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.     

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.     

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.     

Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model

In this paper we consider a modified spatiotemporal ecological system originating from the temporal Holling-Tanner model, by incorporating diffusion terms. The original ODE system is studied for the stability of coexisting homogeneous steady-states. The modified PDE system is investigated in detail with both numerical and analytical approaches. Both the Turing and non-Turing patterns are examined for some fixed parametric values and some interesting results have been obtained for the prey and predator populations. Numerical simulation shows that either prey or predator population do not converge to any stationary state at any future time when parameter values are taken in the Turing-Hopf domain. Prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability. With help of numerical simulations we have shown that Turing-Hopf bifurcation leads to onset of spatio-temporal chaos when predator's diffusivity is much higher compared to prey population. Our investigation reveals the fact that Hopf-bifurcation is essential for the onset of spatiotemporal chaos.     

Emergence of global behaviour in a host–parasitoid model with density-dependent dispersal in a chain of patches

We present a time discrete spatial host–parasitoid model. The environment is a chain of patches connected by dispersal events. Dispersal of parasitoids is host-density dependent. When the host density is small (resp. high), the proportion of migrant parasitoids is close to unity (resp. to zero). We assume fast patch to patch dispersal with respect to local interactions. Local host–parasitoid interactions are described by the classical Nicholson–Bailey model. By using time scales separation methods (or aggregation methods), we obtain a reduced model that governs the total host and parasitoid densities (obtained by addition over all patches). The aggregated model describes the time evolution of the total number of hosts and parasitoids of the system of patches. This global model is useful to make predictions of emerging behaviour regarding the dynamics of the complete system. We study the effects of number of patches and host density-dependent parasitoid dispersal on the overall stability of the host–parasitoid system. We finally compare our stability results with the CV² > 1 rule.     

Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions

Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.     

Host-size-dependent feeding behaviour and progeny sex ratio of Anisopteromalus calandrae (Hym., Pteromalidae)

Host-size related feeding and oviposition behaviour, and allocation of progeny sex by Anisopteromalus calandrae (Howard) were tested on Sitophilus oryzae L. The parasitoid showed a host-size-dependent partition of feeding and oviposition behaviour, preferring small hosts for feeding, but large hosts for oviposition. Neither the mutual interference nor the host density showed any effect on the behaviour of the parasitoid. Allocation of progeny sex by the female parasitoid appeared to be based more likely on absolute than on relative host size encountered. A model for the progeny sex ratio was constructed based on: (1) ovipositional preference of the parasitoid on large hosts; (2) feeding preferentially on small hosts; and (3) host-size-related regulation of progeny sex ratio. The progeny sex ratio of the parasitoid predicted by the model was in close agreement with the observed value.     

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.     

Chemo-orientation responses in hymenopteran parasitoids induced by substrate-borne semiochemicals

Hymenopteran parasitoids can utilize substrate-borne semiochemicals released by conspecifics or by their hosts, increasing the likelihood of successful mating and host location. According to the literature, two substrate-borne chemo-orientation patterns can occur: (1) biased random searching, a non-directional reaction toward the chemicals (kinesis), and (2) trail-following searching, a directional response toward the source emitting the chemical compounds (taxis). These two different strategies can be adopted by parasitoids to locate hosts and mates. In host location, random searching is induced by allelochemicals indirectly associated with the host, whereas trail-following behavior is induced by allelochemicals directly emitted by the target organism. In mate finding, sex pheromones emitted by conspecifics can induce either the random searching or the trail-following behavior, although the spatial distribution of virgin conspecifics could be an important factor driving the evolution of substrate-borne chemo-orientation patterns. The chemical nature of substrate-borne semiochemicals has not yet been fully elucidated. Most studies have shown that crude extracts are biologically active for eliciting parasitoid arrestment response, but few studies have clearly characterized their chemical nature. However, experimental evidence indicates that cuticular lipids located in the external layer of insects’ bodies play a role in parasitoid–parasitoid and host–parasitoid communication. The ecological role of parasitoid chemo-orientation in host and mate location is discussed from a biological control perspective.     

