Time allocation of a parasitoid foraging in heterogeneous vegetation: implications for host-parasitoid interactions

1. Changing plant composition in a community can have profound consequences for herbivore and parasitoid population dynamics. To understand such effects, studies are needed that unravel the underlying behavioural decisions determining the responses of parasitoids to complex habitats. 2. The searching behaviour of the parasitoid Diadegma semiclausum was followed in environments with different plant species composition. In the middle of these environments, two Brassica oleracea plants infested by the host Plutella xylostella were placed. The control set-up contained B. oleracea plants only. In the more complex set-ups, B. oleracea plants were interspersed by either Sinapis alba or Hordeum vulgare. 3. Parasitoids did not find the first host-infested plant with the same speed in the different environments. Sinapis alba plants were preferentially searched by parasitoids, resulting in fewer initial host encounters, possibly creating a dynamic enemy-free space for the host on adjacent B. oleracea plants. In set-ups with H. vulgare, also, fewer initial host encounters were found, but in this case plant structure was more likely than infochemicals to interfere with the searching behaviour of parasitoids. 4. On discovering a host-infested plant, parasitoids located the second host-infested plant with equal speed, demonstrating the effect of experience on time allocation. Further encounters with host-infested plants that had already been visited decreased residence times and increased the tendency to leave the environment. 5. Due to the intensive search of S. alba plants, hosts were encountered at lower rates here than in the other set-ups. However, because parasitoids left the set-up with S. alba last, the same number of hosts were encountered as in the other treatments. 6. Plant composition of a community influences the distribution of parasitoid attacks via its effects on arrival and leaving tendencies. Foraging experiences can reduce or increase the importance of enemy-free space for hosts on less attractive plants.     

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.     

Mathematical and computational modelling of spatio-temporal signalling in rod phototransduction

Rod photoreceptors are activated by light through activation of a cascade that includes the G protein-coupled receptor rhodopsin, the G protein transducin, its effector cyclic guanosine monophosphate (cGMP) phosphodiesterase and the second messengers cGMP and Ca2+. Signalling is localised to the particular rod outer segment disc, which is activated by absorption of a single photon. Modelling of this cascade has previously been performed mostly by assumption of a well-stirred cytoplasm. We recently published the first fully spatially resolved model that captures the local nature of light activation. The model reduces the complex geometry of the cell to a simpler one using the mathematical theories of homogenisation and concentrated capacity. The model shows that, upon activation of a single rhodopsin, changes of the second messengers cGMP and Ca2+ are local about the particular activated disc. In the current work, the homogenised model is computationally compared with the full, non-homogenised one, set in the original geometry of the rod outer segment. It is found to have an accuracy of 0.03% compared with the full model in computing the integral response and a 5200-fold reduction in computation time. The model can reconstruct the radial time-profiles of cGMP and Ca2+ in the interdiscal spaces adjacent to the activated discs. Cellular electrical responses are localised near the activation sites, and multiple photons sufficiently far apart produce essentially independent responses. This leads to a computational analysis of the notion and estimate of 'spread' and the optimum distribution of activated sites that maximises the response. Biological insights arising from the spatio-temporal model include a quantification of how variability in the response to dim light is affected by the distance between the outer segment discs capturing photons. The model is thus a simulation tool for biologists to predict the effect of various factors influencing the timing, spread and control mechanisms of this G protein-coupled, receptor-mediated cascade. It permits ease of simulation experiments across a range of conditions, for example, clamping the concentration of calcium, with results matching analogous experimental results. In addition, the model accommodates differing geometries of rod outer segments from different vertebrate species. Thus it represents a building block towards a predictive model of visual transduction.     

Some effects of energy costs on foraging strategies

We consider the effect of including energy costs on the optimal strategy for animals exploiting a depleting food resource. In the context of central place foraging this leads to the problem of what load size should be brought back to the central place. Two strategies are discussed: (i) maximize gross rate of energy delivery and (ii) maximize net rate of energy delivery. The optimal load size (or optimal patch time) for net maximizers is not always larger than for gross maximizers, as has been claimed. Instead, the difference in optimal load size has the same sign as the difference between metabolic rates of travelling and foraging. We point out that the influence of costs has not always been correctly incorporated in experimental tests of the theory.     

