The Effect of Covert and Overt Infections on Disease Dynamics in Honey-Bee Colonies.

Viral diseases of honey bees are important economically and ecologically and have been widely modelled. The models reflect the fact that, in contrast to the typical case for vertebrates, invertebrates cannot acquire immunity to a viral disease, so they are of SIS or (more often) SI type. Very often, these diseases may be transmitted vertically as well as horizontally, by vectors as well as directly, and through the environment, although models do not generally reflect all these transmission mechanisms. Here, we shall consider an important additional complication the consequences of which have yet to be fully explored in a model, namely that both infected honey bees and their vectors may best be described using more than one infection class. For honey bees, we consider three infection classes. Covert infections occur when bees have the virus under control, such that they do not display symptoms of the disease, and are minimally or not at all affected by it. Acutely overtly infected bees often exhibit severe symptoms and have a greatly curtailed lifespan. Chronically overtly infected bees typically have milder symptoms and a moderately shortened lifespan. For the vector, we consider just two infection classes which are covert infected and overt infected as has been observed in deformed-wing virus (DWV) vectored by varroa mites. Using this structure, we explore the impact of spontaneous transition of both mites and bees from a covertly to an overtly infected state, which is also a novel element in modelling viral diseases of honey bees made possible by including the different infected classes. The dynamics of these diseases are unsurprisingly rather different from the dynamics of a standard SI or SIS disease. In this paper, we highlight how our compartmental structure for infection in honey bees and their vectors impact the disease dynamics observed, concentrating in particular on DWV vectored by varroa mites. If there is no spontaneous transition, then a basic reproduction number [Formula: see text] exists. We derive a condition for [Formula: see text] that reflects the complexities of the system, with components for vertical and for direct and vector-mediated horizontal transmission, using the directed graph of the next-generation matrix of the system. Such a condition has never previously been derived for a honey-bee-mite-virus system. When spontaneous transitions do occur, then [Formula: see text] no longer exists, but we introduce a modification of the analysis that allows us to determine whether (i) the disease remains largely covert or (ii) a substantial outbreak of overt disease occurs.

How ants use quorum sensing to estimate the average quality of a fluctuating resource.

We show that one of the advantages of quorum-based decision-making is an ability to estimate the average value of a resource that fluctuates in quality. By using a quorum threshold, namely the number of ants within a new nest site, to determine their choice, the ants are in effect voting with their feet. Our results show that such quorum sensing is compatible with homogenization theory such that the average value of a new nest site is determined by ants accumulating within it when the nest site is of high quality and leaving when it is poor. Hence, the ants can estimate a surprisingly accurate running average quality of a complex resource through the use of extraordinarily simple procedures.

Parasite species richness and its effect on persistence in food webs.

Parasitic species are likely to have a significant effect on the stability of ecosystems. However, little is known of the nature of this effect, with debate over whether it is positive or negative. In previous work it was observed that a mixture of interaction types increases the local stability of a network. Following this, we investigate the consequences for species persistence of replacing host species with parasitic species. We consider systems with varying mixtures of mutualistic and antagonistic interactions, showing that the effect of parasitic interactions on a system depends on both the interaction types present and the levels of parasitism considered. Higher levels of mutualism make a system vulnerable to destabilisation on the addition of parasite species. However, for systems with antagonistic interactions, persistence in the system decreases primarily due to the failure of parasite species to persist. This increases with increasing proportions of parasite species, leading to a peak number of parasite species able to persist. Increasing parasite species richness does not have as significant an effect on host species richness as we might expect; although parasites have an important role to play in ecological networks, their effect on persistence is seen primarily through their own self-limitation.

On the maintenance of bird song dialects.

Many bird species, especially song birds but also for instance some hummingbirds and parrots, have noted dialects. By this we mean that locally a particular song is sung by the majority of the birds, but that neighbouring patches may feature different song types. Behavioural ecologists have been interested in how such dialects come about and how they are maintained for over 45 years. As a result, a great deal is known about different mechanisms at play, such as dispersal, assortative mating and learning of songs, and there are several competing hypotheses to explain the dialect patterns known in nature. There is, however, surprisingly little theoretical work testing these different hypotheses at present. We analyse the simplest kind of model that takes into account the most important biological mechanisms, and in which one may speak of dialects: a model in which there are but two patches, and two song types. It teaches us that a combination of little dispersal and strong assortative mating ensures dialects are maintained. Assuming a simple, linear frequency-dependent learning rule has little effect on the maintenance of dialects. A nonlinear learning rule, however, has dramatic consequences and greatly facilitates dialect maintenance. Adding fitness benefits for singing particular songs in a given patch also has a great impact. Now rare song types may invade and remain in the population.

