Discrete-time host–parasitoid models with pest control

We propose a simple discrete-time host–parasitoid model to investigate the impact of external input of parasitoids upon the host–parasitoid interactions. It is proved that the input of the external parasitoids can eventually eliminate the host population if it is above a threshold and it also decreases the host population level in the unique interior equilibrium. It can simplify the host–parasitoid dynamics when the host population practices contest competition. We then consider a corresponding optimal control problem over a finite time period. We also derive an optimal control model using a chemical as a control for the hosts. Applying the forward–backward sweep method, we solve the optimal control problems numerically and compare the optimal host populations with the host populations when no control is applied. Our study concludes that applying a chemical to eliminate the hosts directly may be a more effective control strategy than using the parasitoids to indirectly suppress the hosts.     

A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control

A two-component differential equation model is formulated for a host-parasitoid interaction. Transient dynamics and population crashes of this system are analysed using differential inequalities. Two different cases can be distinguished: either the intrinsic growth rate of the host population is smaller than the maximum growth rate of the parasitoid or vice versa. In the latter case, the initial ratio of parasitoids to hosts should exceed a given threshold, in order to (temporarily) halt the growth of the host population. When not only oviposition but also host-feeding occurs the dynamics do not change qualitatively. In the case that the maximum growth rate of the parasitoid population is smaller than the intrinsic growth rate of the host, a threshold still exists for the number of parasitoids in an inundative release in order to limit the growth of the host population. The size of an inundative release of parasitoids, which is necessary to keep the host population below a certain level, can be determined from the two-component model. When parameter values for hosts and parasitoids are known, an effective control of pests can be found. First it is determined whether the parasitoids are able to suppress their hosts fully. Moreover, using our simple rule of thumb it can be assessed whether suppression is also possible when the relative growth rate of the host population exceeds that of the parasitoid population. With a numerical investigation of our simple system the design of parasitoid release strategies for specific situations can be computed.     

Migration frequency and the persistence of host-parasitoid interactions

This paper analyses the effect of migration frequency on the stability and persistence of a host-parasitoid system in a two-patch environment. The hosts and parasitoids are allowed to move from one patch to the other a certain number of times within a generation. When this number is low, i.e. when the time-scales associated with migration and demography are of the same order, host-parasitoid interactions are usually not persistent. When this number is high, however, persistence is more likely. Moreover, in this situation, aggregation methods can be used to simplify the proposed initial model into an aggregated model describing the dynamics of both the total host and parasitoid populations. Analysis of the aggregated model shows that the system reaches a stable steady state for some regions of the parameter domain. Persistence occurs when the movement of the parasitoids is asymmetrical, i.e. they move preferentially to one of the two patches. We show that the growth rate of the host population is a key parameter in determining which migration strategies of the parasitoids lead to persistent host-parasitoid interactions.     

Models for pest control: Complementary effects of periodic releases of sterile pests and parasitoids

Four host-parasitoid models that incorporate the simultaneous or sequential release during each generation of sterile hosts and parasitoids for control or eradication of host species are presented. The models are based on two modifications of the Nicholson-Bailey model which incorporate density regulation either in the host larvae or via parasitoid oviposition. Parasitization of host larvae and adults forms another comparison. The models indicate that the release of sterile hosts alone is more efficient than release of parasitoids alone in controlling the hosts if population regulation is in the parasitoids; otherwise, the release of parasitoids alone is more efficient. The release of both steriles and parasitoids is much more efficient than the release of either alone for either suppressing or eradicating the hosts. This greater efficiency in combination rather than separately appears to be a special case of a more general principle, which is that two pest control methods will mutually complement each other if their optimal actions in reducing host numbers are at very different host densities. This is the case for sterile releases (optimal at low host densities) and parasitoid inundation (optimal at high host densities).     

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.     

Coevolution of Contrary Choices in Host-Parasitoid Systems

We investigate patch selection strategies of hosts and parasitoids in heterogeneous environments. Previous theoretical work showed that when host traits vary among patches, coevolved populations of hosts and parasitoids make congruent choices (i.e., hosts and parasitoids preferentially select the same patches) and exhibit direct density dependence in the distribution of percent parasitism. However, host-parasitoid systems in the field show a range of patterns in percent parasitism, while behavioral studies indicate that hosts and parasitoids can exhibit contrary choices (i.e., hosts avoid patches favored by the parasitoid). We extend previous theory by permitting life-history traits of the parasitoid as well as the host to vary among patches. Our analysis implies that in coevolutionarily stable populations, hosts preferentially select patches that intrinsically support higher host equilibrium numbers (i.e., the equilibrium number achieved by hosts when both populations are confined to a single patch) and that parasitoids preferentially select patches that intrinsically support higher parasitoid equilibrium numbers (i.e., the equilibrium number achieved by the parasitoids when both populations are confined to a patch). Using this result, we show how variation in life-history traits among patches leads to contrary or congruent choices or leads to direct density dependence, inverse density dependence, or density independence in the distribution of percent parasitism. In addition, we determine when populations playing the coevolutionarily stable strategies are ecologically stable. Our analysis shows that heterogeneous environments containing patches where the intrinsic rate of growth of the host and the survivorship rate of the parasitoid are low result in the coevolved populations exhibiting contrary choices and, as a result, promote ecological stability.     

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.     