A parasitoid wasp uses landmarks while monitoring potential resources

Social insects and insects that provision nests are well known to have complex foraging behaviour involving repeated visits to learned locations. Other insects do not forage from a central location and are generally assumed to respond to resources by simple attraction without spatial memory. This simple response to resource cues is generally taken as giving rise to patterns of resource use that correspond directly to resource distribution. By contrast, the solitary parasitoid wasp Hyposoter horticola monitors the locations of multiple potential hosts (butterfly eggs) for up to several weeks, until the hosts become susceptible to parasitism. Essentially all hosts in the landscape are found, and one-third of them are parasitized, independent of host density. Here, we show that the wasps do not relocate hosts using odour markers previously left by themselves or other foragers, nor do they find the eggs anew repeatedly. Instead, the wasps relocate host eggs by learning the position of the eggs relative to visual landmarks. The anticipatory foraging behaviour presented here is a key to the wasp's exceptionally stable population dynamics.     

Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems

The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an ‘intrinsic length’ which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.     

Analysis of oviposition behaviour in the parasitoidEpidinocarsis lopezi (Hym.: Encyrtidae): Comparison of flow charts on factorial maps

Females ofEpidinocarsis lopezi (De Santis) are presented with groups of unparasitized hosts or groups containing different ratios of parasitized hosts (Phenacoccus manihoti). Oviposition behaviour sequences obtained in various situations are filmed and split into units of behaviour or patterns. The succession of patterns is analyzed and translated into flow charts on factorial maps characteristic of each type of sequence. Comparison of flow charts reveals host discrimination by female parasitoids since the relative importance of behaviour patterns is modified according to the level of parasitized hosts encountered; it also gives clues to the strategy adopted by the female parasitoid during the search of a patch. In addition, the analysis divides the oviposition sequence into three separate patterns, specific to host search, oviposition act and post-oviposition behaviour. This series of patterns is closely related to factorial axes which indicates the fundamental trends of oviposition behaviour.     

Global analysis of dynamical decision-making models through local computation around the hidden saddle

Bistable dynamical switches are frequently encountered in mathematical modeling of biological systems because binary decisions are at the core of many cellular processes. Bistable switches present two stable steady-states, each of them corresponding to a distinct decision. In response to a transient signal, the system can flip back and forth between these two stable steady-states, switching between both decisions. Understanding which parameters and states affect this switch between stable states may shed light on the mechanisms underlying the decision-making process. Yet, answering such a question involves analyzing the global dynamical (i.e., transient) behavior of a nonlinear, possibly high dimensional model. In this paper, we show how a local analysis at a particular equilibrium point of bistable systems is highly relevant to understand the global properties of the switching system. The local analysis is performed at the saddle point, an often disregarded equilibrium point of bistable models but which is shown to be a key ruler of the decision-making process. Results are illustrated on three previously published models of biological switches: two models of apoptosis, the programmed cell death and one model of long-term potentiation, a phenomenon underlying synaptic plasticity.     

Simulation modelling of ecological hierarchies in constructive dynamical systems

Organized complexity is a characteristic feature of ecological systems with heterogeneous components interacting at several spatio-temporal scales. The hierarchy theory is a powerful epistemological framework to describe such systems by decomposing them vertically into levels and horizontally into holons. It was at first developed in a temporal and functional perspective and then, in the context of landscape ecology, extended to a spatial and structural approach. So far, most ecological applications of this theory were restricted to observational purposes, using multi-scale analysis to describe hierarchies. In spite of an increasing attention to dynamics of hierarchically structured ecological systems, current simulation models are still very limited in their representation of self-organization in complex adaptive systems. An ontological conceptualization of the hierarchy theory is outlined, focusing on key concepts, such as levels of organization and the compound and component faces of the holons. Various existing formalisms are currently used in simulation modelling, such as system dynamics, discrete event and agent based paradigms. Their ability to express the hierarchical organization of dynamical ecological systems is discussed. It turns out that a multi-modelling approach linking all these formalisms and oriented toward the specification of a constructive dynamical system would be able to express the dynamical structure of the hierarchy (creation, destruction and change of holons) and the functional and structural links between levels of organization.     