Density-dependent parasitoid recruitment per parasitized host: effects on parasitoid-host dynamics

Models of parasitoid-host dynamics are analyzed that include direct density dependence in the host population and either parasitoid- or host-density-dependent variation in parasitoid recruitment per parasitized host (parasitoid "yield"). The principal question addressed is how these forms of density dependence in parasitoid dynamics combine with aggregated parasitism to affect the stability of the models, in relation to suppression of host abundance. When parasitoid yield is an overcompensating function of either parasitoid or host density, stability is enhanced for systems with host equilibria suppressed far below the host carrying capacity. Substantially less aggregation of parasitism is required for stability in this situation than in previous models assuming parasitoid yield is constant. However, this density dependence in parasitoid yield also reduces stability when the host equilibrium is suppressed only moderately below carrying capacity; this is especially true when parasitoid yield is more strongly decreased by high host density than is host per capita reproduction. At present there is little empirical evidence concerning the relationships of parasitoid recruitment to parasitoid and host densities. The substantial effects shown in these models suggest that these relationships should be considered in empirical studies.     

Dietary and foraging strategies of baboons.

As large-bodied savannah primates, baboons have long been of special interest to students of human evolution: many different populations have been studied and dietary comparisons among them are becoming possible. Baboons' foraging strategies can be shown to combine high degrees of flexibility and breadth with selectivity. In this paper we develop and test multivariate models of the basis of diet selection for populations of montane and savannah baboons. Food selection is positively related to protein and lipid content and negatively to fibre, phenolics and alkaloids. Seasonal changes in dietary criteria predicted by these rules are tested and confirmed. Although nutritional bottlenecks occur at intervals, a comparison between long-term nutrient intakes in four different populations indicates convergence on lower degrees of variation than exist in superficial foodstuff profiles.     

Mathematical modelling of extracellular matrix dynamics using discrete cells: fiber orientation and tissue regeneration.

Matrix orientation plays a crucial role in determining the severity of scar tissue after dermal wounding. We present a model framework which allows us to examine the interaction of many of the factors involved in orientation and alignment. Within this framework, cells are considered as discrete objects, while the matrix is modelled as a continuum. Using numerical simulations, we investigate the effect on alignment of changing cell properties and of varying cell interactions with collagen and fibrin.     

Mathematical modelling of dynamics and control in metabolic networks. I. On Michaelis-Menten kinetics

As a starting point for modeling of metabolic networks this paper considers the simple Michaelis-Menten reaction mechanism. After the elimination of diffusional effects a mathematically intractable mass action kinetic model is obtained. The properties of this model are explored via scaling and linearization. The scaling is carried out such that kinetic properties, concentration parameters and external influences are clearly separated. We then try to obtain reasonable estimates for values of the dimensionless groups and examine the dynamic properties of the model over this part of the parameter space. Linear analysis is found to give excellent insight into reaction dynamics and it also gives a forum for understanding and justifying the two commonly used quasi-stationary and quasi-equilibrium analyses. The first finding is that there are two separate time scales inherent in the model existing over most of the parameter space, and in particular over the regions of importance here. Full modal analysis gives a new interpretation of quasi-stationary analysis, and its extension via singular perturbation theory, and a rationalization of the quasi-equilibrium approximation. The new interpretation of the quasi-steady state assumption is that the applicability is intimately related to dynamic interactions between the concentration variables rather than the traditional notion that a quasi-stationary state is reached, after a short transient period, where the rates of formation and decomposition of the enzyme intermediate are approximately equal. The modal analysis reveals that the generally used criterion for the applicability of quasi-stationary analysis that total enzyme concentration must be much less than total substrate concentration, et much less than St, is incomplete and that the criterion et much less than Km much less than St (Km is the well known Michaelis constant) is the appropriate one. The first inequality (et much less than Km) guarantees agreement over the longer time scale leading to quasi-stationary behavior or the applicability of the zeroth order outer singular perturbation solution but the second half of the criterion (Km much less than St) justifies zeroth order inner singular perturbation solution where the substrate concentration is assumed to be invariant. Furthermore linear analysis shows that when a fast mode representing the binding of substrate to the enzyme is fast it can be relaxed leading to the quasi-equilibrium assumption. The influence of the dimensionless groups is ascertained by integrating the equations numerically, and the predictions made by the linear analysis are found to be accurate.(ABSTRACT TRUNCATED AT 400 WORDS)     

Mathematical modelling of dynamics and control in metabolic networks. II. Simple dimeric enzymes

The dynamics of enzyme cooperativity are examined by studying a homotropic dimeric enzyme with identical reaction sites, both of which follow irreversible Michaelis-Menten kinetics. The problem is approached via scaling and linearization of the governing mass action kinetic equations. Homotropic interaction between the two sites are found to depend on three dimensionless groups, two for the substrate binding step and one for the chemical transformation. The interaction between the two reaction sites is shown capable of producing dynamic behavior qualitatively different from that of a simple Michaelis-Menten system; when the two sites interact to increase enzymatic activity over that of two independent monomeric enzymes (positive cooperativity) damped oscillatory behavior is possible, and for negative cooperativity in the chemical transformation step a multiplicity of steady states can occur, with one state unstable and leading to runaway behavior. Linear analysis gives significant insight into system dynamics, and their parametric sensitivity, and a way to identify regions of the parameter space where the approximate quasi-stationary and quasi-equilibrium analyses are appropriate.     

Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies

Angiogenesis, the growth of a network of blood vessels, is a crucial component of solid tumour growth, linking the relatively harmless avascular growth phase and the potentially fatal vascular growth phase. As a process, angiogenesis is a well-orchestrated sequence of events involving endothelial cell migration, proliferation; degradation of tissue; new capillary vessel (sprout) formation; loop formation (anastomosis) and, crucially, blood flow through the network. Once there is blood flow associated with the nascent network, the subsequent growth of the network evolves both temporally and spatially in response to the combined effects of angiogenic factors, migratory cues via the extracellular matrix and perfusion-related haemodynamic forces in a manner that may be described as both adaptive and dynamic. In this paper we present a mathematical model which simultaneously couples vessel growth with blood flow through the vessels--dynamic adaptive tumour-induced angiogenesis (DATIA). This new mathematical model presents a theoretical and computational investigation of the process and highlights a number of important new targets for therapeutic intervention. In contrast to earlier flow models, where the effects of perfusion (blood flow) were essentially evaluated a posteriori, i.e. after generating a hollow network, blood flow in the model described in this paper has a direct impact during capillary growth, with radial adaptations and network remodelling occurring as immediate consequences of primary anastomoses. Capillary network architectures resulting from the dynamically adaptive model are found to differ radically from those obtained using earlier models. The DATIA model is used to examine the effects of changing various physical and biological model parameters on the developing vascular architecture and the delivery of chemotherapeutic drugs to the tumour. Subsequent simulations of chemotherapeutic treatments under different parameter regimes lead to the identification of a number of new therapeutic targets for tumour management.     

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.     

Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial cell migration, and organize themselves into a branched, connected network structure. Subsequent cell proliferation near the sprout-tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumours. In this paper we initially generate theoretical capillary networks (which are morphologically similar to those networks observed in vivo) using the discrete mathematical model of Anderson and Chaplain. This discrete model describes the formation of a capillary sprout network via endothelial cell migratory and proliferative responses to external chemical stimuli (tumour angiogenic factors, TAF) supplied by a nearby solid tumour, and also the endothelial cell interactions with the extracellular matrix. The main aim of this paper is to extend this work to examine fluid flow through these theoretical network structures. In order to achieve this we make use of flow modelling tools and techniques (specifically, flow through interconnected networks) from the field of petroleum engineering. Having modelled the flow of a basic fluid through our network, we then examine the effects of fluid viscosity, blood vessel size (i.e., diameter of the capillaries), and network structure/geometry, upon: (i) the rate of flow through the network; (ii) the amount of fluid present in the complete network at any one time; and (iii) the amount of fluid reaching the tumour. The incorporation of fluid flow through the generated vascular networks has highlighted issues that may have major implications for the study of nutrient supply to the tumour (blood/oxygen supply) and, more importantly, for the delivery of chemotherapeutic drugs to the tumour. Indeed, there are also implications for the delivery of anti-angiogenesis drugs to the network itself. Results clearly highlight the important roles played by the structure and morphology of the network, which is, in turn, linked to the size and geometry of the nearby tumour. The connectedness of the network, as measured by the number of loops formed in the network (the anastomosis density), is also found to be of primary significance. Moreover, under certain conditions, the results of our flow simulations show that an injected chemotherapy drug may bypass the tumour altogether.     

Interactions between ants attendingAphis fabae ssp.cirsiiacanthoidis on thistles and foraging parasitoid wasps

We describe the behavioral interactions between honeydew-collecting workers of the ants Lasius nigerand Myrmica ruginodisand females of three species of aphidiid wasps (Lysiphlebus cardui, Lysiphlebus testaceipes, Trioxys angelicae)foraging for their aphid host, Aphis fabaessp. cirsiiacanthoidis,on thistles. Using field and laboratory experiments, we show that the ant-parasitoid interactions are species specific. Workers of both ant species generally attacked and killed females of T. angelicae,but they ignored those of L. cardui.This pattern was not altered when we anesthetized the wasps slightly with carbon dioxide to reduce their mobility. Prior contacts between L. carduiand either conspecific L. nigerfrom a different nest or workers of a different ant species (M. ruginodis)did not influence L. niger'snonaggressive behavior. The number of aphids parasitized by L. testaceipeswas significantly reduced in aphid colonies attended by L. niger,although this parasitoid was rarely attacked by ants. In encounters between these species of ants and wasps, ant aggression is consistent with differences in wasp behavior. We suggest that, in addition, chemical cues located in the cuticula may enable L. carduito avoid detection by honeydew-collecting ants.     