Time sharing between host searching and food searching in parasitoids: state-dependent optimal strategies

By varying the time spent searching for food, parasitoids modifytheir expected lifespan, and therefore their total lifetimereproductive success. Using a stochastic dynamic approach, wedefine the best choice between searching for food and searchingfor hosts as a function of the state of the parasitoid and theavailability of food when hosts and food are found in differentparts of the environment A first model deals with the influenceof food availability and survivorship conditions on the behaviorof a single parasitoid. Our results suggest that under conditionsof very low food availability, parasitoids should never searchfor food. When food availability is moderate, parasitoids shouldnot wait until their reserves are low before searching for food.When food is abundant and survivorship is independent of foodconsumption, parasitoids should search for food only when theirreserves are almost exhausted. They should not wait so longif survivorship depends on the energy reserves. By finding thestate-dependent ideal free distribution for a population ofparasitoids, we are able to predict their distribution betweenthe feeding area and the host living area at equilibrium. Theproportion of parasitoids in each area is altered by the numberof competitors and interference. Finally, the model predictsthat optimal time sharing between food searching and host searchingmay promote the stability of the host-parasitoid system.     

Mathematical modelling of host-parasitoid systems: effects of chemically mediated parasitoid foraging strategies on within- and between-generation spatio-temporal dynamics.

In this paper we develop a novel discrete, individual-based mathematical model to investigate the effect of parasitoid foraging strategies on the spatial and temporal dynamics of host-parasitoid systems. The model is used to compare na?ve or random search strategies with search strategies that depend on experience and sensitivity to semiochemicals in the environment. It focuses on simple mechanistic interactions between individual hosts, parasitoids, and an underlying field of a volatile semiochemical (emitted by the hosts during feeding) which acts as a chemoattractant for the parasitoids. The model addresses movement at different spatial scales, where scale of movement also depends on the internal state of an individual. Individual interactions between hosts and parasitoids are modelled at a discrete (micro-scale) level using probabilistic rules. The resulting within-generation dynamics produced by these interactions are then used to generate the population levels for successive generations. The model simulations examine the effect of various key parameters of the model on (i) the spatio-temporal patterns of hosts and parasitoids within generations; (ii) the population levels of the hosts and parasitoids between generations. Key results of the model simulations show that the following model parameters have an important effect on either the development of patchiness within generations or the stability/instability of the population levels between generations: (i) the rate of diffusion of the kairomones; (ii) the specific search strategy adopted by the parasitoids; (iii) the rate of host increase between successive generations. Finally, evolutionary aspects concerning competition between several parasitoid subpopulations adopting different search strategies are also examined.     

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 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.     

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.     

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.     

Influence of parasitized adult reproduction on host-parasitoid dynamics: an age-structured model

In this work, we develop an age-structured model (based on delay-differential equations) to investigate the dynamics of host-parasitoid systems in which adults are the target of parasitism. The rare previous work dealing with such interactions assumes that hosts are functionally dead as soon as they are attacked. We relax this assumption and show that low reproduction rates of parasitized hosts can promote stability at the expense of cyclic behavior (either long term or generation cycles). Higher reproduction rates make the regulation of the host population by parasitoids impossible. As it is the case in models in which adults are subjected to attacks but do not reproduce, our model generates generation cycles for a larger set of parameter values than in models in which juveniles are attacked.     

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.     

Population outbreaks in a discrete world

We present and analyze a simple three-patch host-parasitoid model where population growth is discrete. The model gives solutions that are qualitatively similar to the stable large-amplitude patterns in space found in reaction-diffusion theory. In the context of host-parasitoid interactions, the large-amplitude portions of the solution can be thought of as spatially localized host population outbreaks. Here, we show that the biological requirements for localized population outbreaks in a discrete world are identical to those found in reaction- diffusion theory. Furthermore, the model conveniently allows investigation into the robustness of these population outbreaks under the influence of density-dependent dispersal behavior. We find that localized population outbreaks in space can still occur with modest amounts of pursuit and aggregative behavior by parasitoids. We end by showing that evidence from a real host-parasitoid system is consistent with the predictions of the model.     

The effects of parasitoid fecundity and host taxon on the biological control of insect pests: the relationship between theory and data

1. A simple, intuitive argument and the tenets of the biological control literature both suggest that, in general, parasitoids with a greater fecundity will provide better control of their hosts, and will thus be better biological control agents. 2. A model of host-parasitoid dynamics, based on the standard Thompson–Nicholson–Bailey approach and incorporating the effects of parasitoid fecundity-limitation and host density-dependence, also indicates that as parasitoid fecundity decreases so does local stability and the degree of host suppression. 3. A taxonomically diverse data set obtained from the biological control record failed to support this theoretical prediction, but at the same time indicated a strong effect of host taxon on the outcome of biological control. 4. The hypothesis that the fecundity of parasitoids is correlated positively with their ability to suppress host populations is supported by data exclusively from the host order Lepidoptera. 5. Possible explanations for the divergence between the fecundity-limitation hypothesis and the complete data set include: the ability of parasitoids to provide long-term control of pests without the presence of a stable host–parasitoid equilibrium; differences between the concepts of successful control in theory and practice; evolutionary trade-offs between fecundity and other parasitoid life-history features, such as search efficiency, leading to better pest control by parasitoids with low fecundity; and differing windows of vulnerability to parasitoid attack between host taxa.     

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.