Chemical information from the fungus Amylostereum areolatum and host-foraging behaviour in the parasitoid Ibalia leucospoides

Abstract.  Parasitoids locate hosts using reliable and predictable cues such as smells derived from host plants or from the hosts themselves. For host species that live with mutualistic organisms, such as several wood boring insects, cues derived from the symbionts are likely to be exploited by specific parasitoids. Through a set of bioassays, the behaviour of the parasitoid Ibalia leucospoides Hochenwarth (Hymenoptera: Ibaliidae) is studied in response to the fungus Amylostereum areolatum Boidin (Basidiomycotina: Corticiaceae), a symbiont of its host, the wood wasp Sirex noctilio Fabricius (Hymenoptera: Siricidae). The results show that parasitoids are attracted to the fungus when growing naturally within pine logs, and also when growing on an artificial medium. Fungal volatiles also elicit increased parasitoid activity and may provide information on relative densities of hosts available for parasitization. It is speculated that the the chemical information derived from the host fungal symbiont comprises reliable and detectable host-locating cues used by parasitoids to search for concealed hosts.     

Potential for climate effects on the size-structure of host–parasitoid indirect interaction networks

Communities of insect herbivores are thought to be structured mainly by indirect processes mediated by shared natural enemies, such as apparent competition. In host–parasitoid interaction networks, overlap in natural enemy communities between any pair of host species depends on the realized niches of parasitoids, which ultimately depend on the foraging decisions of individuals. Optimal foraging theory predicts that egg-limited parasitoid females should reject small hosts in favour of future opportunities to oviposit in larger hosts, while time-limited parasitoids are expected to optimize oviposition rate regardless of host size. The degree to which parasitoids are time- or egg-limited depends in part on weather conditions, as this determines the proportion of an individual''s lifespan that is available to foraging. Using a 10-year time series of monthly quantitative host–parasitoid webs, we present evidence for host-size-based electivity and sex allocation in the common secondary parasitoid Asaphes vulgaris. We argue that this electivity leads to body-size-dependent asymmetry in apparent competition among hosts and we discuss how changing weather patterns, as a result of climate change, may impact foraging behaviour and thereby the size-structure and dynamics of host–parasitoid indirect interaction networks.     

Patterns of diversity for aphidiine (Hymenoptera: Braconidae) parasitoid assemblages on aphids (Homoptera)

We used aphids (Aphidae) as a representative hemimetabolous host family to investigate patterns of parasitoid (Aphidiine) assemblage size. The aphidiine assemblages from 477 aphid species were used to estimate average assemblage size and the influence of eight ecological and taxonomic variables. Aphids species support an average of 1.7 aphidiine species. Aphid subfamily and invasion status (native or exotic) were the most important determinants of parasitoid richness, explaining 28% of the deviance in aphidiine assemblage size. Aphids within the largest aphid subfamily, the Aphidinae, support larger parasitoid assemblages than those in other subfamilies. Parasitoid diversity was also highest on exotic aphid hosts (within the Aphidinae) and on hosts in developed habitats (agricultural or urban), though the latter effect is weak. Patterns related to aphid food plant architecture were influenced by an interaction with aphid invasion status; parasitoid diversity drops with increasing architectural complexity on exotic aphids, whereas the diversities on native aphid hosts are similar on different plant types. Weak effects were also found for aphid food plant alternation (whether or not aphids switch hosts seasonally) and climate (annual range in temperature); alternating aphids support more parasitoids than non-alternating hosts, and parasitoid assemblage size is lowest in warm climates. Taxonomic isolation of aphids at the generic level showed no significant relationship with parasitoid diversity. Finally, in contrast to parasitoid assemblages on holometabolous hosts, sample size effects were weak for aphids, possibly due to the narrow host ranges of aphidiines. Received: 22 November 1997 / Accepted: 7 March 1998     

Nutritional ecology of endoparasitic insects and their hosts: An overview

The impact of insect endoparasites (parasitoids) on the physiology and behaviour of their hosts is reviewed within the context of the nutritional ecology of the parasitoids and their hosts. Alterations in the consumption, utilization and allocation of food by parasitized hosts are common, as are internal changes in their metabolic physiology. Gregarious parasitoid species frequently increase feeding by their host larvae whereas solitary parasitoid species often reduce feeding and growth of their hosts. Many parasitoid-associated changes in host physiology and behaviour are interpreted to be of adaptive significance to parasitoids. Substantial circumstantial evidence suggests, and a few direct tests of such adaptive significance indicate, that parasitoids alter their hosts in ways beneficial to their own fitness. However, most of the changes in parasitized hosts are of unknown cause and undocumented significance to the parasitoids. Several relevant hypotheses are presented, and these require extensive evaluation (often requiring novel experimental approaches) before a thorough understanding of parasitoid nutritional ecology is established.