Dynamic heterogeneous spatio-temporal pattern formation in host-parasitoid systems with synchronised generations

In this paper we develop a general mathematical model describing the spatio-temporal dynamics of host-parasitoid systems with forced generational synchronisation, for example seasonally induced diapause. The model itself may be described as an individual-based stochastic model with the individual movement rules derived from an underlying continuum PDE model. This approach permits direct comparison between the discrete model and the continuum model. The model includes both within-generation and between-generation mechanisms for population regulation and focuses on the interactions between immobile juvenile hosts, adult hosts and adult parasitoids in a two-dimensional domain. These interactions are mediated, as they are in many such host-parasitoid systems, by the presence of a volatile semio-chemical (kairomone) emitted by the hosts or the hosts food plant. The model investigates the effects on population dynamics for different host versus parasitoid movement strategies as well as the transient dynamics leading to steady states. Despite some agreement between the individual and continuum models for certain motility parameter ranges, the model dynamics diverge when host and parasitoid motilities are unequal. The individual-based model maintains spatially heterogeneous oscillatory dynamics when the continuum model predicts a homogeneous steady state. We discuss the implications of these results for mechanistic models of phenotype evolution.P. Schofield gratefully acknowledges the financial support of the BBSRC and The Wellcome Trust.     

Developmental variability and stability in continuous-time host-parasitoid models

Insect host-parasitoid systems are often modeled using delay-differential equations, with a fixed development time for the juvenile host and parasitoid stages. We explore here the effects of distributed development on the stability of these systems, for a random parasitism model incorporating an invulnerable host stage, and a negative binomial model that displays generation cycles. A shifted gamma distribution was used to model the distribution of development time for both host and parasitoid stages, using the range of parameter values suggested by a literature survey. For the random parasitism model, the addition of biologically plausible levels of developmental variability could potentially double the area of stable parameter space beyond that generated by the invulnerable host stage. Only variability in host development time was stabilizing in this model. For the negative binomial model, development variability reduced the likelihood of generation cycles, and variability in host and parasitoid was equally stabilizing. One source of stability in these models may be aggregation of risk, because hosts with varying development times have different vulnerabilities. High levels of variability in development time occur in many insects and so could be a common source of stability in host-parasitoid systems.     

Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications

A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered.     

Dynamical consequences of optimal host feeding on host-parasitoid population dynamics

This study examines the influence of various host-feeding patterns on host-parasitoid population dynamics. The following types of host-feeding patterns are considered: concurrent and non-destructive, non-concurrent and non-destructive, and non-concurrent and destructive. The host-parasitoid population dynamics is described by the Lotka-Volterra continuous-time model. This study shows that when parasitoids behave optimally, i.e. they maximize their fitness measured by the instantaneous per capita growth rate, the non-destructive type of host feeding stabilizes host-parasitoid dynamics. Other types of host feeding, i.e. destructive, concurrent, or non-concurrent, do not qualitatively change the neutral stability of the Lotka-Volterra model. Moreover, it is shown that the pattern of host feeding which maximizes parasitoid fitness is either non-concurrent and destructive, or concurrent and non-destructive host feeding, depending on the host abundance and parameters of the model. The effects of the adaptive choice of host-feeding patterns on host-parasitoid population dynamics are discussed.     

The effect of evolution on host-parasitoid systems

It is well known that a simple first-order difference equation can exhibit complex population dynamics, such as sustained oscillations and chaos. An interesting problem is whether such oscillatory dynamics are expected to occur in real populations. This paper assumes that the resident system is composed of 1-host and 1-parasitoid and that only the host is allowed to evolve, but not the parasitoid. Based on the invasibility of a host to host-parasitoid systems, we investigate the dynamics of the host-parasitoid system favored by natural selection. We consider two cases. In the first case, the host's evolution involving both the intrinsic growth rate and the sensitivity to density is considered. In the second case, the host's evolution involving both the intrinsic growth rate and the vulnerability to the parasitoid is considered. In both cases, we see that the dynamics with a stable equilibrium will not be favored by natural selection without the trade-off between the host's traits which are allowed to evolve. The host-parasitoid system with a stable equilibrium will be eventually invaded by a host type that develops an unstable equilibrium with the parasitoid. If there is a trade-off between the host's traits which are allowed to evolve, a host-parasitoid system with a stable equilibrium can be favored by natural selection.     