Network dynamics contribute to structure: nestedness in mutualistic networks.

Both ecological and evolutionary timescales are of importance when considering an ecological system; population dynamics affect the evolution of species traits, and vice versa. Recently, these two timescales have been used to explain structural patterns in host-parasite networks, where the evolution of the manner in which species balance the use of their resources in interactions with each other was examined.One of these patterns was nestedness, in which the set of parasite species within a particular host forms a subset of those within a more species-rich host. Patterns of both nestedness and anti-nestedness have been observed significantly more often than expected due to chance in host-parasite networks. In contrast, mutualistic networks tend to display a significant degree of nestedness, but are rarely anti-nested. Within networks with different interaction types, therefore, there appears to be a feature promoting non-random structural patterns, such as nestedness and anti-nestedness, depending on the interaction types involved.Here, we invoke the co-evolution of species trait-values when allocating resources to interactions to explain the structural pattern of nestedness in a mutualistic community. We look at a bipartite, multi-species system, in which the strength of an interaction between two species is determined by the resources that each species invests in that relationship. We then analyze the evolution of these interactions using adaptive dynamics.We found that the evolution of these interactions, reflecting the trade-off of resources, could be used to accurately predict that nestedness occurs significantly more often than expect due to chance alone in a mutualistic network. This complements previous results applying the same concept to an antagonistic network. We conclude that population dynamics and resource trade-offs could be important promoters of structural patterns in ecological networks of different types.

Trophic structure, stability, and parasite persistence threshold in food webs.

Food web structure of free-living species is an important determinant of parasite species richness. Downwardly asymmetric predator-prey interactions (where there are more prey than predator species) have been shown, both theoretically and empirically, to harbour more trophically transmitted parasite species than expected due to chance. Here, we demonstrate that this could be due to the increase in the basic reproductive ratio that the addition of non-host prey species to a system creates. However, we note that the basic reproductive ratio is only increased by those prey that stabilise oscillations in a predator-prey system, and is decreased by those that do not.

Analysis of a vector-bias model on malaria transmission.

We incorporate a vector-bias term into a malaria-transmission model to account for the greater attractiveness of infectious humans to mosquitoes in terms of differing probabilities that a mosquito arriving at a human at random picks that human depending on whether he is infectious or susceptible. We prove that transcritical bifurcation occurs at the basic reproductive ratio equalling 1 by projecting the flow onto the extended centre manifold. We next study the dynamics of the system when incubation time of malaria parasites in mosquitoes is included, and find that the longer incubation time reduces the prevalence of malaria. Also, we incorporate a random movement of mosquitoes as a diffusion term and a chemically directed movement of mosquitoes to humans expressed in terms of sweat and body odour as a chemotaxis term to study the propagation of infected population to uninfected population. We find that a travelling wave occurs; its speed is calculated numerically and estimated for the lower bound analytically.

The persistence of parasitic plasmids.

The conditions under which plasmids are predicted to persist remain controversial. Here, we reevaluate the ordinary differential equations used previously to model plasmid persistence and conclude that the parameter space required for maintenance is far less stringent than has been supposed. Strikingly, our model demonstrates that purely parasitic plasmids may persist, even in the absence of heterogeneity in the host population, and that this persistence is expressed by oscillations or damped oscillations between the plasmid-bearing and the plasmid-free class.

Information flow, opinion polling and collective intelligence in house-hunting social insects.

