasite species that share the same host. Our work extends an earlier framework for a host species that would grow exponentially in the absence of parasitism, to one where an uninfected host population is regulated by factors other than parasites. The model consists of one differential equation for each parasite species and a single density-dependent nonlinear equation for the host. We assume that each parasite species has a negative binomial distribution within the host and there is zero covariance between the species (exploitation competition). New threshold conditions on model parameters for the coexistence and competitive exclusion of parasite species are derived via invadibility and stability analysis of corresponding equilibria. The main finding is that the community of parasite species coexisting at the stable equilibrium is obtained by ranking the species according t! o th e minimum host density H ^* above which a parasite species can grow when rare: the lower H ^* , the higher the competitive ability. We also show that ranking according to the basic reproduction number Q ₀ does not in general coincide with ranking according to H ^* . The second result is that the type of interaction between host and parasites is crucial in determining the competitive success of a parasite species, because frequency-dependent transmission of free-living stages enhances the invading ability of a parasite species while density-dependent transmission makes a parasite very sensitive to other competing species. Finally, we show that density dependence in the host population entails a simplification of the portrait of possible outcomes with respect to previous studies, because all the cases resulting in the exponential growth of host and parasite populations are eliminated.. Received: 24 June 1996 / Revised version: 28 April 1998