Host-Parasitoid Dynamics and the Success of Biological Control When Parasitoids Are Prone to Allee Effects

In sexual organisms, low population density can result in mating failures and subsequently yields a low population growth rate and high chance of extinction. For species that are in tight interaction, as in host-parasitoid systems, population dynamics are primarily constrained by demographic interdependences, so that mating failures may have much more intricate consequences. Our main objective is to study the demographic consequences of parasitoid mating failures at low density and its consequences on the success of biological control. For this, we developed a deterministic host-parasitoid model with a mate-finding Allee effect, allowing to tackle interactions between the Allee effect and key determinants of host-parasitoid demography such as the distribution of parasitoid attacks and host competition. Our study shows that parasitoid mating failures at low density result in an extinction threshold and increase the domain of parasitoid deterministic extinction. When proned to mate finding difficulties, parasitoids with cyclic dynamics or low searching efficiency go extinct; parasitoids with high searching efficiency may either persist or go extinct, depending on host intraspecific competition. We show that parasitoids suitable as biocontrol agents for their ability to reduce host populations are particularly likely to suffer from mate-finding Allee effects. This study highlights novel perspectives for understanding of the dynamics observed in natural host-parasitoid systems and improving the success of parasitoid introductions.     

How parasitism affects critical patch-size in a host-parasitoid model: application to the forest tent caterpillar

Habitat structure has broad impacts on many biological systems. In particular, habitat fragmentation can increase the probability of species extinction and on the other hand it can lead to population outbreaks in response to a decline in natural enemies. An extreme consequence of fragmentation is the isolation of small regions of suitable habitat surrounded by a large region of hostile matrix. This scenario can be interpreted as a critical patch-size problem, well studied in a continuous time framework, but relatively new to discrete time models. In this paper we present an integrodifference host-parasitoid model, discrete in time and continuous in space, to study how the critical habitat-size necessary for parasitoid survival changes in response to parasitoid life history traits, such as emergence time. We show that early emerging parasitoids may be able to persist in smaller habitats than late emerging species. The model predicts that these early emerging parasitoids lead to more severe host outbreaks. We hypothesise that promoting efficient late emerging parasitoids may be key in reducing outbreak severity, an approach requiring large continuous regions of suitable habitat. We parameterise the model for the host species of the forest tent caterpillar Malacosoma disstria Hbn., a pest insect for which fragmented landscape increases the severity of outbreaks. This host is known to have several parasitoids, due to paucity of data and as a first step in the modelling we consider a single generic parasitoid. The model findings are related to observations of the forest tent caterpillar offering insight into this host-parasitoid response to habitat structure.     

Periodicity, persistence, and collapse in host-parasitoid systems with egg limitation

There is an emerging consensus that parasitoids are limited by the number of eggs which they can lay as well as the amount of time they can search for their hosts. Since egg limitation tends to destabilize host-parasitoid dynamics, successful control of insect pests by parasitoids requires additional stabilizing mechanisms such as heterogeneity in the distribution of parasitoid attacks and host density-dependence. To better understand how egg limitation, search limitation, heterogeneity in parasitoid attacks, and host density-dependence influence host-parasitoid dynamics, discrete time models accounting for these factors are analyzed. When parasitoids are purely egg-limited, a complete anaylsis of the host-parasitoid dynamics are possible. The analysis implies that the parasitoid can invade the host system only if the parasitoid's intrinsic fitness exceeds the host's intrinsic fitness. When the parasitoid can invade, there is a critical threshold, CV*>1, of the coefficient of variation (CV) of the distribution of parasitoid attacks that determines that outcome of the invasion. If parasitoid attacks sufficiently aggregated (i.e., CV>CV*), then the host and parasitoid coexist. Typically (in a topological sense), this coexistence is shown to occur about a periodic attractor or a stable equilibrium. If the parasitoid attacks are sufficiently random (i.e. CV1. When CV<1, the parasitoid exhibits highly oscillatory dynamics. Alternatively, when parasitoid attacks are sufficiently aggregated but not overly aggregated (i.e. CV>1 but close to 1), the host and parasitoid coexist about a stable equilibrium with low host densities. The implications of these results for classical biological control are discussed.     

Strange limits of stability in host-parasitoid systems

The classical Nicholson-Bailey model for a two species host-parasitoid system with discrete generations assumes random distributions of both hosts and parasitoids, randomly searching parasitoids, and random encounters between the individuals of the two species. Although unstable, this model induced many investigations into more complex host-parasitoid systems. Local linearized stability analysis shows that equilibria of host parasitoid systems within the framework of a generalized Nicholson-Bailey model are generally unstable. Stability is only possible if host fertility does not exceede ⁴=54.5982 and if superparasitism is unsuccessful. This special situation has already been discovered by Hassell et al. (1983) in their study of the effects of variable sex ratios on host parasitoid dynamics. We discuss global behaviour of the Hassell-Waage-May model using KAM-theory and illustrate its sensitivity to small perturbations, which can give rise to radically different patterns of the population dynamics of interacting hosts and parasitoids.     