Mathematical modelling of residue allelopathy: the effects of intrinsic and extrinsic factors

The mechanistic model of residue allelopathy that was constructed before only simulates allelopathic phenomena caused by decaying plant residues and does not include factors controlling expression of such phenomena. As both intrinsic and extrinsic factors are affecting the potential of residue allelopathy, the model has been developed further to include these factors. The concept of a constant for residue-allelopathic potential, functions for effects of temperature, water, and their combination on allelochemical concentrations were added to the model. As a result, the improved residue model is able to examine the responses to these phenomena under a wide variety of conditions. Several sets of published experimental data were compared with prediction from the model, and a good agreement has been achieved. Its significance in helping better understanding of residue allelopathy is discussed.     

Within- and between-generation effects of temperature on life-history traits in a butterfly

Non-genetic parental effects may largely affect offspring phenotype, and such plasticity is potentially adaptive. Despite its potential importance, little is known about cross-generational effects of temperature, at least partly because parental effects were frequently considered a troublesome nuisance, rather than a target of experimental studies. We here investigate effects of parental, developmental and acclimation temperature on life-history traits in the butterfly Bicyclus anynana. Higher developmental temperatures reduced development times and egg size, increased egg number, but did not affect pupal mass. Between-generation temperature effects on larval time, pupal time, larval growth rate and egg size were qualitatively very similar to effects of developmental temperature, and additionally affected pupal mass but not egg number. Parental effects are important mediators of phenotypic plasticity in B. anynana, and partly yielded antagonistic effects on different components of fitness, which may constrain the evolution of cross-generational adaptive plasticity.     

Mathematical modelling and numerical simulations of actin dynamics in the eukaryotic cell

The aim of this article is to study cell deformation and cell movement by considering both the mechanical and biochemical properties of the cortical network of actin filaments and its concentration. Actin is a polymer that can exist either in filamentous form (F-actin) or in monometric form (G-actin) (Chen et al. in Trends Biochem Sci 25:19–23, 2000) and the filamentous form is arranged in a paired helix of two protofilaments (Ananthakrishnan et al. in Recent Res Devel Biophys 5:39–69, 2006). By assuming that cell deformations are a result of the cortical actin dynamics in the cell cytoskeleton, we consider a continuum mathematical model that couples the mechanics of the network of actin filaments with its bio-chemical dynamics. Numerical treatment of the model is carried out using the moving grid finite element method (Madzvamuse et al. in J Comput Phys 190:478–500, 2003). Furthermore, by assuming slow deformations of the cell, we use linear stability theory to validate the numerical simulation results close to bifurcation points. Far from bifurcation points, we show that the mathematical model is able to describe the complex cell deformations typically observed in experimental results. Our numerical results illustrate cell expansion, cell contraction, cell translation and cell relocation as well as cell protrusions. In all these results, the contractile tonicity formed by the association of actin filaments to the myosin II motor proteins is identified as a key bifurcation parameter.     

The effects of metapopulation structure on indirect interactions in host-parasitoid assemblages.

The interaction between two species that do not compete for resources but share a common natural enemy is known as apparent competition. In the absence of other limiting factors, such three-species interactions are impermanent, with one species being excluded from the assemblage by the natural enemy. Here, the effects of metapopulation structure are explored in a system of two hosts that experience apparent competition through a shared parasitoid. A coupled-map lattice model is developed and used to explore species coexistence and patterns of patch occupancy at the metapopulation scale. Linking local and regional dynamics favours coexistence by uncoupling the dynamics of the three species in space. Coexistence is promoted by the inferior species being either a fugitive or a sedentary species. The occurrence of these two mutually exclusive mechanisms of coexistence is influenced by the relative dispersal of the inferior apparent competitor.     

Effects of patch scale on density-dependence and species-dependence in two host-parasitoid systems

Summary Data from two host-parasitoid communities were analyzed to ascertain whether patch scale affected the kinds of correlations existing between 1) spatial differences in host density and the intensity of parasitism (density-dependence) and 2) number of species of parasitoids and the intensity of parasitism (species-dependence). We concluded that parasitization rates are usually independent of both host density and number of parasitoid species present regardless of patch scale. Therefore, the responses of parasitoids to host density and the addition of parasitoid species to a community are equally unpredictable in outcome.