The sharing and collective processing of information by certain insect societies is one of the reasons that they warrant the superlative epithet 'super-organisms' (Franks 1989, Am. Sci. 77, 138-145). We describe a detailed experimental and mathematical analysis of information exchange and decision-making in, arguably, the most difficult collective choices that social insects face: namely, house hunting by complete societies. The key issue is how can a complete colony select the single best nest-site among several alternatives? Individual scouts respond to the diverse information they have personally obtained about the quality of a potential nest-site by producing a recruitment signal. The colony then deliberates over (i.e. integrates) different incoming recruitment signals associated with different potential nest-sites to achieve a well-informed collective decision. We compare this process in honeybees and in the ant Leptothorax albipennis. Notwithstanding many differences - for example, honeybee colonies have 100 times more individuals than L. albipennis colonies - there are certain similarities in the fundamental algorithms these societies appear to employ when they are house hunting. Scout honeybees use the full power of the waggle dance to inform their nest-mates about the distance and direction of a potential nest-site (and they indicate the quality of a nest-site indirectly through the vigour of their dance), and yet individual bees perhaps only rarely make direct comparisons of such sites. By contrast, scouts from L. albipennis colonies often compare nest-sites, but they cannot directly inform one another of their estimation of the quality of a potential site. Instead, they discriminate between sites by initiating recruitment sooner to better ones. Nevertheless, both species do make use of forms of opinion polling. For example, scout bees that have formerly danced for a certain site cease such advertising and monitor the dances of others at random. That is, they act without prejudice. They neither favour nor disdain dancers that advocate the site they had formerly advertised or the alternatives. Thus, in general the bees are less well informed than they would be if they systematically monitored dances for alternative sites rather than spending their time reprocessing information they already have. However, as a result of their lack of prejudice, less time overall will be wasted in endless debate among stubborn and potentially biased bees. Among the ants, the opinions of nest-mates are also pooled effectively when scouts use a threshold population of their nest-mates present in a new nest-site as a cue to switch to more rapid recruitment. Furthermore, the ants' reluctance to begin recruiting to poor nest-sites means that more time is available for the discovery and direct comparison of alternatives. Likewise, the retirement of honeybee scouts from dancing for a given site allows more time for other scouts to find potentially better sites. Thus, both the ants and the bees have time-lags built into their decision-making systems that should facilitate a compromise between thorough surveys for good nest-sites and relatively rapid decisions. We have also been able to show that classical mathematical models can illuminate the processes by which colonies are able to achieve decisions that are relatively swift and very well informed.

Coexistence of sexual and asexual conspecifics: a cellular automaton model.

Most previous models of populations mixed for reproductive mode have omitted important local interactions between sexual and asexual individuals. We propose a cellular automaton model where local rules focus on fertilization and colonization. This model produces rich sets of data which are then studied by means of spatial statistics. Results point to the fixation of one of the two reproductive modes in the landscape. However, some examples of coexistence of sexual and asexual conspecifics over long periods of time are also found. This model is an example of a CA that diverges from its mean field approximation. The formation of sexual and asexual clusters reduces effective colonization rate in the CA and may account for this behaviour.

Chimpanzees and the mathematics of battle.

Recent experiments have demonstrated the importance of numerical assessment in animal contests. Nevertheless, few attempts have been made to model explicitly the relationship between the relative number of combatants on each side and the costs and benefits of entering a contest. One framework that may be especially suitable for making such explicit predictions is Lanchester's theory of combat, which has proved useful for understanding combat strategies in humans and several species of ants. We show, with data from a recent series of playback experiments, that a model derived from Lanchester's 'square law' predicts willingness to enter intergroup contests in wild chimpanzees (Pan troglodytes). Furthermore, the model predicts that, in contests with multiple individuals on each side, chimpanzees in this population should be willing to enter a contest only if they outnumber the opposing side by a factor of 1.5. We evaluate these results for intergroup encounters in chimpanzees and also discuss potential applications of Lanchester's square and linear laws for understanding combat strategies in other species.

The adaptiveness of defence strategies against cuckoo parasitism.

Host bird species of the Eurasian Cuckoo, Cuculus canorus, often display egg-discrimination behaviour but chick-rejection behaviour has never been reported. In this paper, we analyse a host-cuckoo association in which both population dynamics and evolutionary dynamics are explored in a discrete-time model. We introduce four host types, each with their own defence behaviour, displaying either egg or chick rejection, neither or both, We also introduce fitness functions for each of these host types. Although we can characterize the long term behaviour in many cases by a simple heuristic argument which is in accordance with common views in ecology, there are a number of other phenomena that are not explained within this framework: we describe stable oscillatory behaviour and coexistence of two defensive host types. We analyse the scenarios in which chick rejection may establish itself and give a first explanation as to why this defence trait has never been recorded in nature. We find that chick rejectors generally are at an intrinsic disadvantage with respect to a host type that rejects eggs. Hosts benefit more from rejecting cuckoo eggs than cuckoo chicks, and our model suggests that this is chiefly responsible for the absence of chick rejection. Moreover, even though it seems that chick rejection must be useful as an extra defence, it is shown that hosts with both defence strategies are less likely to establish themselves in competition with egg-rejectors than hosts which reject chicks only. These results provide insight in the extent to which adaptations may be perfected by natural selection.