An implicit approach to model plant infestation by insect pests

Various spatial approaches were developed to study the effect of spatial heterogeneities on population dynamics. We present in this paper a flux-based model to describe an aphid-parasitoid system in a closed and spatially structured environment, i.e. a greenhouse. Derived from previous work and adapted to host-parasitoid interactions, our model represents the level of plant infestation as a continuous variable corresponding to the number of plants bearing a given density of pests at a given time. The variation of this variable is described by a partial differential equation. It is coupled to an ordinary differential equation and a delay-differential equation that describe the parasitized host population and the parasitoid population, respectively. We have applied our approach to the pest Aphis gossypii and to one of its parasitoids, Lysiphlebus testaceipes, in a melon greenhouse. Numerical simulations showed that, regardless of the number and distribution of hosts in the greenhouse, the aphid population is slightly larger if parasitoids display a type III rather than a type II functional response. However, the population dynamics depend on the initial distribution of hosts and the initial density of parasitoids released, which is interesting for biological control strategies. Sensitivity analysis showed that the delay in the parasitoid equation and the growth rate of the pest population are crucial parameters for predicting the dynamics. We demonstrate here that such a flux-based approach generates relevant predictions with a more synthetic formalism than a common plant-by-plant model. We also explain how this approach can be better adapted to test different management strategies and to manage crops of several greenhouses.     

Host-parasitoid dynamics of a generalized Thompson model

A discrete-time host-parasitoid model including host-density dependence and a generalized Thompson escape function is analyzed. This model assumes that parasitoids are egg-limited but not search-limited, and is proven to exhibit five types of dynamics: host failure in which the host goes extinct in the parasitoid's presence or absence, unconditional parasitoid failure in which the parasitoid always goes extinct while the host persists, conditional parasitoid failure in the host and the parasitoid go extinct or coexist depending on the initial host-parasitoid ratio, parasitoid driven extinction in which the parasitoid invariably drives the host to extinction, and coexistence in which the host and parasitoid coexist about a global attractor. The latter two dynamics only occur when the parasitoid's maximal rate of growth exceeds the host's maximal rate of growth. Moreover, coexistence requires parasitism events to be sufficiently aggregated. Small additive noise is proven to alter the dynamical outcomes in two ways. The addition of noise to parasitoid driven extinction results in random outbreaks of the host and parasitoid with varying intensity. Additive noise converts conditional parasitoid failure to unconditional parasitoid failure. Implications for classical biological control are discussed.     

Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems

When searching for hosts, parasitoids are observed to aggregate in response to chemical signalling cues emitted by plants during host feeding. In this paper we model aggregative parasitoid behaviour in a multi-species host-parasitoid community using a system of reaction-diffusion-chemotaxis equations. The stability properties of the steady-states of the model system are studied using linear stability analysis which highlights the possibility of interesting dynamical behaviour when the chemotactic response is above a certain threshold. We observe quasi-chaotic dynamic heterogeneous spatio-temporal patterns, quasi-stationary heterogeneous patterns and a destabilisation of the steady-states of the system. The generation of heterogeneous spatio-temporal patterns and destabilisation of the steady state are due to parasitoid chemotactic response to hosts. The dynamical behaviour of our system has both mathematical and ecological implications and the concepts of chemotaxis-driven instability and coexistence and ecological change are discussed. I. G. Pearce gratefully acknowledges the financial support of the NERC.     

Host-parasitoid association and diffuse coevolution: when to be a generalist?

In host-parasitoid communities, hosts are subjected to selective pressures from numerous parasitoid species, and parasitoids may attack several host species. The specificity of host resistance and parasitoid virulence is thus a key factor in host-parasitoid coevolution. A continuum of strategies exists, from strict specificity to a generalist strategy. The optimal level of specificity may differ in host and parasitoid. I investigated the optimal level of resistance specificity using a model in which the host could be attacked by two parasitoid species, with variable levels of defense specificity. The fitness of a parasitoid attacking two host species with different levels of virulence specificity was also modeled. Finally, a fluctuating environment was simulated by introducing variable probabilities of encounters between antagonistic species over several generations. If the frequency of encounters with the antagonistic species is fixed, then both host and parasitoid gain from a strategy of exclusive specialization toward the most frequent antagonist. If the frequency of encounters fluctuates between generations, generalist host resistance and partially specialist parasitoid virulence are favored. Generalist host resistance may be considered to be a bet-hedging response to an unpredictable environment. This asymmetry in host-parasitoid coevolution may account for some of the genetic structures observed in the field for host-parasitoid associations.     

Host-parasite interactions between whiteflies and their parasitoids

There is relatively little information available concerning the physiological and biochemical interactions between whiteflies and their parasitoids. In this report, we describe interactions between aphelinid parasitoids and their aleyrodid hosts that we have observed in four host-parasite systems: Bemisia tabaci/Encarsia formosa, Trialeurodes vaporariorum/E. formosa, B. tabaci/Eretmocerus mundus, and T. lauri/Encarsia scapeata. In the absence of reported polydnavirus and teratocytes, these parasitoids probably inject and/or produce compounds that interfere with the host immune response and also manipulate host development to suit their own needs. In addition, parasitoids must coordinate their own development with that of their host. Although eggs are deposited under all four instars of B. tabaci, Eretmocerus larvae only penetrate 4th instar B. tabaci nymphs. A pre-penetrating E. mundus first instar was capable of inducing permanent developmental arrest in its host, and upon penetration stimulated its host to produce a capsule (epidermal in origin) in which the parasitoid larva developed. T. vaporariorum and B. tabaci parasitized by E. formosa initiated adult development, and, on occasion, produced abnormal adult wings and eyes. In these systems, the site of parasitoid oviposition depended on the host species, occurring within or pressing into the ventral ganglion in T. vaporariorum and at various locations in B. tabaci. E. formosa's final larval molt is cued by the initiation of adult development in its host. In the T. lauri-E. scapeata system, both the host whitefly and the female parasitoid diapause during most of the year, i.e., from June until the middle of February (T. lauri) or from May until the end of December (E. scapeata). It appears that the growth and development of the insects are directed by the appearance of new, young foliage on Arbutus andrachne, the host tree. When adult female parasitoids emerged in the spring, they laid unfertilized male-producing eggs in whiteflies containing a female parasitoid [autoparasitism (development of male larvae utilizing female parasitoid immatures for nutrition)]. Upon hatching, these male larvae did not diapause, but initiated development, and the adult males that emerged several weeks later mated with available females to produce the next generation of parasitoid females. Thus, the interactions that exist between whiteflies and their parasitoids are complex and can be quite diverse in the various host-parasitoid systems.     

Comparative dynamics of three models for host-parasitoid interactions in a patchy environment

Phenomenological models represent a simplified approach to the study of complex systems such as host-parasitoid interactions. In this paper we compare the dynamics of three phenomenological models for host-parasitoid interactions developed by May (1978), May and Hassell (1981) and May et al. (1981). The essence of the paper by May and Hassell (1981) was to define a minimum number of parameters that would describe the interactions, avoiding the technical difficulties encountered when using models that involve many parameters, yet yielding a system of equations that could capture the essence of real world interactions in patchy environments. Those studies dealt primarily with equilibrium and coexistence phenomena. Here we study the dynamics through bifurcation analysis and phase portraits in a much wider range of parameter values, carrying the models beyond equilibrium states. We show that the dynamics can be either stable or chaotic depending on the location of a damping term in the equations. In the case of the stable system, when host density dependence acts first, a stable point is reached, followed by a closed invariant curve in phase space that first increases then decreases, finally returning to an asymptotically stable point. Chaos is not seen. On the other hand, when parasitoid attack occurs before host density dependence, chaos is inevitably apparent. We show, as did May et al. (1981) and stated earlier byWang and Gutierrez (1980), that the sequence of events in host-parasitoid interactions is crucial in determining their stability. In a chaotic state the size of the host (e.g., insect pests) population becomes unpredictable, frequently becoming quite large, a biologically undesirable outcome. From a mathematical point of view the system is of interest because it reveals how a strategically placed damping term can dramatically alter the outcome. Our study, reaching beyond equilibrium states, suggests a strategy for biological control different from that of May et al. (1981).     

Spatial synchrony in host-parasitoid models using aggregation of variables

We consider a host-parasitoid system with individuals moving on a square grid of patches. We study the effects of increasing movement frequency of hosts and parasitoids on the spatial dynamics of the system. We show that there exists a threshold value of movement frequency above which spatial synchrony occurs and the dynamics of the system can be described by an aggregated model governing the total population densities on the grid. Numerical simulations show that this threshold value is usually small. This allows using the aggregated model to make valid predictions about global host-parasitoid spatial dynamics.     

Sex ratio genetics and the competitiveness of parasitic wasps

In the first part of the paper we analyse dynamics of the genetic mechanisms responsible for maintaining biased sex ratios in host-parasitoid interactions. We begin by reviewing recent results relating to the maintenance of sibmating in haplo-diploid populations. We then investigate the evolutionary stable sex ratio in populations in which all or some of the females mate with their brothers. In particular, we derive a diallelic one-locus model for studying evolutionary stable sex ratios in partially sibmating haplo-diploid populations. In the second part of the paper we review the impact of sex ratio on host-parasitoid populations. We then analyse how the sex ratio strategy of one parasitoid species may affect its interaction with another parasitoid species competing for the same host. In particular we show that, although a female biased sex ratio may enhance the inherent competitiveness of one species, it may also destabilize the ecological interaction of the three species so that all become extinct.     

Evolutionary character changes and population responses in an insect host-parasitoid experimental system

In an insect host (the cowpea weevilCallosobruchus maculatus)- parasitoidHeterospilus prosopidis) experimental system, the population densities of the component species oscillated for the first 20 generations and then abruptly stabilized as the parasitoid density decreased. Examination of the host and parasitoid after the 40th generation in the long-term experiment showed that (1) host larvae exhibited contest-type competition (killing other larvae inhabiting the same bean), in contrast to the founder population being scramble-type competitors and (2) the parasitoid attack rate on the host did not change. There was also an evolutionary trade-off between body size and the rates of larval survival and development, suggesting a cost of contest competition on larval survivorship and development. I tested model predictions (Tuda and Iwasa 1998) that (1) host equilibrium population size should gradually decrease as the proportion of the contest type increases and that (2) random attacks of the parasitoid on the host should reduce the rate of increase in proportion of the contest type, and the effect should become manifest especially during the first 20 generations. Two of three host-only replicates showed significant decrease in population sizes. Although the density of emerging adults per bean did not differ between replicates of the host-only and host-parasitoid systems, comparison of the host body size between them on day 270 (at the 13th generation) showed that the host was more contest-type in the host-only system than in the host-parasitoid system, as the model predicted, and later on day 650 the effect of the parasitoid had disappeared.     

Delayed emergence and the success of parasitoids in biological control

Delayed emergence, a life history feature of many insects living in unpredictable environments, can have major consequences for the dynamics of host–parasitoid interactions, which vary according to their physiological interactions. We studied, through simple modeling, the significance of prolonged diapause on the suppression levels achieved by parasitoids and illustrate our case with a system involving a major forest pest, the woodwasp Sirex noctilio and two of its parasitoid species that have been introduced into different geographical regions through classical biological control programmes. Our findings suggest that the physiological relationship between parasitoid and host delayed emergence patterns may help understand observed variable success in several bio-control programs. We conclude that for given environments, host delayed emergence and the way in which parasitoids deal with it, should be included in the list of selection criteria of natural enemies of many pests, especially those affecting forests.     

Incomplete control of the diamondback moth,Plutella xylostella,by the parasitoid Cotesia vestalis in a cabbage field under tropical conditions

Immature Plutella xylostella (Lepidoptera: Plutellidae) and parasitoids were sampled for 39 months in an unsprayed cabbage field near Cotonou, Benin, to determine how and when host-parasitoid interactions influence the population dynamics of the moth in a tropical environment. Eighty-three samples were taken at approximately two-week intervals. There were no seasonal patterns in the abundance of immature moths, which was not correlated with weather variables, although heavy rainfall during the principal rainy season may have temporarily affected the population. The host-parasitoid system consisted almost exclusively of P. xylostella and its larval parasitoid Cotesia vestalis (Hymenoptera: Braconidae), both species occurring at similar levels of abundance (on average 7.5 ± 0.3 and 7.2 ± 0.3 individuals per plant, respectively). The tendency for host-parasitoid dynamics to cycle was apparent in the field. P. xylostella and C. vestalis showed coupled oscillations in abundance, with a time lag of about two weeks between host and parasitoid peaks. High parasitoid abundance resulted in significant decreases in moth abundance over several weeks. However, the parasitoid population in turn decreased, could not prevent the moth from rebounding, and there was no stable control of the pest. We conclude that under tropical conditions in which P. xylostella populations grow rapidly, combined with a high probability of recolonization from surrounding areas, biological control by a well-established specialist parasitoid reaches its limits and additional control measures are necessary.